aa r X i v : . [ phy s i c s . g e n - ph ] F e b Unification in One Dimension
David J. Jackson
February 15, 2016
Abstract
A physical theory of the world is presented under the unifying principle that allof nature is laid out before us and experienced through the passage of time. The one-dimensional progression in time is opened out into a multi-dimensional mathematicallyconsistent flow, with the simplicity of the former giving rise to symmetries of thelatter. The act of perception identifies an extended spacetime arena of intermediatedimension, incorporating the symmetry of geometric spatial rotations, against whichphysical objects are formed and observed. The spacetime symmetry is contained asa subgroup of, and provides a natural breaking mechanism for, the higher generalsymmetry of time. It will be described how the world of gravitation and cosmology, aswell as quantum theory and particle physics, arises from these considerations. ontents Symmetry on h O Transformations on a Form of Time . . . . . . . . . . . . . . . . . . . 1286.5 Lie Algebra of E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1401 Review of the Standard Model 148 Symmetry Breaking 170 O . . . . . . . . . . . . . . . . . . . . . . . . . 1708.2 Internal SU(3) c × U(1) Q Symmetry . . . . . . . . . . . . . . . . . . . . . 1768.3 Elements of Electroweak Theory . . . . . . . . . . . . . . . . . . . . . . 1858.3.1 SU(2) Transformations and SU(3) s Symmetry . . . . . . . . . . . 1858.3.2 SU(2) × U(1) Mixing Angle . . . . . . . . . . . . . . . . . . . . 1958.3.3 Origin of Mass and Higgs Phenomena . . . . . . . . . . . . . . . 204 O and Further Weyl Spinors . . . . . . . . . . . . . . . . . 2139.2 E Symmetry and the Freudenthal Triple System . . . . . . . . . . . . . 2249.3 E Symmetry and the Standard Model . . . . . . . . . . . . . . . . . . . 238
10 Particle Physics 250
11 A Novel Conception of HEP Processes 288
12 Cosmology 357
13 A Novel Perspective on Cosmological Structure 380
14 The Origin of Time 426
15 Towards a Complete Theory 467 hapter 1
Introduction
In establishing a conceptual framework for a physical theory of the world one of themost fundamental questions concerns the nature of the ultimate entity out of whichthe world is made. From the earth, water, air and fire of the ancient Greeks, throughvarious manifestations of elementary extended or point-like particle theories, to thequantum fields of 20 th century high energy physics, the notion of a fundamental formof matter behaving according to laws of nature, to be identified empirically or throughpowers of reason, has strongly influenced the development of scientific theories of theworld. The general trend has been to dig deeper into the layers of matter such thatmacroscopic objects are taken to be composed of discrete particle-like entities whichin turn are composed of more basic particles which have become themselves to beconsidered as merely the states detected in physics experiments as a manifestation ofyet deeper underlying entities, such as fields or strings. It is a trend which ever posesthe question of what may be uncovered at the next layer down, or whether we mayreach the ultimate bedrock of the world.The view taken in the present investigations is that the world can be builtout of the one entity within which all our experiments, experiences, perceptions andindeed our thoughts in general are conducted, that is through progression in time.This universal nature of time applies both to inner thought experiences in the mind,as well as outer thoughts of the physical world; for example, perception of a cloudpassing by or of a book on the table. With the basic structure of time being identifiedwith, or isomorphic to, that of the real numbers R this gives an immediate connectionto a purely mathematical world. The mathematical possibility to express an inner one-dimensional sense of time in the form of an outer multi-dimensional space as anintrinsic and elementary property of the real numbers provides a significant motivationfor this study. The aim will be to demonstrate how the external world of experiencecan result directly from the mathematical structure of temporal flow itself without theneed to interpose or postulate the notion of an underlying ‘material’ substratum ofany form.It may be helpful to begin with an analogy using a familiar example in whichmulti-dimensional structures are enfolded within a lower-dimensional entity, namelya child’s ‘pop-up’ book of cartoon zoo animals, although, of course, it should not be3aken too literally to represent the theory to be presented here. We can considersuch a book, when closed, to be an essentially 2-dimensional object in space. Whenopened fully on a given page a figure will ‘pop-up’, perhaps an elephant, extended in3-dimensional space; on another page a 3-dimensional crocodile may appear, and soon. It is down to the creativity and origami skills of the bookmakers to form such 3-dimensional structures that can be perfectly folded away into the 2-dimensional plane(when the book closes) within the fixed constraints of the possibilities allowed by thelaws of Euclidean geometry.It is the contention of these investigations that the 1-dimensional flow of timeitself naturally opens out, according to necessary mathematical and geometrical forms,into a higher-dimensional space. The mechanism will be somewhat different to thatin the above metaphor since time, unlike a book in space, is not experienced ‘all atonce’ and indeed space itself will need to be unfolded out of the temporal flow. How-ever, within the 1-dimensional flow of time we shall find implicitly contained not onlythe appropriate mathematical structures for 3-dimensional space and 4-dimensionalspacetime but also still higher-dimensional possibilities. The intermediate 3 and 4-dimensional cases can be interpreted as subspaces of the higher-dimensional forms,with the properties of physical objects perceived in spacetime being largely deter-mined by the nature of the general higher-dimensional structures. It is claimed thatthe opening out of the progression of time in this way into a mathematically deter-mined multi-dimensional flow is responsible for not only our perceptions of objects,from books to real elephants themselves, but of the entire physical universe around us.In a similar way that the laws of geometry constrain the design of pop-upbooks, so mathematics will constrain the way in which the physical world can openout from the flow of time and hence determine the laws of physics. It is the main aim ofthis paper to show how far the consequences of this idea resemble the observed laws ofnature of the actual world. In traditional theories properties are assigned to underlyingparticles or fields, out of an enormous range of conceivable choices of such properties,largely for pragmatic reasons to match the empirically observed world. Here, on thecontrary, we expect the present theory to make a much more thorough and directcontact with the structure of the physical world.In aiming for an inclusive and complete theory, as well as accounting for thebasic observed scientific phenomena from particle physics to cosmology, the theorymight also address the everyday direct manner through which we actually encounterand experience the world. We shall touch upon all these areas, all of which wouldbenefit from further study, in an attempt to gain an overall consistent worldview.Although this paper is lengthy all of the contents relate to a single unifiedtheory, rather than to a collection of independent ideas, as will be clear from theprogression of sections and the mutual cross-referencing within the text. Here wereview the contents of the paper to guide the reader towards the sections which maybe of most interest. While the overall order of the text has been designed to introducethe various facets of the theory in a reasonably logical sequence there are four mainareas in which progress on the theory has been made essentially in parallel. Each areaaddresses a particular question and related set of issues which might be asked of anycandidate for a unified physical theory. The four areas correspond generally to subsetsof the subsequent chapters of this paper: 4 Chapters 2–5: The main goal here is to describe how an extended spacetimearena may be identified together with an external and internal curvature and themanner in which they are mutually constrained. First, beginning with a one-dimensional temporal interval in chapter 2, we make precise the notion of themulti-dimensional flow of time by deriving in elementary terms what is consideredto be its general mathematical form and give several relevant examples. Wealso emphasise the fact that we are dealing here with a general symmetry oftime, in contrast to the symmetry of a higher-dimensional spacetime found ina different class of theories. The notion of perception as identified with theinterpretation of certain mathematical forms, implicit in the multi-dimensionalflow of time, in the shape of a geometrical spacetime. That is, we describehow an extended external spacetime arena for the world can ‘pop-out’ from thetemporal flow. This structure motivates the employment of more sophisticatedmathematical tools, and in chapter 3 standard textbook material on differentialgeometry and general relativity is reviewed. Papers in the literature from aroundthe mid-1970s to mid-1980s regarding non-Abelian Kaluza-Klein theories, whichalso describe a unified approach relating external and internal curvature within asimilar mathematical framework, are then reviewed in chapter 4. In chapter 5 wethen pick up the thread from end of chapter 2, in light of the mediating chapters,using the constraints of the present theory to study the relationship between theexternal and internal curvature in 4-dimensional spacetime and consider furtherimplications such as constraints on the equations of motion. • Chapters 6–9: Here we consider higher-dimensional forms of temporal flow withthe main aim of establishing a connection with the structures of the StandardModel of particle physics in the breaking of the full symmetry of time over the4-dimensional base manifold. Crucial to this investigation are the referencesconcerning the structure of the Lie group E acting on the space h O of 3 × on the elements of h O interpreted as a particularly richsymmetry of a 27-dimensional cubic form of temporal flow. In chapter 7 theprinciple features of the Standard Model, and their relation to mathematicalmodels based on unification groups, are reviewed in order describe the physicalstructures to be accounted for by the present theory and the kind of theoreticalstructures which may be relevant. In chapter 8 we investigate the extent to whichthe E symmetry action on h O in the context of the present theory can accountfor the properties of the Standard Model. Several successes are noted in terms ofa correlation between the transformation properties of components of h O underthe external and identified internal symmetries and corresponding propertiesof Standard Model particle states. The need to incorporate further particleproperties and the natural extension to the larger structure of an E actionpreserving a quartic from on the space F (h O ), interpreted as a symmetry of a56-dimensional form of temporal flow, leads to some further success in chapter 9and also the suggestion of investigating yet higher-dimensional forms.5 Chapters 10–11: In addition to uncovering Standard Model features here theambition is to understand how the present theory might accommodate the em-pirical observations of HEP experiments, in terms of cross-sections and decayrates for example, and incorporate quantum phenomena in general. The essen-tial textbook aspects of quantum field theory are reviewed in chapter 10, within particular the structure of cross-section calculations analysed into its basicelements in order to establish a correspondence with the present theory. Thiscorrespondence is described in chapter 11 in which the conceptual origins ofquantum phenomena within the context of the present theory are established.In one sense this involves generalising the relation between the external geome-try and a particular internal gauge field, as assessed in relation to Kaluza-Kleintheories in the earlier chapters, for an external geometry expressed in terms ofa degeneracy of underlying field solutions. These underlying fields include bothgauge fields deriving from the symmetry of time and fields deriving from com-ponents of the multi-dimensional form of temporal flow itself, mutually relatedby a set of implicit constraint equations rather than via a dedicated Lagrangianfunction. Within the scope of these investigations two main and related pointsconcern the conceptual nature of physical particle states, as analysed in labora-tory experiments, and the manner in which the phenomena of quantum theoryand general relativity coexist as aspects of the same unified theory. • Chapters 12–14: The theory has been developed with not only laboratory phe-nomena in mind but also the large scale structure of the universe with the goalof understanding the extent to which observations in cosmology might also beaccounted for. To this end in chapter 12 textbook material on both the standardcosmological model and inflationary theory is reviewed. A new feature of thepresent theory is described in chapter 13 concerning the possibility of non-trivialintrinsic curvature for the spacetime manifold arising from the elementary prop-erties of its projection out of the multi-dimensional form of time. The extentto which this, combined with additional features of the theory identified ear-lier, might account for both the phenomena of the dark sector in cosmology andthe structure of the very early universe is then considered, with the question ofuniqueness for the theory in general also discussed. The study of the Big Bangepoch raises broader questions, in addition to the need to describe physical prop-erties, relating to the reason why the universe should exist at all. In the contextof the present theory, with everything constructed through a multi-dimensionalform of progression in time, this inevitably leads to the question concerning theorigin of time itself. In addressing this issue the speculations of chapter 14 in-clude areas which are not necessarily within the traditional bounds of physicsbut touch upon other scientific fields of study. This detour is however of value inproviding an opportunity to elaborate upon the possibility of identifying a firmfoundation for the full physical theory.Following the discussion of the foundations of the theory in chapter 14 in theconcluding chapter 15 we look outwards to the prospects for the further developmentof the theory in the four main areas outline above, which are also depicted as the fourfronts in figure 15.1. In section 15.1 the mutual relations between all aspects of this6nified theory will also be described.Broadly, the four subsets of chapters for the four main branches of the theorylisted above are each presented with a structure to some extent analogous to a PhDthesis, in terms of combination of the presentation of new ideas and reviews of es-tablished material. In particular summaries of standard textbook material and othercited literature are presented mainly in the subsections, sections and chapters: 2.2.2,3, 4, 6, 7, 9.2 (up to equation 9.41), 10 and 12, although always in the context of thepresent theory. On the other hand the main novel theoretical developments follow atrail through the sections and chapters: 2, 5, 8, 9, 10.1, 11, 12.1, 13, 14 and 15, withreference to the standard material of the above intervening chapters and section andfurther citations discussed in the course of the presentation.Generally speaking the central chapters 6–11 deal more with the microscopicand laboratory scale while the outer sections through to chapter 13 pursue a threadmore closely associated with the macroscopic and large scale features typically stud-ied under general relativity. However all aspects of the present theory are relevantfor all scales, as further discussed in the concluding chapter. The current point ofclosest approach between the present theory and empirically established features ofthe physical world is in terms of a relationship with the Standard Model of particlephysics, regarding in particular transformation properties under the external Lorentzsymmetry group and the internal SU(3) c × SU(2) L × U(1) Y gauge group, as alludedto in the synopsis of Chapters 6–9 listed above. The shortest path from introducingthe basic ideas of the theory to an elaboration of this connection with the StandardModel is to follow sections: 2.1, the opening of 2.2, 8.1, 8.2, 9.2 (after equation 9.41)and with further discussion in section 9.3.The purpose of these investigations can be described as an enquiry into theextent to which the form of the physical world can be determined purely from the factthat it is perceived in time; that is, the extent to which the world can be constructedout of the pure mathematical nature of the progression in time itself. To this endwe establish in the following section an expression for the potential multi-dimensional flow of time in the appropriate general mathematical form. This will later provide themeans to incorporate 4-dimensional spacetime together with the structures of ‘extradimensions’ in a naturally unified way. We begin with a particular example of a multi-dimensional expression implicit in a finite interval of time which exhibits an apparentgeometric symmetry. 7 hapter 2 The Symmetry of Time
A finite interval of time represented by the real number s ∈ R can be algebraicallyexpressed in terms of other real numbers x a ( a = 1 , , . . . ) in an endless variety ofways. For example s = x + x composes time intervals in series, while s = x x might represent a rescaling of the temporal unit, or more generally we can have s = x ( x x + x ) and so on simply by employing the basic arithmetic structure of thereal line. More specifically, writing the square of the interval in the form, familiarsince Pythagoras, s = ( x ) + ( x ) + ( x ) , the interval s is then invariant undertransformations of the set of real numbers { x , x , x } ∈ R by the orthogonal rotationgroup O(3), as depicted in figure 2.1. In this case the set of possible numbers { x a } consistent with this form for s exhibits the mathematical symmetry of a vector in3-dimensional Euclidean space that maintains a fixed length under rotations. This isexactly the mathematical structure within which the physical objects of our perceptionsappear to us spatially; thus providing a simple example of how the geometric propertiesof space can be algebraically embedded within structures implicit in the arithmeticproperties of time as represented by the real line.The broad range of possible expressions for a finite interval s in terms of anarbitrary number of variables { x a } , a = 1 . . . n , will be constrained to a more restrictivestructure in the limit of infinitesimally small temporal intervals. We first consider thislimit for the trivial case with the flow of time s expressed in terms of a single realvariable x only for which we have simply s = x . This can symbolically be written as δs = δx as we approach the limit of infinitesimal intervals. We then express the rateof change of x with respect to s in this limit as: v = dx ds ≡ δx δs (cid:12)(cid:12)(cid:12)(cid:12) δs → = 1 (2.1)For the case with multiple real numbers { x a } ∈ R n representing the flow of time s each will be associated with a corresponding rate of change v a = dx a /ds with respect8igure 2.1: With a fixed finite 1-dimensional interval of time s expressed in theform s = p ( x ) + ( x ) + ( x ) the numerical morphisms of the three numbers { x , x , x } under which s is invariant can be interpreted as mapping out a spheri-cal shell in a 3-dimensional geometrical space.to pure time. For example, we may consider the propagation of time expressed for aninfinitesimal interval as:( δs ) = ( δx ) + ( δx ) + ( δx ) (2.2)= η ab δx a δx b with η ab = diag(+1 , +1 , +1) (2.3)where { a, b } = { , , } (and with the conventional summation over repeated indicesimplied throughout this paper). Dividing by ( δs ) and taking the limit δs → η ab v a v b = 1 or ( v ) + ( v ) + ( v ) = 1, which is invariant underthe group, O(3), of orthogonal transformations in three dimensions applied to v =( v , v , v ) ∈ R . This is simply the infinitesimal case of the situation depicted infigure 2.1, with s → δs/δs = 1 and x a → δx a /δs = v a (for δs → { v a } ∈ R . Thequestion is then how to express the general case for the composition and symmetriesof a multi-dimensional set of velocities { v a } ∈ R n .The infinitesimal elements of time can be written most generally, taking careto balance the order of the vanishing elements in each term, as: δs = α a δx a + p α bc δx b δx c + q α def δx d δx e δx f + . . . . . . (2.4)Here the coefficients α abc... are each equal to ± δs purely in terms of simple arithmetic relations of the δx a . In equation 2.4 each termdivides δs into a separate portion of time: δs = δs + δs + δs + . . . (2.5)9here each term δs p is the p th -root of a homogeneous polynomial of order p in the { δx a } . Taking each term in turn, dividing by the interval δs p in each case and takingthe limit { δs p , δx a } → δs p = p p α abc... δx a δx b δx c . . . (2.6)divide by δs p : 1 = p p α abc... v a v b v c . . . (2.7)that is: α abc... v a v b v c . . . = 1 (2.8)which we write: L ( v ) = 1 (2.9)where L is a homogeneous polynomial of order p in the components v a ; it can beconsidered as a map from the elements of a real n -dimensional vector space v ∈ R n onto the unit 1 ∈ R .The p th -root is dropped in stepping from equation 2.7 to equation 2.8 since,trivially, 1 p = 1. If the equality in equation 2.7 involved a variable quantity on theleft-hand side rather than unity, such as in the case of finding a ‘path of extremallength’ on an extended manifold for a quadratic form, or metric, using a variationalmethod then the root would be needed, as will be described later for equation 3.78.Further, the components of a local ‘metric’ η ab = α ab ∈ {± , } may be mapped ontoa general metric involving components g µν ( x ) / ∈ {± , } under a transformation from‘local coordinate’ variables { x a } to a ‘general coordinate system’ on such an extendedmanifold, as we shall describe leading up equation 2.16. (In principle this observationcould also apply to the other coefficients α abc... of equation 2.4 considered as generalised‘metrics’ for the corresponding extended dimensions).Equation 2.9 is taken to express the general mathematical form of multi-dimensional temporal flow and it is the central equation of this paper. The symmetriesof L ( v ) = 1 will be represented by groups acting on the vector space R n such thatfor all elements g of the group G and all vectors v ∈ R n satisfying L ( v ) = 1 we have L ( σ g ( v )) = L ( v ′ ) = 1 where σ g ( v ) represents the action of the group element g ∈ G on the vector v ∈ R n . As G acts on R n over a continuous range of elements beginningat the identity g = e ∈ G we can think of this as a continuous morphism of the realnumbers v a . (This is equivalent to the symmetry over the 2-sphere in the examplewith finite intervals { x a } in figure 2.1). This morphism is always consistent with thedissolving of the fundamental temporal flow s into the possible rates of change v a of the multi-dimensional real quantities x a conforming to the requirement L ( v ) = 1and hence may be termed an isochronal symmetry, of which we next describe severalexamples.Quadratic forms in general, including the 4-dimensional example of the ex-pression L ( v ) = η ab v a v b , with v ∈ R , Minkowski metric η ab = diag(+1 , − , − , − { a, b } = { , , , } , and the norm of an element of a division algebra ( R , C , H or O as introduced below), together with their symmetry groups, are expected to beparticularly significant forms of L ( v ) = 1. This is due to their close relation to Clif-ford algebras and Euclidean spatial geometry, describing for example the space withinwhich we perceive objects. Other possible forms of L ( v ) = 1 include the determinantsof matrices, which are homogeneous polynomials in the matrix elements.The complex numbers C had been studied by Hamilton in the 1830s in amanner consistent with his view of algebra as the science of pure time. This program10n part led to his discovery of the quaternions in the 1840s, which also however ledhim to essentially abandon the notion of a close relation between algebra and timeowing to the non-commutative property of the quaternion algebra. Subsequently an8-dimensional algebra, the octonions O , was discovered independently by Graves andCayley in the mid-1840s and completed the unique series, R , C , H and O , of normeddivision algebras [1], as will be reviewed in section 6.2. In fact division algebras onlyexist over vector spaces of dimension 1 , , A is a division algebra if ab = 0 implies a = 0 or b = 0, with a, b ∈ A ; it is a normed division algebra if A is also anormed vector space with | ab | = | a || b | . This latter property naturally provides a sourceof structures of form of equation 2.9 together with the corresponding symmetries.For example, the quaternion algebra H may be used to compose a possiblemulti-dimensional form of progression in time. On the space of unit norm elements v ∈ H , with L ( v ) = | v | = 1, the symmetry group G composed of quaternions of unitnorm operating on v under left and right algebra multiplication forms the two-to-onecover of SO(4). The 1-dimensional character of temporal flow is represented by the‘norm’ function L applied collectively to the components of v ∈ H ≡ R ; with the non-commutative behaviour of the σ g symmetry operations within L ( . . . σ g ′′ σ g ′ σ g ( v )) = 1describing the properties of the multiple apparently ‘internal’ temporal dimensions.For the case in which G is homomorphic to an orthonormal rotation group (as is thecase for H representing three or four dimensional space, with for example the threeimaginary units of the quaternions associated with 3-dimensional Euclidean space)the non-commutative algebraic properties correlate directly with the non-commutativeproperty of spatial rotations for n > O , being non-associative, do notthemselves form a group in such a direct way as for the complex numbers or thequaternions; they will however play a significant role in the symmetry of time andhence in physics as will be explained in this paper. Here the division algebras will becombined with matrix algebras in considering the 27-dimensional real vector space of3 × O over the octonions with the determinant required to beunity: L ( v ) = det( X ) = 1, with X ∈ h O . The group G of determinant preservingsymmetry transformations on h O is the exceptional Lie group E . This group is wellknown to be of interest for unification models and will be discussed in detail in thecontext of the present investigations in chapters 6–8.With various different forms of progression in time to be considered, in generalthe subscript n in the notation L ( v n ) = 1 indicates collectively the vector space R n ,the implied form L and the corresponding symmetry group G (respectively v ≡ X ∈ h O ≡ R , L ( v ) = det( X ) = 1 and G = E in the above example for n = 27), whereany case of ambiguity will be clarified in the text.Given a possible n -dimensional form of progression in time, L ( v n ) = 1, thevector v n ∈ R n may be written as the ordered set of velocities: v n = { v , v , . . . v n } (2.10)= (cid:26) dx ds , dx ds , . . . dx n ds (cid:27) (2.11)the values of which are unchanged by a numerical translation of the real variables, x a → x a + r a (2.12)11or any constant set { r a } = r n ∈ R n , or for a subset of R n . Above we described apossible symmetry of L ( v ) = 1 with the action of a group G mixing the numericalcomponents v a , which represent elements of the temporal flow dx a /ds . Here we havea further symmetry implicit in L ( v ) = 1 with respect to translations of the numericalvariables as x a → x a + r a . That is, we also have trivially: v n = (cid:26) d ( x + r ) ds , d ( x + r ) ds , . . . d ( x n + r n ) ds (cid:27) . (2.13)satisfying L ( v n ) = 1. For the 1-dimensional case of equation 2.1 the symmetry v = d ( x + r ) ds can be readily visualised as a flow v present everywhere on the real lineparametrised by r ∈ R (rather than at a single arbitrary point for example). In thegeneral case since equation 2.13 is equally valid for all possible r n ∈ R n the temporalflow, under the condition L ( v n ) = 1, effectively occupies the entire R n manifold asdepicted in figure 2.2.Figure 2.2: Since the real variables { x a } ∈ R n are arbitrary, the flow described within L ( v n ) = 1 applies equally for the particular value x ∈ R n as for x ′ = x + r n andover the range −∞ < r a > ∞ , for a = 1 . . . n . This ‘translation’ symmetry is impliedwithin the form L ( v n ) = 1 and is depicted here for n = 3.This n -dimensional freedom in R n forms a continuous n -dimensional parameterspace, which may be considered to form an implicit ‘base manifold’ M n , upon which thevector v n naturally resides in the tangent space T x M n at every point x ∈ M n . Hencethe internal structure of the form L ( v n ) = 1 and its symmetries contain the skeletalform of a mathematical framework for the description of an apparently external andextended spatial structure.In other theories and models a higher-dimensional symmetry of spacetime isconsidered, extending beyond our familiar 4-dimensional spacetime arena to one witha total of, for example, five or ten spacetime dimensions. Such models, initiated byKaluza and Klein, will be described in more detail in chapter 4. In these theories it is12ecessary to explain how our 4-dimensional spacetime world is embedded in the largerarena, and the means by which the ‘extra dimensions’ are compactified or otherwiseevade direct observation.As described in the introductory chapter we are familiar with the idea that notonly all of our scientific experiments but also everything we experience in the worldtakes place in time. Relative to 4-dimensional spacetime the flow of pure time is anapparently ‘lower-dimensional’ structure which pervades all observations and eventsin the universe. This is in contrast to hypothetical extra dimensions, above the fourof space and time, which are beyond our domain of experience. Here we begin on afirm footing by treating one-dimensional temporal flow as the fundamental entity ofthe world.Hence, in contrast with Kaluza-Klein theory, for the theory presented in this pa-per we deal instead with a general higher-dimensional symmetry of time , and it is herenecessary to explain how the large scale extended perception . It is the means throughwhich time experienced as a purely 1-dimensional progression can be experienced si-multaneously as a multi-dimensional flow of physical objects in an extended spacetime.The mathematical basis for obtaining such an extended base manifold will be found inthe application of the symmetry described in figure 2.2 to a 4-dimensional spacetimesubset of the translational degrees of freedom of a higher-dimensional temporal form.For the case considered for the real world, in addition to the 27-dimensionalspace h O described above another important example of a form of time involvingboth a matrix and a division algebra is identified in the determinant of elements ofthe 4-dimensional real vector space h C ≡ R , that is the 2 × , C ). This group is the double cover of the Lorentz group and will also besignificant in this paper since h C is naturally embedded as a subspace of h O , withthe symmetry group SL(2 , C ) being a subgroup of E .Applying the translation symmetry of equation 2.13 in four dimensions only,corresponding to the h C components, provides a natural mechanism for breaking thesymmetry of the larger group through the necessary identification of a 4-dimensionalbackground manifold M upon which the Lorentz group acts locally, and to a goodapproximation globally over extended regions of spacetime. Under the overall normali-sation L ( v ) = 1 the 4-dimensional form will take more general values L ( v ) = h ∈ R for the subcomponent v ⊂ v local tangent vectors on M (in this paper the rela-tion v ′ ⊂ v between two vectors will denote the projection of v ′ out of v ). Furtherconsequences of the symmetry breaking are associated with the necessary choice ofa particular direction for the vector field v ( x ) ∈ h C , locally a 1-dimensional flowembedded within a 4-dimensional manifold. Comparisons between these symmetrybreaking structures and the Standard Model of particle physics will be made in chap-ter 8. The relation between the ‘translation symmetry’ of L ( v ) = 1 and the ‘rotationsymmetry’, more generally denoted by the action σ g ( v ) for g ∈ G , is key to thedevelopment of the geometrical structure of the theory and motivates the review of13lements of textbook geometry in chapter 3. We begin in the following two sectionsby describing a simple model universe, based on a small number of dimensions inorder to elaborate upon the nature of the geometric structures involved, in particularconcerning the identification of the base manifold. The geometric properties of thismanifold, which are significant in general relativity, are intrinsically related to thegeometry and symmetries of the residual dimensions, which are significant for gaugetheories, resulting from the projection of a higher-dimensional form of L ( v ) = 1 overthe base manifold and corresponding symmetry breaking pattern, as will be describedin section 2.3. This development of the theory will be continued in chapter 5 wherethe relation between the external gravitational field and internal gauge fields over a 4-dimensional spacetime manifold in the context of the present theory will be described. The fact that all of our experiences in the world are encompassed within the passageof time motivated the formulation of the general expression for temporal progression, L ( v ) = 1, presented as equation 2.9 of the previous section. However it is also necessaryto account for the fact that all of our experiences of such a physical world appear to bedistributed through an extended manifold , with the immediate and necessary locationof observed physical objects in space , as well as in time . While the general mathematicalform for the flow of time may be exemplified by a wide range of mathematical structuresand symmetry groups it is the identification of relatively simple structures, those whichmay be most readily suited to the organisation and understanding of experiences inthe world with respect to a background arena of space as well as time, that will bedesignated by the term perception .The apparent physical form of the world is shaped out of the interplay betweenthese two basic notions: that of the mathematical form of temporal flow and that of anecessary form of perception. It is the act of interpreting algebraic structures withinthe temporal flow v in terms of an extended coherent geometrical structure that breaksthe symmetry of the general flow of time described by L ( v ) = 1.In this and the following section the discussion will be maintained largely ata general level with a simplified model universe, a world with two spatial dimensionsonly, being used to make the presentation more concrete for a case which is math-ematically simpler than our own world and, in particular, one which may be morereadily visualised. The notion of a base manifold may be introduced by consideringhow it would be possible for physical objects in a spatially 2-dimensional world tobe perceived propagating in time. This situation brings to mind the image picturedin figure 2.3. (Such illustrations clearly also serve by analogy to represent our ownworld, with one spatial dimension being suppressed. Indeed, throughout this chapterthe model universe described should be considered both as a metaphor for the generalcase and for our own world in particular).Objects in such a world are here depicted by figures in a 2-dimensional planewhich are animated, presumably according to certain laws of physics in the form ofequations of motion, as they propagate through the third dimension on a 3-dimensionalbase manifold M . The geometrical structure of the 2-dimensional plane may be con-14igure 2.3: A representation of a model universe with 2-dimensional physical objectspropagating through a 3-dimensional base manifold.sidered to be compatible with the notion of spatial perception of objects by beings inthis model world if it possesses, at least to a good approximation, an SO(2) rotationalsymmetry about any point as well as translational symmetry in this plane. Hence thelocal symmetry group G of the manifold M must:i) contain as a subgroup the symmetry of the purely spatial structure of the world;here the group SO(2),ii) act on a space of one dimension higher than that of the spatial geometry; in thiscase 3-dimensional, andiii) be a possible symmetry group or subgroup of a form L ( v ) = 1 in order to conformwith the present conceptual ideas.For our model universe we begin with the 3-dimensional form of temporal flow: L ( v ) = ( v ) + ( v ) + ( v ) = 1 (2.14)that is with L ( v ) = η ab v a v b = 1 and the 3-dimensional metric η ab of equation 2.3 asintroduced in the previous section. The full 3-dimensional translational symmetry ofthis form depicted in figure 2.2 provides the framework for an extended 2-dimensional‘spatial’ environment, in addition to the temporal one, constituting the backgroundmanifold M . Ultimately a metric with a ‘spacetime’ signature will be required inorder to incorporate causal structure on the base manifold, however this feature isneglected for the simple model presented in this chapter. For the case of the modelworld an unbroken external symmetry SO(3) will be described in this section, beforeextending to a larger symmetry SO(5) over the same 3-dimensional base manifold inthe following section. 15 .2.1 The Base Manifold The metric η ab implies the existence of an orthonormal basis { e a } with respect to whichthe pure temporal flow v can be expressed in terms of the components { v a } ∈ R as: v = v a e a = dx a ds e a = d ( x a + r a ) ds e a (2.15)With a 3-dimensional translation symmetry −∞ < r a > ∞ as depicted in figure 2.2the orthonormal basis projects over the base manifold as an orthonormal frame fieldon M . This smooth differentiable manifold naturally possesses a tangent space T x M at each point x ∈ M , that is the space M has the properties of a 3-dimensional basemanifold of a tangent bundle space, as we shall discuss further in section 3.3 for thegeneral and 4-dimensional spacetime cases.The assignment v ( x ) = v a e a is valid for a local orthonormal coordinate basisor a frame field (with index a for such an orthonormal frame, here a = { , , } ).General coordinates on the manifold naturally give rise to a coordinate basis for thetangent space { ∂ µ } , with ∂ µ ≡ ∂/∂x µ , (with index µ for general coordinates, here µ = { , , } ). Relabelling the parameters { r a } ∈ R in equation 2.15 as a particularset of ‘general coordinates’ x µ = δ µa r a on M , there is an implied coordinate frameon the base manifold { ∂ µ } such that v = v a e a of equation 2.15 can be expressed as v = v µ ∂ µ = v a e µa ( x ) ∂ µ , with the ‘triad’ components e µa ( x ) = δ µa .More generally under a passive reparametrisation to any general coordinates { x µ } ∈ R in a region of M a frame field consists of a triad of vector fields e a = e µa ( x ) ∂ µ (for a = 1 , ,
3) with components with respect to the general coordinateframe given by the matrix function e µa ( x ) which points to the local Euclidean metricstructure at any x ∈ M . The set of components e µa ( x ) contains the same informationas its matrix inverse e aµ ( x ), and either of these matrices are sometimes referred to as the‘triad’ itself. These matrices transform both under general coordinate transformationsand local, or gauge , SO(3) transformations.The kernel symbol v will usually denote a vector or vector field correspondingto the fundamental flow of time in the form L ( v ) = 1, while the kernel symbol u will denote arbitrary tangent vector fields, such as u ( x ) = u µ ( x ) ∂ µ , as indicated infigure 2.4. Either type of vector field may be expressed either in a local orthonormalframe or in a general coordinate frame. The components a vector field u ( x ) belongto the space R whether presented in a local or a general coordinate basis; these twopossibilities are related by the matrix e aµ ( x ) ∈ GL(3 , R ) such that u a ( x ) = e aµ ( x ) u µ ( x ).Through a frame field e a ( x ) on M the flow of time described numericallyby v ( x ) is isomorphic to an external tangent vector field which may be described interms of general coordinates on M , and may be considered to be a flow of time on thismanifold space itself, even for the case in which the global geometry is not Euclidean.This latter situation will arise when the local tangent space on M is embedded withina higher-dimensional form of temporal flow, as described in the following section. Inthis case M will necessarily be treated as a differentiable manifold with finite curvaturein general for which only the local geometry at any point x ∈ M will be isomorphicto the Euclidean geometry of R .Via the triad field e aµ ( x ) the internal space constant metric η ab of equation 2.3implied in equation 2.14, similarly as for the vector components v a , may be expressed16igure 2.4: A local orthonormal basis e a ≡ ∂/∂x a for the vector field v ( x ) = v a ( x ) e a ( x ) is related to any other such basis of the same orientation at each point x ∈ M by the action of the symmetry group G = SO(3), which can vary arbitrarilyover M . In general a tangent vector field u ( x ) on the manifold may be expressed interms of an arbitrary frame field, or a particular orthonormal or general coordinateframe.on the tangent space for a general coordinate basis. This determines the metric tensor: g µν ( x ) = e aµ ( x ) e bν ( x ) η ab . (2.16)In the theory of general relativity it is the freedom of the metric field g µν ( x ), orequivalently the tetrad field e aµ ( x ), on a 4-dimensional spacetime base manifold as willbe described for equation 3.50, with respect to an arbitrary coordinate system thatdescribes gravitation in the world, as we shall review in section 3.4. While in generalthe components of g µν ( x ) differ from those of η ab in a general coordinate system, evenfor a flat spacetime, in general relativity it is the absence of any global coordinate basissuch that g µν ( x ) = δ aµ δ bν η ab everywhere that is responsible for gravitational effects.In order to consider the curvature of the base manifold it is necessary to for-malise the notion of parallelism . The question concerns the way in which the basemanifold M originates out of the flow of time as depicted in figure 2.2, specificallywith n = 3 for the model case here, such that the symmetry G = SO(3), acting uponindividual vectors v ( x ) ∈ R in the equation L ( v ) = 1 can act as an approximately global symmetry over scales that are large compared with the objects being perceived.In section 2.1 we began with finite intervals of multi-dimensional time, as depicted infigure 2.1, and then went on to the infinitesimal case s → δs in order to derive the rela-tion L ( v ) = 1 of equation 2.9. We here need to understand how the symmetry of suchinfinitesimal intervals can apply coherently over finite distances on the manifold M .This is required on the manifold in order to frame stable perceptions of 2-dimensionalspatial objects propagating through such a world as depicted in figure 2.3.17n terms of the model world the point is that since we are locally free tochoose an orthonormal frame within which to specify the numerical values v a ( x ) forthe components of v ( x ) ∈ T M , the values themselves have no absolute meaning. Inparticular, no conclusion concerning the equality, or parallelism, of two sets of vectorcomponents v a ( x ) and v a ( x ) at two different points x , x ∈ M in figure 2.4 maybe drawn since the bases of local frames at x and x may be chosen independently,within the local SO(3) freedom of the relation L ( σ g ( x ) ( v ( x ))) = 1. A triad frame(such as e a described in equation 2.15 for the coordinate frame x µ = δ µa r a ) could bedeclared to specify a parallelism on the manifold (that is, v ( x ) is parallel to v ( x )if each of the components agree, v a ( x ) = v a ( x ), in the specified triad frame e a ( x )).However for the case of a curved space or spacetime no global frame field e a ( x ) existsin a manner compatible with the parallelism, since the latter now depends on the pathtaken between x and x .The underlying notion of parallelism is more generally defined in terms of a connection x , x ∈ M , and hence determines whether any two vectors at these locationsare parallel with respect to the path. With such a structure the relative values of thecomponents of two vectors at differing locations does acquire meaning. Hence we wishto identify an SO(3) connection form A ( x ) on the base manifold M . We describehow a flat connection arises canonically on M through it’s relation to the symmetrygroup G in subsection 2.2.3 (and further in section 3.2 in the context of the principlebundle structure). In the following subsection we first review the standard geometryon a group manifold itself. While the { r a } ∈ R translational symmetry of L ( v ) = 1 gives rise to the basemanifold M , the algebraic structure of the rotational symmetry constitutes a seconddifferentiable manifold which is identified with the Lie Group G = SO(3) itself. Thismanifold is also intimately related to the temporal flow v through the expression L ( v ) = L ( σ g ( v )) = 1, with the action g ∈ G realised on the subspace of unit normvectors in R .Elements of a general Lie group g ∈ G also act as diffeomorphisms on themanifold G itself [2, 3, 4]. An example is the diffeomorphism L g : G → G , mappingthe point h → gh with g, h ∈ G , called ‘left translation’ on the manifold. Due tothe nature of the algebraic properties of a symmetry group a Lie group manifold G exhibits distinctive canonical geometrical structures. The significance of ‘canonical’ (inthe sense of intrinsic or naturally existing) structures, where relevant, is that they carrythe mathematical development of a theory forward in a necessary and non-arbitraryway. As for the base manifold described above, the group G as a manifold also has atangent space T g G at each point g ∈ G , through which a tangent vector field V ( g ) maybe described on G . Smooth vector fields X ( g ) belonging to the subset which satisfythe relation: L g ∗ X ( h ) = X ( gh ) (2.17)18or all g, h ∈ G , where L g ∗ is the ‘tangent mapping’, or differential, of the left trans-lation L g (acting upon objects defined on the tangent space of G ), are said to beleft-invariant. The set of left-invariant vector fields together with their multiplicationin terms of the commutator [ X, Y ] (considering the vector fields
X, Y as mappings inthe space of scalar functions f ( g ) on G ), which itself describes a left-invariant vectorfield, defines the Lie algebra L ( G ) of the Lie group G . As a vector space L ( G ) isisomorphic to the set of tangent vectors at any location on G , and in particular tothe space T e G , where e ∈ G is the identity element of the group. Given any point h ∈ G the orbit of left translation for all g ∈ G covers the entire group manifold, as aconsequence of the transitive property of multiplication within a Lie group, and hencethe corresponding tangent mapping L g ∗ carries any vector V ( h ) into a left-invariantvector field on G .In general a 1-form ω ( x ), or covector field, on a differentiable manifold M mapsa vector field V ( x ) into the space of real functions on M ; this map may be denotedby: h ω ( x ) , V ( x ) i = f ( x ) (2.18)at any point x ∈ M . Over the manifold G a linearly independent set of left-invariantvector fields, { X α } with α = 1 . . . n G = dim( G ), forms a global frame field on G . Adual basis of 1-forms { θ α } with α = 1 . . . n G such that h θ α , X β i = δ αβ , constitutes acoframe field on G . These covector fields are also left-invariant with L ∗ g θ α ( gh ) = θ α ( h )for the ‘pull-back’ L ∗ g of the left translation by g ∈ G .The ‘exterior algebra’ of differential forms includes the exterior product ‘ ∧ ’ andexterior derivative ‘d’ which act on 1-forms such as ω ( x ) = ω µ d x µ and σ ( x ) = σ µ d x µ to produce 2-forms such as: ω ∧ σ = ω ⊗ σ − σ ⊗ ω (2.19)and d ω = ∂ω µ ∂x ν d x ν ∧ d x µ (2.20)For any diffeomorphism between manifolds, f : M → N (where it may be that M = N ), the pull-back map f ∗ is a structure preserving homomorphism of the exterioralgebra. Hence, as for the 1-form basis covectors θ α ( α = 1 . . . n G ), the 2-forms d θ α and θ β ∧ θ γ are also left-invariant on G and are therefore related via left-invariant, thatis constant, scalar coefficients c αβγ as defined in:d θ α + 12 c αβγ θ β ∧ θ γ = 0 . (2.21)This is the Maurer-Cartan equation which also serves to define the Lie algebrastructure constants c αβγ , with respect to the basis { θ α } . It is equivalent to the defini-tion of the structure constants in terms of the dual basis of vector fields { X α } , whichrepresents the Lie algebra itself, in the relation:[ X β , X γ ] = c αβγ X α . (2.22)The Maurer-Cartan 1-form θ is a single, basis independent, canonical object onthe manifold G that expresses the properties of the collective set of n G { θ α } .It is a Lie algebra-valued 1-form defined by its action on a general tangent vector field19 = V α ( g ) X α on G with h θ, V ( g ) i := V α X α ∈ L ( G ), where V α are the componentvalues of V at g and V α X α is hence the Lie algebra element corresponding to theleft-invariant vector field on G with the tangent vector V ( g ) at the given point g ∈ G .The Maurer-Cartan form can be written as θ = θ α X α in terms of the dualbases. The canonical form θ encapsulates the parallelisable nature of any Lie groupmanifold by defining a consistent global parallelism on G . That is, θ represents a singlereference frame for each of the tangent spaces which resolves each vector V ( g ) at any g ∈ G into its components with respect to a left-invariant frame field { X α } . For amatrix group such as SO(3), with a matrix basis { E α } for L ( G ), the Maurer-Cartanform can also be expressed as the left-invariant matrix of 1-forms θ = g − d g . In thiscase h θ, V ( g ) i is the Lie algebra element V α E α represented in matrix form in terms ofthe components of the corresponding left-invariant vector field at T e G .In terms of θ the Maurer-Cartan equation 2.21 can be written as:d θ + 12 [ θ, θ ] = 0 (2.23)where the bracket denotes the ‘exterior product’ for Lie algebra valued 1-forms. Such aproduct may be defined on vector-valued p -forms in general provided there is a productdefined on the vector space of the values. This is the case for Lie algebra valued formswhere, with θ = θ α X α and φ = φ β X β and with { X α } a basis for L ( G ), the product isdefined as: [ θ, φ ] := θ α ∧ φ β [ X α , X β ] . (2.24)For a matrix basis { E α } and the case of a single 1-form as for equation 2.23 the product [ θ, θ ] = θ ∧ θ which implicitly incorporates the multiplication of the E α matrices.On the group manifold G each left-invariant field X generates a one-parametersubgroup described by the flow φ t = exp( tX e ), where X e ∈ T e G denotes the tangentvector belonging to the field X at the identity e ∈ G and ‘exp’ is the ‘exponential map’from L ( G ) into the manifold G . The action of this one-parameter group on any point h ∈ G is by right translation, as indicated in figure 2.5.Alternatively, a left-invariant field X A on G can be induced by the right action R g : h → hg of elements of the one-parameter group g ( t ) = exp( tA ), with A ∈ T e G such that: X Ah ( f ) = ddt f ( h exp( tA )) | t =0 (2.25)where f is a real-valued function on the manifold G . Left-invariant fields are sometimesdenoted by a label ‘ R ’ since they are generated by right translations; hence X R denotesa left-invariant field.Since a right-invariant field Y L (which can be generated by left translation) isby definition invariant under right translations the Lie derivative of Y L with respectto the vector field X R vanishes: L X R Y L = ddt φ ∗ t Y L | t =0 = 0 (2.26)that is: [ X R , Y L ] = 0 (2.27)For each g ∈ G a further diffeomorphism on the group manifold called theadjoint map can be defined by Ad g ( h ) = ghg − for all h ∈ G , that is Ad g = L g ◦ R g − ≡ G generate right translations. R g − ◦ L g by the associative property of group composition. The adjoint map is an automorphism of the group composition. Since Ad g ◦ Ad g = Ad g g this is a leftaction of G on itself.The adjoint map applied to elements near e ∈ G gives rise to the group rep-resentation Ad g = L g ∗ ◦ R g − ∗ | e ≡ R g − ∗ ◦ L g ∗ | e acting upon the Lie algebra of thegroup. For a group represented by matrices this takes the form Ad g ( Y ) = gY g − forany Y ∈ L ( G ). The adjoint representation g → Ad g is a group homomorphism of G into GL( L ( G )). The ‘derived homomorphism’ of this representation induces the corre-sponding adjoint representation for the Lie algebra elements with ad X Y = [ X, Y ] for
X, Y ∈ L ( G ), as an automorphism of the Lie bracket algebra, which naturally involvesthe structure constants of the group through equation 2.22.Finally we note that a left-invariant field X R , which also generates right trans-lations, itself transforms under right translation as R g ∗ X R = Ad g − X R by the defi-nition of left-invariance and the adjoint representation (while L g ∗ X L = Ad g X L for aright-invariant field X L ). These group properties will be important for the structureof principle bundles described in section 3.1. As for the basis { X α } on the manifold G , a frame field { e a } , with a = 1 . . . n , maybe introduced on any n -dimensional differentiable manifold M n , forming a linearlyindependent set of tangent vectors at each point of the manifold. The real quantities c abc ( x ) in the relation: [ e b , e c ] = c abc ( x ) e a (2.28)or equivalently in: d e a = − c abc ( x ) e b ∧ e c (2.29)in terms of the dual coframe field { e a } , are here variables called structure coefficients (or ‘coefficients of anholonomy’) rather than constants as for equations 2.21 and 2.22.21iven a general coordinate chart { x µ } on M n and a holonomic frame e µ = ∂ µ thecorresponding coefficients c ρµν ( x ) are all zero, while c abc ( x ) = 0 implies that a non-coordinate frame is being employed.On the manifold G frames composed of left-invariant vector fields { X α ( g ) } wereidentified as being particularly important owing to the group structure. On the basemanifold M , possessing the metric of equation 2.16, basis vectors forming orthonormal frames { e a ( x ) } are particularly significant. As described in subsection 2.2.1 such atriad frames the components of v subject to the pure numerical relation L ( v ) = 1 ofequation 2.14, which implicitly contains the local 3 × η as expressedin equation 2.3. (In this paper indices a, b, c . . . for a basis { e a } will denote an arbitrarysmooth frame field, as for M n above, or an orthonormal frame field, as for M here, oreven a coordinate basis depending on the context; while indices µ, ν, ρ . . . for a basis { e µ } ≡ { ∂ µ } will always denote a coordinate frame).The two manifolds M and G = SO(3), representing the translational androtational symmetries of the form L ( v ) = 1, as described in section 2.1 and in thetwo subsections above, are linked through the mapping g ( x ) : M → G . An initialorthonormal frame field { e a ( x ) } can be transformed to any other orthonormal framefield { e ′ b ( x ) } by the matrix action e ′ b = e a g ab via the group element g ( x ) ∈ SO(3) atevery x ∈ M . The map g ( x ) : M → G , as depicted in figure 2.6, expresses the localchoice of an orthonormal frame field { e a ( x ) } , essentially the choice of local { x , x , x } axes of figure 2.1 at each point x ∈ M , as a basis for tangent vectors v ∈ T M . Itis this ‘gauge’ freedom g ( x ) ∈ SO(3) in the choice of local orthonormal frames thatprevents a particular frame { e a ( x ) } from directly representing parallelism on the basemanifold, as described towards the end of subsection 2.2.1.Figure 2.6: The gauge choice of a frame at each x ∈ M described as a map intoelements g ∈ G between the two manifolds.22ince the operations of the exterior algebra of p -forms are preserved under thepull-back of forms through diffeomorphism maps on manifolds the Lie algebra-valued1-form: A = g ∗ θ (2.30)on M captures the structural properties of the Maurer-Cartan 1-form θ on G relativeto the map g . While on G we have the linear map h θ, V ( g ) i ∈ L ( G ) from V ( g ) ∈ T g G into the Lie algebra of G , on M we have the linear map h A ( x ) , u ( x ) i ∈ L ( G ) from u ( x ) ∈ T x M into the same Lie algebra. The Lie algebra-valued 1-form A ( x ) maybe written as A ( x ) = A αµ ( x ) X α d x µ where { d x µ } is a coordinate basis of 1-formson M and { X α } is a basis for L ( G ). In the appropriate 3 × { E α } ≡ { L pq } , labelled by a single indexmnemonic double letter symbol pq = { , , } , with:( L pq ) ab = δ pa δ qb − δ pb δ qa (2.31)where a and b label the matrix rows and columns respectively, that is: L = − , L = − , L = − (2.32)Unlike the canonical 1-form θ = θ α X α on G , the 1-form A on M has variablereal coefficients A αµ ( x ) which, however, are not arbitrary but depend upon the choiceof gauge function g ( x ) as well as upon the choice of coordinates { x µ } on M . Explicitly,for a matrix group G , the 1-form A = g ∗ θ on M can be expressed as: A ( x ) = g − d g = g − ∂g∂x µ d x µ (2.33)It is this canonical mathematical object that serves as a connection 1-form on thebase manifold M , formalising the notion of parallelism in manner which will naturallygeneralise for the case of finite curvature. Here it is possible to choose a gauge with A ( x ) = 0 everywhere on M , simply by taking g ( x ) to be constant in equation 2.33,and hence we have a flat connection. Indeed, this connection can always be writtenin terms of ‘pure gauge’, as it is in equation 2.33, which is one way of defining a flatconnection (to be described in more detail in section 3.2). Given a connection A ( x ) = 0a gauge transformation via any g ( x ) transforms the connection in the standard wayas: A → A ′ = g − Ag + g − d g. (2.34)which can be expressed as pure gauge A ′ = g ′− d g ′ , that is in the form of equation 2.33,in terms of an appropriate gauge function g ′ ( x ).By the homomorphism of exterior algebra relations across the pull-back mapthe Lie algebra-valued 1-form A = g ∗ θ is also subject to a structure equation corre-sponding to equation 2.23, that is:d A + 12 [ A, A ] = 0 (2.35)23n general the curvature 2-form F on the base manifold can be expressed as: F = d A + 12 [ A, A ] (2.36)which transforms under a gauge change g ( x ) as F → F ′ = g − F g . Equations 2.35and 2.36 then immediately show that the curvature is equal to zero, with F = 0 inany gauge, and further expresses the global parallelism implied by the canonical flatconnection of equation 2.30.While the connection 1-form can be written as A ( x ) = A αµ ( x ) E α d x µ andcurvature 2-form can be written as F ( x ) = F αµν ( x ) E α d x µ ∧ d x ν (where the factorof arises from the convention of equation 2.19 and the double counting implicit onthe right-hand side since the set of asymmetric d x µ ∧ d x ν { µ, ν } = { , , } , does not describe a linearly independent basis).Interest in the group G = SO(3) arose as a symmetry action on the form L ( v ) = 1 and hence the Lie algebra values of A ( x ) and F ( x ) are composed of elements { E α } ≡ { L pq } of equation 2.32 in a representation of L ( G ) acting naturally upon thevectors u ∈ T M , that is on the tangent space of the base manifold, and in particularon the vector v ∈ T M originating in the form L ( v ) = 1. The mathematical objectsinvolved are hence intimately associated with each other, with the base space M andthe flat connection A ( x ) = g ( x ) ∗ θ of equation 2.30 upon it arising out of the translationand rotation symmetry properties of L ( v ) = 1, with the vectors v themselves beingtangent to M .As an example of this association the constancy of the scalar function L ( v ) = 1on M can be expressed as ∂ µ L ( v ) = 0, or, consistent with the gauge transformationsof v ′ = g − v and equation 2.34, covariantly as: D µ L ( v ) = D µ ( v · v ) = 2 v · D µ v = 0 (2.37)with D µ v = ∂ µ v + A µ v (2.38)The ‘covariant derivative’ D µ relating to a connection A µ will be defined more preciselyin the following chapter, leading to equation 3.6. These above two equations show howthe connection field A µ ( x ) explicitly acts on the vector field v ( x ) in a constrainingrelation, and hence there is a ‘coupling’ between these fields over the base manifold M . A vector field u ( x ) which satisfies D µ u = 0 everywhere represents a parallelvector field on the manifold. A frame field { e a } that satisfies D µ e a = 0 for each valueof a defines a parallel frame field – in which case the frame field itself may be usedto define the parallelism on the manifold, which is only possible for a flat connection.With respect to the original global coordinates defined in terms of a parametrisationof the translation symmetry of equation 2.15 on M the triad field with components e µa ( x ) = δ µa was identified. The covariant derivative of the corresponding orthonormalbasis vectors e a is constant with respect to the connection form A ( x ) = 0, with constant g ( x ) in equation 2.33, and hence { e a } defines the parallelism in this case. The geometricobjects e µa ( x ) = δ µa and A ( x ) = 0 may both be associated with the constant gaugefunction g ( x ) on M taken as the identity element e ∈ G = SO(3) of the group. A gaugechange by a constant g ( x ) = e ∈ G changes the frame { e a } but not the connection24 ( x ) = 0. Under a general gauge change g ( x ) the connection, with A ( x ) = 0 in general,can be written as an explicit function of the triad field.In summary the canonical connection A = g ∗ θ , constructed as depicted infigure 2.6, defines a global parallelism on M (as θ does on the manifold G ), suchthat the parallel transport of a vector u ( x ) from x to another point x on the basemanifold, see for example figure 2.4, results in a definite vector u ( x ) independent ofthe path taken. This formalises the notion of parallelism on the base manifold in amanner which can be generalised for the case of non-global parallelism.Since A ( x ) is SO(3)-valued in acting on the tangent space T M it also describesa ‘metric compatible’ connection. In being completely determined by the triad fieldthe connection A ( x ) is also torsion-free, where torsion will be defined in section 3.3. Infact since A ( x ) is a particular case of a linear connection acting on the tangent spacethe curvature 2-form F of equation 2.36 may be identified with the Riemann tensorand hence denoted R . The so(3)-valued curvature tensor R = R αµν E α d x µ ∧ d x ν on M , with α = 1 . . . n G , then has components R abµν = R pqµν ( L pq ) ab . Via the triad fieldthe same tensor can be expressed either fully in a local orthonormal frame or fully ina general coordinate frame – in the latter case with four general coordinate indices as: R ρσµν = e aρ e bσ R abµν (2.39)Hence we have constructed a zero Riemann curvature tensor with all components R ρσµν = 0 as implicit in the identification of a canonical flat connection A = g ∗ θ .In section 3.1 we shall review the geometry of a principle fibre bundle, henceincorporating M as the base manifold and G as the structure group together in asingle manifold, before reviewing Riemannian geometry itself. A non-zero Riemanniancurvature will ultimately be obtained on the original base manifold by expanding theform L ( v ) = 1 into a higher-dimensional temporal flow with a larger symmetry group,as we provisionally describe in the following section. In the previous section the construction of a model world required that we drew atten-tion to the particular form of temporal flow L ( v ) = 1, as expressed in equation 2.14,as an example of the general n -dimensional case. It was shown how v could be in-terpreted as a tangent vector field over a base manifold M , represented in figure 2.4,which in turn may be parametrised by a set of real number coordinates x µ ∈ R , andwith a choice of a local orthonormal reference frame { e a ( x ) } determined within thefreedom of the local SO(3) symmetry.However, in general there are many higher symmetry groups acting upon vectorspaces of a larger dimension, with elements conforming to L ( v ) = 1, which we have nomathematical reason to neglect. Indeed, the reasoning of section 2.1 is consistent withthe flow of time being channelled into a space of arbitrarily large dimension. Hence,mathematically, there is nothing to prevent the 3-dimensional space of parameters v ∈ R , representing a 3-dimensional flow of time, from further dividing into a largermulti-dimensional space of parameters described by the vector v n ∈ R n ( n >
3) subjectto a new form L ( v n ) = 1 with a higher symmetry group G . (In later chapters the25xpression L (ˆ v ) = 1 will denote the full form of temporal flow being considered, whilethe full symmetry group, excluding translations, may be denoted ˆ G for clarity, as forthe remainder of this section).The original SO(3) geometric symmetry group may now be identified as asubgroup H ⊂ ˆ G , with the ‘overline’ denoting an external symmetry, acting on the3-dimensional flow v which is projected onto the tangent space of the base manifold M out of the higher-dimensional temporal flow. In this section we begin to considerthe conceptual implications and mathematical possibilities of this generalisation forthe necessary existence of such a higher symmetry group ˆ G acting upon a higher-dimensional form of temporal flow L ( v n ) = 1.Since it will ultimately be required to mathematically support the kind ofsituation depicted in figure 2.3, in which the smaller symmetry H ⊂ ˆ G is treated in adistinctive way in giving rise to the global geometrical nature of a perceived universe ofphysical objects, we shall expect to be dealing with a natural mechanism for breakingthe higher symmetry. The full ‘rotation’ symmetry of the form L ( v n ) = 1, as ageneralisation of that depicted in figure 2.1, is now broken since only the degrees offreedom of a subset of the possible dimensions of translation symmetry, depicted infigure 2.2, is employed to locally construct the base manifoldIn subsequent chapters, for the real world, we shall motivate the choice of ˆ G asthe Lie group E , acting on a 27-dimensional form L ( v ) = 1, with a Lorentz subgroupacting on the local tangent space of the 4-dimensional spacetime base manifold M . Inthe meantime here, for the model world, we shall take ˆ G to be a symmetry group ofa form of L ( v n ) = 1 large enough to contain SO(3), the orthonormal frame symmetrygroup, as a subgroup of ˆ G , while retaining the 3-dimensional base space M .For the case of the model universe H = SO(3), acting upon v ∈ R , could betaken to be embedded within various kinds of larger groups, for example ˆ G = SU(3)acting upon the 6 real components of v corresponding to a 3-dimensional complexvector c ∈ C with L ( v ) = c † c = 1. However, here we consider the vectors v ∈ R of section 2.2 to be vectors in a subspace of R n with n > n ), the latter being a perfectly acceptablesymmetry of L ( v n ) = 1, acting upon the vectors v n ∈ R n . In particular we choose n = 5 and consider the Lie group ˆ G = SO(5) acting on the form L ( v ) = 1: L ( v ) = v · v = ( v ) + ( v ) + ( v ) + ( v ) + ( v ) = 1 (2.40)= η ab v a v b + ( v ) + ( v ) = 1 (2.41)where a, b = 1 . . . η ab = diag(+1 , +1 , +1) represents the 3-dimensional Euclideanmetric, which was introduced in equation 2.3 of section 2.1.The 5-dimensional vector v has components v a = dx a /ds , with a = 1 . . . L ( v ) = 1 implicitly contain the 5-dimensional translationalfreedom x a → x a + r a , { r a } ∈ R , as a particular example of equation 2.13 and figure 2.2which generalises the 3-dimensional case of equation 2.15. However, we consider onlythe 3-dimensional freedom of this parameter space and continue to take M to be thebase space as we did in section 2.2 and as depicted for the present case in figure 2.7.This choice will ultimately be justified by the identification of geometrical structures on M which may then be interpreted as the base space for perception of physical events assketched in figure 2.3. The model described in figure 2.7 provides a convenient picture26or a provisional discussion of the symmetry breaking structure which will be pickedup again in section 5.1 for the more realistic case over a 4-dimensional spacetime basemanifold M .Figure 2.7: (a) The gauge choice at each x ∈ M depicted in figure 2.6 is extended forˆ G = SO(5) with (b) only the subgroup SO(3) ⊂ SO(5) now acting on
T M .A basis for the 10-dimensional Lie algebra so(5) (with the lower case ‘so’ de-noting the Lie algebra corresponding to the SO(5) Lie group), as represented on a5-dimensional vector space as the generators of a symmetry of L ( v ) = 1, is providedby the set of ten 5 × L pq ) ab of the type described in equation 2.31, now withten distinct labels composed out of p, q = 1 . . . , p < q . The three so(5) Lie algebraelements: L = − , L = − , L = − (2.42)generate an SO(3) ⊂ SO(5) subgroup, as can be see directly as guided by the horizontaland vertical lines drawn into the matrices in equation 2.42 and by comparison withequation 2.32. This SO(3) subgroup can be taken to act on the tangent space of M and hence upon the subspace of vectors v ⊂ v projected onto the base space. Ofthe other seven SO(5) generators one acts purely on the complementary 2-dimensional27ubspace v ⊂ v , namely: L = − (2.43)With the subspace of vectors v ⊂ v with v ∈ T M in the external space the SO(2)generator L can be said to act upon the internal v ∈ R , which can be considered to represent extra dimensions over those requiredto describe the extended base space M . In general while an ‘overline’ denotes anexternal object an ‘underline’ will denote an object defined in the internal space. Inthis representation basis the remaining six so(5) Lie algebra elements are: L = − , L , L , L , L , L = − (2.44)This final set of six matrices generate SO(5) group elements that mix the external v and internal v parts of the full 5-dimensional temporal flow v = (cid:0) v v (cid:1) . The vectors v and v are physically distinct with respect to the M base space. The full SO(5)symmetry is hence broken down to:SO(3) × SO(2) ⊂ SO(5) (2.45)as the external SO(3) symmetry, represented by the generators of equation 2.42 ‘lockson’ to the tangent space
T M leaving the residual internal symmetry SO(2), repre-sented by the generator in equation 2.43, as depicted in figure 2.7(b). Hence only fourof the original ten generators of SO(5) survive the symmetry breaking.For the original unbroken full symmetry the constancy of L ( v ) = 1 can beexpressed, in comparison with equations 2.37 and 2.38, as the vanishing of the covariantderivative of L ( v ) = 1 on the base manifold:ˆ D µ L ( v ) = 0 ⇒ v · ∂ µ v + v · ˆ A µ v = 0 (2.46)where the ‘hat’ on ˆ D µ and ˆ A µ ( x ) signifies that the unbroken 10-component so(5)-valuedconnection 1-form on M is being considered. With the identification of a Riemanniancurvature tensor on M six of the gauge field generator degrees of freedom are lost andthe broken, physical, form of equation 2.46 can be written as: D µ L ( v ) = 0 ⇒ v · ∂ µ v + v · A µ v + v · Y µ v = 0 (2.47)where A µ ( x ) represents the external so(3)-valued connection 1-form and the gaugefield Y µ ( x ) describes an internal so(2)-valued connection 1-form. The final term in28quation 2.47 expresses an ‘interaction’ between the gauge field Y µ ( x ) and the internaltemporal components v ( x ) which follows directly from the ‘minimal coupling’ betweenthem implicit in the covariant derivative (as a generalisation from the purely externalfield coupling described for equations 2.37 and 2.38). The structure of the pattern ofinteractions for the breaking of the E symmetry of L ( v ) = 1 over the base manifold M for the real world will be described in chapter 8.Here for the model, a 3-dimensional projection of the full temporal flow v ⊂ v will form a tangent vector field on the base manifold M . While | v | = p L ( v ) is fixedin equation 2.40 the quantity | v | in principle has a variable magnitude, however herewe mainly focus on the breaking of the group symmetry action and in particular therelation between the geometry of the resulting external and internal curvature.Considering first the case of figure 2.7(a) with an unbroken set of ten SO(5)Lie group generators the algebra product is given by the following expression, whichis valid in general for the orthogonal SO( n ) groups:[ L pq , L rs ] = δ qr L ps − δ qs L pr − δ pr L qs + δ ps L qr (2.48)From this the so(5) algebra structure constants c pqrs tu in this basis can be read off,for example c = − L , L ] = − L . Since the group SO(5) is connectedand compact any element g ∈ SO(5) in this matrix representation can be expressed as g = exp( α pq L pq ) (2.49)with ten real coefficients α pq (summation is implied over repeated index combinations,for the ten labels { p, q = 1 . . . , p < q } , however the ‘upper’ or ‘lower’ location of theseindices is of no significance). The ‘exponential map’ was described in the discussionaround figure 2.5 in the context of left-invariant vector fields on the group manifold.Here with the Lie algebra represented by real matrices L = α pq L pq ∈ L ( ˆ G ) the expo-nential map may be explicitly written as exp( L ) = P ∞ k =0 1 k ! L k which converges to amap from L ( ˆ G ) → ˆ G .The elements of equation 2.49 satisfy the relation gg T = (where ∈ ˆ G is the identity element of the group as represented by the 5 × g ) = 1, as required for the special orthogonal group SO(5). As an example, withthe notation c pq = cos α pq and s pq = sin α pq the element g = exp( α L ) has theform: g = c s
00 1 0 0 00 0 1 0 0 − s c
00 0 0 0 1 (2.50)The Maurer-Cartan 1-form θ , now defined on the manifold of the full gaugesymmetry group ˆ G = SO(5), can be pulled-back onto the base manifold M as de-scribed in subsection 2.2.3. In this way we canonically identify a flat so(5)-valued con-nection 1-form ˆ A ( x ) = g ∗ θ and an so(5)-valued curvature 2-form ˆ F ( x ) = d ˆ A + [ ˆ A, ˆ A ] =0, following equations 2.35 and 2.36, where the ‘hat’ on A and F here again denotequantities involving the full SO(5) symmetry group.29eginning with a choice of constant gauge g c ( x ) : M → SO(5), with fixed g c ∈ SO(5), we have ˆ A ( x ) = g ∗ c θ = 0 from equation 2.33. Under a more general gaugetransformation of the form of equation 2.49 but with in particular g = exp( α L )and with small values of the function α ( x ), we have from equation 2.34:ˆ A ′ ( x ) = g − ˆ A ( x ) g + g − d g (2.51) ≃ d α ( x ) L for ˆ A ( x ) = 0 (2.52)This can be written ˆ A ′ ( x ) = A ( x ) L with A ( x ) = d α ( x ) = ∂ µ α ( x )d x µ as the1-form coefficient of the Lie algebra element L . Applying a full sequence of all six‘mixing’ actions in the order g = g g g g g g (that is with g first and g last, each generated by an element of equation 2.44 and all being functions of x ∈ M )a 1-form coefficient for each of the ten Lie algebra basis elements L pq may be found.This is equivalent to taking ˆ A ( x ) = g ∗ θ = g − d g . To order O ( α pq d α rs ) for smalltransformations the ten coefficients of this connection ˆ A ( x ) = A pq ( x ) L pq are found as: A = − α d α − α d α , A = − α d α − α d α , A = − α d α − α d α ,A = d α , A = d α , A = d α , A = d α , A = d α , A = d α ,A = − α d α − α d α − α d α (2.53)The full so(5)-valued connection 1-form ˆ A ( x ) = A pq ( x ) L pq , summing over the ten 1-form coefficients in equation 2.53, is ‘unphysical’ in the sense that it is ‘pure gauge’with respect to the full SO(5) symmetry, and merely presents the same original flatconnection, for which we had all ten A pq ( x ) = 0, in a different choice of gauge, namely g ( x ) ∈ SO(5).Of more significance is the structure and interpretation of the ten componentsof the curvature 2-form ˆ F in the new gauge. These can be written using the gen-eral expression for the curvature 2-from coefficients in terms of the connection 1-formcoefficients (consistent with equation 2.36):ˆ F α = d A α + 12 ˆ c αβγ A β ∧ A γ (2.54)The Lie algebra basis indices for L (SO(5)) are here denoted by { α, β, γ } = 1 . . . α = 1 , . . . corresponding to pq = 12 , . . . ) and the structure constants may beread off from equation 2.48. For the internal curvature 2-form coefficient ˆ F ( x ) in thegauge g ( x ), using the asymmetry in the βγ indices of the structure constants and ofthe exterior product of 1-forms, we find:ˆ F = d A + 12 ˆ c β γ A β ∧ A γ = d A + ˆ c A ∧ A + ˆ c A ∧ A + ˆ c A ∧ A = d A − A ∧ A − A ∧ A − A ∧ A = d α ∧ d α + d α ∧ d α + d α ∧ d α − d α ∧ d α − d α ∧ d α − d α ∧ d α = 0 (2.55)where in the penultimate line the connection coefficients from equation 2.53 have beensubstituted into this expression. This result is as expected and indeed zero curvature,30 F pq = 0 for all ten 2-form coefficients, is associated with the connection 1-form A pq coefficients, in any gauge choice such as that for equation 2.53, so long as the full10-dimensional Lie algebra of SO(5) is retained.At each point on M the Lie algebra so(5) acts partly on the external space T x M of the base manifold in figure 2.7(a), via an SO(3) ⊂ SO(5) subgroup, and partlyon the internal space through the complementary SO(2) ⊂ SO(5), while the remaininggenerators of equation 2.44 straddle the external and internal parts of v ∈ R . Bychoosing an SO(5) gauge in figure 2.7(a) and then breaking the symmetry througha projection onto the structure in figure 2.7(b) the aim here is to demonstrate hownon-zero curvature may be associated with the SO(3) and SO(2) subgroups.While an so(5)-valued connection ˆ A ( x ) provides a means for the parallel trans-port of a 5-dimensional vector v ( x ) over M for figure 2.7(a), a single componentsuch as A ( x ) L can be interpreted in the restricted sense of a representation in thesubgroup SO(2) ⊂ SO(5), that is ˆ A → A = A L with L ∈ L (SO(2)), acting on thesubspace of 2-dimensional vectors v ⊂ v . In this case A ( x ) is a 1-form connectiondescribing the parallel transport of vectors v ( x ) in the internal vector space over M for figure 2.7(b).Hence treating the generator L in isolation from the other nine generatorsas a purely SO(2) action on the internal space of vectors v ∈ R a curvature 2-form F = F L may be identified for this restricted internal SO(2) symmetry. Incomparison with equation 2.55, for the restricted SO(2) subgroup a finite curvaturemay be obtained: F = d A = d α ∧ d α + d α ∧ d α + d α ∧ d α = 0 in general. (2.56)That is, under a suitable choice of gauge parameters α pq ( x ) in the full SO(5) symmetryit is possible to identify non-zero curvature components with respect to subgroupssuch as SO(2), which may be interpreted as a structure with finite internal physicalcurvature over the base space manifold.To obtain equation 2.56 we effectively took a restricted set of structure con-stants, which is trivial for the Abelian subgroup SO(2) ⊂ SO(5) with a single generatorand hence there are no c αβγ terms in place of the ˆ c αβγ terms of equation 2.54. Thisresults in a non-zero internal curvature F = 0 for the subgroup H under what waspurely a change of gauge from the point of view of the full group ˆ G . In this way a finiteSO(2) curvature is essentially carved out of the degrees of freedom implicit within thestructure of the unbroken flat SO(5) connection.Complementary to the subgroup SO(2) ⊂ SO(5) the SO(3) subgroup is gen-erated by the Lie algebra elements of equation 2.42. Acting on the external vectorcomponents v ( x ) ∈ T x M the connection 1-forms A L , A L and A L , asso-ciated with the three generators of SO(3), define parallelism on the external tangentspace of the manifold M . However here we temporarily follow the same approachapplied to the SO(2) case in leading to equation 2.55, hence treating this SO(3) as an‘internal’ symmetry in order to examine any new features that arise for a non-Abeliangauge group such as SO(3). Again, initially substituting the full set of SO(5) connec-31ion coefficients from equation 2.53 into equation 2.54, we find ˆ F = ˆ F = ˆ F = 0as for all ten so(5)-valued curvature components as discussed above.A purely SO(3) Lie algebra-valued curvature 2-form can be obtained, similarlyas for the SO(2) case, by using only the restricted Lie algebra values of the connectionacting purely on the subspace of vectors v ⊂ v to identify the curvature tensor F = F pq L pq , with pq = 12 , ,
23, over M . The three components are obtained fromequation 2.54 by curtailing the summations to include only the so(3)-valued parts witha restricted set of structure constants { c αβγ } ⊂ { ˆ c αβγ } describing the external SO(3)symmetry only. Following a similar procedure that led to equation 2.56 a set of threeso(3)-valued curvature coefficients is found to lowest non-trivial order: F = d α ∧ d α + d α ∧ d α F = d α ∧ d α + d α ∧ d α F = d α ∧ d α + d α ∧ d α (2.57)Hence the SO(3) curvature is also non-zero in general and clearly correlatedwith the SO(2) curvature of equation 2.56, with the correlation mediated through thesix mixing gauge functions α pq ( x ), with pq = 14, 15, 24, 25, 34, 35 under the fullSO(5) symmetry. Either the SO(3) or the SO(2) curvature may be non-zero while theother remains zero for a suitable choice of the α pq ( x ), while both F = 0 and F = 0are simultaneously attained under any choice of SO(5) gauge with constant α pq ( x ) forexample.The six full SO(5) curvature components ˆ F pq = 0 for pq = 14, 15, 24, 25, 34,35 are not directly associated with subgroup restrictions giving rise to further finitecurvature components. Rather these six mixing degrees of gauge symmetry are lostor broken in the projection of the full SO(5) symmetry over M through which finitephysical curvature for the four components in equations 2.56 and 2.57 is identified. Thefour functions α pq ( x ), with pq = {
12, 13, 23 } and 45, corresponding to four actions g pq = exp( α pq L pq ) of the gauge symmetry, survive the symmetry breaking and areretained as the gauge symmetries associated with the non-Abelian SO(3) and AbelianSO(2) subgroups respectively.However the mechanism of symmetry breaking over M itself implies that thereis a more fundamental difference between the SO(3) and SO(2) subgroups in the contextof the model world we are considering. The former acts externally on the tangent spaceof the base manifold, that is on v ∈ T M as depicted in figure 2.7(b), and is thereforeclosely related to the geometry of the background space itself and to the existenceof a linear connection on the base manifold. Hence the correlation observed abovebetween the SO(3) and SO(2) curvature, with both effectively treated as internal symmetries, merely provides a provisional motivation for seeking a unified frameworkin which SO(2) remains as an internal gauge symmetry while SO(3) is consideredas an external symmetry on T M . A non-zero curvature for the internal symmetry F = 0 in equation 2.56 was obtained by considering the generator of the subgroup H = SO(2) within the full gauge symmetry group ˆ G = SO(5) over a fixed base space M . A non-zero Riemannian curvature with components R ρσµν ( x ) = 0 on M willtransform locally under the complementary external subgroup SO(3) ⊂ SO(5) actingon the tangent space
T M . 32n this chapter the base manifold M and group manifold G (where here G maybe the full symmetry ˆ G of the full form L ( v n ) = 1 or either the internal H or external H subgroups) have been treated as largely independent geometric objects, however theirmutual relationship is more precisely defined in terms of a single manifold in the formof a principle bundle with base space M and structure group G . Hence the standardproperties of these geometric objects, together with a review of Riemannian geometryand the Lagrangian approach to obtaining equations of motion for the correspondingfield entities, with be presented in the following chapter.A relationship between the internal gauge curvature and external Riemanniancurvature might be determined through such a principle fibre bundle P = ( M , ˆ G ) withbase space M and structure group ˆ G , based on the picture in figure 2.7(a), with thecanonical zero full curvature ˆ F = 0 providing the constraint that relates the internaland external geometry. However the physical situation is represented by figure 2.7(b)which leads to a consideration of two detached bundle spaces, P = ( M , SO(3)) and P = ( M , SO(2)). While P directly only contains information about the externalsymmetry and curvature, the bundle space P explicitly contains both the structureof the external geometry on M and that of the internal H = SO(2) curvature in thebundle space. As a unifying framework for combining external and internal symmetriesthis latter structure is very similar to that employed in non-Abelian Kaluza-Kleintheories, which are hence reviewed in chapter 4.In chapter 5 we consider how the above structures of principle bundles andKaluza-Klein theory might be adapted for the present theory. There we shall upgradethe model presented in this chapter by considering the real world situation with a4-dimensional base space M with local Lorentz symmetry. This will be embeddedin the ‘full’ symmetry group taken as ˆ G = SO + (1 , L ( v ) = 1, which will be broken to SO + (1 , × SO(6) in theprojection onto the spacetime base M . This symmetry breaking structure naturallyembeds in the further higher-dimensional extensions considered from chapter 6, whichwill provide a more realistic framework for the details of the internal structures also.33 hapter 3 Review of Geometry andEquations of Motion
In section 2.2 we introduced two independent differentiable manifolds, the base space M and Lie group G = SO(3) with points labelled by x ∈ M and g ∈ G respectively,both of which are associated with the 3-dimensional form of temporal flow L ( v ) = 1through the respective ‘translational’ { r a } ∈ R and ‘rotational’ σ g : g ∈ G symmetries: L (cid:18) σ g (cid:26) d ( x a + r a ) ds (cid:27)(cid:19) = 1 (3.1)The map between the manifolds g : M → G described in figure 2.6, mapping x → g ( x )(where { x µ } may be taken as general coordinates on M ) represents a local choiceof gauge, or orthonormal frame, in which to express the tangent vector v ( x ) on M . This association between M and G may be examined more precisely throughthe construction of a single differentiable manifold, namely a principle fibre bundle P , which combines the geometric properties of the base space M and Lie group G together with their mutual relation.In the general case the structure group G of a principle fibre bundle P does notneed to be related to a symmetry on the tangent space to the base manifold M , as it isfor the SO(3) model as implied in equation 3.1. Indeed in the case of figure 2.7(a) thesymmetry group G = SO(5) acts only partially on the tangent space of M , while forfigure 2.7(b) the group G = SO(2) does not act on the external tangent space at all.Hence it is the generalisation in which G and M are initially introduced independentlythat we shall review here for the benefit of the subsequent application to the case of ahigher symmetry group such as presented in section 2.3.A principle bundle P is a G -manifold, that is a differentiable manifold uponwhich the transformation group G acts, with a particular structure as described, with34eference to figure 3.1, by the following properties (see for example [2], [3] chapterVbis, [4]):Figure 3.1: The relations between three differentiable manifolds: a principle fibrebundle P , the base space M and the structure group G .1: There is a surjective map π : P → M projecting from the bundle space onto thebase manifold. Given a section σ ( x ) : M → P then π ◦ σ ( x ) = x is the identitymap on points x ∈ M .2: For each x ∈ M the submanifolds π − ( x ) ⊂ P , called the fibres of P , are diffeo-morphic to each other and to the Lie group G .3: The right action of g ∈ G on points p ∈ P , that is R g : P → P with R gh = R h ◦ R g ,preserves the fibres of P , that is π ◦ R g = π , and is free and transitive on eachfibre.4: There exist local trivialisations over each open subset U r ⊂ M , consisting ofmaps ψ r : π − ( U r ) → U r × G with ψ r : p → ( x, h ), such that ψ r : pg → ( x, hg ) –that is the right action on P is compatible with the right action on G .While the right action of G on itself induces left-invariant fields as describedin equation 2.25 of subsection 2.2.2, the right action of G on the manifold P induces‘vertical’ vector fields in the tangent space TP as (where f is now a real-valued functionon the bundle space): V Ap ( f ) = ddt f ( p exp( tA )) | t =0 (3.2)where A ∈ L ( G ) and V Ap is a tangent vector to the fibre of P at the point p . Themap A → V Ap described in equation 3.2 represents an isomorphism of the Lie algebra L ( G ) into the space of vector fields residing in the vertical tangent space VP ⊂ TP .That is, the Lie algebra bracket structure [ X A , X B ] = X [ A,B ] of equation 2.22 for thecorresponding left-invariant fields { X A } on the manifold G is respected by the Liebracket on the P bundle with: [ V A , V B ] = V [ A,B ] (3.3)35his structure relates to VP , the space of vectors tangent to the individualfibres of P . Different fibres may be related by an additional structure on P called a connection which, conceptually, is smooth assignment of a ‘horizontal’ subspace H p P of the full tangent space T p P at each point p ∈ P such that: T p P = V p P ⊕ H p P (3.4) R g ∗ H p P = H pg P (3.5)where compatibility of the horizontal subspaces on P with the right action of G isassured by the latter requirement.At every point p ∈ P a basis for the tangent space of the principle bundle canbe expressed in terms of these complementary subspaces. Such a basis { ´ e i } = { ´ e α , ´ e a } consists of the subset { ´ e α } ∈ VP (that is vectors of the form V Ap in equation 3.2,tangent to the fibres G x over each point x ∈ M ) and the subset { ´ e a } ∈ HP , where´ e a is the ‘horizontal lift’ of the basis vector e a ∈ T x M to the point p ∈ P such that π ∗ ´ e a = e a . The ‘acute’ mark above the kernel symbol, such as for ´ e , denotes an objectdefined on a principle bundle space in the horizontal lift basis. In all cases the indices { i, j, k . . . } correspond to basis elements for TP in the total space; { α, β, γ . . . } in thefibre space on P or on the manifold G ; and { a, b, c . . . } in a complementary subspaceon P or on the base space M .It should be noted that ´ e a and e a are not only different vector fields but arealso defined on two different manifolds, P and M respectively, although there is a one-to-one correspondence between them. Similarly, there is a one-to-one correspondencebetween a vector field ´ e α on P and a vector field X α on G , for example as generatedby the same element A ∈ T e G in equations 3.2 and 2.25 respectively. The relationsbetween these vector fields are indicated in figure 3.2.Figure 3.2: Vertical and horizontal basis vectors in a local trivialisation U × G of aprinciple bundle P , together with their associated basis vectors on the group space G and the base manifold M respectively.The specification of a connection on the principle bundle allows ‘parallel trans-port’ between the fibres to be defined by a path in P for which the tangent vector at36ny p ∈ P always lies within the horizontal subspace H p P . This notion of parallelismover M is used in turn to define a covariant derivative for associated fields φ ( x ) on thebase space that transform under a representation of the structure group G , by trackinga parallel basis for the field φ ( x ) over any curve C on the base manifold (technically, φ ( x ) is a section in a fibre bundle associated with P ).As depicted in figure 3.3 given a point p ∈ P with π ( p ) = x and a curve C on M from x to x a connection on P specifies a unique horizontal lift of the curve C to the curve C ′ on P , by advancing locally within the horizontal subspace HP ⊂ TP .The path C ′ then represents the ‘parallel transport’ of p mapped to the unique point p ∈ P , with π ( p ) = x .Figure 3.3: Parallel transport C ′ between fibres on a principle fibre bundle P abovethe curve C on the base manifold from the point x ∈ M to x ∈ M .The geometric structure developed in section 2.2 corresponds to a particularkind of principle bundle, namely a frame bundle with structure group G = SO(3)over the base space M , which may be denoted P = ( M , SO(3)). In this case themapping from p to p in figure 3.3 provides a unique, path C dependent, transportof an orthonormal basis frame from x to x on the base manifold (such basis frames { e a } are shown in figure 2.4 for the model on M ). With respect to such a parallelframe the difference between the values of the vector field v ( x ) (belonging to thevector representation of SO(3)) at the two base points of the associated vector bundlecan be determined. In particular vectors v ( x ) and v ( x ), as originally depicted infigure 2.4, are defined to be parallel with respect to a given path C and connection HP if each of their components coincide in an orthonormal reference frame transportedfrom x to x along C via the horizontal lift C ′ . This definition of parallelism isindependent of the choice p ∈ π − ( x ) of initial frame. In addition, any vector v ( x )may be ‘parallel transported’ to any point of the curve C by maintaining constant37ector components in the corresponding transported frame of the principle bundle ateach point along C .This construction generalises for an arbitrary structure group G acting viaa group representation on the field φ ( x ) over the base manifold M . The covariantderivative of the field φ ( x ) is defined in a such a way as to quantitatively indicatedeviations of the value of the field function from that of the parallel transported fieldfor infinitesimal displacements on the base manifold – i.e. the extent to which thefield is not self-parallel along a path on the base manifold. That is, if c is a vectorat x tangent to the curve C parametrised by λ → C ( λ ) on M with C (0) = x , seefigure 3.3, then the covariant derivative of the field φ ( x ) along C at x is defined as: D c φ | x = lim λ → ( T λ, φ ( x λ ) − φ ( x )) /λ (3.6)where T λ, φ ( x λ ) is the field value φ ( x λ ) ≡ φ ( C ( λ )) parallel transported along C from x λ to x . The covariant derivative for the SO(3) connection applied to vectors v ( x )in equation 2.37 and 2.38 was denoted D µ corresponding to derivatives with respectto general coordinate parameters { x µ } on M .If the connection is such that, for all p ∈ P and all X, Y ∈ H p P , the bracketcomposition on H p P is closed, that is:[ X, Y ] ∈ H p P (3.7)then Frobenius criterion is satisfied and P is ‘foliated’ into a family of integrable‘leaves’. Each leaf is ‘horizontal section’ of P , with tangent space HP , that is a smooth n -dimensional submanifold of P , where n is the dimension of the base space M . In thiscase all horizontal lift curves C ′ of figure 3.3 effectively follow the contours of a singlehorizontal section submanifold, globally defined over M , and parallelism is independentof the path C taken between any two points on the base manifold. A ‘flat’ connectionis defined by this property, as will be described in more detail in the following section.Generally a horizontal subpace can be specified by a connection 1-form ω on P . The defining structure for HP of equation 3.4 and 3.5 can be attained via a smoothLie algebra-valued 1-form ω ∈ L ( G ) ⊗ T ∗ P , mapping vectors X ∈ TP into elements of L ( G ), with the properties (the first of which is essentially the reverse of equation 3.2):(i) ω ( V A ) = A with A ∈ L ( G ) (3.8)(ii) R ∗ g ω = Ad( g − ) ω i.e. R ∗ g ω pg ( X ) = g − ω p ( X ) g with X ∈ T p P (3.9)where H p P ≡ { X ∈ T p P | ω ( X ) = 0 } (3.10)is the horizontal subspace. A set of trivialisations { U r , ψ r } consists of an atlas { U r } covering the base manifold M together with a mapping ψ r of each π − ( U r ) ⊂ P onto U r × G such as depicted figure 3.1 and described in the subsequent ‘item 4:’.Each trivialisation ψ r is canonically associated with a section in π − ( U r ) which can bewritten as σ r ( x ) = ψ − r,x · ι ( x ), where the map ι : U r → U r × G sends x → ( x, e ) r , with e ∈ G being the identity element of the group and ψ r,x is the map ψ r restricted to thespace π − ( x ). That is, σ r ( x ) ∈ P corresponds to the identity element e ∈ G under thelocal trivialisation map ψ r , as depicted in figure 3.4. More generally we have the map ψ r,x : P → ( U r × G ) mapping between the points σ r ( x ) h → ( x, h ) r , with h ∈ G .38igure 3.4: Two trivialisations, denoted { U r , ψ r } and { U s , ψ s } , with an overlap region,on a principle bundle P over a base space M .In the overlap regions on M , for x ∈ U r ∩ U s , transition mappings g rs : U r ∩ U s → G between such trivialisations are defined as the functions on M : x → g rs ( x ) = ψ r,x ◦ ψ − s,x ∈ G (3.11)The transition functions act on the left on a fibre such that ψ − r,x ( x, g rs h ) r = ψ − s,x ( x, h ) s .These relate the corresponding canonical sections σ r ( x ) via the right action of the struc-ture group on P , which commutes with the left action, in a way that is consistent withboth ψ s ( σ s ( x ) h ) = ( x, h ) s and ψ r ( σ r ( x ) h ′ ) = ( x, h ′ ) r in the respective trivialisations,as: σ s ( x ) = σ r ( x ) g rs ( x ) (3.12)Given a general connection 1-form ω on P and a set of trivialisations { U r , ψ r } aunique family A r of connection 1-forms may be defined on M . Under a particular sec-tion σ r the connection A r ( x ) = σ ∗ r ω ( p ) on M can be expressed as A r ( x ) = A α ( x ) X α = A αµ ( x ) X α d x µ where { X α } is a basis for L ( G ). The field of connection coefficients A αµ ( x ) link the basis { X α } for the Lie algebra, typically expressed in the appropriaterepresentation (such as the set of matrices { E α } of equation 2.32 for the case of thevector representation of G = SO(3)), with indices α = 1 . . . dim( G ), to a coordinate ba-sis of 1-forms { d x µ } with indices µ = 0 , , , M . Further, given an L ( G )-valued 1-form A ( x ) on M and any section σ ( x ) then there exists a unique connection 1-form ω ( p ) on P such that A = σ ∗ ω .In a gauge theory, that is a theory which is invariant under transformations ofthe gauge group G which describes a local internal symmetry, the local Lie algebra-valued 1-form A ( x ) on the base manifold M is also known as a Yang-Mills field or ‘gaugepotential’. Such fields will be generically denoted Y ( x ) in this paper, as for examplein equation 2.47. The notation A ( x ) may refer to a general connection 1-form, asdescribed above, the gauge field associated with an internal U(1) gauge symmetry, asfor electromagnetism, or a connection associated with an orthonormal frame in the39xternal space, as was the case in subsection 2.2.3 and as will be the case in relationto general relativity, depending on the context. A gauge theory bases upon an internalsymmetry, through the notion of a connection 1-form, involves similar mathematicalstructures as found in general relativity based upon an external symmetry.For the present theory it will be assumed that the structure of principle bundleswith a trivial global topology will be sufficient. In this case the bundle P = ( M, G ) canbe expressed as P ≡ U × G where a single ‘subset’ U of figure 3.4 may be identifiedwith the entire base manifold M . This triviality is implied in deriving the bundlestructure through the symmetries of L ( v ) = 1 as described for example in figures 2.2,2.6 and 2.7. In this case the ‘overlap region’ for a change of trivialisation, or gaugetransformation, may consist of the entire volume of the base space M , rather than alimited patch as depicted in figure 3.4.In order to study the dynamics of the gauge fields it is helpful to introducethe exterior covariant derivative on the bundle space, which will be important forequations in physics. This derivative essentially combines the properties of the exteriorderivative, introduced for equation 2.20, with the structure of the partial derivative ∂ µ as augmented to the covariant derivative D µ , as described for equation 3.6.More explicitly, on a principle bundle P the exterior covariant derivative Dmaps a V -valued r -form φ , which acts upon r vector fields { X . . . X r } on P , to a V -valued ( r + 1)-form D φ , where V is the representation space associated with G . Theaction of D is defined as: D φ = (d φ ) ◦ hor (3.13)where ‘hor’ first maps vectors X on the tangent space of P to their horizontal compo-nents (that is, hor : X → X h , such that X h ⊂ H p P of equation 3.4 and ω ( X h ) = 0,with ω the connection 1-form on P ), and d is the exterior derivative map acting onthe r -form φ .In general, for a V -valued r -form φ on P which is horizontal (that is if any of the r vectors X on which φ acts is purely vertical then the map is zero, φ ( X, . . . ) = 0) and equivariant of type ρ (that is φ in the associated bundle transforms as R ∗ g φ = ρ ( g − ) φ under the right action by g ( x ) in the ρ representation) then the exterior covariantderivative of φ , equation 3.13, takes the simplified form:D φ = d φ + ρ ′ ( ω ) ∧ φ. (3.14)where ρ ′ ( ω ) denotes the appropriate representation of the Lie algebra acting on V . The curvature 2-form Ω can be defined as the exterior covariant derivative of theconnection 1-form, that is Ω = D ω , on the principle bundle. The connection 1-form ω is equivariant, of type Ad as seen in equation 3.9, but it is clearly not a horizontalform, as seen in equation 3.8. However, for this particular case the exterior covariantderivative of the connection 1-form can also be expressed in a simplified form directlyin terms of ω = ω α X α itself (with { X α } a basis for L ( G )) through the Cartan structure40quation for the curvature 2-form Ω = Ω α X α on P :Ω( X, Y ) = D ω ( X, Y ) = d ω ( X, Y ) + [ ω ( X ) , ω ( Y )] (3.15)= d ω ( X, Y ) + 12 [ ω, ω ]( X, Y ) (3.16)that is Ω α ( X, Y ) = d ω α ( X, Y ) + 12 c αβγ ω β ∧ ω γ ( X, Y ) (3.17)acting upon any pair of tangent vectors
X, Y ∈ TP (the meaning of [ ω, ω ] is explainedin the discussion around equation 2.24). The Lie algebra-valued curvature 2-formΩ on P is defined in such a way as to be quantitatively sensitive to deviations of theconnection 1-form ω , and hence horizontal subspace HP , from the condition of flatness.This can be seen by substituting any X, Y ∈ HP as arguments for the curvature 2-formΩ in equation 3.15, or equivalently for any X, Y ∈ TP in terms of Ω = D ω we have:Ω( X, Y ) = D ω ( X, Y ) (3.18)= d ω ( X h , Y h ) (3.19)= X h h ω, Y h i − Y h h ω, X h i − h ω, [ X h , Y h ] i (3.20)= −h ω, [ X h , Y h ] i (3.21)where equation 3.19 follows directly from equation 3.13, equation 3.20 follows from thestandard definition of the exterior derivative of a 1-form and the map h , i was definedin equation 2.18. Hence it follows that Ω is non-zero only if the local horizontalsubspaces on P defined by ω are non-integrable, that is the Frobenius criterion ofequation 3.7 is not satisfied, and hence a non-zero curvature Ω indeed indicates anon-flat connection.The Lie algebra valued curvature 2-form Ω on P is equivariant of type Ad, thatis it transforms under the adjoint representation of G as R ∗ g Ω = Ad( g − )Ω. However,unlike the connection ω , the curvature Ω is also a horizontal form on P . Hence Ω, unlike ω , is a tensorial form meaning that, for a given choice of gauge or cross-section σ overa region of the base manifold, it can be mapped via the pull-back σ ∗ to a geometricalobject on the base manifold that transforms homogeneously as a representation of thegauge group. Such quantities may be more naturally equated in the expressions ofphysics. For the model universe of the previous chapter the curvature form of theSO(3) connection on P is tensorial of type (Ad, so(3)), that is it takes values in theSO(3) Lie algebra and transforms under the adjoint representation, as indicated afterequation 2.36 for the curvature form F on the base manifold.In a trivialisation P ≡ U × G on the principle bundle a direct product basis { ¨ e i } = { ¨ e α , ¨ e a } for the tangent space consists of the subset { ¨ e α } ∈ VP , tangent tothe fibres G x over each point x ∈ M , and the subset { ¨ e a } with ¨ e a = σ ∗ e a for eachbasis vector e a ∈ T x M such that π ∗ ¨ e a = e a . Each vector ¨ e a defined on the section σ ( x ) is Lie transported via the right action of G on P such that the basis covers theentire principle bundle. The ‘double dot’ mark above the kernel symbol, such as for ¨ e ,denotes an object defined on a principle bundle space in the direct product basis.Since P itself is a differentiable manifold equation 2.28 applies for any framefield { e i } on P and is here expressed as:[ e j , e k ] = c ijk ( p ) e i (3.22)41ith real-valued structure coefficients c ijk ( p ). In the direct product basis the bracketrelations [¨ e j , ¨ e k ] = ¨ c ijk ( p )¨ e i are simply:[¨ e α , ¨ e β ] = c γαβ ¨ e γ (3.23)[¨ e α , ¨ e b ] = 0 (3.24)[¨ e a , ¨ e b ] = 0 (3.25)where c γαβ are the structure constants of the group G . The zero coefficients for thesecond equation follow as the vector fields ¨ e α generate the right translations whichLie transport the vectors ¨ e b over P , and those in the final equation correspond to thechoice of a coordinate basis on M .By contrast the horizontal lift basis ´ e i = (´ e α , ´ e a ) for the tangent space TP ,introduced after equation 3.4, is adapted to a given connection ω such that ´ e α ∈ V p and ´ e a ∈ H p , as was depicted in figure 3.2, with ω (´ e α ) = X α and ω (´ e a ) = 0, by thedefinition of the horizontal lift basis. Given a trivialisation the horizontal lift basis { ´ e i } can be expressed in terms the direct product basis { ¨ e i } on P via the coefficients ω αa ( x, g ) with: ´ e α = ¨ e α , ´ e a = ¨ e a − ω αa ¨ e α (3.26)´ e α = ¨ e α + ω αa ¨ e a , ´ e a = ¨ e a (3.27)where { ´ e i } = { ´ e α , ´ e a } is the dual basis defined as usual such that h ´ e i , ´ e j i = δ ij . Therelation between the horizontal lift basis { ´ e i } on P and a direct product basis { ¨ e i } on U × G is indicated in figure 3.5.Figure 3.5: The adapted tangent space basis { ´ e i } on P with respect to a particularlocal trivialisation ψ : P → U × G and the corresponding direct product basis { ¨ e i } .Acting on both sides of the second expression in equation 3.26 with the 1-formcoefficients ω β of the Lie algebra-valued connection 1-form ω = ω β X β on P determinesthe connection coefficients ω βa ( x, g ) = ω β (¨ e a ) on U × G , as depicted in figure 3.5. Fromthe transformation property of the connection 1-form ω on P under R ∗ g in equation 3.9and with the vector field ´ e α generating right actions on P , it follows that:´ e α ω βa = ¨ e α ω βa = − c βαγ ω γa (3.28)42s the infinitesimal form of the adjoint transformation under the right action of thegroup. Covariant differentiation on the base space is intimately related to the direc-tional derivative ´ e a on the principle bundle. Using equation 3.26 the bracket [´ e a , ´ e b ]may be expressed in a direct product basis as:[´ e a , ´ e b ] = [(¨ e a − ω αa ¨ e α ) , (¨ e b − ω βb ¨ e β )]= [¨ e a , ¨ e b ] − [ ω αa ¨ e α , ¨ e b ] − [¨ e a , ω βb ¨ e β ] + [ ω αa ¨ e α , ω βb ¨ e β ]= 0 + ¨ e b ( ω αa )¨ e α − ¨ e a ( ω βb )¨ e β + ω αa ω βb c γαβ ¨ e γ + ω αa (¨ e α ω βb )¨ e β − ω βb (¨ e β ω αa )¨ e α = ¨ e b ( ω γa )¨ e γ − ¨ e a ( ω γb )¨ e γ + ( ω αa ω βb c γαβ ¨ e γ − ω αa c γαβ ω βb ¨ e γ + ω βb c γβα ω α a ¨ e γ )= (cid:0) ¨ e b ω γa − ¨ e a ω γb − ω αa ω βb c γαβ (cid:1) ¨ e γ = − Ω γab ´ e γ (3.29)using the first of equations 3.26 and whereΩ γab ( x, g ) = ¨ e a ω γb − ¨ e b ω γa + c γαβ ω αa ω βb (3.30)are the curvature components on the principle bundle expressed in a particular trivial-isation, as can be shown explicitly by substituting (¨ e a , ¨ e b ) for ( X, Y ) in equation 3.17.At any point p ∈ P the components of Ω( p ) are numerically the same in the horizontallift basis as for a direct product basis, that is Ω γab = Ω γ (´ e a , ´ e b ) = Ω γ (¨ e a , ¨ e b ), since Ω isa horizontal form and ´ e a and ¨ e a differ only by a vertical vector, as seen in the secondof equations 3.26 and figure 3.5. From equation 3.30, using equation 3.28, it can beshown that: ´ e α Ω βab ( x, g ) = ¨ e α Ω βab ( x, g ) = − c βαγ Ω γab ( x, g ) (3.31)again transforming infinitesimally under the adjoint representation, as for the gaugefield ω αa ( x, g ), on the principle bundle.In summary in the horizontal lift basis the full set of structure coefficients on P are considered with: [´ e α , ´ e β ] = c γαβ ´ e γ (3.32)[´ e α , ´ e b ] = 0 (3.33)[´ e a , ´ e b ] = ´ c αab ´ e α = − Ω αab ´ e α (3.34)Equation 3.33 follows directly from equations 3.24 and 3.26. Since right translationsinduce the basis vectors of the subspace VP , via equation 3.2, equation 3.33 expressesthe right-invariance of the fields ´ e b ∈ HP , consistent with equation 3.5, and may becompared with equation 2.27 in which Y L is right-invariant. For the third equation thestructure coefficients ´ c dab are set to zero since here a coordinate basis is taken for { e a } on the base manifold M in order to simplify the expressions. The fibre dependenceof the structure coefficients ´ c αab may be deduced by application of the Jacobi identitywith: [´ e α , [´ e a , ´ e b ]] + [´ e a , [´ e b , ´ e α ]] + [´ e b , [´ e α , ´ e a ]] = 0= [´ e α , ´ c βab ´ e β ] + 0 + 0 = 0 ⇒ (´ e α ´ c βab )´ e β + ´ c γab c βαγ ´ e β = 0 ⇒ ´ e α ´ c βab = − c βαγ ´ c γab (3.35)43he final expression describes the directional derivative of the coefficients ´ c βab withrespect to the vector field ´ e α , and hence expresses the transformation of ´ c βab underthe action of right translation, that is the gauge transformation generated by ´ e α . Thisis consistent with the transformation property in equation 3.31, for the componentsthe curvature 2-form under infinitesimal gauge transformations, as expected since byequations 3.29 and 3.34 we have simply:´ c αab = − Ω αab (3.36)Given a curvature 2-form Ω( p ) on a principle bundle P and a local section σ ( x )on P , for x ∈ U ⊂ M , the local representative of Ω on the base space is defined bythe pull-back map as the 2-form F ( x ) = σ ∗ Ω( p ), which also takes values in the Liealgebra, that is F ( x ) = F α ( x ) X α .Another significant property of the curvature on the principle bundle P is thatthe exterior covariant derivative of Ω itself vanishes as a consequence of the definitionsused to construct it, that is DΩ = 0, which is called the Bianchi identity. The objectDΩ = 0 is also a tensorial form on P , like Ω itself, and since the exterior algebrastructure pulls back through a section map σ ( x ) we have a similar property for thecorresponding object on M , that is on the base space we have D F = 0, which is alsoreferred to as the Bianchi identity.Through the section map σ the structure equation for the curvature 2-form Ωon P , for example in equation 3.16, pulls back to the base space M as: F = d A + 12 [ A, A ] (3.37)which was introduced in equation 2.36. In a particular trivialisation the componentsof the ‘Yang-Mills field strength’ on the base manifold M are F αab ( x ) = Ω αab ( x, e ),while the ‘gauge potentials’ are A αa ( x ) = ω αa ( x, e ). Consistent with equation 3.30 theabove expression for F can be written in components, in a coordinate basis on M , as: F αµν ( x ) = ∂ µ A αν − ∂ ν A αµ + c αβγ A βµ A γν (3.38)while the 2-forms F α are related to the 1-forms A α according to equation 2.54.For a connection on a principle bundle for which the structure group G as asubgroup of GL( m, R ) exhibits a matrix representation acting upon objects v ( x ) ∈ V of an m -dimensional vector space (where m is not necessarily equal to the dimension n of the base manifold) the vector and curvature fields transform under a change ofgauge g ( x ) ∈ G on the base space M as: v → v ′ = g − v (3.39) F → F ′ = g − F g (3.40)This form of transformation follows from the choice of a right action of G on P , asfeaturing for example in equation 3.12, and in turn ultimately on the choice for L ( G )to be represented by left -invariant vector fields on G as described in subsection 2.2.2.Connection 1-forms A r ( x ) = σ r ( x ) ∗ ω on the base manifold with respect todifferent trivialisations are related under the local gauge transformations by g rs ( x )between the sections of equation 3.12 as: A s ( x ) = Ad( g − rs ( x )) A r ( x ) + ( g ∗ rs θ ) x A r ( x ) and θ is the Maurer-Cartan 1-form on the group manifold G , which here ispulled back onto M via the transition function map g rs ( x ) : M → G . For a matrixrepresentation, dropping the subscript labels, this transformation can be written as: A → A ′ = g − Ag + g − d g (3.41)where the second term is needed to take into account general gauge changes g ( x )between sections over M since ω is not a horizontal form on P . Under a change ofsection σ ′ ( x ) = σ ( x ) g ( x ) via the local gauge function g ( x ), the transformations ofequations 3.39–3.41 are considered a passive symmetry from a physical point of view.The connection 1-form ω on the principle bundle, which is a Lie algebra valuedmap on the tangent space T p P of equation 3.4, may be restricted to a mapping onelements of V p P tangent to the fibres of the bundle space, as it is in equation 3.8for example. Under this restriction the properties of ω are equivalent to the Maurer-Cartan 1-form θ , described in subsection 2.2.2, which maps left-invariant vector fieldson the manifold G as θ ( X A ) = A and which transforms under right translation as R ∗ g θ = Ad( g − ) θ , to be compared with equations 3.8 and 3.9.Indeed, for a trivial bundle we have P = M × G and through the naturalprojection π : M × G → G , the canonical Maurer-Cartan 1-form θ on G can bepulled back to ω = π ∗ θ on P . Since the pull-back map captures the structure of theexterior algebra as seen through the map itself the Maurer-Cartan equation, that isequation 2.23, pulls back to: d ω + 12 [ ω, ω ] = 0 (3.42)By comparison with equation 3.16 it can be seen that for this connection the curvaturevanishes, Ω = 0, that is ω is the canonical flat connection on P .In general for a continuous map between two differentiable manifolds f : M → N , with a vector field u on M and a 1-form ξ on N , the pull-back of the 1-form ξ onto M can be defined as h f ∗ ξ, u i x = h ξ, f ∗ u i f ( x ) . For the present case the canonical flatconnection on the base manifold M , expressed as A ( x ) = σ ∗ ω = A αµ ( x ) X α d x µ is aLie algebra-valued map on tangent vectors u ∈ T x M and we have: h A, u i x = h σ ∗◦ π ∗ θ, u i x = h θ, π ∗◦ σ ∗ u i g = π ◦ σ ( x ) (3.43)where in the latter expression the vector u ∈ T x M has been ‘pushed forward’ throughthe two maps to a vector in the tangent space of the group manifold. In general h A, u i 6 = 0, even for a flat connection, since an arbitrary trivialisation can be used todefine the section map σ r ( x ) ≡ ψ − r ( x, e ) r . However, for the canonical flat connectionon P the horizontal subspace is everywhere tangent to a submanifold M × { g } for some g ∈ G and the Frobenius criterion of equation 3.7 is satisfied. Hence in this case thesection map from M to P may be chosen to coincide with the horizontal section of thecanonical flat connection and we have: h A, u i x = h σ ∗ ω, u i x = h ω, σ ∗ u i p = σ ( x ) = 0 (3.44)since for all u ∈ T x M we have σ ∗ u ∈ H p P in this case, and hence we have A ( x ) = 0 inthis choice of gauge section. In general the cross-section σ and horizontal subspace H p P P , as indicated for example in figure 3.5, relating to the gaugechoice g ( x ) and connection ω respectively. As can be seen from equations 3.26 and3.27 if it is possible to choose a direct product basis to coincide with the horizontal liftbasis on P then ω αa ( x, g ) = 0, that is all connection coefficients vanish for this choiceof section.Here we have described the flat connection that was introduced in equations 2.30and 2.35 directly on the base manifold without constructing the principle fibre bundle.The use of the principle bundle will be more significant for the case of an enlargedsymmetry group of L ( v ) = 1 as introduced in section 2.3 and studied further in sec-tion 5.1. Any n -dimensional differentiable manifold M is canonically associated with the prin-ciple fibre bundle of frames FM , with structure group GL + ( n, R ), which preserves theorientation of the frames, over M as the base manifold. A linear connection e ω can bedefined on a frame bundle as a gl( n, R )-valued 1-form on FM which may be written e ω = e ω ab E ba . The quantities e ω ab = e ω abi e i (with { e i } a basis of 1-forms on the framebundle) are a set of n FM . Each 1-form e ω ab is associated with a basiselement of gl( n, R ) represented by the n × n matrix E ba for which the only non-zeroentry is a ‘1’ in the a th -row and b th -column, that is ( E ba ) dc = δ bc δ da (where { a, b } labelthe matrices and { c, d } label the matrix elements. By comparison the generators ofSO( n ), as described in equation 2.31, form a subalgebra of gl( n, R ) with matrices ofthe form L pq = E qp − E pq ).The frame field { e a } on the base space M is a general basis which in somesituations may be taken to be an orthonormal or coordinate basis. A section σ on FM corresponds to a choice of frame, that is a basis { e a } , at each point of the basespace M , with the pull-back Γ = σ ∗ e ω being the representative of e ω under this section.This linear connection 1-form Γ on M has components Γ ab = Γ abc e c , where { e a } is acoframe basis for T ∗ M .In general for a gauge symmetry group with generators represented by m × m matrices E α ∈ L ( G ) the connection components, for an arbitrary coframe { e a } on thebase manifold, may be written A rs = A αa ( E α ) rs e a = A rsa e a , with { r, s } = 1 . . . m ,composing a matrix of 1-forms. In the case of a linear connection on M , with u ( x ) asany tangent vector field, Γ ab ( u ) = Γ abc u c is a matrix element with Γ abc ( x ) being thecomponents of the linear connection.The covariant derivative D a for the case of a linear connection on the externaltangent space will be denoted by the kernel symbol ∇ . With respect to a generalframe field { e a } , the components of the corresponding linear connection Γ abc satisfythe relation ∇ e b = Γ abc e c ⊗ e a , that is: ∇ c e b = Γ abc e a and hence, Γ abc = h e a , ∇ c e b i (3.45)where in the final term the angular brackets, defined in equation 2.18, denote the46-form e a mapping the vector field ∇ c e b into the space of real numbers, that is thecoefficients Γ abc .The linear connection coefficients Γ abc transform under a general change ofbasis to e b ′ = e a e ab ′ ( x ), with primed indices denoting the new frame and the matrix e ab ′ ( x ) ∈ GL + ( n, R ), as:Γ a ′ b ′ c ′ = ( e − ) a ′ d e eb ′ e fc ′ Γ def + ( e − ) a ′ d e c ′ e db ′ (3.46)Compared with the gauge transformation of equation 3.41 an extra e fc ′ factor appearshere for the 3-index affine connection to reflect the tensor-like transformation law ofthe 1-form part of the connection under a local change of frame on the manifold M .A subset of frames is provided by a general coordinate chart on the patch U ⊂ M for which a section of the general frame bundle σ ( x ) : U → FM is given by thecoordinate basis x → { ∂ µ } x . This defines a holonomic frame { ∂ µ } , with [ ∂ µ , ∂ ν ] = 0,through which a local representative of the linear connection Γ = σ ∗ e ω may be obtained.A second general coordinate chart with coordinate frame section { ∂ µ ′ } defines a furtherrepresentative of the linear connection Γ ′ = σ ′∗ e ω . The transition function j ( x ) : M → GL + (4 , R ) for all x ∈ M relates coordinate frames as: ∂ µ ′ ( x ) = ∂ ν ( x ) j νµ ′ ( x ) (3.47)where j νµ ′ ( x ) = ∂x ν /∂x µ ′ is the Jacobian matrix of the general coordinate transfor-mation. These transformations form a special case for equation 3.46 corresponding toa change of coordinate system { x µ } → { x µ ′ } on M .If M is an n -dimensional Riemannian or pseudo-Riemannian manifold ( M, g ),that is given a metric field with components g µν ( x ) on the manifold, a subset of dis-tinguished frames may be identified which are orthonormal with respect to the metric.This subset of frames over M reduces the total space of FM to a submanifold OM ⊂ FM which is itself a principle fibre bundle with structure group SO + ( p, q ) (or more gen-erally O( p, q )) with p + q = n . There is a one-to-one correspondence between metricfields g µν ( x ) on M and reductions of the structure group GL + ( n, R ) to SO + ( p, q ) on FM , with each choice of field g µν ( x ) isolating one out of the many possible isomorphiccopies of principle SO + ( p, q )-bundles.From the above general case we next consider specifically the spacetime symme-try of a 4-dimensional manifold M . Matrices l ab ( x ) ∈ SO + (1 ,
3) of the Lorentz groupdescribe spacetime orientation preserving gauge transformations between sections ofthe principle bundle of orthonormal frames. With the set of vector fields { e a ( x ) } foreach x ∈ M now representing such an orthonormal frame, any other orthonormal framecan be expressed as: e b ′ ( x ) = e a ( x ) l ab ′ ( x ) (3.48)while the dual coframe transforms as e b ′ ( x ) = ( l − ) b ′ a ( x ) e a ( x ). Equation 3.48 expressesthe right action of elements of the Lorentz group on the frame field. Since the set oforthonormal frames on the tangent space at any one point x ∈ M is isomorphic to theLorentz group, through equation 3.48, a principle fibre bundle over M is obtained,with both the fibre space and structure group being SO + (1 ,
3) itself. It is a reduction ofthe principle bundle of general linear frames FM , the latter having fibres isomorphicto the larger group GL + (4 , R ). 47e can consider a tetrad field e aµ ( x ) as describing an element of a restrictedset of the gauge group GL + (4 , R ) of all possible orientation-preserving frame trans-formations over M or, in bridging local orthonormal frames with general coordinateframes, as a mapping between the principle bundle of Lorentz frames and the principlebundle of coordinate frames. That is, e aµ ( x ) relates a section of orthonormal frames { e a } x with a coordinate frame basis { ∂ µ } x via the right action: ∂ µ ( x ) = e a ( x ) e aµ ( x ) (3.49)with e aµ ( x ) ∈ GL + (4 , R ), which can be directly compared to equation 3.48 with thetransformation l ab ( x ) ∈ SO + (1 , g ( x ) on M any local orthonormal frame { e a } isassociated with the Minkowski metric η ab = g ( e a , e b ) = diag(+1 , − , − , − e aµ ( x ) (similarly as we had in equation 2.16 for the 3-dimensional model): g µν ( x ) = e aµ ( x ) e bν ( x ) η ab (3.50)The SO + (1 ,
3) bundle OM may be extended to the frame bundle FM withan SO + (1 , A ( x ) uniquely inducing a linear connectionΓ( x ) for the extended bundle space. Such a GL + (4 , R )-valued linear connection Γ iscompatible with the metric, that is ∇ g = 0, while Γ and g need not be related inthe general case. The principle bundle of orthonormal frames OM , equipped with aLorentz connection, as a subbundle of the principle bundle of general linear frames FM over the base manifold hence induces a unique metric connection on the latterspace. Expressing the Lorentz connection in a coordinate basis on M as A ( x ) = A µ ( x )d x µ the tetrad components may be considered as a local gauge transformation– that is as a change from a choice of local orthonormal Lorentz frames to the generalcoordinate frames over the base manifold, within the GL + (4 , R ) freedom of the princi-ple bundle FM . In this way, and by comparison with equation 3.41 for example, themetric preserving linear connection Γ for a general coordinate system may be definedby: Γ λµν = e λa A abν e bµ + e λa ∂ ν e aµ (3.51)The identification of the linear connection Γ in this form implies that the co-variant derivative of the tetrad field vanishes identically: ∇ µ e aν = ∂ µ e aν + A abµ e bν − Γ λνµ e aλ = 0 (3.52)This condition itself implies that A and Γ are compatible connections, regardless of thevalue of the torsion (defined below). In this case the tetrad field e aµ ( x ) ‘commutes’with the operation ∇ of covariant differentiation. This means that the operation ofinterchanging between local field components, such as u a ( x ), and general coordinatetangent space field components, such as u µ ( x ), via the tetrad field e aµ ( x ), applies in astraightforward manner even for equations involving covariant derivatives.In particular, since g µν ( x ) has the form of equation 3.50 and the Minkowskimetric is a constant, the metric field g ( x ) is preserved by covariant differentiation48efined in terms of the linear connection Γ( x ), which in turn is defined in terms of theLorentz connection through equation 3.51, that is ∇ g = 0 as cited above. If A abµ ( x ) ischosen to be the unique torsion-free Lorentz connection for a given tetrad field e aµ ( x ),then the corresponding linear connection Γ is the unique torsion-free metric connectionexpressed in a general coordinate system. This is the Levi-Civita connection, significantfor general relativity, which can be written uniquely as a function of the metric tensorcomponents g µν ( x ) as: Γ σµν = 12 g σρ ( ∂ µ g ρν + ∂ ν g µρ − ∂ ρ g µν ) (3.53)On the space of the frame bundle over any n -dimensional differentiable manifold M , even without a metric, a canonical R n -valued 1-form θ C = θ a E a can be identified,with each θ a being a 1-form on FM and { E a } a basis for R n , such that at any point f ∈ FM and for any vector X ∈ T f FM we have: h θ a , X i := h e a , π ∗ X i = ( π ∗ X ) a (3.54)which is just the components of the projection of X onto the base space M in the frame f = { e a } itself. Given a section σ ( x ) = f on FM the pull-back e a = σ ∗ θ a describesthe dual basis vectors of the general GL( n, R ) frame f .The canonical 1-form θ C is therefore horizontal and equivariant and hencea tensorial form on FM . Given a linear connection e ω on FM the exterior covariantderivative Θ = D θ C is called the torsion FM . With Θ = Θ a E a , and followingequation 3.14, the torsion can be expressed as:Θ a = d θ a + e ω ab ∧ θ b (3.55)This object in turn pulls back to the torsion 2-form T = σ ∗ Θ on the base manifold M with coefficients T abc defined in T a = T abc e b ∧ e c , with: T a = d e a + Γ ab ∧ e b (3.56)= − c abc e b ∧ e c + Γ abc e c ∧ e b (3.57)= ( − c abc −
12 (Γ abc − Γ acb )) e b ∧ e c (3.58)where each term above is a 2-form. Hence for a general linear connection on themanifold M the torsion components can be written as: T abc = − a [ bc ] − c abc (3.59)with [ . . . ] denoting n ! times the antisymmetrised sum of the n ! terms obtained throughpermuting the n enclosed indices. Via the vielbein field e aµ ( x ) this may be written ina general coordinate frame as: T ρµν = − Γ ρµν + Γ ρνµ (3.60)since [ ∂ µ , ∂ ν ] = 0 for such a holonomic frame.49he curvature of the linear connection may also be defined on the frame bundle FM as e Ω = D e ω , that is as the exterior covariant derivative of the connection in theusual way, to obtain the tensorial form e Ω of type (Ad , gl( n, R )). However, here wedeal directly with objects on the base manifold M for an arbitrary frame field { e a } and study the Riemannian curvature R = σ ∗ e Ω = R ab E ba , where the matrices E ba were defined in the opening of this section. From the definition of the curvature 2-form in equations 3.15–3.17 and the gl( n, R ) commutators [ E ba , E dc ] = δ bc E da − δ da E bc (which can be compared with the commutators for the L pq matrices describing theso( n ) subalgebra in equation 2.48) the components of curvature R ab may be writtenfor any linear connection Γ in any choice of frame field as: R ab = dΓ ab + Γ ad ∧ Γ db = (dΓ abc ) e c + Γ abd d e d + Γ adc e c ∧ Γ dbe e e = ( e e Γ abc ) e e ∧ e c − Γ abd c dce e c ∧ e e + Γ adc Γ dbe e c ∧ e e = ( e c Γ abe − e e Γ abc + Γ adc Γ dbe − Γ ade Γ dbc − Γ abd c dce ) e c ∧ e e (3.61)In terms of the components of the rank-4 Riemann tensor the curvature can be ex-pressed as R = R abcd e c ∧ e d E ba . Hence the curvature components on the base manifold M can be written in terms of the linear connection and structure coefficients as: R abcd = e c Γ abd − e d Γ abc + Γ aec Γ ebd − Γ aed Γ ebc − c ecd Γ abe (3.62)If a metric g is also defined on M then { e a } may represent a local orthonormalframe field. In the dual covector basis { e a } the Riemann tensor may be written as: R = 12 R pqcd L pq e c ∧ e d = R pqcd L pq e c ⊗ e d (3.63)where the latter follows due to the asymmetric arrangement of the { c, e } indices forthe coefficients in the final line of equation 3.61. Under the group SO + ( p, q ) thisobject transforms as a rank-4 tensor which can be expressed in components in severalequivalent ways, including: R abcd = R pqcd ( L pq ) ab and R abcd = η ae R ebcd (3.64)This latter object is asymmetric in the indices { a, b } as well as in { c, d } . The Riemanntensor in a general coordinate system, as described towards the end of section 2.2 inthe context of the SO(3) model on M , may be obtained through the vielbein field e aµ ( x ), with the resulting components: R ρσµν = e aρ e bσ e cµ e dν R abcd (3.65)Both the curvature and torsion may be considered properties of a linear connec-tion Γ in general. Although they are related through the Ricci and Bianchi identities,respectively: R ρ [ σµν ] = − T ρ [ σµ ; ν ] − T ρκ [ σ T κµν ] (3.66) R ρσ [ µν ; τ ] = − R ρσκ [ τ T κµν ] (3.67)50where ; τ denotes the covariant derivative ∇ τ with respect to the x τ coordinate) thecurvature and torsion are independent geometric concepts where either one may benon-zero while the other is zero. For example for the complete parallelism exhibitedon a Lie group manifold G in terms of the self-parallel frame composed of left-invariantvector fields X α on G , with each Γ αβγ = 0, the curvature vanishes, as can be seen triv-ially from equation 3.62, while the torsion is finite, with T αβγ = − c αβγ , as determineddirectly by equation 3.59. On the other hand for the linear connection Γ αβγ = − c αβγ ,in the same basis on G , the curvature is finite while the torsion vanishes, as can alsobe seen from equations 3.62 and 3.59. This latter case is the unique Levi-Civita con-nection on a group manifold defined in terms of the Killing metric on G . In generalthe identities of equations 3.66 and 3.67 clearly simplify for the torsion-free case.Returning to the case of 4-dimensional spacetime M the quantities R ρσµν ofequation 3.65 are the components of a general coordinate frame rank-4 tensor withtransformations j µν ′ ∈ GL + (4 , R ), introduced in equation 3.47, acting on all indicesunder a change of coordinates. The most general rank-4 tensor on a 4-dimensionalmanifold has 4 = 256 independent components. However the geometric origin andstructure of the Riemann tensor results in considerably less freedom. In components R ρσ µν is asymmetric in the first two indices { ρ, σ } since it derives from a Lorentz-valued metric connection and also asymmetric in the final two indices { µ, ν } since thecurvature originates as a 2-form object. This reduces the number of free componentsdown to (6 ×
6) = 36. For the torsion-free case considered here the Ricci identity in ageneral coordinate system of equation 3.66 reduces to simply: R ρ [ σµν ] = 0 (3.68)or R ρσµν + R ρνσµ + R ρµνσ = 0where the second equation follows from the asymmetry of R ρσµν in the final two indices.This further constraint results in a final total of 20 independent components for theRiemann curvature tensor for the metric and torsion-free case.The Ricci tensor may be defined as the ‘trace’ of the Riemann tensor R σµ = R ρσµρ . This is also termed a ‘contraction’ of upper and lower indices in R ρσµν , whichtransform in a dual manner to each other under the action of GL + (4 , R ). Also for theLorentz curvature tensor components R abµν transformations in the { a, b } indices viathe group SO + (1 ,
3) are closely related to those in the { µ, ν } indices via the holonomicsubgroup of GL + (4 , R ) through the components of the tetrad field e aµ ( x ), and it isthrough the latter field that tensor contractions are again possible. In both cases theLie algebra valued part of the curvature form possesses a transformation symmetryclosely related to that of the r -form part in the tangent space of the base manifold.This, of course, is not the case for curvature forms derived for general principle bundleswith the symmetry group composing the fibres unrelated to the local symmetry of thebase space manifold, and hence an equivalent contraction does not exist for a gaugetheory based on such an internal symmetry.The Ricci tensor is symmetric and hence possesses 10 independent degreesof freedom, including the scalar curvature R = g µν R µν (as distinct from the Riemanntensor denoted by a bold R , as on the left-hand side equation 3.63). The utility of suchexpressions follows from the fact that the operation of contraction maps a tensor objectonto another tensor, that is the contracted tensor also transforms as a representation51f GL + (4 , R ). This tensor preserving property is shared by the operations of thecovariant derivative and exterior algebra as we described earlier, and hence all of theseoperations are useful for identifying the equations of physics.The remaining 10 components of the Riemann tensor, the non-Ricci part, aredescribed by the Weyl tensor C ρσµν , it is the trace-free part of R ρσµν (all contractionsare zero) with which it shares the same symmetries. The trace-free property implies tenrelations C σµ = C ρσµρ = 0 between the components of the Weyl tensor C ρσµν and henceonly ten of them are independent. The Weyl tensor is also the conformally invariantpart of the Riemann tensor, that is it is unchanged under a conformal transformationof the metric g µν ( x ) → f ( x ) g µν ( x ) where f ( x ) is any smooth real function on M . Thetwenty components of the Riemann tensor can be decomposed explicitly in terms ofthose of the Weyl tensor and Ricci tensor as: R ρσµν = C ρσµν + 12 ( g ρµ R σν − g ρν R σµ − g σµ R ρν + g σν R ρµ )+ 16 ( g ρν g σµ − g ρµ g σν ) R (3.69)that is: R ρσµν = C ρσµν + 2 R [ ρ [ µ g σ ] ν ] − Rg [ ρµ g σ ] ν The Bianchi identity of equation 3.67 for the curvature tensor in the torsion-freecase is simply: R ρσ [ µν ; τ ] = 0 (3.70) ⇒ ( R µν − Rg µν ) ; µ = 0 (3.71)where the latter expression follows from the double contraction of the former. TheEinstein tensor is defined as G µν := R µν − Rg µν . Hence the Einstein tensor G µν , unlikeits ‘dual’ geometric object the Ricci tensor R µν , represents an identitically conservedquantity, that is G µν ; µ = 0, which is the origin of its central importance in the fieldequation of general relativity.For general relativity in regions of ‘empty space’ with T µν = 0 by the Einsteinequations 3.75 we also have G µν = 0 and hence R µν = 0 and the manifold is said tobe ‘Ricci flat’. In this Ricci vacuum the Riemann tensor is simply R ρσµν = C ρσµν , ascan be seen explicitly from equation 3.69. The spacetime curvature is then describedin terms of the Weyl tensor C ρσµν , yet in a way dependent upon the matter contentin other spacetime regions as will be reviewed alongside equation 5.44 in section 5.2.We note here that the various possible sign conventions for the expressions ofgeneral relativity can be distilled down to the ± sign used for the right-hand side ofjust three expressions in the Riemannian geometry:1) The metric tensor: η ab = diag(+1 , − , − , −
1) (3.72)With ‘+1’ for the time component this is a natural convention for the presenttheory based on forms of temporal flow.2) The Riemann tensor: R ρσµν = ∂ µ Γ ρσν − ∂ ν Γ ρσµ + Γ ρλµ Γ λσν − Γ ρλν Γ λσµ (3.73)Where the final term of equation 3.62 is zero when expressed in a coordinateframe as is the case here. 52) The Ricci tensor: R µν = R ρµνρ (= − R ρµρν ) (3.74)This is equivalent to choosing the sign convention for the Einstein field equationas G µν = − κT µν with positive normalisation constant κ (as will be justified afterequation 5.35).The convention for these three signs chosen here is the same as used for examplein ([5] p.24) that is with signs ‘( − + − )’ relative to the original discussion of theseconventions in [6]. The Einstein equation, and general relativity itself, will be reviewedin the following section. In his 1854 work ‘On the Hypotheses which lie at the Foundation of Geometry’ Rie-mann, building upon the study of the intrinsic curvature of 2-dimensional surfaces byGauss, considered more generally spaces of n -dimensions and introduced tensor anal-ysis, in particular incorporating the metric tensor and the Riemann curvature tensor.At the same time Riemann also speculated on the possible curvature for the space ofour own world, both on small and large scales, and its possible physical implications.At around the same time (1861,1865) Maxwell, building upon the ‘field’ con-cept introduced earlier by Faraday based on empirical observations, formulated theequations of motion for the electromagnetic field, providing a unified description ofelectric fields, magnetic fields and also the properties of light.The mathematical structure of general relativity was developed leading up to1915 as an application of Riemann’s work in geometry, with the dimension of time now included along with space in a 4-dimensional spacetime manifold. Influenced bythe work of Maxwell on electromagnetism objects such as the metric and Riemanncurvature tensor, as mathematical functions describing the phenomena of gravitation,were now considered as fields in spacetime.In search of a relativistic gravitational field equation consistent with the ‘equiv-alence principle’, defined below, and under the empirical guidance that the Newton-Poisson equation ∇ Φ = 4 πG N ρ (a second order differential equation, with Laplacianoperator ∇ = ∂ x + ∂ y + ∂ z , relating the gravitational scalar potential Φ, via Newton’sconstant G N , to the scalar mass density distribution ρ ) should emerge in the non-relativistic limiting case for small distortions from a flat spacetime, Einstein convergedin 1915 upon the field equation: G µν = − κT µν (3.75)with κ a constant and T µν the energy-momentum tensor for the distribution of matterin 4-dimensional spacetime. From the limit of Newtonian gravity the normalisationconstant is found to be κ = πG N c .In general relativity, it is considered always possible to have a local inertial coordinate system on M that is valid within a sufficiently small region of curved4-dimensional spacetime – strictly an infinitesimal neighbourhood about any point x ∈ M , with local metric η = diag(+1 , − , − , − strong equivalence principle states that within such a local coordinate sys-tem, within a sufficiently small region about the point x ∈ M , all laws of physics,other than gravity, take the same form that applies for special relativity in an unac-celerated Cartesian coordinate system in the absence of gravity. These assumptionsaugment the weak form of the equivalence principle for which the ‘laws of physics’ arelimited to ‘the laws of motion of freely falling particles’ corresponding to the equiv-alence of gravitational and inertial mass, and the observation of the apparent lack ofgravitational effects within a freely falling lift.The motion of a freely falling particle in such a local inertial coordinate system { x a } satisfies the equation d x a /dτ = 0, in choosing the proper time τ to parametrisethe trajectory. Transforming to a general coordinate system { x µ } this becomes: d x λ dτ + Γ λµν dx µ dτ dx ν dτ = 0 (3.76)which is called the geodesic equation of motion and which is valid also in an extendedcurved spacetime. The quantities Γ λµν are the coefficients of the linear connection andthe proper time τ itself can be defined in terms of an integral of the invariant intervals dτ = ( g µν dx µ dx ν ) / along the trajectory. In terms of the 4-velocity u µ = dx µ /dτ theabove geodesic equation can be written as simply: u µ ∇ µ u ν = 0 (3.77)The equivalence principle states that all gravitational effects can be locallytransformed away and can be interpreted to mean that we may always choose a localinertial coordinate frame at any x ∈ M such that all the coefficients Γ λµν = 0.Hence, although the coefficients of the non-tensor object Γ will be frame dependentthe torsion tensor T vanishes in all reference frames, by equation 3.60. This torsion-free assumption for Einstein’s theory of general relativity has the benefit of simplifyingsome of the mathematics of the theory, as for example in equations 3.68 and 3.70 ofthe previous section.Given a metric g µν ( x ) on M the Levi-Civita connection is the unique metric( ∇ g = 0), torsion-free ( T = 0) linear connection. The corresponding connectioncoefficients may be written in a general coordinate frame uniquely in terms of those ofthe metric tensor as described in equation 3.53. For such a connection equations 3.76and 3.77 describe the trajectory which extremises the path length between any givenend points: L = Z ( g µν u µ u ν ) / dτ (3.78)and hence earns the name ‘geodesic’. Further, for this connection with Γ( x ) determineduniquely by g ( x ), as implied by the equivalence principle, the metric alone determinesall gravitational effects and hence can be considered to be the gravitational field forEinstein’s general relativity. Since the tetrad field e aµ ( x ) may be considered to bethe ‘square-root’ of the metric, with g µν = e aµ e bν η ab in equation 3.50, the tetrad fielditself, which everywhere exhibits the presence of the local inertial frames, may also beconsidered to represent the gravitational field.As well as being able to express the metric as g µν = diag(+1 , − , − , −
1) atany spacetime location there is sufficient freedom under coordinate transformations54uch that at any x ∈ M all 40 components of the metric derivatives can be set to zero,that is g µν,ρ ( x ) = 0, corresponding to coordinate frames with Γ = 0 as can be seenfrom equation 3.53. However there is insufficient freedom under general coordinatetransformations to set all 100 second derivative quantities g µν,ρσ ( x ) to zero and thereremain 20 irreducible degrees of freedom which are described by the Riemann curvaturetensor, as deduced earlier after equation 3.68.The components of the metric tensor field g µν ( x ) may be determined by solvingthe second order differential field equation G µν = − κT µν for a distribution of matterdescribed by the energy-momentum tensor T µν , in practice by introducing ‘boundaryconditions’ as described in the following section. For a particular physical state for thegeometry of the world there will be a range of possible solutions for g µν ( x ) and e aµ ( x )in spacetime (over and above the local Lorentz freedom for the latter field) all withequivalent physical content.Essentially there is only one ‘coordinate system’ R through which any re-gion of spacetime may be described, as depicted in figure 3.6(a), as a simple spaceof 4 independent real parameters upon which a solution for the field g µν ( x ) may beinscribed.Figure 3.6: (a) Alternative metric solutions on R for the same physical state and (b)as apparently represented through an ‘alternative’ coordinate system overlaid uponthe ‘original’ coordinates.An alternative expression of the same physical solution then corresponds to adifferent metric function g ′ µν ( x ) inscribed upon the same R space. For example in theSchwarzschild solution for the metric field associated with a single massive body locatedat one point in space, to be presented in equation 5.49, the physical point where thecurvature scalar R is largest, and perhaps even singular, will in general have differentcoordinate values x ∈ R under a ‘coordinate transformation’, as indicated by the twosmall circles in figure 3.6(a). However the transformed solution could be conceivedof as a new set of ‘curvilinear’ coordinates overlaying the same physical configuration(explicitly represented by the same metric field) as shown in figure 3.6(b).In general it is less useful to think of any coordinates as curvilinear, indeed itis always the case that [ ∂ µ , ∂ ν ] = 0 with all structure coefficients c ρµν = 0. In thissense all coordinate systems can be pictured as a ‘flat’ purely mathematical parameterspace, which for the case of R can be visualised as the set of ‘Euclidean’ real numberparameters as represented in figure 3.6(a). Physical curvature is a property of the fields M itself with the Riemann curvature tensor describing the geometrical structureand warping of the corresponding physical spacetime. The set of components R ρσµν ( x )are given at points on the manifold labelled x ∈ M under the coordinate chart map φ : M → R , or on a U ⊂ M subset. A general coordinate transformation is then amapping between solutions represented on different choices of the map M → R onto aunique R (assuming here a non-degenerate Jacobian matrix j µν ( x ), that is neglectingthe artificial difference of a ‘coordinate singularity’ for example for polar coordinatesat the corresponding Cartesian coordinate origin).In general relativity a general coordinate system { x µ } is of no physical signifi-cance; all the physics is in the ‘fields’ on the manifold (see for example [7] chapter 2),with the gravitational field e aµ ( x ) giving rise to the spacetime geometry of the man-ifold. It is the possibility of relating field quantities on M , such as the coincidenceof physical events or the equating of the Einstein tensor with the energy-momentumtensor, that determines the physical content of the theory.While the coordinate system plays a passive unphysical role, in particular cir-cumstances it may be associated with physical structure. This is true in the case of theSchwarzschild solution in which the origin of a polar coordinate system is associatedwith the central massive object. This is an example with non-zero Riemann curvaturein which the exact spherical symmetry of the physical state is assumed to be exhib-ited by the metric for which a solution may be found in a greatly simplified form ina naturally preferred system of spherical polar coordinates. For similar reasons, butwith finite 4-dimensional curvature considered on a much larger scale, cosmologicalmodels also employ a preferred system of coordinates to study solutions of Einstein’sfield equation, as we shall describe in section 12.2.In general, however, there will be no preferred solutions and hence no privilegedcoordinate systems on the base manifold. In this sense all coordinate systems are‘equally bad’, or at least on a equal footing, and this expresses the relevance of generalcovariance for general relativity. Other theories may also be ‘generally covariant’, butif there is always a particular kind of distinguished coordinate reference frame thenthe general covariance may be of no relevance. This is the case for special relativityand also for Newtonian mechanics formulated against a flat absolute background of anindependent space and time.Even for general relativity, if the curvature is very small, as it is in practicein a laboratory on the surface of the Earth or even locally within the solar systemwith respect to the ‘fixed stars’ of the galaxy, then there will be ‘preferred’ solutionswith everywhere e aµ ( x ) ≃ δ aµ and g µν ( x ) ≃ diag(1 , − , − , −
1) found for a coordinatesystem which is then implicitly pseudo-Euclidean to a very good approximation. Inthe limit of flat Minkowski spacetime there is a preferred coordinate systems with g µν ( x ) = diag(1 , − , − , −
1) exactly. The corresponding tetrad field is e aµ ( x ) = δ aµ ,within a global Lorentz transformation (which leaves the metric invariant). In this casea coordinate transformation such that in general g ′ µν ( x ) = diag(1 , − , − , − { ∂ µ } does not allow for arbitrary frame transformationsas elements of GL + (4 , R ). Rather the transition functions j µν ( x ) of equation 3.47 arerestricted to a ‘holonomic subgroup’ of all possible GL + (4 , R ) transformations overthe manifold, sometimes called the ‘Einstein gauge’, and this to some extent disguisesunderlying gauge structure of general relativity.Although the ‘coordinate invariance’ symmetry of the kind implied by generalcovariance is mathematically rather different from the usual concept of a ‘gauge in-variance’ symmetry, there is a close analogy between them. In both cases there is a loosening of a global symmetry or absolute structure that would otherwise be arbitrar-ily imposed. In both cases also the equations of motion, together with their solutions,are mapped on to equally valid equations and solutions under the coordinate or gaugetransformations. Further, while a particular choice of coordinates greatly assists withcalculations for some solutions in general relativity a particular choice of gauge isfrequently employed to assist with calculations in a gauge theory.For general relativity to be considered in terms of a GL + (4 , R ) gauge theoryof gravity, within the framework of general covariance, the equivalence principle isneeded to distinguish the local Minkowski metric as being physically significant inthat it marks the transition to special relativity in local inertial coordinate frames.That is, the metric or tetrad field needs to be introduced everywhere on M (there isno equivalent of such fields for an internal symmetry gauge theory). This implies thepossibility to contract the GL + (4 , R ) structure group down to the Lorentz subgroup(which is then the holonomy group of the general frame bundle). The local Lorentzsymmetry itself has mathematical properties very closely related to those of the localsymmetry of a gauge theory.Indeed, while gravitation in Einstein’s original theory of 1915 is describedthrough the freedom of the metric field g µν ( x ) field, together with its relation to theLevi-Civita connection Γ( x ), an equivalent formulation of general relativity can begiven in terms of the tetrad field e aµ ( x ) together with a Lorentz connection A ( x ). Thislatter approach was introduced in 1956 by Utiyama [8] in which general relativity isconsidered as a type of gauge theory invariant under local Lorentz transformations.Such local transformations are displayed in equation 3.48 and map one local inertialcoordinate frame onto another. As well as tensor representations the Lorentz groupalso has spinor representations. Hence spinor fields can be introduced on a spacetime57anifold with an arbitrary metric g µν ( x ) via the tetrad field e aµ ( x ). This also permitsgravitation to be considered in terms of an SL(2 , C ) gauge theory, where SL(2 , C ) isthe double cover of the Lorentz group, as will be described in section 7.3.The fundamental structures on the base manifold are the local Minkowskispaces, together with their mutual relations through the Lorentz connection on M .With respect to a given coordinate system either the tetrad e aµ ( x ) or metric g µν ( x )field identifies the local inertial frames. In 1920 Einstein postulated that the metricfield should be considered to be the fundamental entity of general relativity, referringto it as the ‘new ether’. However, whichever fields are considered as fundamental, fieldequations are still required in order to determine the nature of the field dynamics.At the same time that Einstein arrived at equation 3.75 via the heuristic argumentsoutlined in the opening of this section Hilbert was in the process of deriving the sameequation via a Lagrangian approach. This latter argument, and the employment ofLagrangian methods more generally, will be reviewed in the following section. In this section we review the standard use of the Lagrangian formalism to derivephysical equations of motion, including those for general relativity and gauge theories.In the 4-dimensional spacetime of general relativity the scalar curvature R is adoptedas the principle geometric contribution to the total scalar Lagrangian function, withthe field equations determined from the Einstein-Hilbert action integral ([9] p.75): I = Z ( α ( R − L ) p | g | d x (3.79)Here Λ is the cosmological constant, L is the Lagrangian function for matter fields and α is a normalisation constant. The magnitude of the metric determinant | g | is employedin the 4-dimensional invariant volume element p | g | d x . The vacuum equations forgeneral relativity, that is with L = 0 and Λ = 0, are obtained by requiring that δI = 0in equation 3.79 under variation of the metric δg µν . With δg µν = − δg µν to first order, δ p | g | = p | g | g µν δg µν and with R = R µν g µν we have: δI = Z α ( R δ p | g | + R µν δg µν p | g | + δR µν g µν p | g | ) d x (3.80)= Z α ( 12 R g µν − R µν ) δg µν p | g | d x (3.81)where the final term in equation 3.80 contributes zero to the integral since g µν δR µν =( g µν δ Γ ρµν − g µρ δ Γ νµν ) ; ρ and δ Γ ρµν vanishes on the boundary of integration ([9] p.75).Requiring the action to be stationary, δI = 0, for any variation of the metric, δg µν ,leads directly from equation 3.81 to the Einstein vacuum equation: G µν := R µν − R g µν = 0 (3.82)For the non-vacuum case the energy momentum tensor T µν for a general matterLagrangian L 6 = 0 can be defined under variations of the metric δg µν through: δI = δ Z L p | g | d x = Z T µν δg µν p | g | d x (3.83)58ence for the full action integral of equation 3.79 stationarity δI = 0 under the metricvariation gives Einstein’s field equation for the general case, with κ ≡ − α adopted asthe normalisation constant: G µν + Λ g µν = − κT µν (3.84)Assuming that the matter Lagrangian may be a function of g µν ( x ), but not of the metricderivatives, the energy-momentum tensor itself, consistent with these equations, canbe expressed directly in terms of the matter Lagrangian as: T µν = 2 p | g | ∂ ( L p | g | ) ∂g µν = 2 ∂ L ∂g µν − L g µν (3.85)A simpler application of the principle of least action in the context of gen-eral relativity was described earlier for equation 3.78 regarding the derivation of thegeodesic equation of motion for a body moving in a gravitational field. Generalis-ing from equation 3.78 for a body with mass m and charge q moving in a curvedspacetime through an electromagnetic field with 4-potential A µ ( x ) an action S may beconstructed including both the kinematic and an interaction Lagrangian term respec-tively in: S = Z (cid:16) m ( g µν u µ u ν ) / + qu µ A µ (cid:17) dτ (3.86)Requiring δS = 0 under variation of the trajectory of the charged body leads to theequation of motion: m (cid:18) du λ dτ + Γ λµν u µ u ν (cid:19) = + F λσ J σ (3.87)where F λσ are components of the electromagnetic field tensor and J µ = qu µ is the4-current of the charged body having 4-velocity u µ with respect to the proper time τ . The above equation hence describes a correction to the purely geodesic trajectoryof equation 3.76. In the limit of a flat Minkowski spacetime, and will respect to aCartesian coordinate frame, equation 3.87 simplifies to: ∂p b ∂τ = + F bc J c (3.88)which is the relativistic Lorentz force law, for the charged body with 4-momentum p b = mu b . Further, in the non-relativistic limit equation 3.88 becomes m a = q ( E + v × B ),the original form of the Lorentz force law, where v and a are the 3-velocity and 3-acceleration of the body respectively.These examples, for the trajectory of a body in a gravitational and/or electro-magnetic field, demonstrate the flexibility and generality of the Lagrangian approach.As well as applying to macroscopic physical bodies the use of Lagrangian functions is astandard tool in classical field theory. In general the form of the Lagrangian L , a func-tion of the fields such as φ ( x ), guided by considerations of symmetry, is constructedin order that the requirement for the action integral S = R L ( φ, ∂ µ φ ) ω (where ω is thevolume 4-form) to be stationary, δS = 0, under variations of the fields, such as δφ ,yields the required equations of motion for the fields via the Euler-Lagrange equation: ∂ µ ∂ L ∂ ( ∂ µ φ ) − ∂ L ∂φ = 0 . (3.89)59n a flat spacetime, in terms of the electromagnetic curvature tensor F µν ,Maxwell’s equations are: F [ µν,ρ ] = 0 (3.90) F µν,µ = + J ν (3.91)which can also be written as d F = 0 and d ∗ F = ∗ J respectively (where ‘ ∗ ’ denotesthe ‘Hodge dual’ as employed in equation 5.24). These equations are equally valid ina curved spacetime on replacing the partial derivatives ‘ , ρ ’ by the covariant deriva-tives ‘; ρ ’, as an application of the strong principle of equivalence. The first of theseequations is simply the Bianchi identity, introduced in section 3.2, for the curvaturetensor of a U(1) gauge theory. Here working in the Lorenz gauge with ∂ µ A µ = 0 theinhomogeneous Maxwell equation 3.91 can be written as: (cid:3) A µ = + J µ (3.92)The Maxwell Lagrangian for the electromagnetic field is constructed as: L em = − F µν F µν (3.93)Under variation of the electromagnetic gauge field A µ ( x ) the Euler-Lagrange equationfor L em yields Maxwell’s equation for the source-free J ν = 0 case, that is F µν,µ = 0. Incombining the Lagrangian of equation 3.93 with the final term of that in equation 3.86,hence including a term coupling the electromagnetic field to a classical charged body,the corresponding Euler-Lagrange equation for δA µ ( x ) yields equation 3.91 with thesource term on the right-hand side.The form of the Lagrangian for non-Abelian gauge theory is guided by theAbelian case of electromagnetism, motivating the Lorentz and gauge invariant Yang-Mills Lagrangian: L YM = − F α µν F α µν (3.94)as a direct generalisation of equation 3.93. For the non-Abelian case there is a furthercontraction over the index α = 1 . . . n G , for the group generators, between the adjointand coadjoint representations, which are related by the Killing metric K αβ (which in asuitable basis is simply − δ αβ for the compact simple Lie groups relevant for the internalgauge symmetries in particle physics). In this case the Euler-Lagrange equation 3.89for L YM under variation of the gauge field components Y αµ ( x ) yields the non-linearsecond order differential equation: D µ F α µν = ∂ µ F α µν + c αβγ Y βµ F γ µν = 0 (3.95)where D µ is the gauge covariant derivative, which also appears in the Bianchi identity D [ ρ F αµν ] = 0 as the non-Abelian generalisation of equation 3.90. The immediatedistinctive feature of equation 3.95, in comparison with the Maxwell equation 3.91, isthe additional non-linear term of the form [ Y, F ] appearing for the non-Abelian case.Such terms are interpreted as self-interactions of the gauge fields Y µ ( x ), which do notoccur for the Maxwell theory. This self-interaction is intrinsically geometric in originand is implied in the Lagrangian of equation 3.94 itself given that the curvature for60 non-Abelian gauge field has the form of equation 3.38 with non-trivial structureconstants.Additional terms in the Lagrangian, either for the Maxwell or Yang-Mills case,may lead to further sources of interactions. In the Standard Model of particle physicsinteractions between fermion and gauge fields in the corresponding equations of mo-tion are introduced through the ‘minimal coupling’ in the covariant derivative termsincluded in a Lagrangian. For example by including L YM alongside the Dirac La-grangian for a massless spinor field ψ ( x ), which transforms as a multiplet under theinternal symmetry, together with the conjugate field ψ = ψ † γ (where the γ -matriceswill be defined in section 7.1), we have combined: L YMD = − F α µν F α µν − ψγ µ D µ ψ (3.96)where D µ = ∂ µ + Y αµ E α and ψγ µ D µ ψ = ψγ µ ∂ µ ψ + Y αµ j µα with j µα = ψγ µ E α ψ (3.97)where in the appropriate representation the E α are n × n matrices acting on the n -dimensional field ψ in the internal space. Here a lower case ‘ j ’ will generally denote acurrent such as the Lorentz vector in equation 3.97 composed of elementary fields, asopposed to the upper case analogue J µ = qu µ for the macroscopic current featuring inequation 3.87 for example. Under variation of the gauge field Y αµ ( x ) the extra term Y αµ j µα in this Lagrangian leads to a modification of equation 3.95 with the source j µα ( x ) now appearing in the right-hand side to give: D µ F α µν = ∂ µ F α µν + c αβγ Y βµ F γ µν = j ν α (3.98)In practice factors of i = √− ± signs in the above equationswill depend upon the conventions adopted, with coupling constants such as g alsoappearing in expressions for specific applications in the Standard Model as will bereviewed in section 7.2. In addition to the requirements of symmetry the form of thescalar Lagrangian function is typically heavily guided by the need to obtain the desiredequations of motion. As a further example the above Lagrangian of equation 3.96,augmented with a fermion mass term + mψψ , under variation of the field ψ ( x ) yieldsthe Euler-Lagrange equation: ( γ µ D µ − m ) ψ = 0 (3.99)which is the Dirac equation for the spinor field ψ (within conventional factors of i ).The interaction between the fermion field ψ ( x ) and the gauge field Y µ ( x ) = Y αµ ( x ) E α is here found in the ‘minimal coupling’ in the action of the covariant derivative D µ ψ = ∂ µ ψ + Y µ ψ in the kinetic term of the Lagrangian in equation 3.96.As for the case of the charged macroscopic body in equation 3.86 here also themass m for the field ψ in equation 3.99 has been introduced through a Lagrangian massterm, in this case with + mψψ appended to equation 3.96. Mass terms are generallyadded to the Lagrangian by hand in this way, although this may not be straightforwardto achieve. For example, a corresponding Lagrangian term such as m Y µ Y µ for a gauge61eld mass is prohibited by the requirement of gauge invariance, and even the fermionmass term mψψ is prohibited in the Standard Model Lagrangian due to the left-rightasymmetry of the SU(2) L -valued gauge field relating to electroweak interactions. Inboth cases mass terms are incorporated into the Lagrangian through interactions withthe Higgs field and spontaneous symmetry breaking, involving the addition of further,apparently ad hoc, terms to the Lagrangian, as will be described in section 7.2.As described above interactions may be introduced into the Lagrangian bythe requirement of invariance under a local gauge symmetry. Such a local symmetryincorporates a corresponding global symmetry of the equations of motion and henceNoether’s theorem applies. The theorem states that each global continuous symmetryis associated with a conserved current, written in terms of the field φ ( x ) as: j να := (cid:18) ∂ L ∂ ( ∂ ν φ a ) (cid:19) ( E α ) ab φ b (3.100)for each generator E α of the global symmetry. For the Dirac Lagrangian with a U(1)gauge symmetry, that is the final term of equation 3.96 for the Abelian case, the globalU(1) symmetry with a single generator is associated with the Dirac current: j µ = ψγ µ ψ (3.101)and the conservation law is simply ∂ µ j µ = 0.In contrast to the case of an internal global symmetry of the Lagrangian ap-plying Noether’s theorem for the external symmetry of global translational invarianceof L ( φ ) in a flat Minkowski spacetime leads to the quantity ([10] p.27): t µν = (cid:18) ∂ L ∂ ( ∂ µ φ ) (cid:19) ∂ ν φ − L η µν (3.102)which satisfies the conservation law ∂ µ t µν = 0, again owing to the Euler-Lagrange fieldequation. In field theory equation 3.102 can be taken as a definition of the energy-momentum tensor. There are four ‘conserved charges’ associated with t µν , namelythe 4-momentum P µ = R d x t µ . These include the Hamiltonian H = P and the3-vector P which is interpreted as the physical 3-momentum carried by the field.However in general the form of t µν defined in equation 3.102 is neither sym-metric nor gauge invariant. For example, with the Lagrangian for the electromagneticfield L em = − F µν F µν of equation 3.93, as a function of A µ ( x ), equation 3.102 yields: t µν = − F µρ ∂ ν A ρ + 14 η µν F ρσ F ρσ (3.103)for which the lack of symmetry is clear in the µν indices in the first term and thelack of gauge invariance is clear from the form of the explicit A ρ in this term. Thestandard interpretation of this observation is that in Lagrangian field theory the energy-momentum t µν is not a directly measurable quantity and the corresponding ambiguityallows for the addition of a extra terms, leading for example to the quantity ([10]p.101): T µν = t µν + ∂ ρ ( F µρ A ν ) (3.104)For the source-free case considered here with ∂ µ F µν = 0 this produces a symmetricgauge invariant form of the Maxwell energy-momentum tensor, in fact in the form62f equation 3.105 below with η µν in place of g µν . However, in addition to the adhoc nature this procedure is clearly flawed in that it is incompatible with generalrelativity. That is, for any T µν = 0 the spacetime geometry, described by the Einsteinequation 3.84, is not flat and hence the assumption of spacetime translation symmetrywhich led to equation 3.102 itself is invalid.On the other hand, the electromagnetic energy-momentum tensor T µν canbe derived directly by a different standard procedure, in general relativity, from thestationarity of the matter Lagrangian δ R L = 0 with respect to variation in the met-ric tensor g µν ( x ), as described towards the opening of this section. Substituting theMaxwell Lagrangian of equation 3.93 into equation 3.85 gives directly: T µν = + F µρ F νρ + 14 g µν F ρσ F ρσ (3.105)This general relativistic method yields an energy-momentum tensor T µν which is sym-metric, gauge invariant and complies necessarily with the Einstein equation 3.84 sinceit derives from the Einstein-Hilbert action of equation 3.79.For general relativity the four relations G µν ; µ = 0 of the contracted Bianchiidentity of equation 3.71, together with the identity (Λ g µν ) ; µ = 0, places four con-straints T µν ; µ = 0 on the energy-momentum tensor for the general case via the Einsteinequation 3.84, which in turn implies that only six of the ten field equations are indepen-dent. Hence the metric g µν ( x ) is not determined uniquely by G µν + Λ g µν = − κT µν ,but rather four degrees of freedom remain for arbitrary coordinate transformations.Indeed, the field equation is only required to define g µν ( x ) up to an equivalence class( M , g ) of geometries on the manifold M related by coordinate transformations θ suchthat ( M , g ) and ( M , θ ∗ g ) are physically equivalent, as described in the discussion offigure 3.6 in the previous section.Within the Lagrangian framework it is also possible to derive the contractedBianchi identity G µν ; µ = 0 itself. Taking L to be the Ricci scalar R the Einstein-Hilbertaction I = R R p | g | d x (equation 3.79 for the vacuum case and setting α = 1) is ascalar quantity and hence invariant under coordinate transformations. Indeed whilethe variational method can be employed, via the Einstein equation, to determine the4-dimensional spacetime geometry it is unable to deduce a specific choice of metricfunction and coordinates, by the principle of general covariance. However, the factthat δI = 0 for coordinate transformations can be shown ([6] p.503) to imply theidentity G µν ; µ = 0 of equation 3.71.The examples of this section have shown some of the great variety of circum-stances in which the Lagrangian method may be employed. These include cases inNewtonian mechanics, special relativity and general relativity as well as for electro-magnetism and non-Abelian gauge theories. However all of these examples also reston the assumption of the validity of the Lagrangian approach. One of the aims of thepresent theory is to derive all equations of motion without employing a LagrangianAlready it has been described for equation 3.71 how the relation G µν ; µ = 0 isa geometric identity which stands alone as a ‘conserved’ geometric quantity without the need for a Lagrangian formulation. In the present theory it stands at the headas central to the derivation of physical equations of motion, as we shall investigatein section 5.2. This is universally true both for equations of motion at the effectivemacroscopic level, relating to classical phenomena such as the Lorentz force law, and63lso at the microscopic level of the fundamental underlying fields, relating to quantumphenomena, where the constraint of the full form of temporal flow L (ˆ v ) = 1 will alsoprove central to the physics.In contrast to the Lagrangian approaches in general relativity via equation 3.85and in field theory via equation 3.102 in the present theory the Einstein equation willessentially be interpreted as the definition of energy-momentum, that is T µν := − κ G µν ,where a possible Λ κ g µν term may be implicitly included in the left-hand side. (In sub-sequent chapters this relation may be written simply as T µν := G µν to emphasisethe equivalence of the two tensors, with the implied normalisation factor of − κ ex-plicitly introduced for practical applications). Since the geometric content of G µν is measurable in general relativity, in principle at least as the gravitational influence ontest bodies, defining the energy-momentum tensor this way does have an unambiguousmeaning. In principle the structure of the energy-momentum tensor in such a theorymay be uniquely specified, distinguishing between equations 3.103 and 3.105 in theexample of the electromagnetic field.This is the case for Kaluza-Klein theory in which equation 3.105, generalised fornon-Abelian internal symmetry, is derived from the structure of a higher-dimensionalgeometry as will be reviewed in the following chapter leading to equation 4.17. Withthe Yang-Mills equation 3.95 also being derived in equation 4.18 within this frameworkthe Kaluza-Klein approach achieves a degree of unification with less dependence uponthe introduction of Lagrangian terms, such as equation 3.94. In section 5.1 we describehow the techniques of Kaluza-Klein theory might be adopted within the present theorybefore continuing in section 5.2 to explore some of the consequences of these structuresin terms of avoiding the need to postulate Lagrangian functions.64 hapter 4 Kaluza-Klein Theory
Theories with an extra spatial dimension were initially proposed [11, 12] within a fewyears of the publication of the general theory of relativity, with the aim of account-ing for non-gravitational forces of nature through the higher-dimensional geometry, ata time when only two fundamental forces were known, namely gravitation and elec-tromagnetism. A generalisation of the original Kaluza-Klein theory for the case of anon-Abelian internal symmetry, incorporating further dimensions, was elaborated indetail around half a century later ([13], see also [14], [15] sections I–V and [16]).This unifying framework for gravitation and gauge theories, reviewed here, isconstructed in the mathematical setting of a principle fibre bundle. Keeping withinthe spirit of Einstein’s original 4-dimensional spacetime theory of gravitation and theextension to a 5-dimensional arena by Kaluza and Klein, the geometric unificationwith non-Abelian gauge theory is founded upon a metric tensor ˇ g , now defined uponthe manifold of the principle bundle P = ( M , G ) itself (with the ‘check’ on ˇ g denotingan object on the bundle space).We note that conventions vary in the literature – in particular with respectto the assignment of index labels such as { a, b, . . . } , { α, β, . . . } , and { i, j, . . . } whichin this paper are associated with objects on the manifolds M , G and P respectively,in the manner described shortly before and in figure 3.2. The conventional order ofthe indices for the linear connection coefficients Γ abc also varies, with the conventionof equation 3.45 adopted here, while the sign of the Ricci tensor R µν = R ρµνρ ofequation 3.74 also differs in some of the references. Hence in turn a number of derivedexpressions here will have signs differing to those in the literature.In addition to the metric g ab on the base manifold M a natural metric for thegroup manifold G is provided by the Killing form K , which as a matrix of componentsis invertible provided G is a semi-simple Lie group and negative definite if G is compact.In the latter case a basis for the Lie algebra can be chosen such that the Killing formhas components K αβ = − δ αβ , is described after equation 3.94. Here we choose metriccomponents g αβ = + K αβ in order to match the signature convention of equation 3.72,65ith spacelike components having a negative norm.The Ad( G )-invariant Killing form defines a bi-invariant metric on the manifold G ; that is with both the left L a and right R a group actions, for any a ∈ G , beingisometries on G , with for example ( R ∗ a g ) b ( X, Y ) = g b ( X, Y ) for all
X, Y ∈ T b G forthe Killing metric g at any point b ∈ G (subsequently the Killing metric will oftenbe denoted by g αβ , rather than simply the kernel letter g , as the notation used forthe indices helps identify the space to which the object belongs). In particular, interms of the group structure constants c αβγ in a left-invariant basis { X α } on the groupmanifold, the components of the Killing metric are: g αβ = K αβ = c ρασ c σβρ (4.1)A gauge connection 1-form ω on a principle bundle P specifies a right-invarianthorizontal subspace H p P for all points p ∈ P , as described in section 3.1. A uniquemetric ˇ g may be defined on such a principle bundle space, aligned with the gaugeconnection structure with:ˇ g ( X, Y ) = g ( π ∗ X, π ∗ Y ) + K ( ω ( X ) , ω ( Y )) (4.2)where X, Y ∈ TP , while here g and K are the metrics on the base space M andgroup space G respectively. This construction yields an intuitively natural metric onthe bundle space in the sense that the vertical VP and horizontal HP subspaces of thetangent space of P , as depicted in figure 3.2, are then orthogonal with respect to ˇ g ,with ˇ g ( X, Y ) = 0 if X ∈ VP and Y ∈ HP for example.Alternatively, and perhaps more in the spirit of the original Kaluza-Klein the-ory, the metric ˇ g rather than the connection ω can be taken as the fundamental entityon P . That is, the bundle is initially endowed with a pseudo-Riemannian metric ˇ g with certain restrictions – namely compatibility with a metric g ab on M and metric g αβ on the fibres G x and the requirement of invariance under the right action of G on P : R ∗ a ˇ g pa ( X, Y ) = ˇ g p ( X, Y ) = ˇ g pa ( R ∗ X, R ∗ Y ) (4.3)for any p ∈ P , a ∈ G and X, Y ∈ TP . This latter property then implies the existenceof a subspace HP , orthogonal to VP , which is right-invariant and hence is equivalentto the existence of a connection 1-form ω on the bundle P , which is related to ˇ g asdescribed in equation 4.2.From either perspective from the relation of ˇ g to ω in equation 4.2 in thehorizontal lift basis ´ e i = { ´ e α , ´ e a } , with ´ e α ∈ VP and ´ e a ∈ HP , for the tangent spaceon P the metric ´ g , and its inverse, take respectively the simple forms:´ g ij = g ab g αβ and ´ g ij = g ab g αβ (4.4)That is with the components of the metric on the base space M being g ab = ´ g (´ e a , ´ e b )and those of the Killing metric on the group space being g αβ = ´ g (´ e α , ´ e β ). The off-diagonal components in equation 4.4 are all zero, with for example ´ g (´ e a , ´ e β ) = 0describing the orthogonality of any X ∈ H p P to any Y ∈ V p P with respect to thisright-invariant metric ´ g . 66nder a change of frame to a direct product basis { ´ e i } → { ¨ e i } , that is thereverse of equation 3.26 with ¨ e α = ´ e α and ¨ e a = ´ e a + ω αa ´ e α for a choice of trivialisation ψ : P → U × G , see figure 3.5, we have:¨ g ij = g ab + g αβ ω αa ω βb ω αa g αβ g αβ ω βb g αβ and ¨ g ij = g ab − g ab ω βb − ω αa g ab g αβ + g ab ω αa ω βb (4.5)In this latter basis the non-Abelian gauge fields ω αa ( p ) on P for the internalsymmetry are found alongside the external spacetime metric elements g ab framed withinthe components of the full metric ¨ g ij on the bundle space. This is a generalisation ofthe original 5-dimensional Kaluza-Klein theory in which the electromagnetic 4-vectorpotential A a appears alongside the components of the spacetime metric g ab within theextended 5 × M n is canonically associated with a principle bundleof linear frames FM n with structure group GL( n, R ), where n is the dimension of thebase manifold M n , as described in the opening of section 3.3. This includes the case inwhich the base manifold is actually the space of a given principle fibre bundle P itself.While the metrics g and K on the manifolds M and G can be naturallyextended to the metric ˇ g of equation 4.2 on the principle bundle P = ( M , G ) with aconnection ω , linear connections on the manifolds M and G may also be generalisedto the domain of the larger manifold P . As described for equation 3.45 such a linearconnection ˇΓ will define covariant differentiation with ˇ ∇ ˇ e i = ˇΓ ji ˇ e j ≡ ˇΓ jik ˇ e k ⊗ ˇ e j in ageneral tangent space basis { ˇ e i } for TP with dual basis { ˇ e i } for T ∗ P . The identificationof the smooth symmetric gauge covariant rank-2 tensor ˇ g everywhere on P endows theprinciple bundle itself with the structure of a pseudo-Riemannian manifold. In turn aconnection ˇΓ compatible with the metric ˇ g , and hence with the geometric structure ofthe underlying manifold P , may be extended from the notion of a metric connectionon M . Indeed, and further guided by Einstein’s general theory of relativity in 4-dimensional spacetime, the unique linear connection which is torsion-free, ˇ T ijk = 0,and compatible with the metric, ˇ ∇ k ˇ g ij = 0, that is the Levi-Civita connection, maybe defined on the bundle space P . The corresponding connection coefficients can beexpressed, with Γ ijk = g il Γ ljk and c ijk = g il c ljk , as:ˇΓ ijk = 12 (ˇ e j (ˇ g ik ) + ˇ e k (ˇ g ij ) − ˇ e i (ˇ g jk )) −
12 (ˇ c ijk + ˇ c kji + ˇ c jki ) (4.6)which expresses equation 3.53 in a general frame. These coefficients take a relativelysimple form in the horizontal lift basis, as employed for the metric in equation 4.4,while a coordinate basis will also be adopted on the base space M . In this basis theconnection coefficients Γ abc on the base space M contribute to the set in equation 4.6with: ´Γ abc = Γ abc = 12 g ad ( e b ( g cd ) + e c ( g bd ) − e d ( g bc )) (4.7)which is simply equation 3.53, since the structure coefficients on P are related to thestructure coefficients on the base manifold with ´ c abc = c abc = 0 in this basis (and with67he corresponding term hence absent in equation 3.34). The connection coefficients ˇΓ ijk are also related to the internal curvature through equation 4.6 since in the horizontallift basis, by equation 3.36, we have ´ c αab = − Ω αab . Here we adopt the convention ofdenoting the components of curvature Ω αab on the principle bundle by F αab , which maythen represent the curvature components on P or M depending on the context, inorder to match the notation in many of the references. Ultimately the curvature F αab will feature in gauge invariant expressions on the base manifold. From equation 4.6 wefind in the horizontal lift basis on the bundle P terms such as (see [13] equation (22)):´Γ αab = + 12 F αab and ´Γ abα = ´Γ aαb = + 12 g ac g αβ F βbc (4.8)The complete set of coefficients for the Levi-Civita connection on P are listed under‘Cho [13]’ as the first case in table 4.1 in the following section.Hence the Levi-Civita connection of equation 4.6 on the total bundle space P is intimately related to the external curvature on the base space as well as the internalcurvature of the gauge group. In turn the components of the Riemann curvaturetensor ˇ R ijkl calculated for this Levi-Civita connection on P according to equation 3.62is intimately related to both the external curvature on M via equation 4.7 and the internal curvature, associated with gauge group G , which is drawn into the Riemanniangeometry through equation 4.8.It is important to clarify the relation between the linear connection ˇΓ( p ) andgauge connection ω ( p ) on the manifold P . In fact from the point of view of framebundles and principle fibre bundles in general a linear connection e ω (see the openingof section 3.3) on FP would be the same kind of object as the gauge connection ω on P . Here we are dealing with Riemannian geometry of the manifold P itself, which ishence the base space upon which the gl( m, R )-valued 1-form ˇΓ( p ) = Σ ∗ e ω is defined,where m = dim( P ) = dim( M ) + dim( G ) and Σ( p ) is a section map P → FP over P .The same manifold P is also the principle bundle upon which the gauge connection ω is defined, with A ( x ) = σ ∗ ω ( p ) being the gauge field on M , for a section map σ ( x ) : M → P over the space M .Having the metric ˇ g ij on P the Ricci tensor ˇ R jk = ˇ g il ˇ R ijkl (equation 3.74) andscalar curvature ˇ R = ˇ g ij ˇ R ij may also be computed, where the latter is found to be(with differing sign convention to [13]):ˇ R = R M + R G + 14 F (4.9)Here R M is the usual scalar curvature on the base manifold (which varies with the point x = π ( p ) ∈ M under p ∈ P ) and R G is the scalar curvature on the group manifold G (which can be interpreted as a, problematically very large, cosmological constant inthis version of Kaluza-Klein theory). The term F = F αab F abα , constructed from thenon-Abelian gauge fields, is also gauge invariant and the curvature components F αab ( p )on P can be interpreted as the corresponding gauge covariant curvature components F αab ( x ) on the base space M , for example in table 4.1.As a scalar ˇ R in equation 4.9 is a quantity which is independent of the basis { ˇ e i } in which it is determined (for example in the direct product or horizontal lift basisrespectively for equations (17) and (24) of reference [13]). The equations of motionfor the theory are then derived by adopting the Lagrangian function ˇ R p | ˇ g | , where | ˇ g |
68s the magnitude of the determinant of the metric ˇ g ij on P , in the Einstein-Hilbertaction integral: A m = Z ˇ R p | ˇ g | d x d n G G (4.10)with m = 4 + n G . The integration over the group manifold G , with volume V G , istrivial and the above expression reduces to the 4-dimensional action integral: A = V G Z ˇ R p | g | d x (4.11)where | g | is here the determinant of the metric g ac on M . The variational princi-ple is then applied under the constraint δA m = 0, and hence δA = 0, with respectto restricted variations of the metric δ ˇ g on the bundle space, consistent with equa-tion 4.3, as explained before equation 4.16 in the following section. Within this re-striction this again follows the prescription for the original theory of general relativityon a 4-dimensional spacetime manifold M with scalar curvature R for which the fieldequations can be determined from the Einstein-Hilbert action integral of equation 3.79.By comparison of equations 4.9 and 4.11 with 3.79 the constant R G of theKaluza-Klein theory indeed appears as a cosmological constant term (albeit too largeby a factor of ∼ if a natural normalisation is used with the length scale of thegroup space G taken to be of order the Planck length [13]), while the term F effectivelycontributes the content for the matter Lagrangian L . Hence, as a particularly elegantfeature of Kaluza-Klein theory, the geometry of the 4-dimensional spacetime manifoldalong with a matter contribution is identified within a single geometrical object in theform of ˇ R on the principle bundle space. One way to remove the problematic cosmological term R G in equation 4.9 would beto redefine the Lagrangian for the Kaluza-Klein theory by simply adding by hand a counter -cosmological constant term Λ c to ˇ R in equation 4.10 to cancel R G . Howeverthis would be an ad hoc measure, similar in spirit to the inclusion of the originalcosmological constant term Λ in equation 3.79, contrived largely to match empiricalobservation.However there is flexibility within the Kaluza-Klein approach on a principlefibre bundle if the metric ˇ g ij is not treated as the fundamental object of the theory(see for example [17, 18, 19, 20]). While the same natural metric ˇ g ij of equation 4.4is employed the linear connection ˇΓ ijk on P may be defined with some independencefrom ˇ g ij , unlike for the Levi-Civita connection of equation 4.6. In this case it is possibleto derive a curvature scalar ˇ R on P such that the cosmological term R G vanishes andequation 4.9 reduces to simply: ˇ R = R M + 14 F (4.12)One way to achieve this is to require the linear connection ˇΓ ijk to incorporatea description of absolute parallelism on the bundle fibres G x . As reviewed in subsec-tion 2.2.2 on the manifold G itself the list of canonical geometric objects include a69asis of left-invariant vector fields { X α } and the Maurer-Cartan 1-form θ = θ α X α aswell as the structure constants c αβγ and the Killing form metric g αβ of equation 4.1.As described below equation 3.67 in the basis { X α } the choice of linear connection co-efficients Γ αβγ = 0 is equivalent to inducing parallel transport on the group manifoldvia the left action L a of G on itself, for any a ∈ G , that is with parallelism definedby the left-invariant vector fields { X α } on G , while Γ αβγ = − c αβγ corresponds to theparallelism described by a right-invariant frame field under R a . In either case the re-sulting Riemann curvature is zero with R αβγδ = 0, as can be shown using equation 3.62together with the Jacobi identity expressed in terms of the structure constants.More generally, employing the derivative action of the left-invariant basis vec-tors { X α } , the right-invariance of the Killing metric implies that the covariant deriva-tive of the metric on G vanishes: ∇ α g βγ = X α g βγ − Γ δβα g γδ − Γ δγα g βδ = 0 (4.13)provided Γ αβγ = − ρ c αβγ for any ρ ∈ R (4.14)since X α g βγ = 0 and by the antisymmetry of the c αβγ indices. Hence for any value of ρ this linear connection is metric compatible, with ∇ g = 0 on G . The torsion is zeroonly for ρ = which hence represents the unique Levi-Civita connection on G . Onthe other hand the Riemannian curvature is zero on G only for the cases of ρ = 0 and ρ = 1, which with finite torsion are not Levi-Civita connections. However these lattertwo cases in describing an absolute parallelism on G can be considered as geometricallynatural metric connections on G .For the linear connection Γ αβγ = 0 or Γ αβγ = − c αβγ employed on the bundlefibres G x a subset of the torsion components on P are also necessarily non-zero, withˇ T αβγ = 0. Hence with the torsion allowed to be non-zero on the bundle space P thisversion of Kaluza-Klein theory resembles the Einstein-Cartan theory on 4-dimensionalspacetime for which Γ and g are treated as independent geometric objects. Here wereview four such approaches in the literature.In Kopczy´nski [17] a G -invariant linear connection ˇΓ is constructed in terms ofthe structure on the principle bundle with a gauge connection ω without reference to ametric and with non-zero torsion. This generalises from the Levi-Civita connection de-scribed in the previous section (as employed by [13] and others) with the ‘gravitationalfield’ on P now being described by the combination of both ˇ g of equation 4.2 and thecomponents of ˇΓ as listed in the corresponding column under ‘Kop [17]’ in table 4.1.With these components the scalar curvature on P is found to be ˇ R = R M + α ( α − K ,with K = K αβ K αβ . For the case β = 0 the connection is metric compatible, resem-bling Einstein-Cartan general relativity in 4-dimensional spacetime. While this refer-ence shows that the connection coefficients can be greatly simplified compared withthe Levi-Civita case, listed under ‘Cho [13]’ in the first column of table 4.1, in orderto achieve the correct dynamics a more complicated Lagrangian function is postulatedwith L = ˇ R + µ ˇ T ijk ˇ T jki including a quadratic torsion term. The cosmological constantΛ obtained in this approach is arbitrary, and may be set to be zero or very small by asuitable choice of the parameters α and µ .In Orzalesi and Pauri [18] the main motivation is to describe a linear connec-tion ˇΓ on the principle bundle which is gauge covariant. In particular requiring theRicci curvature on the fibre space to be gauge invariant implies the adoption of zero70urvature on the group manifold, that is the case ρ = 0 or ρ = 1 as described abovefor equation 4.14. This form differs in a relatively minimal way from the Levi-Civitaconnection, as can be seen by comparing the entries of column [18] with column [13]in table 4.1. Here the simple scalar Lagrangian L = ˇ R on the bundle space is againadopted, with the resulting vanishing of Λ ≡ R G on the base space M interpreted asa consequence of the underlying gauge G -symmetry of the Riemannian geometry on P . Without an R G term the vacuum solution corresponds to zero external curvature R M = 0 together with zero internal curvature F = 0.In Kalinowski [19] the linear connection 1-forms ˇΓ ij = ˇΓ ijk ˇ e k on P are definedas the horizontal part of the Levi-Civita connection 1-forms Γ ij of equation 4.6, thatis ˇΓ ij = hor(Γ ij ) (with ‘hor’ introduced in equation 3.13) which maps the verticalcomponent of tangent vectors on TP to zero. The components of this linear connectionˇΓ ijk in the horizontal lift basis are listed in column [19] of table 4.1. The factors of λ arise as here the metric on G is taken to be g αβ = λ K αβ . This linear connectionˇΓ ijk is metrical, invariant under the G -action, again with non-zero torsion and, whilemotivated in the context of gauge derivatives of spinor fields, again leads to a vanishingcosmological constant.In Katanaev [20] an initially completely general ˇΓ ijk on the principle bundleis considered. Four conditions are postulated for ˇΓ in a geometrically meaningfulway related to the structure group G over P and, as for the previous reference, withemphasis on horizontal propagation. In particular for column [20] of table 4.1 ontaking c = 1 for entry ‘5)’ ´Γ αab = cF αab the change in a tangent vector to P underparallel transport using these linear connection coefficients equals the change in thevector due to the basis transformation under parallel transport of the fibres using thegauge connection, with the latter depicted in figure 3.3. The entry ‘4)’ in this columnis included for metric compatibility. The coefficients listed represent the case presentedin [20] with finite torsion and the absence of a cosmological constant term, althougha different choice of ˇΓ consistent with the postulates is possible. A further possibilitywithin this framework would be to set the first two entries, ‘1)’ and ‘2)’, equal to zero incolumn [20]. This reference is of significance for the present paper in that it highlightsthe possibility of a geometric origin of ˇΓ on P without any appeal to the Levi-Civitaconnection.The complete set of linear connection coefficients for reference [13], augmentingequation 4.8, are collected in the first column of table 4.1. These are listed alongsidethe linear connection coefficients ˇΓ ijk on the bundle space P for the above four caseswith non-zero torsion. Where necessary signs have been aligned to the conventionsused here, with for example linear connection 1-forms ˇΓ ij = ˇΓ ijk ˇ e k . The motivationfor the final column headed ‘minimal’ will be explained in section 5.1.Only the first case in table 4.1 describes a torsion-free linear connection, yeteach of the six cases is a Kaluza-Klein theory providing a unifying framework forgeneral relativity together with gauge field theory. The purpose of collecting togetherthe range of linear connection coefficients is to demonstrate that a significant degreeof flexibility is possible within Kaluza-Klein theory while still maintaining this unifiedframework.The derivation of Einstein’s equations on 4-dimensional spacetime from theEinstein-Hilbert action of equation 3.79 was described in the opening of section 3.5.71Γ ijk Cho [13] Kop [17] O+P [18] Kal [19] Kat [20] minimal1) ´Γ αβγ − c αβγ − αc αβγ − c αβγ or 0 0 − c αβγ
02) ´Γ αγa − ω βa ´Γ αβγ
03) ´Γ abγ g ac g γβ F βbc g ac g γβ F βbc γg ac g γβ F βbc
4) ´Γ aγb g ac g γβ F βbc g ac g γβ F βbc λ g ac g γβ F βbc cg ac g γβ F βbc
05) ´Γ αab F αab βF αab F αab λ F αab cF αab
06) ´Γ abc Γ abc Γ abc Γ abc Γ abc Γ abc Γ abc Table 4.1: Linear connection components ´Γ ijk on a principle bundle extracted from[13] equation (22), [17] p.367, [18] equation (19), [19] equation (29), the case in [20]with non-zero torsion on G and for a ‘minimal’ model. All components are expressedin the horizontal lift basis and ´Γ aβγ = ´Γ αaγ = 0 in all six cases. Each of λ > α , β , c and γ , where used as coefficients, are real constant parameters.In the vacuum case with L = 0 and Λ = 0 variation of the metric δg µν on M leads tothe equation of motion G µν = 0 of equation 3.82. For the Kaluza-Klein extension tothe scalar curvature ˇ R for the Levi-Civita connection on a principle bundle space thesame steps lead to the requirement of the stationarity of the action integral over thefull bundle space in equation 4.10, that is δA m = 0, under variation of the extendedmetric ˇ g ij on P , which results in the expression:ˇ G ij = ˇ R ij −
12 ˇ R ˇ g ij = 0 (4.15)In some versions of Kaluza-Klein theory, in particular for the 5-dimensional case, equa-tion 4.15, which implies ˇ R ij = 0, is quoted as an ansatz at the outset in order to deriveequations of motion for the 4-dimensional world by imposing this higher-dimensional‘vacuum equation’ (see for example [21], in which the 5-dimensional metric ˇ g ij ( x ) maydepend on the 5 th coordinate).However for the extended Kaluza-Klein theories described here the variationsin the metric ˇ g ij are not arbitrary since the structure of the symmetries of ˇ g ij on thebundle space P need to be preserved under the variations δ ˇ g ij . That is, the right-invariance of ˇ g ij of equation 4.3 and the general form of the metric in equation 4.5should be preserved. This limits the metric variations to the components δg ac and δω αa on P and leads to two equations of motion on the base manifold M . Applyingthe variation δg ac under δA m = 0 for the action in equation 4.10, with the curvatureˇ R of equation 4.12, in a general coordinate basis on M leads to ([19] equation 38): G µν = R µν − R g µν = − κT µν (4.16)with − κT µν = − F αµρ F ρνα − g µν F αρσ F ρσα (4.17)The left-hand side of the top line would read G µν + R G g µν if the scalar curvature ofequation 4.9 based on a Levi-Civita connection is used instead. On the other hand the72ariation δω αa leads to D µ F α µν = 0 (4.18)Equation 4.16 is the Einstein field equation with the energy-momentum tensor T µν composed as equation 4.17 purely from the gauge fields, with the latter being subjectto equation 4.18 which is the Yang-Mills field equation (or Maxwell’s equation F µν ; µ = 0in the case of the Abelian internal symmetry group G = U(1), see also the discussionafter equation 3.91). Hence the source-free Yang-Mills field equation 3.95 has beenderived without the explicit introduction of the Yang-Mills Lagrangian of equation 3.94.Rather such a ‘Lagrangian term’ F = F α µν F α µν has been incorporated within theEinstein-Hilbert action based purely on the geometry of the bundle space. In this waythe non-Abelian Kaluza-Klein theory provides a unified framework for the combinedEinstein-Yang-Mills field equations. A further generalisation of Kaluza-Klein theory is also of relevance for the frameworkpresented in this paper. In the present theory the symmetry group G rather thanbeing motivated independently is introduced in terms of the set of symmetry actionson a form L ( v ) = 1 of multi-dimensional temporal flow. This structure is reminiscentof Kaluza-Klein theories with homogeneous fibres in which G acts on a k -dimensionalmanifold S k . A bundle space E is constructed with each fibre being a copy of S k over the base space M . Based on the references [22, 23, 24] this approach will becollectively summarised in this section.In these models the bundle E = ( M , S k ) is constructed over the base space M while the fibres S k may be considered to represent k ‘extra dimensions’. For ourpurposes it is sufficient to consider the trivial bundle with E ≡ M × S k . While eithera left or right action may be considered here we take the gauge group G to act onthe space S k on the left (as for reference [22] for example, and as will be the case forthe E action on the space h O as discussed alongside equation 6.55) such that eachLie algebra element X α ∈ L ( G ) generates a vector field K α on S k with the bracketcomposition exhibiting the negative of the structure constants c αβγ of L ( G ), that is:[ K β , K γ ] = − c αβγ K α (4.19)The group actions may also be considered to be one-to-one with the isometrytransformations for an inner product defined on the tangent space TS k . That is, a G -invariant metric may be defined on S k with Killing vector fields: K α = K ˚ αα e ˚ α (4.20)where K ˚ αα are the components of K α in a linearly independent tangent space basis { e ˚ α } on S k , with indices ˚ α = 1 . . . k and α = 1 . . . dim( G ). Such a G -invariant metric g ˚ α ˚ β on S k may be induced from the Killing metric K αβ on G itself.If G acts upon S k transitively then S k is a homogeneous space. Given anypoint y ∈ S k the elements h ∈ G for which h · y = y under the left action of thegroup form the isotropy subgroup H , with h ∈ H ⊂ G . The homogeneous space S k is73hen diffeomorphic to the space of left cosets gH as identified for varying g ∈ G , thatis S k ≡ G/H where H is the isotropy subgroup of the isometry group G . As a vectorspace the Lie algebra of G may be decomposed as: L ( G ) = L ( H ) + B (4.21)with [ L ( H ) , L ( H )] ⊂ L ( H ) and [ L ( H ) , B ] ⊂ B , where B ≡ T ( G/H ) forms a basis forthe tangent space at y ∈ S k .Such a linearly independent basis { e ˚ α } for TS k forms a basis for the verticalsubspace of the tangent space on the fibre bundle E . A complete ‘horizontal basis’on E , written ´ e i ( x, y ) = { ´ e ˚ α , ´ e a } , in place of the horizontal lift basis for the principlebundle P of figure 3.2, can be expressed as:´ e ˚ α = ¨ e ˚ α , ´ e a = ¨ e a − A αa ( x ) K ˚ αα ( y )¨ e ˚ α ( x, y ) ≡ ¨ e a − A αa ( x )¨ e α (4.22)in terms of a direct product basis ¨ e i ( x, y ) = { ¨ e ˚ α , ¨ e a } on E , by comparison with equa-tion 3.26 and figure 3.5, using the Killing vector components K ˚ αα defined in equa-tion 4.20. As implied in equation 4.22 the construction of such a horizontal basis on E corresponds to the introduction of a connection form ω on the associated principlebundle P ≡ M × G . This connection form is written in terms of the coefficients A αa ( x )rather than ω αa ( x, g ) since the vertical basis is here defined through the left action of G (see the discussion in [22] after equation (7.2) for example).Consistent with the horizontal basis of equation 4.22 a natural metric on thebundle space E may be defined, for which horizontal and vertical vectors are mutuallyorthogonal, and expressed in a direct product basis as:¨ g ij = g ab ( x ) − g ˚ α ˚ β ( y ) K ˚ αα ( y ) A αa ( x ) K ˚ ββ ( y ) A βb ( x ) K ˚ αα ( y ) A αa ( x ) g ˚ α ˚ β K ˚ βα ( y ) A αb ( x ) g ˚ α ˚ β g ˚ α ˚ β ( y ) (4.23)which may be compared with equation 4.5 for the case of a principle fibre bundle.Changes in the vertical coordinates { y } on E described by the infinitesimal isometries ε α ( x ) induce changes in the metric components with respectively: y ˚ α → y ˚ α + ε α ( x ) K ˚ αα ( y ) A αa → A αa + ∂ a ε α ( x ) + c αβγ ε β ( x ) A γa (4.24)Hence such isometries effectively simulate non-Abelian gauge transformations with A αa ( x ) identified as the Yang-Mills gauge field on the base space.Following the Kaluza-Klein prescription described in section 4.1 the Levi-Civitaconnection, that is the unique torsion-free linear connection compatible with the met-ric, and curvature can be constructed on the manifold E based on the metric ¨ g ij ( x, y ) ofequation 4.23. In turn an action principle may be employed on this (4 + k )-dimensionalspace with action A k = R ˇ R p | ˇ g | d x d k y in comparison to equation 4.10 where nowthe curvature scalar ˇ R on the bundle E ≡ M × S k is found to take the form:ˇ R = R M + R S k + 14 g ˚ α ˚ β K ˚ αα K ˚ ββ F αab F βab (4.25)74here R S k is the scalar curvature of the homogeneous space S k . With F αab being thegauge curvature components for the internal symmetry group G the above equationagain demonstrates a relation between the external Riemann curvature with scalar R M and a quadratic term in the internal curvature. This relationship derived for G actingon homogeneous fibres is hence in turn similar to that obtained in equations 4.9 and4.12 with G itself composing the fibres of a principle bundle. A linear connection on E differing from the Levi-Civita connection may be employed to remove the cosmologicalterm S k by analogy with the examples cited in the previous section. The Einstein-Yang-Mills equations also follow from a prescription analogous to that described forequations 4.16–4.18.For models with homogeneous fibres in which the metric g ˚ α ˚ β ( y ) is replaced bythe more general field components g ˚ α ˚ β ( x, y ) = K ˚ αα ( y ) K ˚ ββ ( y )Φ αβ ( x ), which depend on x ∈ M and where Φ is a non-Killing metric on G , a set of scalar fields are introducedinto the theory with a number of further terms featuring the derivatives ∂ a g ˚ α ˚ β ( x, y )appearing in the corresponding generalisation of equation 4.25 (see for example [22]equation (8.6)).On the other hand on constructing ¨ g ij in equation 4.23 for the case of trivialisotropy group H = { e } , where e ∈ G is the identity element, then by equation 4.21 wehave B = L ( G ) and the ˚ α, . . . indices can be replaced by α, . . . indices, with K ˚ αα = δ ˚ αα .In this case the theory simplifies to that described in section 4.1 based on the metricof equation 4.5 with the set of vector fields { ´ e α } spanning the vertical subspace of thetangent space on P ≡ M × G (with care for the convention choice of a left or rightgroup action, see for example [13] equations (8 ′ ) and (12)).Even for the case with H = { e } the full G -symmetry Yang-Mills dynamics isobtained so long as G acts effectively on the fibres S k . This is also the case for G acting on L ( v ) = 1 for the present theory in this paper, and in particular for the E action to be described in chapter 6, and for the broken internal subgroups of G .The action of G on the set of elements v underlying the form L ( v ) = 1 is alsotransitive, and hence this set forms a homogeneous space, motivating the review ofthis section. However with the observation that the Kaluza-Klein unification achievedwith homogeneous fibres, given an effective group action, is closely related to thatachieved on the associated principle bundle, in the following section we apply some ofthe observations of the previous section to the present theory. This in particular picksup from the development of section 2.3 with the goal of relating the external curvatureto the internal curvature in the context of the new theory.75 hapter 5 Geometry Unified throughTemporal Flow
In this section, ultimately guided by the framework of Kaluza-Klein theories describedin the previous chapter, the aim is to determine a relation between the external andinternal geometry over the base manifold arising out of the symmetries of a form oftemporal flow L ( v ) = 1, building upon the structures described in chapter 2. In placeof the base space M with local symmetry SO(3), introduced for the model universe insection 2.2 with the 3-dimensional form L ( v ) = 1 of equation 2.14, here we considerthe form: L ( v ) = ( v ) − ( v ) − ( v ) − ( v ) = 1 (5.1)that is L ( v ) = η ab v a v b = 1 with Minkowski metric η = diag(+1 , − , − , − + (1 ,
3) symmetry, projected into the 4-dimensional base space M . Overthe spacetime manifold M a globally defined orthonormal basis arises in the mannerof equation 2.15 with the natural parallelism on M described by a linear connectionwith components Γ abc ( x ) = 0 in this basis. With the local symmetry group SO + (1 , M the principle bundle P = ( M , SO + (1 , P ≡ M × SO + (1 ,
3) owing to the triviality of the bundle as described towards the endof section 3.1.However, following section 2.3, here we study initially the geometry of theprinciple fibre bundle P ≡ M × ˆ G , where ˆ G = SO + (1 ,
9) is provisionally taken asthe full symmetry group for the form L (ˆ v ) = L ( v ) = 1, which in turn is the 10-dimensional extension of equation 5.1. The extended base manifold M now arisesout of four of the ten translational degrees of freedom of L ( v ) = 1, in the mannerdescribed in equation 2.13. In place of figure 2.7 for the SO(5) model over M describedearlier, for this more realistic model we now have the structures described in figure 5.1.76igure 5.1: (a) The full symmetry group ˆ G = SO + (1 ,
9) over the base space M (b)broken to the internal symmetry SO(6) with external subgroup SO + (1 , ⊂ SO + (1 , TM .The structure of figure 5.1 is associated with a canonical flat connection on M , as described by A ( x ) = g ∗ θ of equation 2.30 where here θ is the Maurer-Cartan1-form on the group manifold ˆ G = SO + (1 , { ´ e i } on the corresponding principle bundle structure P ≡ M × SO + (1 , ijk can be defined in this basis on P .While Γ abc = 0 represents the initial parallelism on M the set Γ αβγ = 0 de-scribes an absolute parallelism on the manifold ˆ G , as described in section 4.2. Extend-ing to the full bundle space P here we provisionally consider the ´Γ ijk set of reference [18]listed in the third column of table 4.1. This set of linear connection coefficients aregauge covariant on P and compatible with the metric of equation 4.2 deriving from thegauge connection ω on P , that is with ∇ ˇ g = 0. On adopting such a linear connectionon P , based on compatibility with the structures of the form L ( v ) = 1 here, we thenconsider the implications of incorporating this element of Kaluza-Klein theory into thepresent framework.The components of the Riemann curvature on the manifold P ≡ M × SO + (1 , ijk described above, directly from equation 3.62 as:´ R ijkl = ´ e k ´Γ ijl − ´ e l ´Γ ijk + ´Γ imk ´Γ mjl − ´Γ iml ´Γ mjk − ´ c mkl ´Γ ijm (5.2)In the present theory we begin with the translational symmetry of the form L ( v ) = 1over the manifold M with a flat Minkowski metric g ac ( x ) = η ac and the canonical flatSO + (1 , ω on P . As described in chapter 2, initially in equa-tions 2.35 and 2.36, this latter property means that the full curvature is zero ˆ F = 0, orin components ˆ F αab = 0. Hence, given that ´Γ αβγ = Γ αβγ = 0 and ´Γ abc = Γ abc = 0, all the linear connection coefficients in column ‘O+P [18]’ of table 4.1 are zero, ´Γ ijk = 0,77nd in turn all components of the Riemann curvature tensor of equation 5.2 vanish onthe principle bundle manifold P .Here the natural absolute parallelism on M and G has been generalised toa natural parallelism on P ≡ M × G with ´Γ ijk = 0 for all coefficients of the linearconnection in the horizontal lift basis. In fact for the canonical zero full curvatureˆ F αab = 0 on the principle bundle all five non-Levi-Civita choices for ´Γ ijk in table 4.1lead via equation 5.2 to ´ R ijkl = 0, which as the components of a tensor vanish for anyframe field on P , expressed generally as:ˇ R ijkl = 0 (5.3)On the other hand there are torsion components with ˇ T ijk = 0 and hence the torsion isfinite, as it is for the case of a self-parallel frame composed out of left-invariant vectorfields on a Lie group manifold G , a copy of which here forms part of the total parallelframe field on P , as described in section 4.2.In this way the full zero gauge curvature ˆ F = 0 for ˆ G over M has beentranslated into zero Riemannian curvature ˇ R ijkl = 0 on the bundle space P . Thequestion then remains regarding how this structure might provide the link throughwhich the external gravitational field will relate to the internal gauge fields over thebase space M when the full symmetry is broken.On the principle bundle P ≡ M × SO + (1 ,
9) a trivialisation may be chosen suchthat the corresponding direct product basis for the tangent space TP is identical to thehorizontal lift basis associated with the canonical flat connection ω , which in turn isderived from the full symmetry group. Such a trivialisation represents a gauge choicefor which the section σ , depicted in figure 5.2, on the principle bundle P coincideswith the submanifolds of the integrable horizontal subspaces of ω on P , and hencewith gauge connection components ω αa ( x, g ) = 0 on the bundle space. The linearconnection components are identical ¨Γ ijk = ´Γ ijk = 0 in the respective direct productand horizontal lift bases for this choice of gauge, describing the absolute parallelismdefined in the frame field adapted to this section on P .While the canonical flat connection ω = π ∗ θ on P describes a unique horizontalsubspace and the corresponding horizontal lift basis, a direct product basis may bedefined in terms of any section on the bundle. Indeed, more generally geometric objectsover the base space M may be described with respect to a choice of gauge on the bundle P , as for example determined by the section σ ′ = σ g , with g ( x ) ∈ ˆ G = SO + (1 , ω αa ( x, g ) = 0 inthe new trivialisation are such that the vectors of the horizontal lift basis { ´ e i } areexpressed as for equation 3.26 with:´ e a = ¨ e a − ω αa ¨ e α with ω αa = 0 (5.4)while [´ e a , ´ e b ] = − ˆ F αab ´ e α with ˆ F αab = 0 (5.5)since the full SO + (1 ,
9) zero curvature is a gauge independent structure. However thefull group SO + (1 ,
9) does not act purely as an internal symmetry but is broken bythe action of the subgroup SO + (1 , ⊂ SO + (1 ,
9) on the external tangent space TM .While the choice of gauge g ( x ) ∈ SO + (1 ,
9) remains arbitrary with respect to the fullunbroken symmetry it will affect the physics of the broken symmetry over M . For the78igure 5.2: Geometric objects ˇΓ and ω on the principle bundle P in relation to thelinear connection Γ on the base space M and Maurer-Cartan 1-form θ on the groupmanifold ˆ G .restricted set of internal SO(6) generators the horizontal lift vectors extracted fromequations 5.4 and 5.5 have the properties:´ e a = ¨ e a − ω αa ¨ e α with ω αa = 0 (5.6)while [´ e a , ´ e b ] = − F αab ´ e α with F αab = 0 (5.7)Here the ω αa are the components of an so(6)-valued connection 1-form, with the sumsover α restricted to the SO(6) generators, resulting in a generally non-zero internalcurvature F αab , as was demonstrated in equation 2.56 for the finite internal SO(2)curvature achieved for small SO(5) gauge transformations over M for the model worldof section 2.3. Here we are reproducing the symmetry breaking approach of section 2.3in the light of the principle bundle structure and Kaluza-Klein theories described inthe previous two chapters.As well as the transformation of the gauge connection ω for a different choiceof basis on P the linear connection ˇΓ also transforms. For any change of frame e i ′ = e i e ii ′ with e ii ′ ∈ GL( m, R ) the transformation of a linear connection, displayed inequation 3.46, can be written as:Γ i ′ j ′ k ′ = e i ′ i e jj ′ e kk ′ Γ ijk + e i ′ l e k ′ e lj ′ (5.8)The gauge choice associated with the section σ ′ = σ g ( x ) on P corresponds to atransformation from the horizontal lift basis to an arbitrary direct product basis ¨ e i ′ =´ e i e ii ′ on a principle bundle, that is the reverse of equation 3.26 or 5.4, and we have: (cid:16) ¨ e a ′ ¨ e α ′ (cid:17) = (cid:16) ´ e a ´ e α (cid:17) δ aa ′ ω αa ′ δ αα ′ (5.9)79nd hence e ii ′ = δ aa ′ ω αa ′ δ αα ′ with inverse ( e − ) i ′ i = δ a ′ a − ω α ′ a δ α ′ α (5.10)As a consistency check the same transformation is applied to the full set ofLevi-Civita connection coefficients in the horizontal lift basis ´Γ ijk as listed in the ‘Cho[13]’ column of table 4.1 (as extracted from [13] equation (22)). The expressions forthe ¨Γ i ′ j ′ k ′ components obtained in the direct product basis using equations 5.8 and 5.10is found to agree with the original reference (the ¯Γ components in the notation of [13]equation (15)).The general aim of this approach is to use equations 5.2 and 5.3, with ˇ R ijkl = 0deriving from the full zero curvature ˆ F = 0, on the principle bundle P ≡ M × SO + (1 , F αab = 0 from equation 5.7 will be introduced quadratically intothe terms of equation 5.2 via relations to the linear connection ˇΓ of the kind listedin table 4.1 (by adopting for example the coefficients of column [18] as provisionallysuggested above) and hence into correlation with the external curvature R abcd = 0 onthe base manifold as identified within the appropriate components of ˇ R ijkl = 0 in asuitable basis.These structures emerge in the symmetry breaking as represented by the transi-tion from figure 5.1(a) to (b). The structure of figure 5.1(a) implies that the total sym-metry SO + (1 ,
9) of L ( v ) = 1 is associated with a canonical flat gauge field A = g ∗ θ with full curvature ˆ F = 0, under which a correlation between the external curvature R and internal curvature F is implied in the symmetry breaking to the structure offigure 5.1(b), in particular with the case of both R = 0 and F = 0 simultaneouslypossible.In this picture a non-zero external curvature R = 0 on M is absorbed underˇ R ijkl = 0 on the extended bundle space as the ‘buckling’ of the geometry of thebase manifold is countered by a corresponding finite internal curvature F = 0. Theexternal and internal curvature is hence generated in a necessarily mutually consistentway under the symmetry of L ( v ) = 1 in a choice of SO + (1 ,
9) gauge over the basespace M . The invariance of the zero Riemann tensor of equation 5.3 under a change offrame adapted to choice of section is analogous to the invariance of the action integralof equation 4.10, defined in terms of a scalar curvature ˇ R , under variations of themetric ˇ g ij of the kind described in section 4.2. This motivates the conjecture thatthis framework leads to a similar unification of the Einstein-Yang-Mills equations ofmotion, that is equations 4.16–4.18, as found for non-Abelian Kaluza-Klein theorybut ultimately without the need to postulate a Lagrangian function, coupled with thevariational principle for the corresponding action integral, to obtain these equations.Resulting from the projection of the structure of figure 5.1(a) over that offigure 5.1(b), with a choice of an SO + (1 ,
9) gauge section over M for the former,two further bundle structures, associated with the latter figure, may be identified andconsidered separately. The subgroup SO + (1 ,
3) is distinguished in that it acts ontangent space vectors v ∈ TM of the base manifold, as depicted in figure 5.1(b),and therefore is designated as an external symmetry, with the residual SO(6) acting80n the remaining components of v ⊂ v of the form L ( v ) = 1 and constitutingan internal symmetry. This results in consideration of the complementary subbundles P ≡ M × SO + (1 ,
3) and P ≡ M × SO(6) which effectively decouple from eachother as mathematical structures, although related through the correlated geometricalstructures they support, as they are mutually carved out of the initial unbroken bundle P = M × SO + (1 , + (1 , ⊂ SO + (1 , + (1 ,
3) identified as the external symmetry and absorbed into the localtangent space geometry on M , that breaks the full SO + (1 ,
9) symmetry. The basespace M is naturally associated with the frame bundle FM , which is itself a particulartype of principle fibre bundle as described in section 3.3. The bundle space P ≡ M × SO + (1 , restriction of the P ≡ M × SO + (1 ,
9) bundle, can alsobe interpreted as a reduction of the FM frame bundle. In turn an so + (1 , P may be extended to a gl(4 , R )-connection on the frame bundle, together with theassociated tetrad e µa ( x ) and metric g µν ( x ) fields on M , as familiar in the theory ofgeneral relativity and also described in section 3.3.As described towards the end of section 3.4 the SO + (1 ,
3) symmetry can betreated by analogy with an ‘internal’ Yang-Mills gauge structure. Indeed, as describedabove for the full SO + (1 ,
9) symmetry, quadratic terms in the external SO + (1 ,
3) ‘gaugecurvature’ F αab will appear in the third and fourth terms of equation 5.2 (essentially asdescribed in [25], which adopts the Levi-Civita connection on the bundle space, leadingto equations (3.14) and (3.15) there). However this same external geometry, from theaction of SO + (1 ,
3) on TM , is represented by the Riemannian curvature R abcd ( x ),which is also contained within the corresponding ˇ R ijkl components in a suitable basis(as also described in [25]). Hence the bundle P appears to incorporate a redundantdescription of the external geometry while lacking an explicit reference to the internalcurvature.On the other hand the subbundle P ≡ M × SO(6) is closely related to both theframe bundle FM , upon which the external SO + (1 ,
3) geometry is expressed in termsof fields such as g µν ( x ), as well as the structures of the internal SO(6) geometry withthe associated gauge field Y αµ ( x ) and curvature F αµν ( x ) components constructed on M . Hence in principle all the necessary geometric structures for relating the externaland internal curvature can be identified on the bundle P .Rather than dealing with a connection form ω over M for the full ˆ G =SO + (1 ,
9) symmetry it is precisely through the symmetry breaking action, with thedegrees of freedom of the SO + (1 , ⊂ SO + (1 ,
9) subgroup part of the gauge connectionbeing converted into the freedom of a linear connection on M , that the bundle space P ≡ M × SO(6) emerges. This in turn implies that the structure of the zero curvatureˆ F = 0 for the full canonical flat connection does not explicitly survive the symmetrybreaking transition from figure 5.1(a) to (b).This motivates the study of a unified framework on the space P ≡ M × SO(6)considered from now as a principle bundle standing independently by itself, and notas subbundle ‘carved out’ of a larger bundle space such as P ≡ M × SO + (1 , + (1 ,
3) geometry in terms of the bundle P . 81arlier in this section an absolute parallelism on the bundle P ≡ M × SO + (1 , ijk = 0, taken from the set ofreference [18] listed in the third column of table 4.1 for the canonical zero full curvatureˆ F = 0, implying the identity of equation 5.3. Now, beginning directly on the bundle P = M × SO(6) in itself, the question arises concerning the possible definition of alinear connection on this space. Since there is a gauge connection (which now derivesfrom the internal SO(6) symmetry and in general is not flat) on P the horizontal liftbasis may be employed, and in turn the natural metric structure with components ´ g ij of equation 4.4 introduced.Hence it is possible to define the unique Levi-Civita connection on this bundle,as described in section 4.1, with the components of equation 4.6 as listed for thehorizontal lift basis in the first column of table 4.1 under ‘Cho [13]’. However in thepresent theory at no stage is P considered to be a physical space or spacetime structure,hence neither the metric ´ g ij nor a linear connection ´Γ ijk on P have a physical geometricmeaning, as they do on the base space M . Hence the unique metric-compatibletorsion-free Levi-Civita connection is not here considered to be a natural structure onthe bundle space as it is for the base manifold, and an alternative argument for theform of ˇΓ on P is sought.In particular the linear connection on P is expected to be closely associatedwith the linear connection Γ on the base space M , which does describe a physicalgeometry. Since this is a gl(4 , R )-valued 1-form Γ( x ) = Γ abc E ba e c on M , with respectto the distinguished horizontal lift basis on P the components ´Γ abc and ´Γ abγ alone maybe favoured for a linear connection Γ on M in some sense lifted onto P , and hencethe only non-trivial coefficients of ´Γ ijk on P might be taken to be:´Γ abc = Γ abc and ´Γ abγ = γ g ac g βγ F βbc (5.11)as listed as the ‘minimal’ set in the final column of table 4.1. The form of ´Γ abγ in theequation above and the third row of the table as adopted from the other models in thetable, consistent with the requirement that ´Γ should transform in a gauge covariantmanner on P as appropriate for any object relating to a physical entity on the basemanifold M . As will be described below this proposal will amount to a minimalstructure on P linking the present theory with Kaluza-Klein theory with a manifestcorrelation between the external Riemannian geometry and internal gauge curvature.While the Levi-Civita connection, Γ = f ( g ) of equation 3.53, on M providesa unique description of the geometry on the base manifold in terms of the metric g µν ( x ), the linear connection ´Γ of equation 5.11 represents an attempt to extend thisstructure onto P while maintaining the character of the connection Γ on M , concerningin particular the gl(4 , R )-valued property. However any linear connection on P isintrinsically a gl( m, R )-valued 1-form (where m = 4 + 15 for the internal SO(6) gaugegroup). For example under the transformation to a direct product basis, as describedin equations 5.8–5.10, the components of equation 5.11 in general give rise to linearconnection coefficients ¨Γ αbc = 0 and ¨Γ αbγ = 0 in addition to ¨Γ abc = 0 and ¨Γ abγ = 0.Since the character of being gl(4 , R )-valued cannot be upheld for a linear connectionon P an alternative proposal, and one for which parallel transport in the horizontaland vertical directions on P more directly reflects the geometry of the base manifold M , will be considered. 82 direct way to obtain a linear connection ˇΓ( p ) on P closely related to Γ( x )on M would be to define ˇΓ = π ∗ Γ as the pull-back of the gl(4 , R )-valued 1-form Γthrough the bundle projection π : P → M , by analogy with the identification of thecanonical Lie algebra-valued 1-form ω = π ∗ θ as the pull-back of the Maurer-Cartan1-form θ through the projection map π : P ≡ M × ˆ G → ˆ G for the full bundle P as described in figure 5.2. Indeed, the gl(4 , R )-valued linear connection Γ on M ,associated with the external symmetry, and L ( ˆ G )-valued 1-form θ on ˆ G , associatedwith the full symmetry, each describe the parallelism on their respective manifolds.The canonical flat connection ω = π ∗ θ on P ≡ M × SO + (1 ,
9) itself is anunambiguous geometric object, completely independent of any particular choice ofsection or gauge over the base manifold. It derives purely from the properties of θ onthe gauge group ˆ G = SO + (1 , π ∗ Γ on P or P hasno physical significance in itself other than that derived from its relation to a linearconnection Γ, and the related Riemannian geometry, on the base manifold M . Forthe case ˇΓ( p ) = π ∗ Γ( x ) on P and for any vector field X ( p ) ∈ T P we have: h ˇΓ , X i p = h π ∗ Γ , X i p = h Γ , π ∗ X i x (5.12)with X ( p ) projected in the final line onto the vector π ∗ X ∈ TM and with π : p ∈ P → x ∈ M . Hence for any vector in the vertical subspace X ( p ) ∈ V P we have h ˇΓ , X i p = 0 since π ∗ X = 0. This structure is related to the linear connection on thebundle described for the case of Kalinowski [19] in section 4.2 for which all tangentvectors are mapped onto their horizontal parts, again with the property h ˇΓ , X i p = 0for any vertical vector X , and hence again with emphasis on the horizontal structure,which in turn is closely associated with the geometry of the base space M . In factconsideration of all cases collected in table 4.1 leads to the following proposal for theproperties of ˇΓ on P appropriate for the present theory:a) parallel propagation via ˇΓ in the vertical directions is taken to be trivial in themanner of [19] in the fourth column of table 4.1.b) parallel propagation via ˇΓ in the horizontal directions on P is taken to relate tothe contours of the gauge curvature over the base space M , following [20] in thefifth column of table 4.1 for the case c = 1.c) with a view to deriving physical equations on the base space M compatibilitywith gauge covariance should be observed, as emphasised in [18].d) consistent with c) the linear connection may be compatible with the natural, butnon-physical, metric ˇ g ij of equation 4.4, although the torsion may be arbitraryas initially emphasised in [17].e) the bundle P serves as an arena to relate the external and internal symmetrystructures compatible with the simultaneous possibility of R = 0 and F = 0, asderived from consideration of figure 5.1 for the present theory.Based on these observations and the broader discussion of Kaluza-Klein theoryin chapter 4 the conjectured linear connection components on P , as extracted fromtable 4.1, can be summarised as:4) ´Γ aγb = g ac g γβ F βbc ,
5) ´Γ αab = F αab ,
6) ´Γ abc = Γ abc (5.13)83ith all other ´Γ ijk = 0. Hence these are essentially the set of [19] in the fourth columnof table 4.1 with λ = 2, with the motivation for employing this latter value derivedfrom the geometrical argument in [20]. This latter argument also has the benefit offixing the geometry of ˇΓ without any reference to the Levi-Civita connection on P .The whole purpose of constructing a linear connection ˇΓ on P , as describedabove, is to provide a means through which a correlation between the external andinternal curvature may be explicitly described. On the spacetime manifold M anyrelationship between the external geometry, expressed in terms of the Einstein tensorwith components G µν ( x ), and the internal geometry, expressed in terms of the gaugecurvature with components F αµν ( x ), must transform covariantly both under generalcoordinate transformations and under gauge transformations, as described in particularin section 3.4. One technique for obtaining such a relation is to first identify a scalar‘Lagrangian’ function which has these invariance properties, as described in section 3.5.This approach, again following the Kaluza-Klein theories, will be adopted provisionallyhere, although a more direct geometric argument leading to equation 4.16–4.17, whichitself has the desired symmetry properties, would ultimately be preferred. (Since inthe following components such as F αµν ( x ) will always refer to the purely internal gaugecurvature we now omit the underscore for these objects).While earlier in this section the Riemannian curvature ´ R ijkl was constructed onthe full bundle P ≡ M × SO + (1 ,
9) we are now focusing on the bundle P ≡ M × SO(6),upon which the gauge curvature is generally finite. For any linear connection on thebundle space P , such as defined by any of the six sets of connection coefficients ´Γ ijk listed in table 4.1, the Riemann curvature tensor can be determined according to equa-tion 5.2, which is specified in the horizontal lift basis. The corresponding Ricci curva-ture components ´ R αβ and ´ R ac are listed here in the first and fourth rows of table 5.1for the six familiar examples. In all cases the entries in this table calculated here agreewith the corresponding equations of the given references – within the sign conventionssuch as that of equation 3.74 and as alluded to near the opening of chapter 4.The scalar curvature constructed in the horizontal lift basis on the principlebundle space can be written as:´ R = ´ g ij ´ R ij = g ac ´ R ac + K αβ ´ R αβ (5.14)owing to the simple form of the metric ´ g in this basis as expressed in equation 4.4.Hence the Ricci curvature components ´ R aα and ´ R αa are not required in order todetermine the scalar curvature on the bundle.If each of the four factors of in the ‘Cho [13]’ column in table 4.1, for thecase of Levi-Civita connection coefficients ´Γ ijk on the bundle, listed in rows 1), 3), 4)and 5) are replaced by the real factors f , f , f and f respectively then the scalarcurvature in the horizontal lift basis is found to be:´ R = R M + R G + χF (5.15)with χ = f − f f + f f − f f (5.16)This expression agrees with the scalar curvature for the Levi-Civita case, with each f i = , as quoted originally in equation 4.9, and with each subsequent case of table 4.1as quoted in the final row of table 5.1. Equations 5.15 and 5.16 show that f is the only84ho [13] Kop [17] O+P [18] Kal [19] Kat [20]/min´ R αβ = R ( G ) αβ + − F αbd F bdβ − F αbd F bdβ R ( G ) αβ = K αβ ( α − α ) K αβ K αβ ´ R αβ = R G − F R G − F R ac = R ( M ) ac + F βad F dβc F βad F dβc λ F βad F dβc c F βad F dβc g ac ´ R ac = R M + F F λ F c F ´ R = ´ g ij ´ R ij = R M + R G + F R M + R G R M + F R M + λ F R M + c F Table 5.1: Composition of the scalar curvature ´ R on the bundle space for the six casesof table 4.1. Contributions to the components of the Ricci curvature on the bundleinclude R ( G ) αβ and R ( M ) ac from the group manifold and base space respectively, with R G = K αβ R ( G ) αβ and R M = g ac R ( M ) ac being the respective scalar curvatures. Theresults for the sixth, ‘minimal’, case in table 4.1 are identical to those listed for [20] inthe final column above with c = γ .coefficient which is sufficient in itself to introduce a non-trivial F term, alongside R M ,into the scalar curvature ´ R , and this observation in part motivated the consideration ofthis simplest set of ´Γ ijk coefficients, as listed in the ‘minimal’ column of table 4.1 anddescribed in equation 5.11 above. While perhaps not developed as a serious physicalproposal this minimal model further demonstrates the flexibility within the Kaluza-Klein framework, obtaining the appropriate link between the external geometry andinternal curvature with a seemingly much simpler linear connection on the bundlecompared with the Levi-Civita case. More generally, equations 5.15 and 5.16 displaythe mutual consequences of the non-zero ´Γ ijk terms for the models listed in table 4.1.Since ´ R ( p ) is a scalar field on the bundle at any given point p it takes the samevalue in any local frame. Hence for example in a direct product basis, correspondingto a section σ on P , the scalar value is simply ¨ R ( p ) = ´ R ( p ). Further, since each of thescalar terms in the bottom line of table 5.1 is gauge invariant, a corresponding scalarfunction on the base space M may be deduced as:˜ R ( x ) = σ ∗ ¨ R ( p ) = R M + R G + χF (5.17)which is equivalent to ´ R ( p ) for any p ∈ P such that π ( p ) = x ∈ M . Hence ˜ R ( x ) isa real scalar function on M which contains information about both the external andinternal geometry, is invariant both under coordinate and gauge transformations onthe base space, and therefore makes a suitable ‘Lagrangian’ candidate on M . Whetheror not R G vanishes and the real value χ in equation 5.17 depend upon the particularmodel, as can be seen for the examples of table 5.1 and via equation 5.16 respectively.For the case of most interest for the present theory, with non-zero linear connectioncoefficients listed in equation 5.13, corresponding to setting λ = 2 in the ‘Kal [19]’columns of tables 4.1 and 5.1, we have simply ˜ R ( x ) = R M + F .The starting point for the Kaluza-Klein theories reviewed in sections 4.1 and4.2 is the mathematical structure of a principle fibre bundle P = ( M , G ), such as85escribed in section 3.1 and pictured in figure 3.1. This structure features an extendedbase space M over which a gauge connection may be introduced on the bundle space P transforming under the internal symmetry gauge group G . In these theories thebundle space is typically interpreted as a higher-dimensional physical spacetime. Forexample in reference ([18] p.190) the authors write: ‘Our general attitude is to regardthe n G vertical dimensions as physically real, and hence the vertical Einstein equationsas true dynamical equations of the ( n + n G )-theory.’A similar perspective is generally adopted for the theories with homogeneousfibres, described in section 4.3, in this case for the bundle space E = ( M , S k ). Inthe introduction of reference [24] the authors write: ‘Kaluza-Klein theories are theo-ries in which the gravitational potential together with the gauge potentials of variousinteractions are interpreted as manifestations of (pseudo-) Riemannian structure ofthe Universe which is 4 + k dimensional.’ The analogy between coordinate transfor-mations in general relativity and gauge transformations in gauge theory, discussed insection 3.4, is more explicitly realised in these theories as demonstrated for examplein equations 4.24.In Kaluza-Klein theories restrictions on the form of the metric ˇ g ij on the higher-dimensional space, in particular a necessary conformity with equation 4.3, induce a‘dimensional reduction’ or ‘spontaneous compactification’ of the larger space. Thelatter is then interpreted as a bundle structure with fibres, corresponding to the n G -dimensional gauge group G or an associated k -dimensional homogeneous space S k ,over the smaller n = 4-dimensional spacetime M .The origin of the bundle structure in Kaluza-Klein theories hence contrastssharply with that for the present theory. Here the geometric structure M × G arisesout of the symmetries of a general form of temporal flow L ( v ) = 1 as described inchapter 2. In particular for the 10-dimensional form L ( v ) = 1, considered in thissection and employed in figure 5.1, the base space M arises out of a parametrisation ofa 4-dimensional subset of the ‘translational’ degrees of freedom of the components v under L ( v ) = 1, with gauge fields drawn over the base space out of the ‘rotational’degrees of freedom of the same temporal form.Here the only physical space is the manifold M , providing the arena for gen-eral relativity in a 4-dimensional spacetime, with no ‘compactification’ from a higher-dimensional extended spacetime required. The spacetime geometry on M derivesfrom the local Minkowski metric η ab implicit in the 4-dimensional temporal form ofequation 5.1, now written L ( v ) = h in the projection out of the higher-dimensionalform L ( v ) = 1. On the other hand the Killing metric g αβ = K αβ does not describethe geometry of a physical space, either on the group manifold G or bundle space P . It relates the Lie algebra adjoint and coadjoint representations as usual, with forexample F αab = g αβ F βab , and it may be employed as a mathematical structure on P in the derivation of scalar quantities, as for example in equation 5.14.The roles of the metric g µν ( x ) and gauge field Y αµ ( x ) in the laws of physicson M are well defined. When lifted to the principle bundle P these objects canbe augmented by the Killing metric g αβ on G to define a metric ˇ g ij in the form ofequation 4.5 on the bundle space. This latter metric could be employed on P , forexample to construct a curvature scalar ˇ R from the Riemann tensor ˇ R ijkl based ona Levi-Civita connection ˇΓ ijk , but no physical significance should be attached to the86eometric connotations of the metric ˇ g ij introduced in this way.Indeed ˇ g ij , as described in equation 4.4, consists of an unnatural marriage withthe external local metric η ab originating within the form of L ( v ) = 1 upon which thegroup G , with Killing metric g αβ , acts. This is the case whether the group describesfull symmetry ˆ G = SO + (1 , G = SO(6) as considered here. Such a hybrid metric ˇ g ij , composed of partsof quite different character, hence seems an unnatural object to endow with a physicalgeometric meaning. Hence here the construction of a Levi-Civita connection on thebundle space as described in subsection 4.1 is not well motivated, with the bundle notconsidered as representing an extension of general relativity to a higher-dimensionalspace. On the other hand with this unifying framework taking the shape of a principlefibre bundle over the base space the present theory is naturally related to Kaluza-Kleintheories, in particular those of the kind reviewed in section 4.2.While the structure of these Kaluza-Klein theories rests on a deliberate exten-sion of the formalism of general relativity into a space with extra dimensions, in thepresent theory the construction of a linear connection on the bundle space is motivatedrather as a mathematical means to relate the physical Riemannian curvature on thebase space M to that of the internal gauge fields. Indeed it is still possible to definea linear connection ˇΓ on the bundle which is closely associated with both the linearconnection Γ abc on the base space M and the internal gauge curvature F αab , howeveronly the linear connection Γ on M has a significance in terms of describing a physicalspace. As for other branches of this theory, including its connections with the StandardModel, quantum theory and cosmology to be presented subsequently in this paper, theaim is to develop the theory naturally out of the basic conceptual ideas. Here it isthe basic geometric structures relating to the symmetries of L ( v ) = 1, in particular inthe symmetry breaking over the M base manifold pictured in figure 5.1, that providesthe unified framework for the external and internal curvature. The resulting geometricstructure, exemplified here by the principle bundle P ≡ M × SO(6), while not forminga physical spacetime itself, provides the mathematical arena for a unification of theexternal and internal geometry arising out of the breaking of the full L ( v ) = 1symmetry over the base space M .The general form of the relation between the external Riemannian geometry R and internal gauge curvature F is conjectured to arise naturally in this framework, ina generally and gauge covariant manner, essentially taking the form of equation 4.16–4.17. This relation is provisionally derived here via the scalar function ˜ R ( x ) of equa-tion 5.17, interpreted as a geometric perturbation to the Einstein-Hilbert action on thebase space M arising from the higher-dimensional form of temporal flow L ( v ) = 1.In particular, from the range of models studied, with linear connection coefficients ´Γ ijk on the bundle listed in table 4.1 and the corresponding scalar curvature ´ R determinedin table 5.1, the argument outlined in points ‘a) – e)’ earlier in this section leads to theproposed set of equation 5.13. This argument focuses on the horizontal transport in P skirting over the base manifold M , and in appealing in particular to references [19]and [20] meets half-way with Kaluza-Klein theory. Further progress might be madefor example by placing more complete emphasis on point ‘b)’ with a full set of ´Γ ijk coefficients defined in terms of the parallel transport associated with the internal gauge87urvature as described for figure 3.3.In standard Kaluza-Klein theory the action A m for the scalar curvature ˇ R defined on the bundle space P in equation 4.10 reduces to the 4-dimensional actionintegral A of equation 4.11 owing to the trivial integration over the fibre degrees offreedom. The point of view adopted here is that the scalar field ˜ R ( x ) = R M + χF of equation 5.17 (with R G = 0 and χ = 1 for the model of equation 5.13 constructedhere) is defined directly on the base space M itself. In turn the action integral isdefined directly on the base space as:˜ I = Z ( R M + χF ) p | g | d x (5.18)as a coordinate and gauge invariant expression with all fields defined on M . As denotedby the ‘tilde’ on ˜ I this function is considered as a perturbation of the Einstein-Hilbertaction for the vacuum case, equation 3.79 with α = 1, Λ = 0 and L = 0, whichwas described in the opening of section 3.5. That is, equation 5.18 incorporates theperturbation to the scalar curvature R M ( x ) → ˜ R ( x ) on the base space M . The fullEinstein-Hilbert action of equation 3.79 can be written: I = Z ( αR M + L ) p | g | d x (5.19)where the cosmological constant Λ has been dropped in correspondence with the lack ofa finite R G term in equation 5.18. Further comparison between the above two equationsshows that equation 5.18 describes a perturbation to general relativity equivalent tothe introduction of a Lagrangian term L = + αχF in the original Einstein-Hilbertaction. While the mathematical conclusion is identical to Kaluza-Klein theory, herethe interpretation involves a more minimal impact on the arena of general relativityin 4-dimensional spacetime, namely without a physical augmentation into a higher-dimensional extended spacetime.The choice of χ = and α = − πG N respectively in the two equations aboverepresents the standard normalisation for the incorporation of gauge fields into theEinstein-Hilbert action, as described in section 3.5. This standard action if also dis-cussed in ([26] section 20.6) where the shortcomings of the Lagrangian approach arehighlighted. The intention of the present theory is ultimately to avoid any directreference to the Lagrangian formalism entirely. For the present case the form of˜ R = R M + χF in equation 5.18, in deriving from equation 5.17, arises from thegeometry on the bundle P = M × SO(6) in a physically meaningful way in terms ofentities on the base space M . This structure can be considered as a perturbation togeneral relativity deriving from the need to take into account the internal space of theform L ( v ) = 1 and the geometric structures entailed.If the 4-dimensional form L ( v ) = 1 of equation 5.1 alone is considered nosymmetry breaking is involved in the identification of the bundle P ≡ M × SO + (1 , canonical flat connection with zerocurvature, that is R ρσµν ( x ) = 0, without any reference to a Lagrangian. This resultis however identical to that achieved in equation 3.82 for the vacuum case using thestationarity of the Einstein-Hilbert action under variation of the metric field g µν ( x )88n M ; since R ρσµν ( x ) = 0 if G µν ( x ) vanishes everywhere in spacetime. Hence theconjecture here is that a perturbation to this Einstein-Hilbert action, in the form ofequation 5.18, carries with it the consequences for the Riemannian geometry on M that follow from an embedding in the structures of a larger form of temporal flow suchas L ( v ) = 1.Here the provisional adoption of a ‘Lagrangian function’ has a direct conceptualmotivation. This is unlike for example the case of the Standard Model Lagrangian forparticle physics, elements of which will be reviewed in section 7.2, for which boththe fields and Lagrangian terms are generally introduced and contrived by hand withthe aim of achieving the desired equations of motion and particle interactions for theknown phenomena of high energy physics. The means of bypassing the Standard ModelLagrangian for the present theory will then be described in subsequent chapters, whilethe avoidance of a necessary Lagrangian to derive classical equations of motion will beconsidered further here in the following section.Within this caveat for the employment of a Lagrangian approach, the equationof motion obtained by requiring δ ˜ I = 0 for equation 5.18, under variations δg µν ( x ) ofthe metric on M , follows the derivation of equation 4.16–4.17 and can be written hereas: G µν = 2 χ ( − F αµρ F ρνα − g µν F αρσ F ρσα ) =: − κT µν (5.20)At the purely theoretical level the factor of χ in this equation arises directly in equa-tions 5.15 and 5.16, which in turn derive from the relation of linear connection ´Γ ijk on the bundle to the gauge curvature F αab as listed in the columns of table 4.1. Forthe present theory the correlation between the external and internal geometry in thebreaking of the full form L (ˆ v ) = 1 over the base space M has been considered provi-sionally in terms of the set of linear connection coefficients of equation 5.13, and hencewith χ = 1.With gravitational and gauge field phenomena historically studied indepen-dently in practice the normalisation factor connecting the left-hand side and centralexpressions of equation 5.20 is a matter for empirical convention, as for the factor of κ = πG N c on the right-hand side of this equation. Here for normalisation in practicewe shall set χ = κ implying a choice of physical units such that the energy-momentumtensor can be expressed directly in terms of the gauge curvature, as will be the casefor the electromagnetic field tensor F µν in the following section (see for example equa-tion 5.28).Equation 5.20 reduces to the vacuum solution G µν ( x ) = 0 for the case in whichcurvature of the internal gauge field vanishes F αµν = 0. More generally, with theEinstein tensor G µν = R µν − R M g µν , contracting the equation 5.20 with g µν leads tothe conclusion R M = 0, the standard vanishing of the scalar curvature associated witha classical gauge field, while the Ricci curvature is generally finite with R µν = G µν = 0.Hence while for a general solution we have ˜ R = χF = 0, the full expression˜ R = R M + χF = 0 is needed in equation 5.18 in order to derive the field equation 5.20through the method of variation. A similar observation applies for the vacuum equa-tions of general relativity, namely the derivation of equation 3.82, and further suggeststhat the Lagrangian approach may not be entirely satisfactory. Ideally the aim herewould be to derive equation 5.20 purely by geometrical means and without referenceto a Lagrangian. In the meantime, by further considering δ ˜ I = 0 for the action in89quation 5.18, now with respect to variation in the gauge fields δY αµ ( x ), leads, asdescribed earlier for equation 4.18, to the Yang-Mills vacuum equation: D µ F α µν = 0 (5.21)For the case of an Abelian internal U(1) symmetry this relation expresses Maxwell’sequation for a source-free electromagnetic field.While the unification has been described here in terms of the principle bundlespace P ≡ M × SO(6), for the broken group symmetry action, a bundle of homo-geneous fibres E ≡ M × S k might also be constructed, with fibres composed of thepurely internal v components of L ( v ) = 1, complementary to the projection ontothe external spacetime with v ∈ TM as pictured in figure 5.1(b). A transitive actionof SO + (1 ,
9) on the space underlying L ( v ) = 1 can be identified, as for the action ofSO(6) on the internal space which hence forms the homogeneous space S k employedfor the fibres. Since these actions are also effective the complete internal gauge sym-metry dynamics will be represented for the theory formulated in terms of a bundlewith homogeneous fibres, rather than the principle fibre bundle, as was reviewed insection 4.3.In the models of section 4.3 the internal group G can be considered as a global isometry , that is a symmetry preserving a metric g ˚ α ˚ β on S k , with H ⊂ G as the isotropy subgroup leaving any point y ∈ S k fixed. By contrast for the present theoryˆ G = SO + (1 ,
9) can be considered as an isochronal symmetry preserving the temporalform L ( v ) = 1 with H = SO + (1 , ⊂ ˆ G as the local isometry subgroup preservingthe metric on TM , while the complementary H = SO(6) ⊂ SO + (1 ,
9) leaves anyvector v ∈ TM fixed. The bundle structures on E ≡ M × S k may ultimately shedfurther light on the derivation of equation 5.20 together with the theoretical value of χ . While a consistent and rigorous mathematical framework needs to be estab-lished a full understanding of the appropriate conceptual picture for the extraction ofthe geometry on the base manifold derived from, and breaking, the symmetries of thefull form L (ˆ v ) = 1 is also required. It is out of the marriage of these mathematicaland conceptual ideas that an ultimate form for the relationship between the externalRiemannian curvature R and internal gauge curvature F on the base space M mightbe arrived at. This section has described the evolution of ideas arising out of thesymmetries of L ( v ) = 1 described in chapter 2, steered by the structures of differentialgeometry and Kaluza-Klein theory as described in chapters 3 and 4, aiming towardssuch a unification. Attempting to justify all the steps along the way, via the linearconnection on the bundle of equation 5.13, scalar function on the base space of equa-tion 5.17 (with R G = 0) and action integral of equation 5.18, the aim has been to arriveprovisionally at the relation of equation 5.20 with minimal assumptions. This equationshows how a relation between the external and internal curvature might be achieved inthe present theory with non-zero values for R = 0 and F = 0 closely correlated. Thepossibility of deriving equation 5.20 via purely geometric means without any referenceto a Lagrangian formulation remains as a conjecture of the theory.It should be further noted that only classical fields have been considered so farand it may be that, given the symmetry of the classical picture described originallyin figure 2.2, a quantum field description of the theory will be required to provide the90echanism through which non-flat structures ultimately arises on the base manifoldin general. This in turn relates to the concept of ‘many solutions’ for the geometry G µν ( x ) on the base space as will be described in chapter 11. In the meantime, giventhe Kaluza-Klein relation of equation 5.20 itself, a number of further equations ofmotion may be deduced without the need for a Lagrangian formalism. Hence theseconsequences are conjectured also to apply in the present theory, as we review in thefollowing section. In standard field theory the Lagrangian, being a scalar, provides a means to introducearbitrary, although generally empirically motivated, symmetries into the theory withsuch symmetries generally preserved in the resulting equations of motion, as reviewedin section 3.5. In the Lagrangian approach the compatibility of the equations of motionwith energy-momentum conservation ∂ µ T µν = 0 is ensured through the Euler-Lagrangeequation if the energy-momentum tensor is defined according to equation 3.102, as anapplication of Noether’s theorem.In the present theory equation 5.20 emerges out of the constraint of the simpleform L (ˆ v ) = 1 projected over the base space M , in principle without the need fora Lagrangian formalism, as described in the previous section for a model based onthe form L ( v ) = 1. The new theory avoids the ambiguity inherent in the choiceof a scalar Lagrangian function and replaces the need to impose the principle of ex-tremal action with a firm conceptual grounding in the physical manifestation of thefull form of temporal flow L (ˆ v ) = 1 and its symmetries. Hence in contrast to theLagrangian approach here we begin with T µν ; µ = 0 as a direct consequence of thedefinition of energy-momentum as T µν := G µν , within a conventional normalisationfactor in relations such as equation 5.20, together with the contracted Bianchi iden-tity G µν ; µ = 0. In the limit of vanishingly small spacetime curvature, with a linearconnection Γ → ∇ µ T µν = 0 → ∂ µ T µν = 0 and interpreted as energy-momentum conservation. Thequestion then regards the extent to which this constraint determines the equationsof motion, both in a curved spacetime and in the limit of flat Minkowski spacetime,for the entities which apparently compose T µν , without appealing to a Lagrangianstructure.This also contrasts with a more standard approach to general relativity, re-viewed in section 3.4, in which the Einstein tensor G µν is first equated with a genericenergy-momentum tensor, G µν = − κT µν in equation 3.75, via a normalisation con-stant κ . In the meantime various examples of possible forms T µν may be postulated,or deduced from a Lagrangian method, for example for the energy-momentum of aperfect fluid or an electromagnetic field, again with appropriate normalisation factors.Only then are the Einstein tensor and the chosen form for T µν linked together viaequation 3.75. This standard approach distances the relation between the externalcurvature G µν and internal curvature F α µν by the insertion of the apparently mediat-ing object T µν , which may be considered to act as a ‘source’ for the gravitational field.It is this structure that motivates the form of equations 4.16 and 4.17. One of the91ain reasons for considering T µν to be the source term in the Einstein equation is thatmaterial phenomena (such as the properties of everyday tables and chairs) are gener-ally more readily observable than their counterparts in the warping of the spacetimegeometry, particularly within the local laboratory environment.In the present theory the more intimate relation of equation 5.20 arises directly from the basic conceptual ideas of the theory, as described in the previous section,with the symmetry groups of both the external and internal geometry mutually relatedthrough the unifying symmetry of the full form L (ˆ v ) = 1. The motivation for the right-hand side of equation 5.20 to subsequently be interpreted as an energy-momentumtensor corresponding to G µν will be found in the empirical usefulness of such a concept.This will be more apparent when ‘quantum effects’ are introduced and augment thepossible forms of T µν beyond that of continuous classical fields, as we alluded to atthe end of the previous section.Here, beginning from the unified point of view for classical fields, the exter-nal and internal curvatures appear on a similar footing in equation 5.20, with thecontracted Riemann curvature on the left-hand side equated identically with termsquadratic in the internal curvature in the central expression. The great differencein the relative strengths of the respective physical forces encountered empirically innature will later need to be accounted for through the respective interactions and cou-plings of the fields to be identified in the theory. These will give rise to a varietyof laboratory phenomena and will lead to normalisation factors replacing χ in rela-tions such as equation 5.20 once practical units are employed for measured quantities.While the bare mathematical relations are needed to understand the theoretical basisof the unification, for a discussion of the empirical consequences here we set χ = κ assuggested following equation 5.20 in the previous section.The tensor T µν is composed of effective macroscopic quantities or as a functionof fundamental fields, to be determined in the theory, which in turn mutually constrainsthe form of G µν . Here the initial aim will be to demonstrate the extent to which theequations of motion for both external gravitational and internal gauge fields are impliedwithin the unifying form of equation 5.20.First we consider the classical field for the particular case of U(1) as the internalsymmetry, that is the case of electromagnetism. In terms of the components F µν = ∂ µ A ν − ∂ ν A µ of the electromagnetic field tensor the components of the Einstein tensor G µν of equation 5.20, with a single generator for the internal group, can be written as: − κ G µν = F µρ F ρν + 14 g µν F ρσ F ρσ (5.22)Hence through this equation direct contact is made between gravitation in the formof the geometric curvature of spacetime and the familiar laboratory phenomena of theelectromagnetic field. The fact that powerful electromagnetic effects may be observedfor which the associated gravitational field is immeasurably small is an indication ofthe need to explain the origin of laboratory normalisation units, as mentioned above.Given the tetrad field components e µa ( x ) of a local orthonormal frame field { e a ( x ) } the components of the electromagnetic curvature tensor in a local Lorentz92rame F ab = e µa e νb F µν may be written out as the 4 × F ] ab = E E E − E − B B − E B − B − E − B B . (5.23)This is also the conventional form for the electromagnetic field tensor defined globallyfor the flat Minkowski spacetime of special relativity. The special symbols E i and B i ( i = 1 , , E i = F i = F ( e , e i ) and − ε ijk B k = F ij = F ( e i , e j )) for the sixindependent components of the electromagnetic curvature 2-form F in a particularLorentz frame { e a } represent the electric and magnetic fields respectively. These sixcomponents transform non-trivially under external Lorentz transformations but aretrivially unchanged under an internal g ( x ) ∈ U(1) gauge transformation, equation 3.40,since g − F g = F for an Abelian group.Historically it was realised that Maxwell’s equations 3.90 and 3.91 exhibit aU(1) symmetry before an understanding of gauge theories had been developed, al-though it was not considered to be a fundamental physical symmetry of nature since itis not a spacetime symmetry. However in the present theory fundamental symmetriesare not of spacetime (in any dimension) but of multi-dimensional forms of temporalflow expressed as L ( v ) = 1. These include both the familiar 4-dimensional spacetimesymmetry associated with perception on an extended manifold M and equally thegauge symmetry groups, including the U(1) of electromagnetism that arises here aswill be described in section 8.2. Here both external and internal symmetries, togetherwith their respective physical phenomena, originate naturally from the fundamentalconcepts of the theory.In an approximately Minkowskian spacetime the electromagnetic field F ab maybe defined and measured operationally by observing the motion of a body of mass m and charge q in the field and using the Lorentz force law of equation 3.88. In thatequation F bc = η ba F ac is a mixed index form of the electromagnetic curvature tensor.The metric is needed to define this tensor, as it is for F cd = η ca η db F ab and hence inturn to define the ‘Hodge dual’ of the electromagnetic curvature tensor: ∗ F ab = 12 ε abcd F cd with [ ∗ F ] ab = − B − B − B B − E E B E − E B − E E . (5.24)In Minkowski spacetime ε abcd = ε [ abcd ] are the components of the completely antisym-metric rank-4 tensor ε ≡ e ∧ e ∧ e ∧ e , with ε = ε ( e , e , e , e ) = +1 implying thechoice of right-handed orientation for the orthonormal basis { e a } , while the cotensorcomponents are simply ε abcd = − ε abcd . In a general coordinate system, including the93ase of a curved spacetime, the metric volume form ω with components: ω abcd = p | g | ε abcd (5.25) ω abcd = (1 / p | g | ) ε abcd (5.26)where g ( x ) is the determinant of the metric g µν ( x ), is employed for the Hodge dualoperator of equation 5.24 since ε , unlike ω , does not transform as a tensor undergeneral coordinate transformations. The Levi-Civita symbol ε abcd is equivalent to thecomponents of the volume form ω in Minkowski spacetime with global coordinatesemployed such that the metric g µν ( x ) = δ aµ δ bν η ab everywhere.In general on an n -dimensional manifold the space of p -forms has the samenumber of degrees of freedom as the space of ( n − p )-forms with a canonical isomorphismbetween the two sets given by the metric volume form ω . The isomorphism map isthe Hodge dual of a form which contains precisely the same information reorganisedinto the components of the dual form. For example the map from F in equation 5.23to ∗ F in equation 5.24 corresponds to a rearrangement of matrix components with( E i , B j ) → ( − B i , E j ).The Einstein tensor G µν = R µν − Rg µν is the ‘trace-reversed’ Ricci tensor, itcan also be defined as the contraction ([6] p.325): G βγ := G – τβγτ with G – αβγδ := 12 ω αβρσ R µνρσ ω µνγδ and in this sense is ‘dual’ to the Ricci tensor R µν . The tensor G – carries exactly thesame information, and possesses the same rank-4 tensor symmetries, as the Riemanntensor R and hence also has 20 independent components. It is analogous to the dualtensor ∗ F for the electromagnetic curvature tensor F .The electromagnetic energy-momentum tensor identified with T µν := − κ G µν for equation 5.22, as guided by the Kaluza-Klein framework, is identical to that ob-tained in equation 3.105 in the Lagrangian formalism since effectively the same matterLagrangian L ∼ F is introduced in both cases, via equations 5.18 and 3.93 respec-tively. This expression can also be written in an equivalent but more symmetric form([26] p.456): T µν = 12 ( F µρ F ρν + ∗ F µρ ∗ F ρν ) (5.27)= F µρ F ρν + 14 g µν F ρσ F ρσ (5.28)From either of these equations the energy density of the electromagnetic fieldis found to be T = ( E + B ), as originally expressed by Maxwell. There are twoLorentz invariants of the electromagnetic field, the scalar norm F µν F µν = − ( E − B )and the pseudo-scalar F µν ∗ F µν = E (cid:5) B , although expressions of the latter kind (com-posing F µν with its dual) do not feature in T µν . Both of these quantities are functionson the spacetime manifold which locally take the same value in any Lorentz frame andare also invariant under (orientation preserving) general coordinate transformations.The energy-momentum tensor for the electromagnetic field is also traceless, T µµ = 0, from which the trace of the Einstein equation implies that the scalar curvature94anishes, R = 0, and hence in this case the Einstein equation can be written G µν = R µν = − κT µν , as described shortly after equation 5.20 in the previous section. Hencein the Einstein-Maxwell theory while the Maxwell tensor F µν and its dual ∗ F µν appearin a symmetric way in equation 5.27 the Einstein tensor G µν is identical to its ‘dual’ R µν . From this underlying theoretical point of view electromagnetism arises as a U(1)gauge theory with the electromagnetic field tensor being the exact F = d A asdefined in terms of the U(1) connection 1-form A ( x ). Hence by the exterior algebraproperty d = 0 the curvature 2-form is in turn necessarily closed d F = 0 as anidentity that gives immediately the homogeneous Maxwell equations summarised inequation 3.90.With the electric current 1-form defined as J := ∗ d ∗ F (that is ∗ J := d ∗ F con-sistent with the inhomogeneous Maxwell equation 3.91) from the property d = 0 italso follows immediately that dd ∗ F = 0 and we also find the identity d ∗ J = 0. InMinkowski spacetime this in turn implies that ∂ a J a = 0 corresponding to the conser-vation of electric charge expressed in terms of the components of the conserved current J associated with the internal U(1) symmetry. This is very closely analogous to thefact that defining the energy-momentum tensor to be T µν := G µν leads immediatelyto the local conservation of energy-momentum T µν ; µ = 0 via the contracted Bianchiidentity for the Einstein tensor G µν . Hence Noether’s theorem, based on a Lagrangianapproach as described in section 3.5, is not needed to identify either of these conservedquantities, which are both purely geometric in origin.It can be shown ([6] p.472) that for the case J = 0 the Einstein equation, inthe form of equation 5.22, mutually constrains the evolution of both the gravitationaland electromagnetic field, with the latter usually expressed by the source-free Maxwellequation d ∗ F = 0, that is equation 3.91 for J = 0, as we review here. Applying theidentity G µν ; µ = 0 to both sides of equation 5.22 gives:0 = F µτ ; µ F ντ + F µτ F ντ ; µ + 12 g µν F ρσ ; µ F ρσ = F ντ F µτ ; µ + g µν F ρσ F σµ ; ρ + 12 g µν F ρσ ; µ F ρσ = F ντ F µτ ; µ + 12 g µν F ρσ ( F σµ ; ρ + F µρ ; σ + F ρσ ; µ ) ⇒ F ντ F µτ ; µ = 0 (5.29)The final term in the penultimate equation vanishes by the identity d F = 0, that isthe homogeneous Maxwell equation 3.90, or F [ σµ ; ρ ] = 0 in components (again here ‘; µ ’is the covariant derivative with respect to the linear connection Γ in a general curvedspacetime). The remaining expression in the bottom line involves a linear combinationof the four quantities F µτ ; µ . The determinant of the coefficients F ντ is the Lorentzpseudo-scalar | F ντ | = − ( E (cid:5) B ) ([6] p.472). For a general electromagnetic field thisquantity is non-zero, except that it may vanish on hypersurfaces, and hence in generalthe source-free form of the Maxwell equation 3.91 does not need to be imposed, ratherit may instead be deduced from the Einstein equation for the electromagnetic fieldthat: F µτ ; µ = 0 (5.30)95n defining J τ = F µτ ; µ this result shows that vanishing current J = 0 isimplied for the relation of equation 5.22 under the Bianchi identity G µν ; µ = 0. For thisvacuum case J = 0 both the curvature F and its dual ∗ F satisfy a similar equation,d F = 0 and d ∗ F = 0 respectively, while for the external curvature there is a greatersymmetry with G µν equal to its ‘dual’ R µν , as described above.A similar argument may be followed for the non-Abelian case, beginning withequation 5.20 and following the sequence of expressions leading to equation 5.29 exceptwith F µν → F αµν and an extra contraction over the index α , representing the groupgenerators, for each quadratic term in the internal curvature. Sandwiched between thetwo complementary constraining identities for the external and internal curvature, thatis the Bianchi identities G µν ; µ = 0 and D F = 0 respectively, this leads to the Yang-Mills equation D µ F α µν = 0, which was derived from a Lagrangian in equation 3.95,and includes self-interaction terms for the non-Abelian gauge field Y αµ ( x ). The sameequation was also derived as a consequence of Kaluza-Klein theory in equation 4.18from the stationarity of the action integral of equation 4.10 on a principle bundle.Generally for the non-Abelian case, as for the Abelian case of Maxwell’s equations, aconserved current can be obtained in terms of a geometric identity.For the present theory the Maxwell and Yang-Mills equations are also pro-posed to arise through a purely geometric argument, similar to that described forequation 5.29, directly from the identity G µν ; µ = 0 as applied to equation 5.20. Thisrelation itself arose in equation 4.16-4.17 under the stationarity of an action integral inKaluza-Klein theory, although in the previous section we described how equation 5.20might be obtained ultimately in the present theory without any appeal to the La-grangian formalism. Here equation 5.20 is considered to arise as a perturbation to theEinstein vacuum equations, derived for equation 3.82 in terms of the stationarity of theEinstein-Hilbert action under variations of the metric δg µν ( x ). Consistent with thisapproach the above discussion suggests that the variation of the gauge field δY αµ ( x )is not needed in order to derive the vacuum Yang-Mills equation 5.21; rather, as forgeneral relativity, only the δg µν ( x ) variation is needed in order to derive equation 5.20,which in turn itself implies the relation of equation 5.21 as a consequence of the geo-metric structure. With equation 5.20 itself conjectured to arise inevitably out of thegeometric constraints implied in the breaking of the full L (ˆ v ) = 1 symmetry over M any explicit reference to the Lagrangian formalism might be avoided entirely.In the present framework non-Abelian symmetries arise, as for the case of U(1)above, within the internal symmetry action on the full form L (ˆ v ) = 1. The symmetrybreaking is pictured in figure 5.1 for the L ( v ) = 1 model, for which the internalsymmetry is identified simply as SO(6). Internal symmetries deriving from yet higher-dimensional forms of L (ˆ v ) = 1 will be considered in chapters 8 and 9.Returning to the Abelian case of electromagnetism, more generally for J = 0, inapplying to the Maxwell tensor F and not to the dual tensor ∗ F the Bianchi identityd F = 0 introduces a clear break in the mathematical symmetry between these twotensors. This in turn is directly associated with the empirical asymmetry between theobserved roles of the electric and magnetic fields. The field components ( E i , B j ) areoriented within the Maxwell tensor in equation 5.23 such that they are distinguishedby the particular properties that ∇ (cid:5) B = 0 while ∇ (cid:5) E = σ , where σ is the chargedensity for the case of static fields . (From the historical empirical point of view the96symmetry between the expressions for d F in equation 3.90 and d ∗ F in equation 3.91is a physical observation in the sense it ‘might have been’ observed that d F = ∗ J M with a ‘magnetic monopole current’ J M , however empirically such a current has neverbeen seen.)Here we next consider how equations of motion describing the broad macro-scopic properties of matter arise. The microscopic details of fields and quantum physicswhich underlie these properties need not be considered in any detail here. Rather thegeneral freedom inherent in the Einstein equation, beyond a specific form such as equa-tion 5.22, will be opened up to a more general structure G µν = − κT µνǫ , where ǫ heredenotes an effective energy-momentum tensor describing coarse macroscopic phenom-ena. This macroscopic form of T µνǫ will include terms for the effective flow of physicalmatter, either charged or uncharged, as well as for the original electromagnetic field,all combinations of which will be collectively subject to T µνǫ ; µ = 0 through the Einsteinequation.Under the symmetry transformations of a higher-dimensional form of temporalflow L (ˆ v ) = 1 the projection over the base manifold M , as described in the previoussection, leads to a relation between classical external and internal fields culminatingin a relation of the form of equation 5.20, which may be written: G µν = f ( Y ) (5.31)The identity G µν ; µ = 0 then leads to constraints on the equations of motion forthe internal gauge fields Y ( x ), that is the Yang-Mills-Maxwell equations, as describedabove. A particular form for the energy-momentum tensor is identified as T µν := − κ G µν , that is via the Einstein equation.So far we have considered only the case in which G µν ( x ) is equated with afunction of the curvature F αµν ( x ), in turn derived from a classical continuous gaugefield Y αµ ( x ), in the form of equation 5.31, which exhibits a relatively even significancefor the external gravitational field on the left-hand side and the internal gauge fieldon the right-hand side. This structure was motivated to obtain G µν on the left-handside of equation 5.20 corresponding to a global continuous external linear connectionfield Γ( x ) as required to define a geometric perceptual arena on the base manifold asdescribed in section 2.2.More generally a continuous internal gauge field Y ( x ) is only a local require-ment so long as the central expression of equation 5.20 can be modified in a mannercompatible with the identity G µν ; µ = 0. With the components of the internal sym-metry gauge fields Y αµ ( x ) coupled with the internal temporal components, througha relation of the form of equation 2.47, only the combined effect is required to becompatible with the necessary smooth geometric structure on the left-hand side ofequation 5.31 and we can write: G µν = f ( Y, ˆ v ) (5.32)implying in turn a more flexible expression for the energy-momentum tensor T µν := − κ G µν . This extra freedom, not tied to the constraint of a continuous internal gaugefield on M , allows for field exchanges between the internal gauge connection Y ( x ) andcomponents of temporal flow ˆ v ( x ), which will be of the kind described in chapters 8and 9 for more realistic forms L (ˆ v ) = 1 in comparison with the observations of high97nergy physics experiments. The possibility of multiple solutions for G µν involvingexchanges between the field values of Y and ˆ v will be interpreted as quantum andparticle phenomena via the local indistinguishability of the field components, as willbe described chapter 11.While equation 5.31 might be expressed as G µν = − κT µν ( Y ) the more generalnon-classical extension to equation 5.32 can also be written as G µν = − κT µν ( Y, ˆ v ) withthe identification of the rank-2 tensor fields on either side of this expression remainingvalid since both sides transform the same way and the contracted Bianchi identity willstill apply to both. While G µν and T µν are identical in form they denote and possess adiffering internal compositions; while the right-hand side can be interpreted as a source in terms of the fragmented temporal flow composed of apparent ‘ matter fields’, Y ( x )and ˆ v ( x ), the left-hand side represents the same mathematical object interpreted asthe Einstein tensor for a linear connection describing the external geometry , as requiredfor perception.Equation 5.32 expresses the relation between the gravitational field describedby the metric g µν ( x ) underlying G µν and the matter fields Y µ ( x ) and ˆ v ( x ), togetherwith the implicit interaction between these latter ‘microscopic’ fields themselves. Al-ternatively the term ‘matter field’ can refer to an effective macroscopic form for theenergy-momentum tensor such as T µνǫ averaging over the microscopic field interactioneffects. We begin by looking more generally at properties of the symmetric Einsteintensor G µν in terms of T µνǫ := − κ G µν . A timelike eigenvector u may be defined forthe energy-momentum tensor such that ([27] p.174): T µνǫ u ν = ρu µ (5.33)with the vector field u ( x ) normalised as | u | = g µν u µ u ν = u µ u µ = +1 such that ρ = T µνǫ u µ u ν (= ρu µ u µ ) which will be identified as the effective ‘proper energy density’or mass density, effectively averaging over underlying microscopic field interactions. Inthe general case: T µνǫ = ρu µ u ν − S µν (5.34)defines the stress tensor S µν ([27] p.175). This is a symmetric tensor with four con-straints S µν u ν = 0 (as can be seen by contracting equation 5.34 with u ν ) and hencewith six degrees of freedom. The simplest example is that in which the effective energy-momentum tensor represents a pressureless perfect fluid (such as a dust cloud) with: T µνǫ = ρu µ u ν . (5.35)In this case G µν = − κρu µ u ν and we have g µν G µν = − κg µν ρu µ u ν = − κρ . With G µν = R µν − Rg µν this in turn implies R = + κρ with the matter density ρ thereforedirectly associated with the spacetime scalar curvature R and hence with gravitationaleffects. The sign convention of equation 3.74, with G µν = − κT µν and positive constant κ determined in the Newtonian limit, is motivated in part by the resulting sign in therelation R = + κρ , that is such that positive scalar curvature is associated with positive matter density.Applying the contracted Bianchi identity G µν ; µ = 0 to the right-hand side of98quation 5.35 we then have ([27] p.175): T µνǫ ; µ = 0 ⇒ ( ρu µ ) ; µ u ν + ρu µ ( u ν ) ; µ = 0Σ ν ( × u ν ) ⇒ ( ρu µ ) ; µ = 0 since u ν ( u ν ) ; µ = 0hence ρu µ ( u ν ) ; µ = 0 . (5.36)Here the continuity equation ( ρu µ ) ; µ = 0, describing the conservation of mass-energy,in the second line is substituted back into the first line to deduce the expression inthe final line. From this we see that the form of equation 5.35, with ρ = 0, impliesthat u µ ( u ν ) ; µ = 0, that is the flow lines of the fluid are geodesics . Such a resultcould be derived from the simple Lagrangian of equation 3.78, with the requirement δL = 0 under variation of the path implying equation 3.77. However here in the caseof a perfect fluid the geodesic law for the motion of bodies in general relativity is aninescapable consequence of the Einstein field equation and the Bianchi identity, whichis a well-known result.More generally the effective energy-momentum tensor T µνǫ can describe a per-fect fluid with non-zero effective pressure p in the form: − κ G µν =: T µνǫ = ( ρ + p ) u µ u ν − p g µν . (5.37)with, by comparison with equation 5.34, S µν = p ( g µν − u µ u ν ) which satisfies S µν u ν = 0.The material flow u is again subject to | u | = 1 with ρ and also now p as effective macro-scopic terms irrespective of the classical or quantum fields underlying this structure.Again here the structure of matter perceived in spacetime is constrained by the geo-metrical properties of G µν . Applying the Bianchi identity G µν ; µ = 0 to the right-handside of equation 5.37, similarly as above for equation 5.35 leading to equation 5.36, wenow find that in general u µ ( u ν ) ; µ is non-zero and proportional to the pressure gradient([27] p.176), as a deviation from pure geodesic flow of the fluid due to the pressureterm. Alternatively we may consider a pressureless fluid carrying charge, that is afluid with energy density ρ and also a charge density σ . Here we are dealing withcontinuous classical fields and bodies corresponding to the motions of macroscopicentities, where T µνǫ may represent charged metal plates, wires and so on and T µν em describes a classical electromagnetic field, for example in a laboratory setting. For theoriginal case with the classical electromagnetic gauge field only and T µν em := − κ G µν from equation 5.22 consistency with G µν ; µ = 0 required that J ν := F µν ; µ = 0, asdescribed for equation 5.30. It is then through the introduction of effective matterterms that the equations for the electromagnetic field allow for a charged current J = 0 in combination with energy-momentum in the form T µνǫ = ρu µ u ν , both of whichare composed in terms of the effective matter content.We have defined T µν := G µν and argued, following the previous section, thatfor an internal U(1) symmetry identified within the full symmetry of L (ˆ v ) = 1 thisnaturally leads to T µν em in the form of equation 5.22. Similarly here with J ν := F µν ; µ we would like to understand the form of J that results as microscopic field transitionsover M are considered such that equation 5.22 breaks down giving: − κ G µν = T µν ( Y, ˆ v ) = F µρ F ρν + 14 g µν F ρσ F ρσ and J ν = F µν ; µ = 0 (5.38)99ith a specific form for the first equation relating to a specific form for the latter. Inthe phenomenological macroscopic limit the effective energy-momentum tensor T µνǫ = ρu µ u ν arose as a possible form for a non-trivial G µν field for the external spacetimegeometry. With charge density defined by σ := ∇ (cid:5) E in the electrostatic limit, under aLorentz transformation we may associate the 4-vector J ν = σu ν with a charged body,such that σ = u ν J ν is closely analogous to ρ = u µ u ν T µνǫ for the matter density of apressureless fluid. Hence in addition to the 4-momentum density ρu µ the fluid carriesan effective charge 4-current J ν = σu ν , which is identified as a possible form of F µν ; µ and with the identity J ν ; ν = 0 implying the conservation of charge. That is we considerthe flow of matter to be simultaneously associated with: − κ G µν =: T µνǫ = ρu µ u ν (5.39)+ F µν ; µ =: J ν = σu ν (5.40)as the respective definitions of matter density ρ and charge density σ . Here the 4-velocity u ( x ) with | u | = 1 represents a fluid carrying both the mass and the charge.The fluid body is interpreted to be immersed in and passing through the electromag-netic field F µν such that the Einstein equation reads: − κ G µν = T µνǫ = ρu µ u ν + F µτ F ντ + 14 g µν F ρσ F ρσ (5.41)That is the form of the energy-momentum tensor for the electromagnetic field fromequation 5.22 has been combined with the pressureless perfect fluid term. Here the 4-velocity u of the fluid differs from the 4-velocity eigenvector U defined in T µνǫ U ν = ρ ′ U µ by equation 5.33. With T µνǫ = ρ ′ U µ U ν − S µν from equation 5.34, the 4-velocity U represents a synthesis of the charged fluid and the electromagnetic field ([27] p.357).Applying T µνǫ ; µ = 0 the effect on the terms on the right-hand side of equa-tion 5.41 has already been worked out separately in equations 5.36 and 5.29 respec-tively. Combined together we find that under the Bianchi identity equation 5.41 be-comes (based on [27] p.358): T µνǫ ; µ = 0 ⇒ ( ρu µ ) ; µ u ν + ρu µ ( u ν ) ; µ + F ντ F µτ ; µ = 0Σ ν ( × u ν ) ⇒ ( ρu µ ) ; µ + 0 + F ντ F µτ ; µ u ν = 0 ⇒ ( ρu µ ) ; µ + g λτ F λν J τ u ν = 0 ⇒ ( ρu µ ) ; µ + F λν σu λ u ν = 0The final term in the fourth line above is asymmetric in the indices of F λν while sym-metric in the indices of u λ u ν and is therefore equal to zero. The same line then impliesthat ( ρu µ ) ; µ = 0 (as for the second line of equation 5.36) which can be substituted intothe first line giving: ρu µ ( u ν ) ; µ + F ντ J τ = 0 . (5.42)Each term in equation 5.42 was found to be zero for the individual cases of a per-fect pressureless fluid alone or an electromagnetic field alone, giving equation 5.36 forgeodesic motion and Maxwell’s vacuum equation 5.30 respectively. However for thecombined case only the total vanishes and hence G µν ; µ = 0 implies that: ρu µ ( u ν ) ; µ = + F ντ J τ (5.43)100his is the relativistic Lorentz force law for a charged fluid in a curved spacetime,which is equivalent to the corresponding law of equation 3.87 for discrete bodies in theappropriate limit ([27] p.359) as is similarly the case for the geodesic motion of equa-tion 5.36 considered above. Again Lagrangian terms, such as those in equation 3.86,are not required.As we described earlier for the effective energy-momentum tensor of equa-tion 5.37 the geodesic flow of an uncharged fluid is modified by the pressure gradient.Similarly for the energy-momentum tensor of equation 5.41 for charged matter thegeodesic law is modified by the presence of an electromagnetic field to a form, equa-tion 5.43, which precisely gives the Lorentz force law of equation 3.87. This law,typically in the flat spacetime limit of equation 3.88 or the further non-relativisticlimit, can be used to determine the strength of charges and electromagnetic fields inthe laboratory and establish appropriate empirical normalisation factors.The possibility of incorporating electromagnetism and the Lorentz force lawwithin a higher-dimensional approach to general relativity is well known and dates backto Kaluza in 1921 ([11] equation 12). There it was shown that the five -dimensionalgeodesic equation automatically incorporates the Lorentz force law in 4-dimensionalspacetime, in the approximation of low 5-velocity. In the present theory the internalgauge fields, such as that for electromagnetism, arise as a higher-dimensional form L (ˆ v ) = 1 is projected onto the base space M , with charged matter arising through theinteraction properties of the internal fields underlying the smooth spacetime geometry.For the case in which there is no electromagnetic field F µν = 0 or in which thematerial flow is uncharged J µ = 0 the geodesic flow is recovered from equation 5.43.On the other hand J ν := F µν ; µ = 0 represents the case for which T µν as a function of F µν only, equation 5.28, itself is not conserved, as can be seen from the inconsistencywith equation 5.29, while the total T µνǫ of equation 5.41, augmented to include the flowof macroscopic charged matter, is conserved. The Lorentz force law results from theconsistency of this total energy-momentum tensor bound together under the require-ment of T µνǫ ; µ = 0, which itself is a direct consequence of the definition T µν := G µν and the Bianchi identity.While the Bianchi identity implies T µν ; µ = 0 further conservation laws followfrom further geometric identities, principally of the form d = 0 which for examplegiven J ν := F µν ; µ implies that J µ ; µ = 0, as described in the discussion followingequation 5.28. This leads to conserved charges associated with the internal symmetriesboth for Maxwell and Yang-Mills theories. However, while the Maxwell equationswith source J = 0 imply the conservation of charge, this conservation law is limitedto physical entities that carry charge. This marks a fundamental difference with theconsequences of the Einstein equation, which can be interpreted as T µν := G µν , inthat, assuming that all fields are associated with energy-momentum defined this way, all fields are covered under the identity G µν ; µ = 0 and in principle ‘no physical entityescapes this surveillance’ ([6] p.475).In the above only the contracted Bianchi identity for the Riemann curvaturetensor has been employed. Further, the Einstein equation G µν = − κT µν only directlyyields certain linear combinations of the Riemann curvature tensor components. How-ever, although the Weyl tensor, introduced before equation 3.69, is that part of theRiemann tensor which is not directly equated with matter T µν in the Einstein equation101t is not arbitrary. Applying the full Bianchi identity R ρσ [ µν ; τ ] = 0 of equation 3.70 toequation 3.69, rearranging the terms and contracting once leads to ([9] p.85): C ρσµν ; ν = R µ [ ρ ; σ ] + 16 g µ [ σ R ; ρ ] =: K ρσµ (5.44)Hence the full Bianchi identity, which contains more information than the contractedform, can be regarded as a field equation for the Weyl tensor in which the source K ρσµ is defined as a function of the Ricci tensor. This is analogous to the Maxwellequation 3.91 for the electromagnetic field, which can be written in a curved spacetimeas F µν ; µ = J ν , with the electromagnetic current J ν as the source. For equation 5.44 thesource K ρσµ depends on R µν which in turn is intimately related to the matter content T µν through the Einstein equation, which can be written R µν = − κ ( T µν − T g µν )where T = g µν T µν . Hence, by substituting T µν into equation 5.44, the Weyl curvatureat any given location on M depends on the matter content elsewhere in spacetime, in asimilar way that the electric and magnetic fields depend on the charges elsewhere. TheWeyl tensor represents the non-flat part of the Riemann tensor in the matter vacuum,this includes the phenomena of gravity waves (in analogy with electromagnetic waves)as well as gravitational tidal forces and lensing effects. Further, since gravitationalwaves carry energy even in regions of spacetime where G µν = 0 the association of T µν := G µν with ‘energy-momentum’ itself has a degree of ambiguity, while being ofgreat value for many practical applications.In this section we have reviewed how a number of equations of motion ariseout of the geometry of the Bianchi identities for the external and internal symmetries,given the relation of equation 5.20 obtained by comparison with Kaluza-Klein theory.However the equations of motion are derived we note that in order to empirically test atheory solutions of the field equations need to be determined and compared with actualobservations in the world. This in turn requires the specification of initial conditions, ormore general boundary conditions, in order to obtain such solutions. With care for therole of the implicit degrees of freedom of gauge and general coordinate transformationsthe ‘initial value problem’ is well posed for both classical electromagnetism and generalrelativity respectively. The evolution of the spacetime geometry is in principle fullyobtainable from Einstein’s equation and the equations of motion for the matter fieldstogether with suitable boundary conditions.It is generally not possible to begin with a given source term on the right-hand side of the Einstein field equation G µν = − κT µν since a coordinate system isrequired in order to specify the components of T µν ( x ), and further the distribution ofmatter itself is dynamically intertwined with the spacetime geometry through which itpropagates. One procedure would be to begin with arbitrary metric functions g µν ( x )and catalogue ( g µν ( x ) , T µν ( x )) pairs via equations 3.53, 3.73 and the field equationwith − κT µν := G µν = f ( g µν ), in an attempt to converge upon a particular physicalsystem.In practice exact solutions for the metric g µν ( x ) have been found for the cases inwhich T µν represents the vacuum ( T µν = 0), a perfect fluid or the electromagnetic field(or a combination of the latter two, as described for a pressureless fluid in equation 5.41)and then only for spaces with a high degree of symmetry with a simple form of mattercontent. All solutions in general relativity consist of a metric description for a completespacetime geometry, which will be relevant for the study of cosmology, while only a102imited region of the manifold may be of physical interest in other cases such as thestudy of planetary orbits using the Schwarzschild solution, described in the followingsection, for example.In summary, many of the equations of motion derived from a Lagrangian insection 3.5 have been shown to arise directly as a consequence of the identity G µν ; µ = 0given a solution for G µν ( x ) for example in the form of equation 5.20. This latter rela-tion itself arose as guided by Kaluza-Klein theory and equation 4.16–4.17 through theemployment of a single ‘Lagrangian function’ on a principle bundle space. As describedin the previous section in the present theory it is conjectured that the Lagrangian ap-proach might be ultimately side-stepped entirely and that this one remaining pivotalLagrangian, in the action of equation 5.18, may also be discarded. In principle it mayalways be possible to work backwards from the present theory to obtain apparent La-grangian functions for the theory, but from the present point of view the Lagrangianmethod is ultimately effective due to its conformity with G µν ; µ = 0 through the com-patibility of the Euler-Lagrange equation with the requirement T µν ; µ = 0, as describedin the opening of this section.In this section the form of the 4-current J ν := + F µν ; µ , in equations 3.91 and5.40, has been taken to emerge macroscopically and does not necessarily apply for‘elementary particles’. The origin and role of 4-currents for microscopic fields of theform j µα = ψγ µ E α ψ in equation 3.97, as well as the Dirac equation 3.99, in the presenttheory will be addressed in section 11.1, in particular as exemplified by the AbelianU(1) case of electromagnetism. In order to consider the properties of microscopicelementary particles (electrons, photons etc.) it will first be necessary to addressthe more fundamental questions concerning the quantisation of the theory and theconcept of an elementary particle itself. In addition to the identity G µν ; µ = 0 thefull form of temporal flow L (ˆ v ) = 1, projected over the M base space, will provideconstraints on possible field interactions which are closely analogous to those providedby the Lagrangian for the Standard Model of particle physics, as will be described inchapters 8 and 9. Here we consider some of the geometric properties on the 4-dimensional spacetimemanifold M as arising in the present theory and in relation to general relativity. Herethe 4-dimensional base manifold M carries the four coordinate degrees of freedom ofour spacetime experience of physical objects in the universe. The symmetry of theLorentz group fits naturally on such a manifold since it acts on a 4-dimensional vectorspace which corresponds to the tangent space of M . Selecting a 4-dimensional basespace in this way is a provisional empirical input. It is empirical for the obvious reasonand provisional since at this point the choice of four dimensions seems theoreticallyarbitrary and there remains the question of whether a base space of a different di-mension could in principle be considered as a background for experience in anotherpossible world. We shall return to this issue, and the question of the uniqueness of thetheory in general, in section 13.3.Hence the study of the Lorentz symmetry is motivated by the fact that it103ontains SO(3), the rotational symmetry of the background space within which weperceive physical objects, together with its respect for temporal causality, as well asits central importance in established physical theories of the world. In conformity withthe present theory SO + (1 ,
3) is also the symmetry of a possible form of progression intime, denoted L ( v ) = 1 and presented explicitly in equation 5.1, over a 4-dimensionalvector space.Here we are considering the proper orthochronous Lorentz group SO + (1 , ↑ + , which is the part of the full Lorentz group that is continuouslyconnected to the identity element. It is hence a continuous symmetry group acting onvectors in the 4-dimensional vector space R , , denoting the space R with Minkowskimetric η ab = diag(+1 , − , − , − l ∈ SO + (1 ,
3) generate the symmetry transformations σ l : v → v ′ such that L ( v ) = L ( v ′ ) as an invariant form of temporal flow in fourdimensions. At any x ∈ M on the spacetime manifold v ( x ) ∈ TM is a vector in thelocal tangent space.The base space M itself originates out of the four dimensions of the translationsymmetry of the form L ( v ) = 1 which is trivially invariant under x a → x a + r a forthe four components v a = dx a /ds with a = { , , , } , as described more generally inequations 2.10–2.13 of section 2.1 and in section 2.2 for the model world. Here theset of four numbers r a ∈ R can be identified with an initial set of four coordinates x µ ∈ R , with x µ = δ µa r a .The Lorentzian structure of the vector space to which v ( x ) belongs is trans-ferred onto the tangent space of the parameter space M and hence the latter acquiresthe properties of a 4-dimensional pseudo-Riemannian manifold. That is, since the flow v ( x ) necessarily exists on the manifold, with components v a = dx a /ds , on M themetric η ab derives locally from the form L ( v ) = η ab v a v b , and it is described by themetric g µν in a general coordinate system via a tetrad field e aµ ( x ) as: g µν = e aµ e bν η ab (5.45)Hence the manifold M inherits its pseudo-Riemannian structure from theLorentz symmetry of L ( v ); with the SO(3) subgroup implying the possibility of asuitable 3-dimensional background space which appears to us to be of a more funda-mental a priori existence than the objects we perceive moving through it.For such a manifold in which there exist global coordinates such that g µν ( x ) =diag(+1 , − , − , −
1) for all x ∈ M , that is the constant Minkowski metric, we havethe 4-dimensional spacetime of special relativity. In this case the local η ab metric hasbeen drawn out and made global through the existence of large scale coordinates withrespect to which the tetrad field can be simply be expressed as e aµ ( x ) = δ aµ . For such aMinkowski spacetime manifold the SO(3) subgroup of SO + (1 , L ( v ) out onto an approximately uniform background spacetime104ithin which objects are perceived. The mathematical expression for such a spacetimestructure, to be utilised by perception, arises spontaneously out of the translationalsymmetry of the form L ( v ).While the identification of the 4-dimensional spacetime manifold through the4-dimensional form of temporal flow L ( v ) = 1 will result in a flat spacetime geometry,as described for the model world in subsection 2.2.3, ultimately the base manifold M will be obtained through a subset of four translational degrees of freedom breakingthe symmetry of a higher-dimensional form of temporal flow L ( v n ) = 1 with n > L (ˆ v ) = 1. A specific expression of for L (ˆ v ) = 1 will beintroduced in the following chapter, extending beyond the case of L ( v ) = 1 describedin section 5.1. This results in general in a non-zero external Riemannian curvature,complemented by a non-zero internal gauge curvature, as described in sections 2.3 and5.1. On the M manifold the Lorentz form of equation 5.1, now embedded within thefull form L (ˆ v ) = 1, locally expresses the relation between the components v a = dx a /ds of tangent vectors in an ordered orthonormal basis of the tangent space. Such a localbasis, or frame field, { e a } satisfies g ( e a , e b ) = η ab , and with v ( x ) = v a ( x ) e a ( x ) thelocal relation of equation 5.1 is replaced by the looser constraint on the four components v a ( x ) projected onto TM with: L ( v ) = ( v ) − ( v ) − ( v ) − ( v ) = η ab v a v b = η ( v , v ) = h . (5.46)with h ∈ R . While local coordinates { x a } necessarily exist to express the form L ( v ) = h we may also introduce an arbitrary global coordinate system { x µ } over M whichnaturally gives rise to a coordinate frame basis, denoted { ∂ µ } , for the tangent spaceat any x ∈ M . The coordinate frame is related to the orthonormal frame with ∂ µ = e a e aµ ( x ) as described in equation 3.49 and section 3.3.In addition to the observation that in general h = 1 in equation 5.46 thefurther consequence of the embedding in the larger form L (ˆ v ) = 1 is the possibilityof finite Riemannian curvature R = 0 as alluded to above. This implies a warpingof the geometry such that global coordinates no longer exist such that e aµ ( x ) = δ aµ in general. The tetrad field e aµ ( x ) now describes the necessarily non-trivial relationbetween global and local coordinates. As described in section 3.4 the unphysical natureof general coordinates is implied under general covariance, while a tetrad field withrespect to a set of coordinates R , as depicted in figure 3.6(a), indicates physicallydistinguished local orthonormal frames as utilised by the equivalence principle.In the present theory ‘general covariance’ is significant since in general theLorentz symmetry of the form L ( v ) = h cannot be expressed globally with respectto a single coordinate chart on the manifold. Without such a preferred global referenceframe all arbitrary coordinate systems are equally valid for the description of theequations of physics on the manifold. In the context of this theory the metric g µν ( x )has particular physical significance for the nature of perception and describes thegeometric form through which we literally see the world, motivating its prominent roleas the gravitational field; described as the ‘new ether’ by Einstein as discussed at theend of section 3.4.The general global coordinates do not correspond to an underlying Euclideanor any other geometric structure on the manifold. However, the manifold exists asa space for the flow v ( x ) of L ( v ) = h itself and we naturally have a frame field105 e a ( x ) } of local orthonormal basis vectors and local coordinates { x a } with respect towhich this flow can be written with the components v a = dx a /ds , corresponding tothe tangent vector components v µ = dx µ /ds in a general coordinate system, and hencewe necessarily have a local Lorentzian structure on M . While in principle the torsionon such a manifold may be finite the geometry described above is compatible withthe ‘equivalence principle’ which may hence be adopted, together with the implicationof vanishing torsion, as a provisional simplifying assumption which will be discussedfurther in section 13.3.With respect to a set of general coordinates { x µ } on the M manifold arbitraryvector fields u ( x ), that is cross-sections of the tangent bundle TM , can be expressed as u µ ( x ) ∂ µ with the numbers u µ ∈ R regarded as the components of a tangent vector onthe 4-dimensional manifold M . The situation is similar to that depicted in figure 2.4,except now for a 4-dimensional manifold. For any vector field u ( x ) on M the quantity g ( u , u ) = g µν ( x ) u µ ( x ) u ν ( x ) may be determined at any point x ∈ M and the vector u ( x ) described as ‘timelike’, ‘null’ or ‘spacelike’ according to whether this quantity ispositive, zero or negative respectively. This range of possibilities is also the origin ofthe name ‘space-time’ manifold. The ‘time’ in ‘spacetime’ refers to the existence oftimelike vectors and coordinates rather than explicitly to the actual pure temporal flow s which underlies the particular field v ( x ) as constrained by the equation L (ˆ v ) = 1.Since the Lorentzian manifold structure arises out of the flow of time the lightcone geometry of the tangent space is time-orientable over the 4-dimensional volumeof the spacetime manifold M . That is, the time-orientation of the light cones isnecessarily continuous on M as determined by the directed line element field v ( x ) oftemporal flow itself as an extension of the original 1-dimensional progression in time.This time-directed vector field is locally SO(3) invariant and provides a local (1 + 3)-dimensional decomposition of spacetime for all x ∈ M with temporal and spatial partsidentified in the local reference frames.Choosing the local coordinate x to be aligned with v ( x ), with components v a = ( dx ds , , , x effectively acts as a parameter for the pure values of time,that is ds = dx /h for L ( v ) = h , which is a particular case of the more general localexpression described in equation 5.47 below. Three spacelike local coordinates x , x and x can also be constructed orthogonal to each other and to x with respect to η ab ,with local spatial frames related via the SO(3) subgroup.Whereas embedding the perceptual background of an effective 3-dimensionalspace and 1-dimensional time within the symmetry structures of the mathematicalform L ( v ) = 1 led to their incorporation into the 4-dimensional Minkowski spacetimeof special relativity, that is with zero Riemannian curvature, extracting the same basemanifold out of a higher-dimensional form of temporal flow L (ˆ v ) = 1 results in a moreflexible and dynamic 4-dimensional spacetime structure as employed in general rela-tivity. With M itself still originating out of a 4-dimensional translational symmetryof L (ˆ v ) = 1, even for the generalisation in which the external geometry is expressed interms of underlying interacting fields as implied equation 5.32, the Minkowski metric η ab implicit in the form L ( v ) = h is sewn into the local tangent space structureeverywhere on the base manifold. This defines a possible metric structure g µν ( x ) on M associated in a one-to-one manner with the existence of an SO + (1 ,
3) orthonormalframe bundle OM within the canonical GL(4 , R ) general frame bundle FM over the106ase manifold, as described in section 3.3.With the external geometry related to the internal geometry via equation 5.20,or more generally with equation 5.31 augmented to equation 5.32, in principle themetric itself might be obtained by adopting the Levi-Civita linear connection on M .The connection is metric compatible, since it derives from the local SO + (1 ,
3) symmetryof the form L ( v ) = h , and assumed to be torsion-free as described above. Hence asfor general relativity the metric itself may be extracted by solving the second orderdifferential equation G µν = − κT µν given a form for the energy-momentum tensor T µν under appropriate boundary conditions, as described towards the end of the previoussection. An example is given in equation 5.49 below.The tetrad field e aµ ( x ) with 16 independent components carries two kinds ofinformation. The 10 degrees of freedom of the symmetric metric field g µν = e aµ e bν η ab correspond to the gravitational field for the torsion-free metric connection in generalrelativity, and hence the tetrad field itself can be considered to represent the grav-itational field. The remaining 6 degrees of freedom correspond to the local choiceof Lorentz frames implicit in e aµ ( x ). This local symmetry provides a link with theframework of local gauge theories as well as with the application of the spinor rep-resentations of the Lorentz group, as also alluded to towards the end of section 3.4,which are important in particle physics as will be described in chapter 7.Here we consider the physical significance of a non-flat Riemannian geometry,described by the metric field g µν ( x ), in particular on the relative passage of time itself.We also consider the relation of the original pure temporal flow s with the proper time τ which may be recorded by physical objects such as clocks in the material flow of theworld. The underlying pure temporal flow s , subject to the full form L (ˆ v ) = 1, existseverywhere on the base manifold M . The v ⊂ ˆ v projection onto the tangent space TM to the base manifold is a timelike vector, as is the tangent to any world line on M , with components v a = dx a /ds restricted under L (ˆ v ) = 1 such that: | v | = η ab v a v b = h implying ds = η ab h dx a dx b (5.47)However gravitational time dilation will not be directly observed from the perspectiveof the microscopic flow v . Indeed the underlying pure temporal flow s is not measureddirectly by physical instruments. Rather it is through the structure and symmetries ofthe form L (ˆ v ) = 1 that the physical world emerges on M through relations such asequation 5.20, and with more general expressions for the apparent energy-momentumtensor as implied in equation 5.32. This more general apparent material world may bedescribed empirically in part by the effective energy-momentum tensor T µνǫ = ρu µ u ν ,as introduced in equation 5.35 of the previous section and leading to the geodesicequation 5.36, where ρ ( x ) is the matter density and the 4-velocity u µ ( x ) = dx µ /dτ isdefined as the tangent vector at x ∈ M to the world line of the physical body, whichmay be an element of a pressureless fluid. It is through the motion of physical bodies,such as the hands of a mechanical clock, that time dilation effects may be observed.With the proper time τ parametrising the motion of the body for a general coordinatesystem { x µ } on M we have: g µν u µ u ν = 1 and with dτ = g µν dx µ dx ν (5.48)107dentifying an interval of proper time dτ . These expressions are invariant under generalcoordinate transformations. The normalisation for the components of the metric g µν ( x )will depend on the choice of empirical units adopted, for example seconds and metresfor temporal and spatial dimensions, in recording the motions of the parts of a physical‘clock’. The local orthonormal coordinates { x a } constructed empirically for the macro-scopic proper time interval with dτ = η ab dx a dx b will in general not be identical tothose of equation 5.47 arising directly out of the mathematical properties of the pureform of temporal flow L (ˆ v ) = 1. However with the physical world unfolding throughthe progression of the fundamental time parameter, and with s and τ represented bythe 4-vectors v and u in TM respectively, both temporal parameters are subject totime dilation effects in the same way. The proper time τ , in 4-dimensional spacetime,is implicitly linearly proportional to the pure underlying temporal flow s , which maybe expressed in any number of dimensions. This proportionality is expressed throughthe fixed parameter γ in equation 13.3 in section 13.1 where the relationship between τ and s is further explored.Hence along a shared world line the fundamental time interval ds is related tothe proper time interval dτ by a constant scaling and the two temporal parameters areequivalent in this sense – that is, within a fixed normalisation factor physical clocks do measure the progression of pure time s . As described in the introductory chapter,and to be expanded in chapter 14, the fundamental underlying mathematical time s is ultimately identified with ‘experienced’ time, while proper time τ is associated withmeasurable empirical phenomena, which include for example ‘physical brain processes’.Hence these subjective and objective temporal phenomena, which might be exemplifiedby an observer located within the same inertial frame as a physical clock, are intimatelyconnected. We next consider a particular example of time dilation effects.The physical manifestation of the metric g µν ( x ) in a general coordinate systemon M resides in observable relative temporal and spatial distortion effects at differentlocations on the manifold itself. For example the Schwarzschild solution for the metricof a spatially spherically symmetric geometry around a single massive body of mass M is given by the line element: dτ = (cid:18) − G N Mr (cid:19) dt − (cid:18) − G N Mr (cid:19) − dr − r dθ − r sin θdφ (5.49)in the 4-dimensional, spatially spherical polar, coordinates { t, r, θ, φ } , where G N isNewton’s gravitational constant. In addition to the assumption of a spatially spheri-cally symmetric metric this solution is obtained by imposing the boundary conditionthat g µν ( x ) approaches the flat Minkowski limit as r → ∞ spatially. This limit can beseen explicitly on taking r → ∞ in equation 5.49 (this example is closely analogous tothe case of the Coulomb field for a central electric charge).The coordinate r parametrises, but does not determine, radial distances. Thisis consistent with the arbitrary nature of coordinates and all coordinate systems ingeneral, as described in section 3.4 and figure 3.6. The actual radial distance, for givenparameters { t, θ, φ } , is measured by the integral of intervals dR = (1 − G N M/r ) − / dr .Similarly a clock at a fixed coordinate location in space records the proper time τ dτ = (cid:18) − G N Mr (cid:19) dt (5.50)relative to the time τ r →∞ = t measured by a clock in the flat spacetime limit at r → ∞ , and is a function of radial distance from the central mass, as parametrised bythe coordinate r . While at any location x ∈ M it is possible to choose local inertialcoordinates { x a } , for which g µν ( x ) = diag(1 , − , − , − global frame for non-zero mass M > s in exactly the same way as for the proper time τ . Hence with the interval ds ≡ dτ the same metric g µν ( x ) represents the relative temporal dilation on M forthe fundamental flow of time s . An observer, named ‘twin A ’, accompanied by aclock measuring the physical temporal flow τ A carries an equivalent universal timeparameter s A through which the entire universe unfolds through the realisation andsymmetry breaking of the full form of temporal flow L (ˆ v A ) = 1, deriving from s A as described for equation 2.9. A second observer, ‘twin B ’, at a separate spacetimelocation carries a second personal temporal parameter s B through which B perceivesthe same universe to unfold through the form L (ˆ v B ) = 1 in a mutually consistent way.This ‘dovetailing’ of the ‘temporalisation’ experienced by twins A and B as manifestedin the same physical world will be described further in section 14.2 in the discussionof figure 14.7.The same metric solution g µν ( x ) for the single consistent universe, expressedin a particular coordinate system (or equivalently a particular metric expression of agiven geometry in terms of a unique set of coordinates R , adopting the perspective offigure 3.6(a)), provides the relation between the intervals ds A and ds B and the equiv-alent gravitational temporal dilation effect observed between dτ A and dτ B measuredby the clocks of twin A and twin B respectively. The time dilation effect is deter-mined by the empirically constructed metric g µν ( x ) in the coordinate system { x µ } since it implicitly determines local inertial coordinates { x a } which are related to thoseof equation 5.47 by a constant scale factor (again, as will be discussed further near theopening of section 13.1 and alongside equation 13.3). In turn the local coordinates ofequation 5.47 directly parametrise the fundamental temporal flow s , within a factor of h , via the projection of the form L ( v ) onto the tangent space of M .So far we have implicitly considered only the case of constant h ( x ) in equa-tion 5.47. In this case all geometric time dilation effects can be considered as having a‘source’ in the right-hand side of the Einstein equation 3.75 in terms of the apparentenergy-momentum T µν ( x ) of ordinary matter. This is the case for the Schwarzschildsolution of equation 5.49 for a central massive body. On the other hand possible vari-ations in the magnitude of h ( x ) in equation 5.47 will act as conformal transformationsof the geometry the possible consequences of which will be considered in section 13.1,initially alongside figure 13.1. 109 .4 Beyond Kaluza-Klein Theory For Kaluza-Klein theory, originating as a pure higher-dimensional spacetime extensionof general relativity, to be interpreted as a unified theory of gravitation and gaugefields in a 4-dimensional spacetime the symmetry group of general coordinate trans-formations in the extended spacetime has to be broken down to 4-dimensional generalcovariance together with the local gauge symmetry. This is equivalent to placing re-strictions on the metric of the extended space which then possesses a set of isometriesdescribed by Killing vector fields which have a one-to-one relationship with the left-invariant vector fields on the manifold of an apparent gauge group G . In this way aprinciple fibre bundle structure emerges on the extended space, exhibiting symmetriessuch that the freedom in variation of the metric ¨ g ij , as expressed in a direct prod-uct basis in equation 4.5, is effectively reduced to the components g ac and ω αa . Theconstruction of an action integral on the bundle space then leads to correspondingequations of motion such as those of equations 4.16–4.18. A dynamical mechanism forthis process in which an extended 4-dimensional base manifold M of general relativ-ity survives while the extra dimensions lose any sense of external spatial significance,sometimes called ‘spontaneous compactification’, then remains to be specified, as al-luded to in section 5.1. That is, the origin of the above restrictions on the metric forthe full space remains to be accounted for.The Kaluza-Klein models, reviewed in chapter 4, contrast with the idea pre-sented in this paper since here the ‘extra dimensions’, beyond four, are not requiredto satisfy an explicitly geometric, or spacetime, symmetry. In turn for the presenttheory there is no need to explain such a ‘compactification’, rather the base manifold M is the only physically extended manifold to consider as it emerges as a backgroundarena for perception through the translational symmetry of the full form of temporalflow L (ˆ v ) = 1. In section 5.1 we presented these ideas as a mathematical possibilitytaking as an example the SO + (1 ,
9) symmetry of L ( v ) = 1 projected over M , butthe significant conceptual question concerning why this situation should be found innature also needs to be addressed. We review here the conceptual motivation that ledto this framework in the context of this provisional SO + (1 ,
9) model world.Out of the purely algebraic symmetries of L ( v ) = 1 the possibility of a localso + (1 , geometric meaning to M as being not just anumerical parameter space for translational degress of freedom but rather implicitlypossessing a Riemannian structure with local metric g ac ( x ) as an arena for the percep-tion of physical objects in time and space . The identification of an extended base spaceis possible since there is a ‘spacetime’ symmetry as a subgroup of the full symmetry of L (ˆ v ) = 1 which acts on the local tangent space of M . This innate possibility of suchan interpretation is sufficient for such structures to ‘freeze out’ from the full symmetryof L (ˆ v ) = 1 as a kind of ‘gestalt’ through which by necessity the physical world iscreated and perceived.Given this geometrical realisation of the perceptual ‘external’ symmetry on thebase manifold, out of the full symmetry there remain ‘internal’ residual gauge fieldsand surplus temporal components which will collectively contribute to the apparent‘matter’ content of the world through which the properties of physical entities will beperceived and identified on the base space. The symmetry of L ( v ) = 1 is broken inthe identification of the extended M parameter space, with a v ⊂ v component of110he temporal flow projected onto the tangent space TM as depicted in figure 5.1(b).Since the v components are distinguished in this way from the residual internal part v ⊂ v the full symmetry of the original SO + (1 ,
9) action on v is lost. The survivingsymmetry, as gauge freedom over M , is resolved into two pieces with correspondingconnection 1-forms identified for both the external and internal spaces.The combination of the general flow of time, expressed as L ( v ) = 1, withthe implied symmetry properties and canonical mathematical structures existing forthese objects, together with the conceptual need for a perceptual base for observationin a world, all taken collectively, has resulted in the identification of a backgroundmanifold. The full symmetry of the temporal flow L ( v ) = 1 has been ‘sacrificed’ inthe creation of the non-trivial external and internal geometrical entities, but remains asa ‘ghostly’ presence through which these entities are related. This correlation betweenthe external and internal curvature tensors R = 0 and F = 0 (while both can bezero together) was described originally for the SO(5) model over M in section 2.3 andfor the SO + (1 ,
9) model over M , in the light of Kaluza-Klein theory, in section 5.1.This latter structure will also apply to the full symmetry action considered for the realworld from the following chapter.The use of geometrical pictures, such as those of figure 5.1, as a visual aid tounderstanding mathematical structures comes very naturally when the space picturedrepresents the way we actually perceive those structures in the world. However, theunderlying properties of a purely mathematical space, such as those demanded hereby the concept of the symmetry of time, need to be worked out within the appropriate algebraic rules, which are not necessarily visualisable even by analogy with lower-dimensional structures. Hence while possibly serving as a guide a reliance on suchgeometric pictures is ultimately likely to prove misleading. This in particular will bethe case in the following chapter in which the internal dimensions will no longer havea spatial interpretation ( unlike the case for the 6-dimensional space of vectors v ∈ R with an internal SO(6) rotational symmetry for the SO + (1 ,
9) model described above).On the other hand it can be asked what the perceived part of the mathematicsactually looks like, and geometric pictures only really make sense in terms of a literalinterpretation in this context. Perception is our window into the world of mathematicalforms. It is a window which is both opened up and limited through the possibility ofthe internal mathematical relations which frame our experiences in a 4-dimensionalspacetime. It is also part of the difficulty in theorising beyond the 4-dimensional worldof general relativity, for which visualisation is a key tool.In conclusion then, here a spacetime geometric symmetry is only required toexist on the base manifold, hence in four dimensions for our world. It is also requiredto be an approximately global symmetry, such that the base manifold may be identifiedas a suitable arena for perception in the world, at least for extended regions on thescale of everyday observations although not necessarily on the larger scales consideredin cosmology.In Kaluza-Klein theory, as described in chapter 4, while a unified framework isprovided for gravity and gauge boson fields, equations 4.16–4.18, there is no energy-momentum tensor for fermion fields – that is the matter fields for the leptons andquarks of our world are absent. These fields may be added by hand as sections offibre bundles over M , associated to the principle bundle P , transforming as spinors111nder the external SO + (1 ,
3) symmetry and in representation multiplets of the internalgauge symmetry group. Coupling between the gauge fields and fermions may then beintroduced through interaction terms, also added by hand for example via ‘minimalcoupling’ involving covariant derivatives, in the Lagrangian constructed for the theory.A more mathematically self-contained approach is through a supersymmetricextension of the Kaluza-Klein framework (see for example [15] section VI, [28], [29] and[30] sections 1 and 2). Fermions may be included for example through generalising thegauge group G of the principle bundle to a ‘supergroup’ by augmenting the Lie algebra L ( G ) into a ‘graded’ Lie algebra. Here the rule for multiplication in the Lie algebra bycommutation of elements, as exemplified in equation 2.22, is extended algebraically toinclude anticommutation which can be used to accommodate the properties of fermionfields. The Einstein-Yang-Mills theory may be extracted as the purely bosonic sectorof such extended supergravity theories.Of the many formulations of supergravity the most attractive model involvesa single supersymmetry generator, ‘ N = 1’, so that each Standard Model particle hasa single superparticle partner forming a supersymmetric doublet, and is constructedin an 11-dimensional spacetime, that is ‘ d = 11’. The pairing of bosons with fermionsthrough supersymmetry also tends to naturally lead to the attainment of finite calcula-tions in the corresponding quantum field theory. However, even for the most favourableversion in 11-dimensional spacetime a fully renormalisable version of supergravity hasnot been realised ([26] p.880). Further generalisation of supergravity to a superstringtheory, obtaining a finite theory of quantum gravity by modifying QFT at the Planckscale, addresses some of the technical difficulties.Through this geometrisation of matter in the spirit of Kaluza-Klein modelsbased on higher dimensions of spacetime, extended to the supersymmetric theories of11-dimensional supergravity and 10-dimensional superstrings, the aim is to incorporatethe degrees of freedom of the full set of Standard Model gauge interactions within thegeometry of the 7 or 6 extra spatial dimensions. In some cases the extra dimensionsare considered to be small and topologically compactified while in other models ourown universe may be conceived as a 4-dimensional brane -world embedded as a 4-dimensional hypersurface within the higher-dimensional spacetime bulk (see [31] for asimpler case with a 5-dimensional bulk).Einstein’s theory of gravitation based on a metric tensor in 4-dimensional space-time hence stimulated a chain of extensions and generalisations that we have brieflyreviewed above and summarise below in table 5.2.One of the attractions of using a symmetry of extra spatial dimensions, as wellas its intuitive appeal as an extension of 4-dimensional spacetime geometry, is that itlimits the set of possible higher symmetries and mathematical structures to consider.In this paper instead of considering arbitrary symmetries, general geometric symme-tries or specifically the symmetry of a spacetime in higher dimensions we considergeneral symmetries of pure time alone, as expressed through the relation L ( v ) = 1 anddescribed in chapter 2. This also greatly limits the choice of symmetry groups andtheir representations. As well as naturally extending to general higher-dimensionalmathematical forms of the progression of time L ( v ) = 1, at the same time we retainthe significance of the (1 + 3)-dimensional metrical manifold as a form of observationin the world as having a necessary and a priori nature.112heoretical Framework Physical ScopeGeneral Relativity GravitationKaluza-Klein in 5-dimensions ElectromagnetismNon-Abelian Kaluza-Klein Theory Non-Abelian Gauge Fields(with non-Levi-Civita Γ on P ) (avoid large Cosmological term)(with G acting on homogeneous fibres) (keep full L ( G )-valued theory)Supergravity Fermions as well as BosonsSuperstrings Finite Quantum GravityTable 5.2: A series of increasingly general frameworks is listed in the first columnwith their cumulative extent of application listed in the second column for the non-parenthetical entries. The means of including fermions and quantum theory withinthe present framework will be described in section 8.1 and chapter 11 respectively.The important point here is that the symmetry of the space part of spacetime,such as that of the SO(3) subgroup of the Lorentz symmetry SO + (1 ,
3) central togeneral relativity, can be experienced in a different, geometrical, way compared withother higher symmetries of L ( v ) = 1. It may be that higher symmetries, such asSO + (1 , could be interpreted in a geometrical way, but this feature is relativelyincidental in comparison with the fundamental requirement that it must describe asymmetry of time.However, through investigating possible symmetries of time a significant ex-ample is identified for the symmetry group SL(2 , O ) acting on the 10-dimensionalspace h O , constructed in terms of the octonion algebra as described in the followingchapter and in particular section 6.3. With SL(2 , O ) being the covering group of the10-dimensional Lorentzian symmetry SO + (1 ,
9) this structure will naturally correlatewith some of the properties of models based on extra spatial dimensions for which10-dimensional spacetime is significant. Further, the 16-dimensional Majorana-Weylspinor representation of the 10-dimensional Lorentz group, highlighted in table 7.1 ofsection 7.3 and here represented by the θ components appearing in the extension to thespace h O introduced in equation 6.26 of section 6.4 and described near the openingof section 8.1, is significant in various branches of string theory.In the present theory by exploring the physical interpretation of the higher-dimensional forms of L ( v ) = 1, together with the associated isochronal symmetrygroups, expressed over a base space M , contact is made with the series of generalisa-tions mid-way down table 5.2, with the items listed parenthetically, with a frameworkvery similar to non-Abelian Kaluza-Klein theories. The use of a non-Levi-Civita G -invariant linear connection Γ such as described for equation 5.13 defined on a principlefibre bundle P , or on a bundle of homogeneous fibres E , makes a significant area ofcontact with the corresponding literature (including [13, 14, 15, 16], [17, 18, 19, 20],[22, 23, 24], [25]). From this point we then immediately diverge away from the pro-113ression towards supersymmetry and string theory in table 5.2 and in this context weshall need to explain how mathematical structures identified in the present theory cor-respond to the inclusion of fermion states as well as the physical concepts of quantumand particle phenomena in general.Here we describe how field interactions arise in the context of the SO + (1 , P ≡ M × SO(6) under the breaking of the fullSO + (1 ,
9) symmetry of the model world, the local SO + (1 , ⊂ SO + (1 ,
9) symmetryacting on the tangent space TM is associated with the linear connection 1-form Γ( x )on M , which is central to the theory of general relativity and is subject to the Bianchiidentity D R = 0, while the so(6)-valued connection 1-form on P is interpreted as thegauge field Y ( x ) on M , central to the gauge theory arising from the internal symmetry,and is subject to the Bianchi identity D F = 0. The structures of the external andinternal geometry are correlated and the corresponding equations of motion constrainedas described in sections 5.1 and 5.2, with self-interactions arising for the gauge fieldsfor the non-Abelian internal symmetry.Further dynamical equations of motion will arise out of the full 10-dimensionaltemporal flow in the broken form of D µ L ( v ) = 0, by a direct generalisation ofequations 2.46 and 2.47 from the SO(5) model. For the SO + (1 ,
9) model the symmetrybreaking leads to interactions between the gauge field Y ( x ) and the internal degreesof freedom deriving from the components of v ⊂ v . That is, in comparison withequation 2.47, we have: D µ L ( v ) = 0 ⇒ v · ∂ µ v + v · A µ v + v · Y µ v = 0 (5.51)where A µ ( x ) is the external Lorentz connection on M . Through the interactionsbetween the internal fields Y µ ( x ) and v ( x ) the apparent matter content of the worldon the base manifold arises, together with its quantum properties, as outlined forequation 5.32 and alluded to near the opening of this section.As described in section 5.1 the principle bundle P ≡ M × SO(6) is not con-sidered here to represent a physical space or spacetime, and neither is the associatedbundle with homogeneous fibres. In the absence of a structure of extra spatial di-mensions in general the full form of purely temporal flow L (ˆ v ) = 1 is not required tobe associated with a metric geometry. The question then concerns the mathematicalstructure of the higher-dimensional forms of L ( v ) = 1 of relevance for the physicalworld. In the following chapter a particular 27-dimensional form L ( v ) = 1 togetherwith its full symmetry group ˆ G = E will be introduced.Given the extra dimensions of the full vector object v ∈ h O the need toidentify a Riemannian curvature parametrised over a locally approximately flat 4-dimensional base manifold M breaks the full E symmetry. The geometry on M drawn out of the underlying structures and symmetries implied in the form L ( v ) = 1can be described generically by the 4-dimensional relation − κT µν := G µν = f ( Y, ˆ v ) ofequation 5.32, with G µν ; µ = 0, shaping the perceptual background of our observableworld. The external symmetry, acting on the extended manifold M itself, is a priori essential for perception in the world as geometrically described by the linear connectionand Riemannian curvature which are smoothly dependent upon x ∈ M . For thepresent theory this natural and necessary mechanism of symmetry breaking over themanifold M forms a significant part of the conceptual framework through which themathematical structures are realised in the physical world.114or the full theory based on the action of E on h O the internal coupling in thefinal term of equation 5.51 will be replaced by an interaction between internal gaugefields and fermion fields, where the latter are identified in the internal componentsof v ∈ h O under the action of the external symmetry on L ( v ) = 1 as will beexplained in section 8.1. Hence a particular form of extra dimensions can be identifiedfor the present theory which ultimately provides the source for the interacting gauge Y ( x ) and fermion ψ ( x ) fields, each of which transforms in the appropriate way underthe Lorentz symmetry on 4-dimensional spacetime, underlying the matter and particleeffects observed in the real world. A ‘supersymmetry’ is not required in order tointroduce gauge fields alongside fermions fields, together with their mutual interactions,in the unified theory presented here.Having identified fermion states the question remains concerning the origin ofmore specific structures of the Standard Model of particle physics, as implementedthrough Lagrangian terms in the form of equation 3.96 for example and as reviewedmore generally in chapter 7. The origin of a series of Standard Model properties inthe context of the present theory through the breaking of the full form L (ˆ v ) = 1 willbe presented in chapters 8 and 9. The constraints implied in the full form L (ˆ v ) = 1augment the surveillance of the external geometry with G µν ; µ = 0, described at theend of section 5.2, and the need to postulate any form of Lagrangian approach willrecede further, implying ultimately that it may be avoided entirely. The full collectionof constraints will also be utilised in order to address the origin of quantum phenomenafor the present theory in chapter 11.In the meantime, before considering the empirical implications for observedlaboratory phenomena, in the following chapter we leave the model worlds behind andmotivate consideration of E as the symmetry group acting upon L ( v ) = 1 as anatural higher-dimensional form of temporal flow.115 hapter 6 E Symmetry on h O In order to determine the physical effects, observable on the base manifold, of more gen-eral morphisms of the flow of time through a higher-dimensional form we shall need toconsider a suitable larger symmetry group acting on an appropriate higher-dimensionalvector space. The motivation leading to the identity L ( v ) = 1 of equation 2.9 as thegeneral mathematical form acting on the n real number components of temporal flow v in an n -dimensional vector space was described in chapter 2. We are particularlyinterested here in finding such an expression with n somewhat larger than four (sincethe case of the external Lorentz symmetry of the form L ( v ) in equation 5.46 corre-sponds to n = 4) and with a significant degree of symmetry. The vector space h O isthe set of 3 × X = p ¯ a ca m ¯ b ¯ c b n ∈ h O (6.1)with p, m, n ∈ R (here the component labels are chosen to conform with the notationin the relevant references, and n here is of course not the dimension of any space), a, b, c ∈ O and ¯ a denotes the octonion conjugate of a reversing the sign of the 7-dimensional imaginary part (the octonion algebra is described in the following section).Hence the vector space h O is 27-dimensional over the real numbers. It is a space withparticularly rich symmetry properties largely owing to the nature of the 8-dimensionaloctonion subspaces.The dimensions of the vector and spinor representations of the rotation groupSO( n ) converge in the case of n = 8. That is, as well as the 8-dimensional vectorrepresentation of SO(8) the 16-dimensional spinor representation reduces to two dis-tinct 8-dimensional spinor spaces, dual to each other. The three 8-dimensional spacesundergo different SO(8) transformations, however mappings may be defined which in-116erchange the transformation behaviour between the three spaces, with a two-to-onemap from a spinor to the vector representation. The existence of such maps is dueto a property known as the ‘principle of triality’ [32, 33] and it is unique to spaces ofeight dimensions.Three such 8-dimensional spaces can be represented by three copies of the octo-nions, in particular under an appropriate SO(8) symmetry operation on the space h O in equation 6.1, as will be described later around equation 6.50. While the actions onthe vector and two spinor representations differ for particular SO(8) transformations,collectively as three sets of transformation actions they are isomorphic by triality andit is a matter of convention which octonion space is assigned as the vector or spinorof either kind. Further, a 14-dimensional subgroup of the rotation group SO(8) (it-self 28-dimensional) acts on the three octonion spaces in exactly the same way. Thisis G , the automorphism group of the octonion algebra. In fact G ⊂ SO(7) as theautomorphisms only act upon the seven imaginary units of the octonions. (In generalan algebra automorphism Φ acts on any two elements a, b of the algebra such thatΦ( a + b ) = Φ( a ) + Φ( b ) and Φ( ab ) = Φ( a )Φ( b ), with the order of the latter productbeing reversed in the case of an algebra anti-automorphism such as the map a → ¯ a ofequation 6.7 described in the following section).The elements of the vector space h O belong to a Jordan algebra for which thealgebra product is given by: X ◦ Y = 12 (
X Y + YX ) (6.2)with X , Y ∈ h O and where X Y is the ordinary multiplication of the 3 × O algebra is knownas the exceptional Jordan algebra since it cannot be expressed in terms of matriceswith associative elements (such as real or complex numbers). The algebra itself iscommutative but non-associative (as is generally the case for all Jordan algebras) withthe exceptional Lie group F being the automorphism symmetry of the algebra.However, there is a larger symmetry group involving another structure whichcan be defined on the space h O which is of particular interest here. This is a cubicnorm, or determinant, det( X ) ≡ ( X , X , X ) for X ∈ h O which will be presentedexplicitly in section 6.4. A subspace of the vectors v ∈ R map onto elements X ∈ h O that satisfy the homogeneous cubic polynomial equation det( X ) = 1 which expressesa form of the principle relation of equation 2.9 denoted L ( v ) = 1. This subspaceis locally 26-dimensional and hence may be denoted S , as a homogeneous space,following the convention in the opening of section 4.3, although here S representsthe full space of temporal flow rather than a purely internal fibre space.The symmetry of this 27-dimensional form L ( v ) = 1 corresponds to a groupof morphisms of the elements of h O which preserve the unit cubic norm; that is theset of actions such as A λ ( X ), parametrised by λ ∈ R , with ( A λ ( X ) , A λ ( X ) , A λ ( X )) =( X , X , X ). With the identity transformation labelled by λ = 0 elements of the cor-responding Lie algebra may be represented by the objects D ≡ ∂A λ /∂λ | λ =0 . Moregenerally the Lie algebra can be defined directly in terms of the set of operators D D X , X , X ) + ( X , D X , X ) + ( X , X , D X ) = 0 (6.3)These elements are found to comprise a 78-dimensional Lie algebra of rank 6 (the Car-tan subalgebra consists of 6 mutually commuting generators); and these properties,together with the fact that it has a 27-dimensional representation, lead to the iden-tification of the Lie algebra E associated with the exceptional Lie group E . In factit is one of four real non-compact forms of this Lie algebra denoted E − (since theKilling form signature is −
26 (= 26 − L (E ) may be usedto emphasise the Lie algebra. Lower case kernel letters are used to denote a classicalLie algebra, such as so( n ), corresponding to a Lie group, such as SO( n ), although againwhether a statement refers to the Lie group, its algebra or both should generally beclear from the context).The first construction of the E Lie algebra in terms of action on the space h O dates from 1950 [34] and combined the 52-dimensional algebra of derivations D R ofthe Jordan algebra h O (that is, the generators of the automorphism group F ) withthe 26-dimensional set D B composed of operations of right action on h O by tracelesselements of h O itself. The total set of elements in this (52 + 26) = 78-dimensionalspace may be written: D R,B = D R : { F automorphism group of h O Jordan algebra } + D B : { maps X ∈ h O → X ◦ x, with x ∈ h O , tr( x ) = 0 } (6.4)It can also be shown that the commutator [ D Bx , D By ] = D Bx D By − D By D Bx ∈ D R . (The D R and D B are analogous to the rotations and boosts, respectively, for the Lorentzgroup, as we shall see later in this chapter). All elements of the set D R,B have theproperty exhibited by D in equation 6.3 and are therefore associated with an E groupaction that preserves det( X ) for any X ∈ h O ([1] pp.44–46).Alternatively the Lie algebra E − can be expressed in terms of mappingsinduced on h O by the 14 generators of G acting on O , supplementing the actionsof a basis of 64 independent tracefree 3 × algebra, dating from the 1960s ([35] pp.162–164), in terms of a(14 + 64) = 78-dimensional decomposition can be denoted by D G,S and is composedof two sets: D G,S = D G : { G automorphism group of O algebra } + D S : { maps X ∈ h O → x X + X x † , with x ∈ sl (3 , O ) } (6.5)where sl (3 , O ) is the 64-dimensional set of traceless 3 × D G is isomorphic to the 14-dimensional Lie algebra G . All elements of thiscombined set satisfy equation 6.3 and hence L ( v ) = 1, in the form of det( X ) = 1, ispreserved by the associated E group action.More generally ([1] p.28, [36]) denoting by sl ( n, K ) the set of traceless n × n matrices with entries in the division algebra K = R , C , H or O for n > ( n, K ) is closed only if K is commutative and associative (i.e. for R and C n, K ) may be defined to be the Lie algebra of operatorson K n generated by the elements of sl ( n, K ) under an appropriate matrix commutationrule (the Lie algebra L (E ) is identified with sl(3 , O ) in [36] p.950). The Lie groupSL( n, K ) of operators on K n , with an associative multiplication even for K = O , maybe generated by the elements of this sl( n, K ) Lie algebra. For the case n = 2 the groupSL(2 , K ) also has a representation on h K that preserves the determinant.However, this approach of first defining the Lie algebra purely in itself is not followed here. Rather finite group transformations will be constructed first [37, 38, 39,40, 41]. Here SL( n, K ) will be defined principally in terms of a set of group transforma-tions that preserve a particular norm on a vector space, essentially by generalisationfrom SL(2 , C ) as a determinant preserving action on h C . The need for such a real-valued ‘norm’ is here motivated by the form L ( v ) = 1. In obtaining the full set ofsymmetry actions on the form L ( v ) = 1 it is partly a matter of convention whether thisgroup of transformations is given a name of the type SL( n, K ). A representation of thecorresponding Lie algebra sl( n, K ) will be defined and derived subsequently throughthe group action on the representation space. In particular the L (E ) ≡ sl(3 , O ) Liealgebra will be described in terms of a basis of vector fields on the tangent space tothe hypersurface S embedded within the space h O , which itself can be consideredas a 27-dimensional manifold.Associativity is required of any group of operations in general and it is thecase for the elements of E which will be described explicitly in section 6.4. While ingeneral multiplication between octonions is non-associative it is possible to use themin the construction of algebraic elements such that the multiplication defined betweenthese latter elements is in fact associative. Indeed it will be possible to conceive ofthe elements of E acting on X ∈ h O of equation 6.1 in a manifestly associative wayrepresented as a subgroup of GL(27 , R ) acting on v ∈ R , with h O ≡ R as vectorspaces, such that L ( v ) = det( X ) = 1 is invariant, similarly as for all forms of L ( v ) = 1under symmetry operations. Since the construction of the Lie group E ≡ SL(3 , O )here relies on the composition properties of the octonions in the following section wefirst turn to the octonion algebra itself. Having introduced the division algebras in section 2.1 in relation to possible multi-dimensional forms of temporal flow for equation 2.9 here we focus on the largest suchalgebra. For the remainder of this chapter we follow references [37, 38, 39, 40, 41]in leading from the properties of octonions through their relation with Lorentz trans-formations to the construction of the symmetry group E and the corresponding Liealgebra. The main reference for these latter structures in particular is ([38] chapters 3and 4). The above references are extensively reviewed in this chapter, owing to theirimportance for the present work, and they also provide the source for much of thenotation adopted here.We begin then with a general octonion which, as an element of an eight-dimensional vector space, has eight real parameters { a . . . a } and can be written: a = a + a i + a j + a k + a kl + a jl + a il + a l (6.6)119he first term could be written as a e with e ≡ a ∈ R are embedded in the octonions as a e ∈ O . Theseven imaginary units in this basis { i, j, k, kl, jl, il, l } , with i = j = . . . = il = l = −
1, are mutually anticommuting, with il j = − j il etc., with their full algebraiccomposition described in figure 6.1. il k jlj l ikl Figure 6.1: The multiplication of any two octonion units is given by ± the thirdoctonion on the same directed line, with a + (or − ) sign for composition aligned with(or against) the arrow on the line, for example kl il = − j .Hamilton’s quaternions are contained as a subalgebra of the octonions withimaginary units { i, j, k } composed as i j = k with cyclic permutations as representedby the arrowed circle in figure 6.1. The six other arrowed lines represent six furtherequivalent H subalgebras embedded in O . As can be seen from examples such as k l = kl the notation for the imaginary units is chosen as a mnemonic for these relations, whereit should be understood that kl is a single imaginary base unit on equal footing withany of the other six, with kl k = l and so on.While multiplication within any of the seven quaternion subalgebras is asso-ciative, for example ( kl k ) l = kl ( k l ) = −
1, multiplication between any three imaginarybase units not situated on the same line in figure 6.1 is anti -associative, with for ex-ample ( i j ) l = − i ( j l ) = + kl . Care needs to be taken due to the possible ambiguityin expressions involving products of octonions due to this lack of general associativity.However, the algebra does satisfy the weaker condition that products involving onlytwo distinct octonions a, b ∈ O are associative, for example ( aa ) b = a ( ab ), and hencethe octonions form an alternative algebra.120ctonion conjugation is defined as a real linear map a → ¯ a on O such that forthe real unit a → a while for the seven imaginary units a h → − a h ( h = 2 . . . a ∈ O in equation 6.6 is therefore:¯ a = a − a i − a j − a k − a kl − a jl − a il − a l (6.7)which applies to any product as ab = ¯ b ¯ a and is hence an algebra anti-automorphism.For a given octonion a the norm | a | is a real number defined by: | a | = a ¯ a = X h =1 a h (6.8)which applies to any product as | ab | = | a || b | since the algebra is alternative. Hencethe norm is compatible with octonion multiplication and it is these properties, whichalso imply the existence of a unique inverse: a − = ¯ a | a | (6.9)for any element a = 0, which make the octonions a ‘normed division algebra’ ([1] p.9).By a theorem of Hurwitz from 1898 only four such algebras, of real dimension 1, 2, 4and 8, exist; these are R , C , H and O which hence form a unique set of algebras, aslisted in section 2.1. With the octonions being the largest normed division algebra andpossessing a rich symmetry structure they naturally find use in the present contextfor identifying possible forms L ( v ) = 1 for temporal flow together with the associatedsymmetries. In the simplest case for a, b ∈ O with | a | = 1 and b ≡ v the composition ab with | ab | = | a || b | provides a set of symmetry transformations leaving the form L ( v ) := | b | = 1 invariant.Octonion conjugation can be used to extract the real and imaginary partsof a ∈ O as Re( a ) = ( a + ¯ a ) and Im( a ) = ( a − ¯ a ) respectively. While for acomplex number z = x + yi ∈ C the imaginary part is usually defined such thatIm( z ) = y is itself a real number, as for example in equation 10.96, for the quaternionand octonion cases the imaginary part is defined as an imaginary number, with forexample Im( a ) = a i + a j + . . . for equation 6.6, since such an object in general involvesseveral distinct imaginary units. An inner product for any two octonions a, b ∈ O maybe defined by: h a, b i = 12 ( a ¯ b + b ¯ a ) = Re( a ¯ b ) = X h =1 a h b h . (6.10)For a single octonion | a | = h a, a i , while for any two octonions h a, b i = h b, a i and geometric orthogonality can be defined by the algebraic property h a, b i = 0. Fromequation 6.10 it can be seen that any real element of O is orthogonal to any imaginaryelement and also any pair of anticommuting octonions (such as { i, j } with ij = − ji etc.)are orthogonal to each other. In general a unit imaginary s is an element s ∈ Im( O )with unit norm | s | = 1 which is not necessarily one of the basis units { i, j, k . . . } . Thereal unit 1 together with any two orthogonal imaginary units s, s ′ define a quaternionsubalgebra with basis { , s, s ′ , ss ′ } ∈ H , which includes any of the seven H subalgebrasas described by the seven lines in figure 6.1. More generally any two non-parallelimaginary units in O generate a basis for a quaternion algebra.121or any octonion a with Im( a ) = 0 the unit imaginary s as the point on the 6-sphere of unit imaginary octonions in the direction of Im( a ) can be identified. Any such s ∈ S , together with the real unit 1, generate a complex subalgebra of O with basis { , s } ∈ C . In particular any a ∈ O of equation 6.6 may be written as a = | a | e sα , with α ∈ R and the Euler identity e sα = cos α + s sin α applying in the complex subalgebra.Since any two octonions involve at most two complex subspaces, with bases { , s } and { , s ′ } , it follows from the previous paragraph that any calculation involving only twooctonions reduces to the case of the quaternion algebra, which being associate henceaccounts for the alternative property of the octonion algebra.As distinct from the ‘octonion conjugation’ a → ¯ a , for each q ∈ O with q = 0a conjugation map on a ∈ O is a linear transformation expressed by the followingalgebraic composition (which is well defined since O is an alternative algebra): φ q : a → qaq − that is φ q : a → qa ¯ q for | q | = 1 (6.11)where the second expression follows using equation 6.9 and in fact describes the com-plete set of possible transformations since the first expression is insensitive to | q | .Selecting | q | = 1 implies not only q − = ¯ q but also q = e s α where s ∈ O is a unitimaginary and α ∈ R . Since | φ q ( a ) | = | q || a || ¯ q | = | a | (6.12)and h a, b i = 12 ( | a + b | − | a | − | b | ) (6.13)(the latter by equations 6.8 and 6.10) the map in equation 6.11 represents an isometry for the elements of O , since geometric relations are preserved. This isometry, leavingRe( a ) invariant and being continuously connected to the identity transformation, rep-resents an action of SO(7) upon the seven-dimensional space of imaginary octonions.For the quaternion subalgebra, with a, q ∈ H in the basis { , i, j, k } , the map φ q : a → qa ¯ q , in H → H , with q = e i α rotates a vector ( a , a , a , a ) ∈ R by the angle α radians in the ( j - k ) plane (with α = 2 πn , n ∈ Z , being the identity transformation).That is, applying equation 6.11 and anticommutation for the imaginary units: φ e i α : a + a i + a j + a k → a + a i + e iα ( a j + a k )and hence a ja k → ( a cos α − a sin α ) j ( a sin α + a cos α ) k (6.14)when considered as an active transformation relative to a set of constant basis elements { , i, j, k } , which is the point of view adopted for such transformations here (in contrastto passive transformations such as exemplified in equations 3.39–3.41 and describedin sections 3.1 and 3.2 for gauge transformations on a principle bundle space). Onemploying left and right multiplication by independent quaternions of unit norm onthe full space H , with 4 real dimensions, the two-to-one cover of SO(4) is obtained, asalluded to in section 2.1.Transformations in the space Im( H ), in the form of equation 6.11, may beconstructed about any unit imaginary element (taken as i in the example of equa-tion 6.14) as the axis of rotation. In this case for quaternions the map φ q is a universal122wo-to-one covering map of S into the group of rotations SO(3) (elements q ∈ H with | q | = 1 describe the 3-sphere, with q and − q mapping to the same rotation). This isa group homomorphism from the algebraic composition of equation 6.11 for quater-nions with group structure φ r ◦ φ s = φ rs into the geometric transformations SO(3) ina 3-dimensional space, which is also the automorphism group of the algebra H . Thisgroup may also be generated by composing several actions of the form in equation 6.11requiring | q | = 1 and q ∈ Im( H ), that is from the 2-sphere S ⊂ S alone. Simi-lar compositions of actions will actually be required for the octonion case in order toconstruct the full symmetry, as described in the following.In the case of the octonions the map in equation 6.11 for φ e i α again fixes the(1- i )-plane but now rotates all three mutually orthogonal planes, corresponding to thethree quaternion subalgebras containing i identified in figure 6.1, simultaneously by α radians. Here the full set of maps φ q , with q ∈ O and | q | = 1, does not form a grouphomomorphism of the 7-sphere S into SO(7) since in general there may be no valueof q ∈ O for which the map φ q ( a ) is equivalent to the composition φ r ( φ s ( a )), with r, s ∈ O , due to the non-associativity of the octonions. However it is precisely throughthis property of octonion composition that the set of maps φ q of equation 6.11 cangenerate the full Lie group SO(7) by including ordered, or nested , combinations suchas φ r ( φ s ( a )) on a ∈ O as elementary symmetry operations.In general a representation R on a vector space V is a structure preserving ho-momorphism from group elements { g , g , g , e } ∈ G with g g = g into representationmatrices with R ( g ) R ( g ) = R ( g ) and R ( e ) = , where the unit matrix representsthe identity transformation on V . Since group structure is associative the above ac-tions φ r , φ s , owing to the octonion non-associativity, do not technically represent theLie group SO(7).The group structure of these transformations can however be seen when theseactions are instead represented by matrices R ( φ ) ∈ GL(7 , R ) acting on the vector space R . Consider the example of φ r ( φ s ( a )) = r ( s ( a )¯ s )¯ r with r = i ∈ O and s = l ∈ O .The map φ l : a → la ¯ l is a linear transformation of the components of Im( a ), thatis { a , . . . , a } of equation 6.6, which can be represented on R by the action of thediagonal 7 × R ( φ l ) = diag( − , − , − , − , − , − , +1) ∈ GL(7 , R ). Similarlythe map φ i : a → ia ¯ i is represented by R ( φ i ) = diag(+1 , − , − , − , − , − , − R ( φ i ) R ( φ l ) = diag( − , +1 , +1 , +1 , +1 , +1 , − a → i ( l ( a )¯ l )¯ i . Hence while i ( l ( a )¯ l )¯ i = ( i l ) a (¯ l ¯ i ), due to the octonion non-associativity, the nested action can berepresentated in GL(7 , R ) with R ( φ i ) R ( φ l ) = R ( φ i ◦ φ l ), and with matrix compositionsin general, which fully represents the SO(7) Lie group structure. (A similar situation isfound for the spinor representation of SO(7) obtained from the one-sided compositionaction a → r ( s ( a )), and also for the dual spinor using right actions alone).The octonion algebra provides a way to express these symmetry transforma-tions in a compact algebraic form, which uses the non-associativity in order to describethe full symmetry, and which may be unfolded into a more explicit group representa-tion in terms of matrices in GL( n, R ). In fact by using these octonion properties thefull SO(7) rotation group can be generated with the elements | r | = 1 and r ∈ Im( O ),that is on the 6 sphere S ⊂ S alone, as described below.123irstly, setting α = ± π for a single action the conjugation map φ e r ± π , with r an imaginary octonion unit, corresponds to rotating the three planes orthogonal tothe (1- r )-plane by ± ◦ , hence reflecting, or ‘flipping’ these three planes. This canbe readily seen since e r ± π = cos π + r sin ± π = ± r and hence equation 6.11 is simplya conjugation map by a unit imaginary element r ∈ S . For example with r = i themap φ e i π ( j ) = ij ¯ i = − j acts on j , as well as each of the other five imaginary units { k, kl, jl, il, l } , as a sign flip.Performing a second reflection based on the same (1- r )-plane naturally cancelsthe first and leaves no total effect. However in performing the second flip with respectto a different plane, namely the (1-( r cos β + s sin β ))-plane with s a unit imaginaryorthogonal to r , while the combined reflections still cancel for most components aresidual rotation by β radians in the ( r - s )-plane remains as the net effect on R . Thatis the two reflections applied to any a ∈ O as: φ r,s, β ( a ) = ( r cos β + s sin β )( − r a ( − ¯ r ) )( r cos β + s sin β ) (6.15)rotates the components of a in the ( r - s )-plane by β radians, corresponding to tworeflections in two mirror lines in this plane, while giving the identity map on theremaining components. Although two discrete flips are involved in this equation thetotal effect on vectors in O is of a rotation in the ( r - s )-plane varying continuously with the parameter β ∈ R , with the identity transformation for β = 0. In the seven-dimensional space of Im( O ) the 21 possible choices of rotation planes from the 21 setsof imaginary base unit pairs for { r, s } describes the full Lie group SO(7). The 14-parameter automorphism group of the octonions, that is the exceptional Lie group G ,is contained as a subgroup of this SO(7) as will be discussed in section 6.4.The first rotation in equation 6.15 is taken as − π , that is with e r − π = − r ,followed by the second rotation by + π . The corresponding minus signs for the − π rotation in the middle brackets on the right-hand side trivially cancel here but theminus sign is needed for the one-sided spinor actions a → ( r cos β + s sin β )( − r ( a )) inorder for β = 0 to correspond to the identity transformation in the spinor representa-tion. This latter expression is also compatible with the identify transformation for thespinor case corresponding to β = 4 πn with n ∈ Z , rather than any multiple of 2 π asfor the vector representation of equation 6.15. As well as rotations in spaces with a Euclidean metric, such as the case of SO(7) above,composition of division algebra elements can also be used to describe transformations inspaces with a Lorentzian metric, such as on the tangent space of a spacetime manifold.The content of this section is largely based on reference [37]. We begin here with anHermitian 2 × X which may be written as: X = t + z ¯ aa t − z ∈ h O (6.16)124ith a ∈ O in the general form of equation 6.6 and { t, z } ∈ R , and hence X is10-dimensional over the real numbers. Since the components of X , involving only asingle octonion a , can be taken to lie within a single complex subalgebra of O thereare no problems with commutativity or associativity in unambiguously defining thedeterminant of the matrix X in the usual way as:det( X ) = ( t + z )( t − z ) − a ¯ a = t − a − a . . . − a − z (6.17)This expression has the same form as the square of an invariant interval represented bya Lorentz 10-vector x (or interval of ‘proper time’ τ ), with 10-dimensional spacetimemetric η = diag(+1 , − , . . . , − | x | = x T η x = x − x − x . . . − x − x (6.18)Closely analogous structures are obtained for all four normed division algebras, K = R , C , H or O , with h K representing Lorentz vectors in ( k + 2)-dimensional spacetimewhere k = dim R ( K ). For example in the familiar case of 4-dimensional spacetime aLorentz 4-vector ( t, x, y, z ) can be represented by: X = t + z x − yix + yi t − z = tσ + xσ + yσ + zσ ∈ h C (6.19)This case will be considered in more detail in the section 7.1 where the σ -matricesare presented in equation 7.14. The above expression can be generalised by replacing σ = (cid:0) − ii (cid:1) in equation 7.14 with σ q = (cid:0) − qq (cid:1) for q = i, j and k for the quaternion caseor q = i, j, k, kl, jl, il and l for the octonion case of equation 6.16.A Weyl spinor can be expressed as the 2-component object θ = (cid:0) ab (cid:1) ∈ K ⊕ K ,with the Hermitian conjugate θ † = (¯ a ¯ b ), and hence each spinor has 16 real componentsfor the octonion case. As an element of h K the square of a spinor: θθ † = | a | a ¯ bb ¯ a | b | (6.20)has det( θθ † ) = 0 (6.21)and hence corresponds to a null-vector in ( k + 2)-dimensional spacetime. The ‘time’component of this null-vector can be expressed in a scalar spinor product t = θ † θ = ( | a | + | b | ), while the time component of a general element of X ∈ h K is given by t = tr( X ), as can be seen in the examples of equations 6.16 and 6.19.Lorentz transformations in ( k + 2)-dimensional spacetime are defined as actionsΛ which preserve proper time intervals, that is with | Λ( x ) | = | x | . The subset of actionscontinuously connected to the identity transformation may be composed together toform the Lorentz group SO + (1 , k + 1). Since | x | ≡ det( X ) the rotations and boostsof these geometric spacetime symmetries can be associated with algebraic composi-tions in the relevant division algebra which preserve the determinant, and Hermitianproperty, of h K . To represent a 10-dimensional Lorentz transformation the Hermitian125equirement can be achieved by a conjugation map on X ∈ h O with the 2 × M which is well defined if there is no associativity ambiguity: R : X → M XM † := ( M X ) M † = M ( XM † ) (6.22)This in turn is achieved if the components of M all belong to a single complex subspaceof O (an alternative possibility is for the columns of Im( M ) to be real multiples of eachother [37] p.21). In this case det( M ) is well defined and the further requirement thatdet( M M † ) = 1 is sufficient to ensure that the conjugation map X → M XM † leavesdet( X ) invariant. These two-sided transformations on the vector X are required tobe compatible with the one-sided actions on the spinor θ and its Hermitian conju-gate θ † , meaning that there should also be no associativity problems in relating theserepresentations as: M ( θθ † ) M † = ( M θ )( θ † M † ) = ( M θ )( M θ ) † (6.23)where on the left-hand side an octonionic vector is composed as θθ † , which is not ageneral element of h O due to equation 6.21. This compatibility, which will be neededin the following section for the 3 × M all belong to the same complex subalgebra of O and also det( M ) ∈ R . Together with the requirement that the vector transformationpreserves det( X ) this implies that det( M ) = ±
1. A complete set of such transformationmatrices is listed in table 6.1.Category 1: Boosts B tz ( α ) , B tx ( α ) and B tq ( α ) with: M tz ( α ) = e + α e − α , M tx ( α ) = cosh α sinh α sinh α cosh α , M tq ( α ) = cosh α q sinh α − q sinh α cosh α Category 2: Rotations R xq ( α ) , R xz ( α ) and R zq ( α ) with: M xq ( α ) = e − q α e + q α , M xz ( α ) = cos α sin α − sin α cos α , M zq ( α ) = cos α − q sin α − q sin α cos α Category 3: Transverse Rotations R r,s ( α ) with: M r,s ( α ) = r cos α + s sin α r cos α + s sin α nested with M r,s = − r − r Table 6.1: Three categories of matrices [37, 38] for conjugation action on X ∈ h O preserving det( X ); with q, r, s ∈ { i, j, k, kl, jl, il, l } there are (1 + 1 + 7) = 9 boosts,(7 + 1 + 7) = 15 rotations and 21 R r,s ( α ) transverse rotations (where the subscript r, s denotes the ordered pair { r, s } of imaginary units) representing the 45-dimensionalgroup of Lorentz transformations on a 10-dimensional spacetime.126n the first two categories det( M ) = +1 for each of the 24 actions. The thirdcategory is a simple 2 × β replaced by α . The action of the category 3 matrices is ordered by nesting theconjugation as: R r,s ( α ) X = M r,s ( M r,s ( X ) M † r,s ) M † r,s (6.24)with det( M r,s ) = det( M r,s ) = −
1. Since the latter matrices are always combined inpairs, and hence are analogous to the action of a single matrix with a determinant of +1,the full group of transformations is denoted SL(2 , O ). It is composed of the 45 actions intable 6.1, each of which describes a one-parameter subgroup with R ( α ) R ( β ) = R ( α + β )and each of which represents transformations in a single 2-dimensional plane in 10-dimensional spacetime.For each of the 45 transformations with α = 0 and α = 2 π it can be seen that M = (cid:0) (cid:1) and M = (cid:0) − − (cid:1) respectively (this is effectively true for the category 3 casesince these can be expressed by conjugation with a single such matrix M for these α values). Hence SL(2 , O ) is the double cover of the 10-dimensional Lorentz groupSO + (1 , , O ) → SO + (1 ,
9) is a two-to-one homomorphism with kernel { M = ± } , where is the 2 × M give theidentity transformation on X due to the two-sided action in equation 6.22.A number of subgroups may also be identified. The subgroup leaving tr( X )invariant, composed of the 36 category 2 and 3 transformations, defines SU(2 , O ) whichis the two-to-one cover of the purely rotational Lorentz subgroup SO(9), leaving the t -component in equation 6.16 invariant. In turn the 21 category 3 transformationsalone form the Spin(7) subgroup as the double cover of SO(7). The structure of thesesubgroups, including the SO(8) obtained by augmenting the SO(7) with an additional7 R xq ( α ) actions from category 2, will also be important for enlarging beyond SL(2 , O )for the 3 × q, r, s ∈ { i, j, k } , there remain 15 transformations(5, 7 and 3 for category 1, 2 and 3 respectively) acting on h H forming SL(2 , H ) asthe double cover of the Lorentz group SO + (1 ,
5) on 6-dimensional spacetime. Here,loosening the restriction det( M ) = ± R i ( α )( X ) = M i XM † i with M i ( α ) = e i α e i α (6.25)which acts on the quaternion component a ∈ H of X = (cid:0) p ¯ aa m (cid:1) ∈ h H by fixing the (1- i )-plane while performing a rotation in the ( j - k )-plane of α radians as was described inequation 6.14. Taking a similar form to equation 6.25 the two actions R j ( α ) and R k ( α )rotate the ( k - i )-plane and ( i - j )-plane respectively. This is possible for the quaternionssince there is only one imaginary plane orthogonal to each imaginary base unit. Thisis unlike the case for the octonions in which the nested transverse rotations are neededto describe all 21 such single plane rotations (as explained towards the end of theprevious section) and hence account for the complete subgroup SO(7) ⊂ SO + (1 , M ) = ± × C by the six category 1 and 2 matrices M in table 6.1 with q takinga single value such as i . This set of actions with det( M ) = +1 forms the groupSL(2 , C ) as the double cover of the Lorentz group SO + (1 , no imaginary orthogonal planes forthe transformation of equation 6.25 to rotate and the action R i ( α ), which may beconsidered as a residue from the 2 × H and O , not only preservesdet( X ) but also leaves each component of any X ∈ h C unchanged. In this sense theaction in equation 6.25 may be interpreted as an internal U(1) symmetry, relative tothe external
Lorentz symmetry of 4-dimensional spacetime, as will be relevant for thecase of the embedding h C ⊂ h O in section 8.2. Transformations on a Form of Time
In this paper the emphasis is on symmetries of forms of multi-dimensional temporal flow L ( v ) = 1, that is isochronal symmetries as introduced in section 2.1, rather than on isometries of a higher-dimensional space or spacetime, as described for equations 6.11–6.13 and in the previous section for example. The SL(2 , O ) action preserving det( X )with X ∈ h O , described in the previous section, can be interpreted in either way,but there is no reason to restrict multi-dimensional forms of L ( v ) to have such aspacetime interpretation, as it does in taking the quadratic form of det( X ) for example.Further, with O being the largest division algebra, there is no clear extension of thisconstruction based on h K to a higher-dimensional spacetime symmetry. This leads tothe consideration of the extension of h O to the 27-dimensional space of 3 × O which has richer symmetry properties while still possessing anunderlying structure appropriate for a form of temporal flow.An element X ∈ h O of equation 6.1 may be written as (again closely following[38] chapters 3 and 4 together with [39, 40, 41] and generally adopting the notationtherein): X = p ¯ aa m c ¯ b ¯ c b n = X θθ † n ∈ h O (6.26)with p, m, n ∈ R and a, b, c ∈ O , while X and θ have the structure of octonionic2 × × X , Y ∈ h O form the ex-ceptional Jordan algebra. However it is the structure of a cubic norm, or determinant,which may be defined on h O , without any ambiguity due to the non-associativity ofthe octonions, that is of interest here. The cubic norm is a homogeneous polynomialform in the components of h O as a mapping X → det( X ) ∈ R into the real numbers,128nd hence has the correct structure for a form of L ( v ) = 1. This determinant may beexpressed in several equivalent ways including:det( X ) = det( X ) n + 2 X · ( θθ † ) (6.27)= pmn − p | b | − m | c | − n | a | + 2Re(¯ a ¯ b ¯ c ) (6.28)where the 10-dimensional Lorentz inner product X · Y = (tr( X ◦ Y ) − tr( X )tr( Y )),with X, Y ∈ h O , in the first line together with equation 6.26 can be used to derivethe second line in which the cubic composition of components, consistent with thehomogeneous form of equation 2.9, is explicitly seen.The 2 × M of SL(2 , O ) actions listed in table 6.1 can be embeddedin the upper-left corner of 3 × M to obtain the conjugation action for the3 × R : X → MX M † with: MX M † = M
00 1
X θθ † n M
00 1 † = M XM † M θθ † M † n (6.29)This expression contains the vector X → R ( X ) = M XM † , spinor θ → R ( θ ) = M θ and scalar n → n representations of SL(2 , O ), each transforming in the appropriateway with the form of the action R determined correspondingly. These transformationsrespect the 3 × × X → R ( X ) = M n ( . . . ( M ( X ) M † ) . . . ) M † n (6.30)which acts, for example, on the spinor as θ → R ( θ ) = M n ( . . . ( M ( θ ))). As well aspreserving det( X ) with X ∈ h O the 45 actions of 2 × M from table 6.1when embedded in the 3 × M for equation 6.29 also preserve det( X ) for X ∈ h O since, from equation 6.27:det( R ( X )) = det( R ( X )) n + 2 R ( X ) · ( R ( θ ) R ( θ † ))= det( R ( X )) n + 2 R ( X ) · R ( θθ † )= det( X ) n + 2 X · θθ † = det( X ) (6.31)where the second equality is a result of ‘compatibility’, and motivates the introductionof this requirement in equation 6.23, and the third equality follows from the Lorentzsymmetry of the SL(2 , O ) action. It is also by compatibility that the 45 SL(2 , O )transformations act as one-parameter subgroups on the spinor θ (given the minussigns for the M components for the transverse rotations, originating in equation 6.15,as for M r,s in table 6.1) as well as on the vector X .These SL(2 , O ) actions, called Lorentz transformations when acting on X ∈ h O representing 10-dimensional spacetime, also identify 45 one-parameter subgroupsacting on X ∈ h O , with R ( α ) R ( β ) X = R ( α + β ) X , preserving det( X ) (where R ( α ) X R ( X ), for example from table 6.1, for a particular trans-formation parameter α ). Hence these 45 actions are one-parameter subgroups ofE := SL(3 , O ) which is defined as the group of symmetry transformations underwhich the determinant on h O is invariant. Again we emphasise that the key here isthe structure of a higher-dimensional form of temporal flow which, while necessarilycontaining a 4-dimensional form perceived as spacetime, does not itself need to possessa higher-dimensional spacetime interpretation.The exceptional Lie group E is 78-dimensional, as described in section 6.1 andhence the 45 actions adopted from SL(2 , O ) represented on h O is only part of thefull symmetry picture. However the scope of the SL(2 , O ) action can be enlarged bynoting that there are three similar and natural ways to embed the vector X , spinor θ and scalar n representations of SL(2 , O ) in the 3 × X ∈ h O . The original‘type 1’ action described in equation 6.29 for the embedding depicted in equation 6.26can be written more explicitly in terms of the matrix components: M (1) = M M M M
00 0 1 acting on X X θ X X θ ¯ θ ¯ θ n (6.32)Maintaining the variables p, m, n ∈ R and a, b, c ∈ O in the same component locationsof the 3 × X ∈ h O in equation 6.26 their placement within the 2 × X = (cid:0) X X X X (cid:1) and 1 × θ = (cid:0) θ θ (cid:1) under SL(2 , O ) may be reassigned by permutingthe components of the matrices M as follows: M ( a ) = T M ( b ) T † for ( a, b ) = (2 , , (3 , , (1 ,
3) with T = (6.33)With T † = T − (= T ) it can be seen that det( T X T † ) = det( X ) and since det( T ) = 1the action X → T X T † can itself be considered as a transformation of the SL(3 , O )symmetry. The matrices M (2) and M (3) then correspond to ‘type 2’ and ‘type 3’transformations respectively with: M (2) = M M M M acting on p ¯ θ ¯ θ θ X X θ X X (6.34)and M (3) = M M M M acting on X θ X ¯ θ m ¯ θ X θ X (6.35)130he three M ( a ) represent three embeddings of the 2 × × O components of equation 6.1.Each type 1 , M → M ( a ) for a = 1 , O under theoriginal type 1 action of equation 6.29, and hence for all three types of transformationdet( X ) is invariant, as was shown for the type 1 case in equation 6.31. (In additionto the discrete actions of equation 6.33 continuous type transformations may also bedefined as described in [38] section 4.4).With three possible embeddings of the 45-dimensional SL(2 , O ) transformationsthere are now a total of 3 ×
45 = 135 det( X )-preserving one-parameter subgroupactions for E := SL(3 , O ), which cannot be independent since E is known to be a78-dimensional group. A basis for the E actions on h O may be obtained by requiringlinear independence at the Lie algebra level. However a G = E manifold is not well-defined in terms of the space of 3 × M upon which to identify tangentvectors with Lie algebra elements. This is in contrast to a case such as G = SO(3)represented by real 3 × R ∈ SO(3) acting on vectors v ∈ R . In this casethe space of matrices R , with RR T = and det( R ) = 1 describing topologically the3-sphere S with antipodal points identified, defines the group space G upon whichtangent vector fields may represent the Lie algebra, as was depicted for the generalcase in figure 2.5. The Lie algebra may be described in terms of left-invariant vectorfields on the group manifold or in terms of the tangent vectors at the identity e ∈ G through the isomorphism L ( G ) ≡ T e G . (An example for the latter case was listed inthe set of Lie algebra elements { L pq } of equation 2.32 for G = SO(3)).This situation can be understood by considering how a Lie group manifold, onthe tangent space of which the Lie algebra may be defined, might also be identifiedfor the Lorentz groups in ( k + 2)-dimensional spacetime, with k = dim R ( K ) and K = C , H , O , represented by the action of SL(2 , K ) on h K matrices. In the first case for K = C since complex 2 × ×
2) = 8 real parameters and det( M ) =1 ∈ C represents two constraints on the matrices M ∈ SL(2 , C ) these actions aredescribed by (8 −
2) = 6 real parameters, which equals the dimension of the Lorentzgroup SO + (1 , × q = i as described at the end of section 6.3. Hence these six degreesof freedom of the matrices SL(2 , C ) fully describe the corresponding group manifold G ≡ SL(2 , C ) (as the double cover of SO + (1 , + (1 , e ∈ G , will be explicitlylisted as the set of 2 × { ˙ M } in equations 8.7 and 8.8 of section 8.1.For 2 × M ) = 1 ∈ H there are(4 × − M ∈ SL(2 , H ), insufficient alone to describe the15-dimensional Lorentz group SO + (1 , × H ) byloosening the constraint on the matrix determinant to | det( M ) | = 1 ∈ R results in atotal of (4 × − × + (1 , ×
8) = 32 parameters available in a 2 × , O ) in table 6.1cannot immediately be related to a Lie group manifold as they could for the complex(SL(2 , C ) with det( M ) = 1) and quaternion (SL(2 , H ) with | det( M ) | = 1) cases. Indeedthis is why nested SL(2 , O ) actions are required in the octonion case to make up theextra transformations. Similarly for SL(3 , O ), with a maximum (9 ×
8) = 72 realparameters available in the 3 × .The nested action X → M ( M X M † ) M † , in the form of equation 6.30, for thecase in which the elements of the matrices M and M belong to the same C ⊂ O sub-space is an associative composition, that is it is equal to ( M M ) X ( M † M † ). This isbecause each matrix element of X involves at most only one further complex subspace,and hence each multiplicative action on these elements in the linear transformation on X takes place in an associative quaternion subalgebra. Hence these particular cases ofnested transformations do behave like a group representation. More generally however,and as for the case of SO(7) generated by composition of the φ q maps with q ∈ Im( O )in equation 6.11 as described in section 6.2, here there does not exist a group homo-morphism of the full set of E transformations into the set of 3 × M ( a ) . However, associative group matrices could be constructed here by representingthe linear transformations of the E symmetry by 27 ×
27 matrices in GL(27 , R ) actingon the space R ≡ h O , as was the case for SO(7) represented by matrices in GL(7 , R )acting on R ≡ Im( O ) in section 6.2. Indeed with such large matrices there is plenty offreedom in which to express the full symmetry with elements R ( g ) ∈ GL(27 , R ) whichnaturally form an associate algebra and with R ( g ) R ( g ) = R ( g g ) composing as atrue representation of E .Given the 135 one-parameter subgroup actions on h O , collectively impliedin equations 6.32, 6.34 and 6.35, it would be straightforward, although laborious, toconstruct 135 matrices in GL(27 , R ) acting upon R , with the latter containing the 27parameters of an element of h O in equation 6.1 drawn out into the real column vector( p, m, n, a , . . . , c ) T . All such actions would preserve the cubic norm of equation 6.28considered as a map R → R . The multiplication of such elements of E representedas matrices in GL(27 , R ) is clearly associative, as only R -valued matrices are involved.Together with the identity element given by the unit matrix and an inverse obtainedfor any matrix by reversing the transformation with real parameter α → − α , the Liegroup structure is evident. Combinations of the 135 one-parameter subgroup actionswould carve out a G ≡ E submanifold (for this non-compact real form of E ) embeddedwithin the (27 × , R ).In principle left-invariant tangent vector fields, generated by right translationson G and associated with the one-parameter subgroups, could be constructed upon thisE group manifold, as depicted generically in figure 2.5 on a group manifold, and lineardependency used to reduce these to a basis set of 78 vector fields to describe the E Liealgebra. Hence in this representation the Lie algebra may also be identified in terms132f the transformation matrices themselves, in the form of elements D ≡ ∂A λ /∂λ | λ =0 as described before equation 6.3, here with A ∈ GL(27 , R ). Alternatively the left translations of these symmetry transformations on R may be associated with vectorfields in the tangent space T R which also represent the Lie algebra generators of thesymmetry. This construction applies generally (see also the discussion in the openingof section 4.3) – for example in the case of SO(3) acting on R the Lie algebra L (SO(3))may be represented by vector fields in the space T R tangent to the 2-sphere S .This latter possibility of employing the left or right action of G on the represen-tation space itself to construct the Lie algebra can be employed for the representationof E of relevance here, that is on the space h O . Indeed the theoretical motivationfor studying E here is precisely owing to its representation on h O , together withthe subgroup representations on subspaces of h O obtained under symmetry breaking,rather than the pure E group structure in itself. It is the fact that the space h O withunit determinant has the appropriate structure for a form of temporal flow L ( v ) = 1that provides the primary motivation, with E identified in turn as the correspondingsymmetry group. This symmetry is expressed in a very compact cubic form as thedeterminant preserving actions on h O , such as described in equation 6.29, and indeedthe origin of the very high degree of symmetry, involving the triality relation for thelargest division algebra O , is evident explicitly in this form. These structures wouldbe far from manifest in a 27 ×
27 real matrix representation. The non-associativity ofthe octonion algebra is employed in folding the full set of E actions into this highlycompact 3 × , C ) subgroupalready described above) and internal symmetry groups, as we shall study in chapter 8in comparison with the Standard Model of particle physics as reviewed in chapter 7.Since the representations of these subgroups are to be identified in the components ofh O , which is ultimately motivated as the space underlying L ( v ) = 1, the tangentspace T h O provides an apt arena for describing the E Lie algebra.The homomorphism of the Lie algebra L (E ) ≡ sl(3 , O ) into the space of vectorfields in T h O , the tangent space to the 27-dimensional manifold of h O , is in fact anisomorphism since the group action of SL(3 , O ) on h O is effective. This isomorphismis used both to identify individual E generators and also, as described in the followingsection, the Lie algebra structure itself in terms of the commutators of the algebraelements. Clearly the broken subgroups also act effectively on the components ofh O and hence, following the discussion toward the end of section 4.3, the gaugefield dynamics for the full internal symmetry group will be obtained. More generallythe breaking of the isochronal symmetry of L ( v ) = 1 over the base space M willultimately need to be incorporated into the unification scheme described in section 5.1with a structure in principle resembling Kaluza-Klein theory based on homogeneousfibres as reviewed in section 4.3.With the space h O considered as a manifold the map R ( α ) X , for any point X on h O , is a left action on the manifold with R (0) X = X . This action also describesa curve as a mapping from α ∈ R into h O which sends the real number α = 0 to thepoint X . Acting on all values of X ∈ h O this one-parameter group R ( α ) is associated133ith the tangent vector field:˙ R = ∂ ( R ( α ) X ) ∂α (cid:12)(cid:12)(cid:12) α =0 ∈ T h O (6.36)where a ‘dot’ over the kernel symbol such as for ‘ ˙ R ’ will generally denote a tangentvector field on the space h O . The local tangent space on h O under the one con-straint det( X ) = 1 is 26-dimensional, however the space of vector fields over the26-dimensional manifold S is infinite. The task is then to identify the E Lie algebrathrough a one-to-one isomorphic correspondence with a subset of 78 linearly indepen-dent vector fields in T h O , of the form of equation 6.36, within this ∞ -dimensionalspace. The search is narrowed down by adopting a starting point based on the 135one-parameter subgroup actions on h O obtained through the three types of SL(2 , O )conjugation described in equations 6.29 and 6.32–6.35.The first stage, at the level of these one-parameter subgroups, is to find aconvenient new basis for the 21 category 3 transverse rotations R r,s ( α ) described intable 6.1. For each imaginary base unit q in figure 6.1 the three pairs of imaginary unitseach describing a quaternion subalgebra with q also form a right-handed 3-dimensional‘coordinate frame’ with q . That is, for example with q = i , we have jk = + i , kljl = + i and lil = + i , matching the pairs listed in the top row of table 6.2. q ∈ Im( O ) 1 st pair 2 nd pair 3 rd pair i j, k kl, jl l, ilj k, i il, kl l, jlk i, j jl, il l, klkl jl, i j, il k, ljl i, kl il, k j, lil kl, j k, jl i, ll il, i jl, j kl, k Table 6.2: (Adopted directly from [38] p.107, table 4.2). The 3 right-handed quater-nion subalgebras for each imaginary octonion base unit q ordered as 1 st , 2 nd and 3 rd (associated with the rotations R q , R q and R q respectively) as appropriate for thenew basis for transverse rotations listed in equations 6.37–6.39.For each choice of q the associated 1 st , 2 nd and 3 rd planes, from the same row ofthe table, are mutually orthogonal and rotated independently by R q ( α ), R q ( α ) and R q ( α ) respectively, where for example R i ( α ) = R j,k ( α ) by taking the appropriatepair, here { r, s } = { j, k } from table 6.2, to construct the corresponding category3 transverse rotation R r,s ( α ) from table 6.1. Adopting the point of view of active transformations these individual plane rotations are in a clockwise sense about the q -axis for positive α and counterclockwise for negative α . They are then composed134ogether in the following combinations: A q ( α ) = R q ( α ) ◦ R q ( − α ) (6.37) G q ( α ) = R q ( α ) ◦ R q ( α ) ◦ R q ( − α ) (6.38) S q ( α ) = R q ( α ) ◦ R q ( α ) ◦ R q ( α ) (6.39)Since in all cases each of the two or three plane rotations are independent of eachother their order may be interchanged. (The three actions A q , G q and S q may alsobe recombined to recover the original single plane rotations, for example R q ( α ) = A q ( α/ ◦ G q ( α/ ◦ S q ( α/ ×
7) = 21 actions defined in equations 6.37–6.39 hence provide a newbasis for the Spin(7) transverse rotations applied in table 6.1 on the space h O . Sinceeach of these actions is represented by diagonal 2 × O itself, as the double cover of SO(7) acting on Im( O ).However the mathematical motivation for introducing the new basis is seen whenapplied to the 3 × O . Indeed when embedded in the type 1, 2 and3 actions of equations 6.32, 6.34 and 6.35 respectively and determining the tangentvectors of the new transverse rotations in T h O using equation 6.36 it can be shownby direct comparison that: ˙ A q = ˙ A q = ˙ A q (6.40)˙ G q = ˙ G q = ˙ G q (6.41)˙ S q + ˙ S q + ˙ S q = 0 (6.42)for each of the seven cases of q ∈ { i, j, k, kl, jl, il, l } . The superscript a on tangentvectors, as for ˙ A a here, will always denote the type and since raising such a vectorto a power has no meaning the a is not placed inside brackets (in cases of ambiguitybrackets will be used for the type index ‘( a )’ as for M in equations 6.32–6.35). Thenew choice of equations 6.37–6.39 for the category 3 actions on h O is hence justifiedby the manifest clarity of the linear dependencies seen in this basis.Each of the 14 independent generators { ˙ A q ≡ ˙ A aq , ˙ G q ≡ ˙ G aq } (for any a = 1 , , a, b, c ∈ O in equation 6.1 in exactly the same way(while vanishing on the p, m, n ∈ R elements as for all transverse rotations). The 14corresponding group actions of equations 6.37 and 6.38 preserve the multiplication ta-ble for O continuously as a function of the parameter α , forming the proper continuousautomorphism group of the octonions (this group is SO(3) for the quaternion case).Hence, taken together A q and G q compose the exceptional group G , which justifiesthe notation ‘ G q ’ introduced in equation 6.38.The notation ‘ A q ’ in equation 6.37 is introduced owing to the similarity ofthe kernel symbol to ‘ λ ’ which denotes the Gell-Mann matrices, as listed in table 8.5,which generate the Lie group SU(3), a basis for the Lie algebra of which can also becomposed of the 8 generators { ˙ A q , ˙ G l } , as will be described in section 8.2. In fact theautomorphism group of the octonions may be reduced to the subgroup SU(3) ⊂ G byfixing an imaginary unit such as l ∈ O . The identification of this SU(3) subgroup alsoprovides a significant motivation for adopting the basis of equations 6.37–6.39 fromthe potential physical perspective (see also the discussion following equation 6.58).135he notation ‘ S q ’ in equation 6.39 originates from the symmetric action ofthree synchronised rotations of α radians in three different planes. Applied to the2 × X ∈ h O this synchronised action is identical to the original single actionof equation 6.25 (with i generalised to any q ∈ { i, j, k, kl, jl, il, l } ) in rotating threeplanes of the a ∈ O component of X , although due to the transformation of the spinor θ = (cid:0) c ¯ b (cid:1) , as described below, the action of S (1) i for example, that is with q = i inequation 6.39, on h O in the 3 × not equivalent to the action of equation 6.25embedded in equation 6.29 or 6.32.On the other hand equation 6.25 can be augmented to a single unnested 3 × R ( α ) X = M ( a ) S \ q X M ( a ) † S \ q with M ( a ) S \ q , for the type a = 1 case, expressedas: M (1) S \ q ( α ) = e q α e q α
00 0 e − qα (6.43)with a corresponding permutation of the diagonal entries for the type 2 and type 3cases. These actions are denoted by kernel symbol S \ with the ‘ \ ’ as a mnemonic for thediagonal form of equation 6.43. The action M (1) S \ q ( α ) X M (1) † S \ q ( α ) may be consideredas a ‘phase transformation’ in rotating three orthogonal imaginary planes of the a component of X in equation 6.26 by the same angle α . These actions are also relatedto a demonstration of triality in h O involving SO(8) transformations on the threeoctonion subspaces of h O (see the discussion below alongside equations 6.49 and 6.50and in [39] around equation 43).Each of these three 3 × q , contains entries in a single complex subalgebra, satisfies det( M ( a ) S \ q ) = 1 andpreserves the form det( X ) of equation 6.27 or 6.28, consistent with the requirements foran SL(3 , O ) action. However, these actions will not lead to elements of the preferred E algebra basis under construction here since they are not of the form of equations 6.32–6.35 with det( M ) = ± × M xq in table 6.1 and the type 1, 2 and 3embeddings of equations 6.32, 6.34 and 6.35 the diagonal matrix of equation 6.43 canbe expressed by the matrix product: M (1) S \ q ( α ) = M (1) R xq ( − α ) × M (2) R xq ( − α ) (6.44)and hence ˙ S \ q = − ˙ R xq − R xq (6.45)The group action S (1) q ( α ) of equation 6.39 on the 10-dimensional subspaceh O ⊂ h O , consisting of three independent rotations of imaginary planes of the octo-nion a , is precisely the same as the S \ (1) q ( α ) action using equation 6.43. On the spinorcomponents θ = (cid:0) c ¯ b (cid:1) ∈ O ⊂ h O these actions are only equivalent for small trans-formations to order α and diverge at O( α ) and higher powers. However since all 27components of h O transform the same way to O( α ) we have ˙ S q = ˙ S \ q (and similarlyfor the type 2 and 3 cases) as vector fields in the space T h O , and hence these twoobjects are interchangeable in expressions of linear dependence.136ince at the group level S ( a ) q ( α ) and S \ ( a ) q ( α ) differ at O( α ) the Lie bracket, tobe described in the following section, of the corresponding generators with the sameLie algebra element ˙ R will also differ with [ ˙ S, ˙ R ] = [ ˙ S \ , ˙ R ] in general even though˙ S = ˙ S \ as elements of a vector space. The Lie bracket in these two cases agrees for theh O ⊂ h O subspace but differs for the spinor components. The transformations on thecomponents of the spinor θ are expected to be important for the internal symmetries incomparison with the Standard Model and hence care will need to be taken in choosingan appropriate Lie algebra basis. The E Lie algebra table in [38] uses the actions of S (1) q ( α ) from equation 6.39 rather than S \ (1) q ( α ) based in equation 6.43, which is hencesignificant for the Lie algebra structure, and in turn it is the former transformationswhich are also used in this paper.Adding the 7 actions S ( a ) q for a = 1 , { A q , G q } ≡ G completes a Spin(7) double cover of SO(7) for the type 1, 2 or 3 transverse rotationsrespectively. These three SO(7)s are mutually related by equation 6.42. In addition toequations 6.40–6.42 further linear dependencies amongst the generators expressed on T h O are found (the second of which is equivalent to equation 6.45):˙ R xq + ˙ R xq + ˙ R xq = 0 (6.46)˙ R xq = − ˙ R xq − ˙ S q (6.47)˙ S q = + ˙ R xq − ˙ S q (6.48)Appending the set of 7 actions R ( a ) xq to the SO(7) of type a (for a = 1 , same SO(8), that is they are composed of the same subset ofE transformations on h O , due to the triality relation between the h O components.The triality symmetry is described explicitly in [38, 39, 40]. The transformations ofthe SO(8) subgroup of the type 1 SL(2 , O ) action in equation 6.32 can be obtained bya nested composition with 3 × M = q q
00 0 1 (6.49)with q ∈ O and | q | = 1. The action of such type 1 transformations on an element X ∈ h O of equation 6.1 leaves the diagonal elements { p, m, n } invariant while thethree off-diagonal octonion elements transform non-trivially as (see [40] equation 46and discussion): a → qaqb → bqc → qc (6.50)These generate and correspond to the three SO(8) 8-dimensional representations ofvector, dual spinor and spinor exhibited via symmetric, right and left and octonion137ultiplication respectively, with an implicit triality mapping between the above threeoctonion actions identified by simply employing the same q for each of the three actions,as alluded to in the opening paragraphs of section 6.1. Corresponding to the trialityisomorphism the three actions of SO(8) are permuted into each other via the action ofthe matrices T in equation 6.33, such that we effectively have the same copy of SO(8)in common within each of the three types of SL(2 , O ) a actions on h O .This subgroup SO(8) ⊂ E is in fact precisely the subgroup of E transfor-mations on X ∈ h O that leaves invariant the diagonal entries, that is { p, m, n } ofequation 6.1. This unique SO(8) then contains three different SO(7)s, each built inturn on a unique G . Only this subset of 14 G transformations needs to be describedin the form of nested actions while the remaining SO(8) transformations may be com-posed of seven unnested actions from S \ ( a ) q (as for example from equation 6.43, andreplacing the nested S ( a ) q actions to obtain Spin(7) from G ) together with seven R ( a ) xq actions for type a = 1, 2 or 3 [39].Here in this paper the initial importance of triality lies in the fact that itexplains in part the rich symmetry of E on h O as an expression of L ( v ) = 1. Indeedthe triality symmetry is responsible for the large degree of redundancy in the set of(3 ×
45) = 135 generators for three types of SL(2 , O ) a transformation described above.The relations in equations 6.46–6.48 show that given the type 1 actions it is possibleto exclude ˙ S q (and hence, from equation 6.42, also ˙ S q ) as well as ˙ R xq and ˙ R xq from alinearly independent basis for the E Lie algebra.Building on the 28 generators of SO(8) (taking a type 1 basis) with any oneof the three sets of 8 generators { ˙ R axz , ˙ R azq } , for type a = 1 , ×
8) = 52 rotations preserves tr( X ),with X ∈ h O , and they collectively define the group F := SU(3 , O ). The trace onh O is analogous to the time component of the Lorentz vector represented by h K (described after equation 6.21 for equations 6.16 and 6.19), but does not itself have asimple temporal interpretation here for the 3 × X ) ≡ L ( v ) = 1 itself expresses a multi-dimensional form of temporal flow,having the form of equation 2.9 as introduced in section 2.1.Extending further to reproduce the type a = 1 , , O ) a by including the 9 boost generators { ˙ B atz , ˙ B atx , ˙ B atq } for each case, and takinginto account the further linear dependence:˙ B tz + ˙ B tz + ˙ B tz = 0 (6.51)a total of 52 + (3 × − := SL(3 , O ) transformations of h O . The entire groupis then described in terms of the actions of complex matrices M on the space h O ,with the preferred basis for the Lie algebra represented on T h O reproduced below intable 6.3.The generators, as described above, of the subalgebras corresponding to thevarious stages of the subgroup chain:E ⊃ SO + (1 , ⊃ SO(9) ⊃ SO(8) ⊃ SO(7) ⊃ (G ) ⊃ SU(3) (6.52)(here, other than for the 78-dimensional E , the subscripts give the dimension of thealgebra) can be identified within the three type 1 lines of table 6.3. These can be built138ategory 1: Boosts B tz ˙ B tx ˙ B tq B tz ˙ B tx ˙ B tq B tx ˙ B tq R xq ˙ R xz ˙ R zq R xz ˙ R zq R xz ˙ R zq A q ˙ G q ˙ S q , in terms of tangent vectorfields on T h O , reproduced from ([38] p.177, table A.1). The actual tangent vectorfields are determined and listed in tables 6.6 and 6.7 in the present paper at the endof the following section.up from su(3) ≡ { ˙ A q , ˙ G l } to so + (1 , which includes { ˙ A q , ˙ G q } together with all ofthe type 1 generators in table 6.3.The rotation subgroup of E , as the compact real form of F := SU(3 , O ), isgenerated by the 52 category 2 and 3 transformations in table 6.3. The generatorcomposition of a subalgebra chain leading down from E ⊃ F is presented in ([38]p.119, table 4.4). However, although both preserving tr( X ) (for any X ∈ h O ) andbeing the automorphism group of the exceptional Jordan algebra (equation 6.2), thegroup F is not of great significance in the present paper.At the group level in equation 6.52 each ‘SO’ might more strictly be replaced bythe corresponding double cover ‘Spin’ group. As the group SL(3 , O ) necessarily includesthe one-sided spinor actions θ → M θ in equation 6.32 (as well as in equations 6.34 and6.35) the action for M = − (obtained for any of the category 2 rotations in table 6.1with α = 2 π ) on h O does not give the identity transformation. However SL(3 , O )is not a double cover, rather it is a real simply connected form of E := SL(3 , O )itself ([40] section 2, with the same situation applying for F := SU(3 , O ) acting onh O ). On the other hand the action of SL(2 , O ) is a double cover of the rotation groupSO + (1 , ≡ SL(2 , O ) / Z , and similarly for the further rotation subgroups. With anawareness of these issues of group manifold topology groups such as SO + (1 ,
9) andSO + (1 ,
3) can be considered to be embedded within the full group E .139 .5 Lie Algebra of E At the group level the E action on h O is composed of 52 rotations, that is the unitary3 × MM † = , and 26 boosts, that is the Hermitian actions with M = M † , as can be deduced from the embedded 2 × M listed in table 6.1for the category 2 and 3 rotations and category 1 boosts respectively. At the Lie alge-bra level, in a normalised basis for which the Killing metric K is diagonal with entriesin {− , +1 } (or more generally negative or positive entries for a diagonal but unnor-malised Killing form, explicit values for which will be determined in subsection 8.3.1),base vectors X for which K ( X, X ) = − <
0) are called compact generators, cor-responding to ‘rotations’ of the Lie group, while those with K ( X, X ) = +1 (or > −
26, also denoted (52 ,
26) for 52 rotations and26 boosts, the non-compact real form of E constructed in the previous section maybe denoted as E − , and describes the generator composition D R,B introduced insection 6.1 and displayed in equation 6.4. The Killing form employed in equations 4.1and 4.2 of section 4.1 for Kaluza-Klein theory was chosen with components K αβ = − δ αβ corresponding to the choice of a compact gauge group. In the symmetry breakingof E − over the base space M such compact internal symmetry groups will beidentified.An alternative description of E in terms of 14 G actions together with 64non-G transformations, composed from the actions of the 64 tracefree octonion 3 × D G,S denoted the generator com-position as displayed in equation 6.5. In the previous section the G subgroup wasidentified explicitly as the set of 14 { A q , G q } transverse rotations. As described shortlyafter equation 6.50 the remaining 64 actions may be expressed with unnested compo-sitions consisting for example of the 57 group actions corresponding to the category1 and 2 generators of table 6.3 together with seven S \ q actions from equation 6.43 (inplace of S q ). Hence both the (52+26) and (14+64) decompositions can be clearly seenin table 6.3 in terms of the respective subsets of generators.Here all 78 generators are explicitly presented in tables 6.6 and 6.7, for thecategory { , } and 3 transformations respectively, as vector fields ˙ R ∈ T h O which,from equation 6.1, are of the form:˙ R = ˙ p ˙¯ a ˙ c ˙ a ˙ m ˙¯ b ˙¯ c ˙ b ˙ n ∈ T h O (6.53)These 78 matrices are themselves Hermitian and hence also belong to the spaceh O . While there is no constraint on the determinant of any ˙ R ∈ T h O the matricesare tracefree for all of the category 2 rotations and category 3 transverse rotations.The type 1 transformations act on the { p, m, n ; a, b, c } components on h O in the sameway that the type 2 transformations act on the { m, n, p ; b, c, a } components and type 3transformations act on the { n, p, m ; c, a, b } components as can be seen in equations 6.32,6.34 and 6.35, for example by following the explicit invariant components n , p and m respectively in these three equations. This same cyclic permutation, consistent with140he action of T in equation 6.33, is reflected in the tangent vectors in table 6.6 and for˙ S aq in table 6.7.These tables describe in intimate detail the anatomy of the E action as ex-pressed on the tangent space T h O . With p = t + z and m = t − z , embedding equa-tion 6.16 into h O , each type 1 tangent vector can be seen to ‘point’ in the appropriatedirection in the relevant T h O components for the subspace h O plane transformationsresulting from the action of the matrices in table 6.1, with a similar correspondenceidentifiable for the type 2 and 3 cases. For example the non-zero components of thecategory 1 boost and category 2 rotation generators for the type 1 actions on the10-dimensional subspace h O are simply:˙ B tz : ˙ t = + z, ˙ z = + t, ˙ B tx : ˙ t = + x, ˙ x = + t, ˙ B tq : ˙ t = − a q , ˙ a q = − t, ˙ R xq : ˙ x = − a q , ˙ a q = + x, ˙ R xz : ˙ x = − z, ˙ z = + x, ˙ R zq : ˙ z = + a q , ˙ a q = − z (6.54)where, here and in tables 6.6 and 6.7, x ≡ a x ≡ a and similarly a q refers to the realcoefficient in equation 6.6 corresponding to the imaginary unit q , (that is a l ≡ a etc.).The category 3 transverse rotations of equations 6.37, 6.38 and 6.39 each act on severalplanes in h O . The transformations of the spinor components of h O induced by the3 × G was defined in terms of the setof left-invariant vector fields on the group manifold G , as also recalled in the paragraphsleading to equation 6.36 in the previous section. Through any point h ∈ G each suchvector field X generates a one-parameter group of right translations φ t ( h ) = h exp( tA )where A = X e ∈ T e G is the vector of the field X at the identity e ∈ G , as depictedin figure 2.5 . If G acts by right translation on another manifold M this realisation of G induces vector fields V A ∈ TM such that V Ax ( f ) := ddt f ( x exp( tA )) | t =0 , at x ∈ M where f ( x ) is a real function on M , represents a homomorphism of the Lie algebrawith [ V A , V B ] = V [ A,B ] . If the action of G on M is effective there is a one-to-oneisomorphism between the Lie algebra L ( G ) and the set of such vector fields { V A } in TM (as is the case for the action of G on a principle fibre bundle P as described insection 3.1, see equations 3.2 and 3.3).If G acts on the manifold M by left translations then this relationship is ananti-homomorphism. This is the case for right-invariant vector fields on G itself, whichare generated by left translations. The structure constants c αβγ for the Lie bracketof such right-invariant fields { Y Lα } on G are precisely the negative values of the Liealgebra structure constants defined in terms of the corresponding left-invariant fields { X Rα } (which match the right-invariant fields as elements of the tangent vector spaceat the identity e ∈ G , that is each Y Lα ( e ) = X Rα ( e )). This anti-homomorphism was alsonoted for left translations applied to the space of homogeneous fibres for equation 4.19in the opening of section 4.3.In the present case the group manifold for G = E is not constructed itself butrather the group acts transitively on h O , which is hence a homogeneous space, suchthat det( X ) = 1 is preserved for X ∈ h O . The action of E on the underlying spaceh O is also effective and hence the Lie algebra L (E ) may be constructed in terms ofvector fields on the tangent space T h O . The E transformations composed as X →MX M † are left translations as opposed to right translations, as has been described141n the previous section, and as will be seen explicitly for subgroups such as SL(2 , C )in section 8.1. The Lie algebra commutator, which determines the structure constantsof the E Lie algebra, for any two elements ˙ R , ˙ R is defined through the action of therespective one-parameter subgroups R ( α ) and R ( α ) at any point X ∈ h O :[ ˙ R , ˙ R ] = ∂∂ ( α ) [ R ( − α ) ◦ R ( − α ) ◦ R ( α ) ◦ R ( α ) X ] (cid:12)(cid:12)(cid:12) α =0 (6.55)Here the four ± α signs inside the square brackets are chosen so that this Lie algebrastructure deriving from left translations is isomorphic to the standard definition of L ( G ) of equation 2.22 described in subsection 2.2.2. In the general case for a Lie group G equation 6.55 holds with the opposite signs for α in the square brackets for the right translation mapping of one-parameter subgroup curves R → G to the manifold of theLie group space itself. These curves passing through the identity point e ∈ G allowa bracket to be constructed on the vector space T e G isomorphic to the Lie algebra ofthe group, leading for example to the basis of equation 2.32 for the case G = SO(3).In acting upon a representation space with a lower dimension than G , as isthe case here for the group E acting in the space h O , the Lie bracket is constructednecessarily in terms of vector fields on the representation space. The choice of signsin equation 6.55 means that the various subalgebras will be defined in the usual way,equivalent to left-invariant fields on the broken subgroup manifolds. Indeed in principlethe same E Lie algebra could be constructed in terms of left-invariant fields on thesubmanifold of GL(27 , R ) identified as an E group representation acting on R asdescribed in the previous section.Here the term in square brackets on the right-hand side of equation 6.55 rep-resents a curve that passes through any chosen point X ∈ h O for α = 0. Whilethe first derivative ∂∂α of this same term vanishes identically at α = 0 the secondderivative (cid:0) ∂ ∂α , or equivalently ∂∂ ( α ) (cid:1) is non-zero and yields a tangent vector fieldas X varies over h O corresponding to the Lie bracket of the two vector fields ˙ R and˙ R . For example, by direct calculation taking the type 1, 2 or 3 embeddings of theappropriate matrix actions from table 6.1, applying equation 6.55 and by comparisonwith tables 6.6 and 6.7 the twelve brackets listed in table 6.4 are determined explicitly.1) [ ˙ R xi , ˙ R xz ] = ˙ R zi
5) [ ˙ R xi , ˙ B tx ] = − ˙ B ti
9) [ ˙ R xi , ˙ R xl ] = − ˙ G il + ˙ S il
2) [ ˙ R xi , ˙ R zi ] = − ˙ R xz
6) [ ˙ R xi , ˙ R xz ] = − ˙ R zi
10) [ ˙ R xi , ˙ R xj ] = ˙ A k + ˙ G k + ˙ S k
3) [ ˙ B tz , ˙ B tx ] = ˙ R xz
7) [ ˙ R zi , ˙ B ti ] = − ˙ B tx
11) [ ˙ S i , ˙ R xj ] = ˙ R xk
4) [ ˙ B tz , ˙ B ti ] = − ˙ R zi
8) [ ˙ R xz , ˙ R zl ] = ˙ R xl
12) [ ˙ R xz , ˙ R zi ] = ˙ R xi = − ˙ R xl − ˙ S l = − ˙ R xi + ˙ S i Table 6.4: The Lie algebra bracket composition determined for twelve cases by applyingequation 6.55 to the left-hand sides and tracking terms to O ( α ) for the sequence of R ( ± α ) X compositions and matching the right-hand sides with elements in tables 6.6and 6.7.All cases in table 6.4 were calculated in full with two exceptions: in ‘case 10)’the b component only on the right-hand side was determined and for ‘case 11)’ the a S \ (1) i ( α ) was used in place of S (1) i ( α )in the calculation since these actions are identical on the h O ⊂ h O subspace, asdescribed in the discussion following equation 6.43. The purpose of these calculationsis to cross-check the notation and conventions used here. This is useful since thereare several sign differences between quantities in this paper and the correspondingexpressions in reference [38] as listed for example in table 6.5. (We also note that theconventions used in the present paper differ in the sign of ± α for R zq ( α ) and B tq ( α )with respect to ([41] table 1)).Action: R xq ( α ), R xz ( α ), R zq ( α )Sign: e ± q α , ± sin α , ± q sin α Table 6.5: Sign differences between table 6.1 in the present paper and ([38] p.90,table 3.1).Of the 78 basis tangent vectors listed here in tables 6.6 and 6.7 one is explicitlypresented in reference [38]. The calculation of this tangent vector, namely for ˙ A l in table 6.7 here, differs by an overall ± sign from that presented in ([38] p.112,equation 4.1), but agrees with the sign convention for the same components quoted on([38] p.121). There is also a factor of two difference between the expression for [ ˙ R , ˙ R ]displayed here in equation 6.55 and that described in the equations of ([38] p.109).Whether each of these discrepancies is due to a typographical error or the con-ventions used in [38] the choice of signs and factors adopted in this paper is necessaryin order that the calculations here in table 6.4 are both self-consistent and agree withthe corresponding twelve entries in the full E Lie algebra commutation table availablein [38] for which the full set of (78 × − / action on h O in a tractable form which may be dissected for the analysisof symmetry breaking patterns. The few inconsistencies in the notation as describedabove may be accounted for and will not affect the conclusions for physics. In thispaper these conventions have been tuned for internal consistency and to be able toconsistently read off entries from the full L (E ) table [38] as the principal point ofreference. This in turn means that the correspondence between the generators of sub-groups of E , such as an external Lorentz group or an internal SU(3) gauge symmetrygroup, may not neatly match the conventions generally employed in physical theories,as will be seen in chapter 8, for example in equation 8.9. Hence ultimately a newbasis for L (E ) may be desired as tuned through a foreknowledge of the details of thephysical application in the context of the present theory.In general applying equation 6.55 for any two basis vectors on T h O will itselfresult in a basis vector field as listed in table 6.3 (or tables 6.6 and 6.7) or a linearcombination of such elements as is the case for the brackets numbered 8), 9), 10) and 12)in table 6.4 above. While elements such as ˙ R zi on the right-hand side of ‘case 1)’ in thistable may be ‘integrated up’ to the group action R (1) zi ( α ) on X ∈ h O in general it is notstraightforward to associate an element, or linear combination of elements, of the E Liealgebra with a one-parameter action of the Lie group describing curves on h O . This is143ue to the non-associativity of the octonions and the necessary employment of a nestedstructure to describe the transverse rotations. This is hence unlike the case in generalfor Lie algebra elements defined on the tangent space of a group manifold such as G = SO(3) which may be associated with Lie group elements by an ‘exponential map’,as described alongside figure 2.5 and exemplified in equation 2.49 for SO(5). However,of interest here will be broken subgroups, such as the Lorentz group for 4-dimensionalspacetime and the SU(3) colour symmetry which may be expressed without the abovedifficulties. (Again, alternatively, the full E action could in principle be expressed interms of G ⊂ GL(27 , R ) actions on R and the consequences of non-associativity andnested actions sidestepped completely).In the full Lie algebra table [38] with the basis vectors listed in table 6.3 a totalof six mutually commuting elements, that is with [ ˙ R , ˙ R ] = 0 for any pair of these sixelements, can be identified as the set: { ˙ B tz , ˙ B tz , ˙ R xl , ˙ A l , ˙ G l , ˙ S l } (6.56)which hence forms the Cartan subalgebra for the rank-6 Lie algebra E . There is someflexibility in this choice, due for example to equation 6.51, with ( ˙ B tz − ˙ B tz ) replacingthe second element ˙ B tz in ([41] equation 3.8(15)).In [39, 40] ‘symmetry breaking’ is considered in terms of making a choice ofa preferred h O ⊂ h O together with a preferred imaginary unit for the octonionelement in h O . Here we take a subspace h C ⊂ h O , using the isomorphism of the4-dimensional space h C to the space of the Lorentz vectors which in turn we haveidentified with tangent vectors on M , the base space for our perception of objects inthe world; as described in the previous chapters. The Lorentz group, SO + (1 , . With SO + (1 ,
3) being an external symmetry on M , which is also a global symmetry to a very good approximation ina laboratory setting, this will then provide the mechanism for the breaking of the E symmetry down to local gauge symmetries which may be compared with the SU(3) c × SU(2) L × U(1) Y gauge group and representations in the Standard Model of particlephysics.The Lorentz subgroup for 4-dimensional spacetime can be taken to be generatedby the subset of Lie algebra elements in L (E ): { ˙ B tz , ˙ R xl , ˙ B tx , ˙ B tl , ˙ R xz , ˙ R zl } (6.57)acting on h = t + z x − ylx + yl t − z ∈ h C ⊂ h O (6.58)where here the first two generators for this rank-2 subgroup are taken from the Cartansubalgebra for E in equation 6.56. The octonion unit l of the component a in equa-tion 6.1 (rather than i as for equation 6.19 and as discussed at the end of section 6.3)is chosen to represent an external spatial component since then the internal symmetryis more readily identified using the preferred basis of table 6.3, which in turn derivedfrom the conventions of equations 6.37-6.39 and table 6.2 in which l only appears inthe 3 rd pair column. The use of the unit l , rather than i , in this way also serves asa reminder that the Lorentz transformations here are embedded within expressions144ased on the octonion algebra. This external Lorentz symmetry will be studied indetail in section 8.1.In section 8.2 an internal symmetry will be provisionally defined here as anyoperation that fixes the external spacetime components ( t, x, y, z ) of equation 6.58 forany Lorentz 4-vector. This will include in particular the subgroup SU(3) ≡ { A q , G l } ,which from table 6.2 and equations 6.37 and 6.38 leaves the l component invariant,highlighting the significance of this basis choice for physics. The full E Lie algebracommutation table in [38] can be used to identify further internal symmetry groups,as we shall explore in chapter 8.In the following chapter we first review the Standard Model, and in particularthe relationship between the external and internal symmetries found there, beforeturning to the group E in general in section 7.3 as a candidate for unification of thesesymmetries as employed in particle physics. Then in chapter 8 the detailed structureof the action of E on h O , as reviewed in this chapter, will be applied to deduce theproperties of the external Lorentz symmetry in relation to the complementary internalsymmetry for the present theory.Through the historical development from the real numbers to the complexnumbers, continuing on through the quaternions to the octonions, composed then in2 × × as a determinantpreserving action on h O has been presented as an expression of the symmetry oftemporal flow in the form of L ( v ) = 1. It is of course possible that there may beother, higher-dimensional, forms for L ( v ) = 1, with yet higher symmetry groups thatwill have consequences for the physics of the world. Further generalisation should be,however, a well defined mathematical problem.In chapter 9 higher-dimensional forms of temporal flow and the possible role ofthe largest exceptional Lie groups E and E will be considered. For such cases the E symmetry will be an intermediary on the way up to, or operate in some way parallelto, the larger symmetries for the higher-dimensional forms of L ( v ) = 1. Even in thiscase, given that the richness of h O and its symmetries, as 3 × symmetry, assuming that theoverall conceptual framework that we are considering here broadly corresponds to thereal physical world. This, in the very least, would provide a proof of principle for theconceptual scheme being developed in this paper.145 B tz ˙ B tx ˙ B tq + p c − m − ¯ b + ¯ c − b + a x ( p + m ) + ¯ b ( p + m ) + a x + c + b + ¯ c − a q ( p + m ) q + q ¯ b − ( p + m ) q − a q − qc − bq + ¯ cq ˙ B tz ˙ B tx ˙ B tq ¯ a − c + a + m − ¯ c − n c + ¯ a + ¯ c + b x ( m + n )+ a ( m + n ) + b x − cq + ¯ aq + q ¯ c − b q ( m + n ) q − qa − ( m + n ) q − b q ˙ B tz ˙ B tx ˙ B tq − p − ¯ a − a ¯ b b + n + c x + b ( n + p )+ ¯ b a ( n + p ) + ¯ a + c x − c q − qb − ( n + p ) q + ¯ bq − aq ( n + p ) q + q ¯ a − c q ˙ R xq ˙ R xz ˙ R zq − a q − a x q − qc − a q + a x q q ¯ b + ¯ cq − bq + a x − ( p − m ) + ¯ b − ( p − m ) − a x − c + b − ¯ c + a q ( p − m ) q − q ¯ b − ( p − m ) q − a q − qc + bq + ¯ cq ˙ R xq ˙ R xz ˙ R zq ¯ aq − cq − qa − b q − b x q + q ¯ c − b q + b x q c − ¯ a + ¯ c + b x − ( m − n ) − a − ( m − n ) − b x cq + ¯ aq − q ¯ c + b q ( m − n ) q − qa − ( m − n ) q − b q ˙ R xq ˙ R xz ˙ R zq q ¯ a − c q + c x q − aq ¯ bq − c q − c x q − qb − c x − b − ( n − p ) − ¯ b a − ( n − p ) + ¯ a + c x − c q − qb − ( n − p ) q + ¯ bq aq ( n − p ) q − q ¯ a + c q Table 6.6: Vector fields on T h O generated by the 26 Category 1 Boosts and 31Category 2 Rotations from table 6.3 in the form of equation 6.53 ( ˙ B tz , ˙ R xq , ˙ R xq arenon-basis elements). 146 A i : ˙ a = − a j + a k + a kl − a jl ˙ A j : ˙ a = + a i − a k − a kl + a il ˙ A k : ˙ a = − a i + a j + a jl − a il ˙ A kl : ˙ a = + a i + a j − a jl − a il ˙ A jl : ˙ a = − a i − a k + a kl + a il ˙ A il : ˙ a = + a j + a k − a kl − a jl ˙ A l : ˙ a = + a i − a j + a jl − a il ˙ G i : ˙ a = − a j + a k − a kl + a jl − a il +2 a l ˙ G j : ˙ a = + a i − a k + a kl − a jl − a il +2 a l ˙ G k : ˙ a = − a i + a j − a kl − a jl + a il +2 a l ˙ G kl : ˙ a = + a i − a j +2 a k − a jl + a il − a l ˙ G jl : ˙ a = − a i +2 a j + a k + a kl − a il − a l ˙ G il : ˙ a = +2 a i + a j − a k − a kl + a jl − a l ˙ G l : ˙ a = + a i + a j − a k +2 a kl − a jl − a il ˙ S q : ˙ a = q P r = 1 , q a r r ˙ b = + b q − b q − q P r = 1 , q b r r ˙ c = − c q + c q − q P r = 1 , q c r r ˙ S q : ˙ a = − a q + a q − q P a r r, ˙ b = q P b r r, ˙ c = + c q − c q − q P c r r ˙ S q : ˙ a = + a q − a q − q P a r r, ˙ b = − b q + b q − q P b r r, ˙ c = q P c r r Table 6.7: Vector fields on T h O generated by the 21 Category 3 Transverse Rotationsfrom the lower section of table 6.3. In the case of ˙ A q and ˙ G q the form of ˙ b = f ( b )and ˙ c = f ( c ) is identical to ˙ a = f ( a ). With reference to equation 6.53, in all cases˙ p = ˙ m = ˙ n = 0 with { ˙¯ a, ˙¯ b, ˙¯ c } implied from { ˙ a, ˙ b, ˙ c } . ( ˙ S q and ˙ S q are non-basis elements,with P := P r =1 ,q here). 147 hapter 7 Review of the Standard Model
Having introduced the higher 27-dimensional form for the flow of time with symmetrygroup E acting on h O in the previous chapter we shall address the embedding of theLorentz symmetry SO + (1 , C ⊂ h O , within the larger structure in the openingsection of the following chapter. Here we first consider the properties of the groupSO + (1 ,
3) itself together with its representations.In general symmetries implicit in the form L ( v ) = 1 may include rotationgroups, such as SO(3) ⊂ SO + (1 , L ( v ) = 1.Although most of the discussion below applies to Clifford algebras in generalhere we focus on the case of the 4-dimensional vector space R , , with real Cliffordalgebra C (1 ,
3) represented by 4 × γ -matrices satisfying the relations: γ a γ b + γ b γ a = 2 η ab (7.1)with indices { a, b } = 0 . . .
3, Minkowski metric η ab and where denotes the 4 × v, w ∈ R , the associated algebra product with v = v a γ a and w = w b γ b as elements of C (1 ,
3) satisfies vw + wv = 2 η ( v, w ) , implyingfor example the relation v = | v | for all v ∈ R , which is also sufficient to generate148he full algebra. A general element u of the Clifford algebra C (1 ,
3) has the form: u = u + u a γ a + u ab γ a γ b + u abc γ a γ b γ c + . . . ∈ C (1 ,
3) = C ⊕ C ⊕ C ⊕ C ⊕ . . . (7.2)with u , u a , u ab . . . ∈ R , with index values ordered as a < b < c . . . , and where C i denotes the subspace of C (1 ,
3) formed by the product of i basis elements { γ a } in thisrepresentation. Owing to equation 7.1 the Clifford algebra itself has dimension 2 n ,with n = 4 here, that is the elements of the C (1 ,
3) algebra describe a vector spacewith 16 linearly independent elements.The Clifford algebra itself does not form a group since in general an inverseelement may not exist for any given u ∈ C (1 , C (1 ,
3) generated by elements v ∈ C with η ( v, v ) = ± , v ∈ C ∩ Pin(1 ,
3) the map φ v from w ∈ C into C : φ v : w → vwv − (7.3)= 2 η ( v, w ) v − − wvv − = 2 η ( v, w ) vη ( v, v ) − w (7.4)is a reflection of w through the line containing the origin and v in the (psuedo-)Euclidean space R , . These reflections may be combined to describe a representationof Pin(1 ,
3) as orthogonal transformations on the space R , (which is equivalent to C (1 ,
3) as a vector space). The application of Clifford algebra composition to inducerepresentations of the rotation groups via equation 7.3 is similar to the use of theconjugation action for elements of a division algebra such as the octonions, describedby equation 6.11 in section 6.2, also to represent rotations.In fact the Lie group Pin(1 ,
3) is the two-to-one cover of the full Lorentz groupO(1 , ,
3) tothose in the even subalgebra C e (1 ,
3) := { C i (1 , i even } of equation 7.2 identifiesthe subgroup Spin(1 , R , as the group of special orthogonal transformations SO(1 , π with: π : Pin(1 , → O(1 ,
3) (7.5) π : Spin(1 , → SO(1 ,
3) (7.6)Hence the respective Lie algebras are isomorphic, for example spin(1 ,
3) = so(1 , ,
3) as a manifold connected to the identity is in fact ‘simplyconnected’ and is denoted Spin + (1 , + (1 ,
3) –which in turn is the part of the full Lorentz group (described above equation 6.22 forthe k = 2 case) which preserves both the time and the space orientations, as well asthe metric relations, of Lorentz 4-vectors.The set of matrices: σ ab = 14 ( γ a γ b − γ b γ a ) = 14 [ γ a , γ b ] (7.7)149ith a < b and γ a γ b ∈ C (1 , + (1 ,
3) = so + (1 , σ ab elements. This algebra generatesgroup elements R ( ω cd ) = exp( ω cd σ cd ) with ω cd ∈ R (summing over the set of six indexpairs with c < d , in a similar way to the group actions described in equation 2.49).These describe SO + (1 ,
3) vector transformations on the γ a matrices themselves: φ R : γ a → R ( ω cd ) γ a R − ( ω cd ) ≡ ( A − ) ab γ b (7.8)with A ∈ SO + (1 , + (1 ,
3) on 4-componentDirac spinors ψ ∈ C : ψ → ψ ′ = e ω cd σ cd ψ = R ( ω cd ) ψ (7.9)For a complex Clifford algebra in any dimension n this Dirac representation isirreducible. However for the real forms of these algebras with even n = p + q the space ofthe Dirac representation for the group Spin + ( p, q ) decomposes into two halves, knownas chiral ( left and right ) spinors, upon which inequivalent representations act. Thismay be shown by defining the matrix Γ := γ γ . . . γ n ∈ C n ( p, q ) which anticommuteswith each γ a and hence (by equation 7.7 for the general case) commutes with allelements of Spin + ( p, q ), and hence in turn by Schur’s lemma the Dirac representationis reducible (unless Γ is a proportional to the unit 2 n × n matrix, which is generallynot the case).In the case of (1 ,
3) spacetime, with 4 × γ a acting on the elements ψ ∈ C of the spinor space, Γ is denoted γ and the usual convention is to take: γ = iγ γ γ γ for which ( γ ) = + (7.10)Due to the factor of i this object does not belong to the real Clifford algebra. Howeveras a 4 × γ does commute with each element of Spin + (1 ,
3) and can be usedto extract the chiral spinors ψ L and ψ R via the projection operators P L and P R : ψ L = P L ψ with P L = (1 − γ ) (7.11) ψ R = P R ψ with P R = (1 + γ ) (7.12)By Schur’s lemma this decomposition into left and right-handed spinors ψ = ψ L + ψ R ∈ C is maintained under the 4 × + (1 , + (1 ,
3) invariant and irreducible piecescalled Weyl spinors. A suitable explicit representation for the γ -matrices is the Weylbasis with: γ = + , γ a = σ a − σ a , γ = −
00 + (7.13)where each entry is a 2 × σ a for a = 1 , , σ = , σ = , σ = − ii , σ = − (7.14)150n the γ -matrix basis of equation 7.13 the Spin + (1 ,
3) action of equation 7.9 can beexpressed on the Weyl spinors ψ L , ψ R ∈ C simply as: ψ = ψ L ψ R → R L R R ψ L ψ R (7.15)For particle states chirality itself is an observable only for massless fermions,that is m f = 0, in which case it is equivalent to the particle helicity.The ‘spin’ group for the Clifford algebra of the real pseudo-Euclidean vectorspace R , may also be approached directly via the group SL(2 , C ), which is closelyrelated to the representations R L and R R in equation 7.15. The 6-dimensional LorentzLie algebra so + (1 ,
3) can be expressed in a conventional basis of anti-Hermitian rotationgenerators { J , J , J } and Hermitian boost generators { K , K , K } in terms of a 2 × , C ) in the form: J a = − i σ a and K a = − σ a (7.16)for a = 1 , ,
3. The signs are chosen such that the following algebra commutators hold:[ J a , J b ] = ε abc J c (7.17)[ K a , K b ] = − ε abc J c (7.18)[ J a , K b ] = ε abc K c (7.19)with ε = +1. In other conventions the signs may vary, and factors of i = √− J is defined to be Hermitian, as is the case in quantum mechanics in order toidentify real observable quantities for angular momentum. In the standard treatmenta general element of the group SL(2 , C ) is represented by the 2 × S = e r a J a + b a K a = e ( − ir a − b a ) σ a (7.20)with the rotations parametrised by r a ∈ R , a = 1 , ,
3, and the boosts parametrisedby b a ∈ R , a = 1 , ,
3. For the complex linear combinations A a = ( J a + iK a ) and B a = ( J a − iK a ) the Lie bracket reads:[ A a , A b ] = ε abc A c (7.21)[ B a , B b ] = ε abc B c (7.22)[ A a , B b ] = 0 (7.23)demonstrating that the complexified Lie algebra of sl(2 , C ) is isomorphic to su(2) ⊕ su(2)(as will be represented in figure 7.2(d) and described in the accompanying text) whichis used to label the representations of the Lorentz group by the half-integer values( j A , j B ). After the trivial (0,0) scalar case the two lowest-dimensional possibilities arethe representations of SL(2 , C ) denoted R L ( S ) and R R ( S ) with:( j A , j B ) = ( , ⇒ R L ( S ) = e ( − ir a − b a ) σ a (7.24)( A a = − σ a , B a = 0; J a = − i σ a , K a = − σ a )and ( j A , j B ) = (0 , ) ⇒ R R ( S ) = e ( − ir a + b a ) σ a (7.25)( A a = 0 , B a = − σ a ; J a = − i σ a , K a = + σ a )151he first of these representations R L ( S ) can be identified with the original setof 2 × S ∈ SL(2 , C ), that is { S ∈ C (2) : det( S ) = 1 } , as parametrised in theform of equation 7.20. The representation R R ( S ) in equation 7.25 is a different mapfrom the same complete set of SL(2 , C ) elements, considered as an abstract group,into 2 × C . Thetwo representation spaces are given different subscript labels L and R to denote thatthey belong to different SL(2 , C ) representations with the left-handed Weyl spinortransforming as ψ L → R L ( S ) ψ L and the right-handed Weyl spinor transforming as ψ R → R R ( S ) ψ R .Under a discrete parity transformation the sign of a Lorentz boost is reversedwhile the sign of a rotation is invariant. The naming convention of ‘left’ and ‘right’representations originates since R L ( S ) and R R ( S ) are related by the sign of the boostgenerator contributions in equations 7.24 and 7.25 and are hence interchanged undera parity transformation. Indeed in general the parity operation switches between thetwo Lorentz representations ( j , j ) and ( j , j ).Since there is no 2 × D such that R L ( S ) = DR R ( S ) D − for all S ∈ SL(2 , C ) the representations R L ( S ) and R R ( S ) are inequivalent. However the followingrelationships between equations 7.24 and 7.25 hold (with σ defined in equation 7.14): R ∗ L ( S ) = σ R R ( S ) ( σ ) − (7.26) R † − L ( S ) = R R ( S ) (7.27) R L T ( S ) = σ R − L ( S ) ( σ ) − (7.28)showing respectively that the complex conjugate of R L ( S ) is equivalent to R R ( S ), thecontragredient of R L ( S ) is equal to R R ( S ) and the transpose of R L ( S ) is equivalentto its inverse.The Dirac representation R D ( S ) of SL(2 , C ) has the reducible form ( , ⊕ (0 , ), acting on spinors in the space C , and via equation 7.27 it can be written as: R D ( S ) = R L ( S ) 00 R R ( S ) = S S † − (7.29)which is the same action as described in equation 7.15, there derived from the Cliffordalgebra structure, with R L = R L ( S ) and R R = R R ( S ). Hence S ∈ SL(2 , C ) acts onthe left-handed components of the C Dirac spinor and S † − acts on the right-handedcomponents as an inequivalent representation of SL(2 , C ). Equation 7.29 describeshow the Spin + (1 ,
3) Dirac representation can be constructed by combining left andright spinors as the ( , ⊕ (0 , ) representation of SL(2 , C ). In fact Spin + (1 ,
3) isisomorphic to the group SL(2 , C ) (such isomorphisms for the spin groups only existin low dimensions and for a handful of cases), each expressing the two-to-one cover ofSO + (1 , + (1 ,
3) via the Clifford algebra or as acombination of two representations of SL(2 , C ) expresses the relation between the 4-component and 2-component spinor formalism. The 2-component Weyl spinors aremore fundamental in the sense that ψ L and ψ R are treated differently in importantfeatures of the Standard Model, as we shall describe in the following section.152he two-to-one relationship between SL(2 , C ) and SO + (1 ,
3) may be exhibitedby mapping a Lorentz vector v ∈ R , into the space of 2 × v = ( v , v , v , v ) → h = v · σ = v + v v − v iv + v i v − v ⊂ h C (7.30)where σ denotes the 2 × σ together with the three Pauli matrices σ a of equation 7.14. This is the same object introduced in equation 6.19 of section 6.3and also in equation 6.58 of section 6.5, based on the imaginary unit l in the lattercase. We see from this equation, and in comparison with section 6.3, that det( h ) =( v ) − ( v ) − ( v ) − ( v ) = h , with h ∈ R , which may be expressed as the form L ( v ) = h (as employed in equation 5.46). While the fundamental representation ofSL(2 , C ) acts on the space C , the group action for elements S ∈ SL(2 , C ) on the spaceh C provides another representation given by: h → h ′ = S h S † (7.31)This maps h → h ′ onto a new 2 × v a → v ′ a according to a Lorentz transformationof the real 4-vector v ∈ R , . With S ∈ { , − } giving the identity transformation, h ′ = h , the group SL(2 , C ), isomorphic to Spin + (1 ,
3) as described above, is thetwo-to-one covering spin group for SO + (1 , + (1 ,
3) = SL(2 , C ) / Z .In fact the components of v transform under the 4-dimensional vector ( , )representation of SL(2 , C ). The matrix h , and hence the vector v , can be consideredto be constructed out of two 2-component left-handed Weyl spinors χ and φ such that: h = χχ † + φφ † (7.32)as implied in equations 6.20 and 6.21 of section 6.3, with the elements of the groupSL(2 , C ) acting on the spinor components in the appropriate way.This spinor substructure of vectors v has some similarity to the situation dis-cussed in section 5.2 for the relation − κ G µν = ρu µ u ν − S µν , as implied in equation 5.34via the Einstein equation, which describes the possibility of composing the rank–2 Ein-stein tensor in terms of a substructure involving the apparent 4-dimensional macro-scopic vector flow u ( x ) on the base manifold. The natural algebraic substructure of thefield v ( x ) in terms of the spinor decomposition of equation 7.32 may in turn be inti-mately related to the possible field interactions implied within the higher-dimensionalform of time L ( v ) = 1 at the microscopic level, underlying the composition of theEinstein tensor G µν = f ( Y, ˆ v ) as expressed in equation 5.32.Equation 7.31 describes the determinant preserving action h → S h S † ofthe elements S ∈ SL(2 , C ) upon elements of the vector space of matrices h ∈ h C that was extended in equations 6.16 and 6.22, by augmenting the complex numbersto the octonions, to identify an SL(2 , O ) action on h O as the covering group of the10-dimensional Lorentz group, as an intermediary for the E action on h O . Forinfinitesimal transformations we write S = exp( a ) ≃ a , where a ∈ sl(2 , C ) is an153nfinitesimal element of the Lie algebra of SL(2 , C ), and we have: h → (1 + a ) h (1 + a † ) ≃ h + δ h (7.33)with δ h = a h + h a † (7.34)where δ h has the same form as the 64 D S actions on h O of equation 6.5 in section 6.1,and may here be considered to represent the Lie algebra sl(2 , C ) on the tangent space T h C . This corresponds to a possible substructure embedding of h C ⊂ h O ⊂ h O with respective group actions SL(2 , C ) ⊂ SL(2 , O ) ⊂ SL(3 , O ). Before moving to theaction of SL(2 , C ) on full space h O in the following chapter (see equation 8.4), we firsthere consider the action of SL(2 , C ) on the space h C . The 2 × S ∈ SL(2 , C )can be embedded in 3 × X ∈ h C as: X → S
00 1 h ψ L ψ † L n S †
00 1 (7.35)This combines the vector representation of SL(2 , C ) on h ∈ h C and the spinorrepresentation on ψ L ∈ C , together with the scalar n ∈ R , in a single symmetrytransformation which preserves det( X ) ∈ R . In section 8.1 the spinor ψ L will beidentified with θ l ∈ C in a complex subspace of θ ∈ O under the full E action onh O , compatible with the embedding of the SL(2 , C ) action of equation 7.35 withinthe SL(2 , O ) ⊂ E action of equation 6.29. Together with the external Lorentz symmetry internal gauge symmetries are key to theproperties of particle states observed in the laboratory. In this section we review theinternal symmetries of the Standard Model with a particular emphasis on electroweaktheory and the phenomenon of symmetry breaking (see for example [42]).The quarks and leptons of one generation of Standard Model fermions trans-form under an internal symmetry described by the group product SU(3) c × SU(2) L × U(1) Y (with the subscripts ‘ c ’, ‘ L ’ and ‘ Y ’ denoting colour, left-handed and hyper-charge respectively). The corresponding representation of SU(3) c × SU(2) L × U(1) Y iscomposed as a sum of five irreducible pieces each labelled according to their transfor-mation properties by ( n , n , n ) L,R with the subscript L or R denoting left or rightchiral Weyl spinors, represented as four-component Dirac spinors, under the externalLorentz group. The five pieces are of dimension 6, 3, 3, 2 and 1 respectively (withoutan extra piece (1 , , R for a right-handed neutrino ν R ):(3 , , ) L + (3 , , ) R + (3 , , − ) R + (1 , , − ) L + (1 , , − R q L = (cid:0) u L ( ) d L ( − ) (cid:1) u R ( ) d R ( − ) l L = (cid:0) ν L (0) e L ( − (cid:1) e R ( −
1) (7.36)with the corresponding set of 15 particle states named on the second line alongsidetheir electromagnetic charges. The components of particle multiplets transforming as154riplets under SU(3) c , ( n = 3), couple to the strong interaction and consist of u -typeand d -type quarks, while the SU(3) c singlet components consist of the neutrino ν andelectron e leptonic states.In the Standard Model electroweak theory weak eigenstates, that is fields trans-forming according to definite SU(2) L representations, are composed as left-handeddoublets ( n = 2) and right-handed singlets ( n = 1), transforming for example in thecase of leptons as l L → l ′ L = e − iω α τ α l L , with ω α ∈ R and τ α = 12 σ α (7.37)for α = 1 , , e R → e ′ R = e R . With left and right-handed fermions hence under-going different interactions with the SU(2) L gauge field this construction describes theempirical observation of parity violation in weak interactions. When additional gen-erations of fermions are considered the weak eigenstates generally consist of a linearcombination of physical mass eigenstates leading to the phenomena of mixing betweenthe generations, as will be described towards the end of this section.The electromagnetic charge of each particle in a multiplet is given by: Q = T + Y T is the eigenvalue under the third, diagonal, SU(2) L generator and the hy-percharge Y labels the U(1) Y representations, ( n = Y /
2) in equation 7.36, which areall one-dimensional for this Abelian group. For the right-handed states T = 0 andthe hypercharge is simply the electric charge of the fermion Q ( ψ R ) = Y ( ψ R ). Allfields transform as ψ → ψ ′ = e − iω Y ( ψ ) ψ , with ω ∈ R , under the hypercharge gaugesymmetry U(1) Y . The hypercharge Y ( ψ ) itself is ultimately defined to give the correctelectromagnetic charge Q , via the relation in equation 7.38, which is the same for the L and R parts of each fermion type with Q ( e L ) = Q ( e R ) = − Q determines the coupling to the electromagnetic field corresponding to theU(1) Q gauge symmetry that survives electroweak symmetry breaking. Equation 7.38may be considered as a relation either between the eigenvalues or the operators Q , T and Y , depending on the context.The representations of equation 7.36 can be expressed purely in terms of left-handed fields by applying ‘charge conjugation’ to the right-handed cases, under which(3 , , ) R → (¯3 , , − ) L for example. Having all fields expressed in terms of the sameLorentz representation in this way is useful for unification models, in which individualpieces of equation 7.36 are combined in a larger representation of a single unifyinggauge group. Since gauge transformations commute with Lorentz transformations,without interchanging L and R states, such a unifying gauge group then respectsLorentz invariance in the theory. While the states in equation 7.36 are all considered as‘particles’ the action of charge conjugation also introduces ‘antiparticle’ states. Henceboth particle and antiparticle states may be combined in unified multiplets, as for thecase of the SU(5) model [43] cited regarding figure 7.3 in the following section.The dynamics of the Standard Model fields is heavily based on a Lagrangianformalism. The Standard Model Lagrangian includes kinetic terms for the fermions155n the form of the final term of equation 3.96, which for the lepton doublet l L , with aconventional factor of i and covariant derivative D µ , can be expressed as: L kin = i ¯ l L γ µ D µ l L (7.39)with D µ = ∂ µ + ig W αµ ( x ) τ α + ig ′ B µ ( x ) Y ( l L ) (7.40)where τ α is defined in equation 7.37 (and with an additional D µ = . . . + ig s G βµ ( x ) λ β term, with β = 1 . . . λ β matrices listed in table 8.5, for SU(3) c gauge inter-actions in the case of quarks). Hence the interaction between the gauge fields W αµ ( x ), B µ ( x ), with respective couplings g , g ′ , and left-handed leptons has the Lagrangianform: L int = − g (cid:0) ¯ ν L ¯ e L (cid:1) γ µ W µ W µ − iW µ W µ + iW µ − W µ − g ′ g B µ B µ ν L e L (7.41) L ν = − g ν L γ µ ( W µ − g ′ g B µ ) ν L (7.42)where the part L ν describes the gauge coupling to the neutrino alone, as implied inequation 7.39. Physical gauge boson fields A µ ( x ) and Z µ ( x ) are defined as a linearcombination of B µ ( x ) and W µ ( x ) via the orthogonal transformation: A µ = cos θ W B µ + sin θ W W µ Z µ = − sin θ W B µ + cos θ W W µ (7.43)that is with: B µ = cos θ W A µ − sin θ W Z µ W µ = sin θ W A µ + cos θ W Z µ (7.44)where θ W is the weak mixing angle. Hence from equation 7.42 the coupling of theneutrino to the physical gauge field A µ is: L νA = − g ν L γ µ (sin θ W A µ − g ′ g cos θ W A µ ) ν L (7.45)which is zero for: tan θ W = g ′ g (7.46)This value of the weak mixing angle θ W hence describes the electric charge neutrality ofthe neutrino with A µ ( x ) interpreted as the electromagnetic field, the quanta of whichare photons. More generally the coupling terms for the photon can be extracted fromthe relevant part of the covariant derivative D µ , of the form in equation 7.40, actingon any field ψ ( x ) as (with T representing the third component of su(2) L and thehypercharge Y as operators acting on the field ψ ): D µ ∼ ig W µ T ( ψ ) + ig ′ B µ Y ( ψ ) retaining only W µ , B µ field parts= ig sin θ W A µ T + ig ′ cos θ W A µ Y by equation 7.44, dropping Z µ parts= ig sin θ W A µ T + ig sin θ W A µ Y using equation 7.46= ig sin θ W A µ ( T + Y ) ≡ ie A µ Q (7.47)156ence the electromagnetic coupling of any particle state to the photon is always pro-portional to eQ where the particle charge Q is defined in equation 7.38 and the elec-tromagnetic coupling e is given by: e = g sin θ W (7.48)As described after equation 7.38 the different values of Y compensate for the different T values for the L and R states of a given particle such that the respective couplingof each chiral component to the gauge field A µ ( x ) is the same, as can be seen for eachparticle type in equation 7.36. Following the same lines of reasoning in equation 7.47except instead retaining the gauge field Z µ ( x ) and dropping the A µ ( x ) field parts inthe second line leads to: D µ ∼ ig cos θ W Z µ T − ig ′ sin θ W Z µ Y = ig cos θ W Z µ T − ig sin θ W cos θ W Z µ Y = ig Z µ (cid:18)(cid:18) cos θ W + sin θ W cos θ W (cid:19) T − sin θ W cos θ W (cid:18) T + Y (cid:19)(cid:19) = ig cos θ W Z µ (cid:0) T − Q sin θ W (cid:1) (7.49)Hence there are two terms for the weak neutral interactions. The second term isproportional to the electromagnetic charge Q and is hence the same for L and R particle states. However since the eigenvalues of T are only non-zero for the left-handed states the first term only couples to the ψ L components. The combination ofthe two terms in equation 7.49 implies that parity violation is only partial for neutralweak interactions. On the other hand for the charged weak interactions mediated viathe W ± µ ( x ) gauge fields, introduced in equation 7.57 below and involving only SU(2) L components, parity violation is maximal. In the Standard Model Lagrangian the left-handed chiral states ψ L are projected out of the Dirac spinor states for the fermionsusing the P L operator of equation 7.11, as seen for example in equations 7.77 and 7.78at the end of this section.In addition to the spin- fermions and spin-1 gauge bosons the Standard Modelalso introduces a spin-0 Higgs field, which is massive itself and closely associated withthe origin of mass for the W ± and Z gauge bosons as well as the fermion states. Indeedelectroweak theory is inextricably linked to the Higgs sector with the breaking of theelectroweak symmetry SU(2) L × U(1) Y to the U(1) Q of electromagnetism mediatedthrough the action of the gauge group on the Higgs field: φ = φ + φ = 1 √ φ + iφ φ + iφ (7.50)Transforming as a scalar under the external Lorentz symmetry the Higgs field is alsoinvariant under the internal SU(3) c symmetry. On the other hand the above complexdoublet of scalar fields φ transforms as a doublet under SU(2) L while also possessinghypercharge with Y = + , which also accounts for the notation φ + and φ in equa-tion 7.50 by reference to equation 7.38. This collection of properties may be denoted1571 , , ) by comparison with the list of Standard Model fermions in equation 7.36.The Lagrangian for the Higgs sector is: L H = ( D µ φ ) † D µ φ − V ( φ ) (7.51)where D µ φ = (cid:18) ∂ µ + i g W αµ σ α + i g ′ B µ σ (cid:19) φ (7.52)is the gauge covariant derivative which is similar in form to equation 7.40 except with Y = + here, and also σ and σ α = { σ , σ , σ } have been adopted directly fromequation 7.14 rather than via equation 7.37. The fields W αµ ( x ) and B µ ( x ) are theSU(2) L and U(1) Y gauge fields, with couplings g and g ′ respectively, as introduced inequation 7.40. The breaking of the electroweak symmetry relies on the ‘Mexican hat’potential term in the Lagrangian of equation 7.51 with: V ( φ ) = − µ φ † φ + λ ( φ † φ ) (7.53)with real coefficients µ > λ >
0. From equation 7.50 it can be seen that thepotential V ( φ ) is a function of φ † φ = P i =1 φ i only. The vacuum expectation valuefor this field h φ i , that is the minimum in the potential, can be taken without loss ofgenerality (in the ‘unitarity gauge’) to be: h φ i = 1 √ v with v = µ √ λ (7.54)This charge neutral component of the Higgs field φ = v √ is invariant under theaction of the charge generator Q = T + Y = (cid:0) − (cid:1) + (cid:0) (cid:1) = (cid:0) (cid:1) , from equation 7.38applied for the Higgs field, which remains unbroken. Hence the gauge symmetry isbroken from SU(2) L × U(1) Y down to U(1) Q , identified in a linear combination of thethird component of su(2) L and the hypercharge generator u(1) Y , as the symmetrywhich leaves the vacuum value h φ i in equation 7.54 invariant.Masses arise for the gauge fields corresponding to the broken SU(2) L × U(1) Y generators from the kinetic term in the Higgs Lagrangian of equation 7.51. Acting onthe vacuum state the covariant derivative of equation 7.52 can be written as: D µ φ = ∂ µ + i gW µ + i g ′ B µ i g ( W µ − iW µ ) i g ( W µ + iW µ ) ∂ µ − i gW µ + i g ′ B µ √ v (7.55)Hence L H in equation 7.51 contains the expression (for now neglecting fluctu-ations about the vacuum value v ):( D µ φ ) † D µ φ = g W µ + iW µ )( W µ − iW µ ) v
2+ 14 (cid:16) W µ B µ (cid:17) g − gg ′ − gg ′ g ′ W µ B µ v W ± µ ( x ) associated respectively with 2 × σ ± in the complexified SU(2) L Lie algebra defined in turn as: W ± µ = 1 √ W µ ∓ iW µ ) (7.57) σ ± = 12 ( σ ± iσ ) (7.58)the relation: 1 √ W + µ σ + + W − µ σ − ) = 12 ( W µ σ + W µ σ ) (7.59)may be substituted in for the W µ , W µ piece of the covariant derivative in equation 7.52.In turn the first term in equation 7.56 explicitly takes the form of a mass term for the W ± µ ( x ) fields in the Lagrangian: L H = g v W + µ W + µ + W − µ W − µ ) + . . . hence with M W = 12 gv (7.60)being the W ± mass.The second term in equation 7.56 contains a 2 × g, g ′ . Applying the same orthogonal transformationof equations 7.43 and 7.44 to the fields W µ and B µ with the weak mixing angle θ W asspecified in equation 7.46 diagonalises the mass matrix with respect to the fields Z µ and A µ such that: L H = . . . + 12 (cid:16) Z µ A µ (cid:17) M Z
00 0 Z µ A µ (7.61)with M Z = 12 p g + g ′ v = M W cos θ W (7.62)Hence the same weak mixing angle θ W that accounts for the electromagnetic chargeneutrality of the neutrino ν through the covariant derivative D µ acting on the leptonfield l L in equation 7.45, deriving from the kinetic term in the Lagrangian for thelepton field in equations 7.39 and 7.40, also diagonalises the above mass matrix andleaves the photon field A µ massless through D µ acting on the Higgs field φ , derivingfrom the kinetic term in the Lagrangian for the Higgs field in equations 7.51 and 7.52.Considering fluctuations about the vacuum value with v → v + H ( x ) in equa-tion 7.54 (as neglected in writing down equation 7.56) in the quantum theory thereal field H ( x ) is associated with a massive scalar particle known as the Higgs bo-son. In terms of the parameters of the theory the Higgs mass is determined to be M H = √ µ = √ λ v . While the vacuum value is empirically constrained to the orderof the weak scale with, via equation 7.60, v = 2 M W g ∼ ( √ G F ) − ∼
246 GeV, where G F is the Fermi constant, this does not determine the two parameters of the potentialin equation 7.53. These latter parameters can now be deduced given the discovery ofthe Higgs at the LHC and the empirical measurement of M H ≃
125 GeV [44].159t tree level the relations in the quantum field theory described in equa-tions 7.60 and 7.62 lead to the definition of the parameter: ρ = M W M Z cos θ W = 1 (7.63)The fact that this expression holds approximately for the corresponding empiricallymeasured values can be explained in terms of a further symmetry associated with theHiggs sector. Expressing the Higgs field components in the form of a bi-doublet, thatis the 2 × √ ǫφ ∗ , φ ) = 1 √ φ ∗ φ + − φ + ∗ φ (7.64)with ǫ = (cid:0) − (cid:1) , the Higgs potential term of equation 7.53 may be rewritten as: V (Φ) = − µ tr Φ † Φ + λ (tr Φ † Φ) (7.65)This is invariant under the L ∈ SU(2) L action Φ → L Φ and U(1) Y action Φ → Φ e − i θσ with θ ( x ) ∈ R as local gauge transformations. While φ and ǫφ ∗ transform in the sameway under SU(2) L , they have opposite hypercharge, with Y ( φ ) = + and Y ( ǫφ ∗ ) = − ,and hence the generator for U(1) Y transformations here is σ rather than σ (see forexample [45] section 3). The Higgs Lagrangian of equation 7.51, which is also invariantunder these gauge transformations, can be written in the form: L H = tr ( D µ Φ † D µ Φ) − V (Φ) (7.66)where D µ Φ = ∂ µ Φ + i g W αµ σ α Φ − i g ′ B µ Φ σ (7.67)is the gauge covariant derivative for the bi-doublet. In the limit g ′ → global , symmetry denoted SU(2) R with action Φ → Φ R † forany R ∈ SU(2) R , as can be seen by cyclic permutation of the arguments under thetrace, with tr( R Φ † Φ R † ) = tr( R † R Φ † Φ) = tr(Φ † Φ) for example. This symmetry in theStandard Model is considered to be ‘accidental’ in the sense that it was not explicitlyintroduced in constructing the Higgs field to break the electroweak symmetry. Itenlarges the complete global symmetry of the Higgs field to the action of SU(2) L × SU(2) R , as Φ → L Φ R † (where L here represents a global action of the local SU(2) L symmetry), which is simply the SO(4) symmetry of the quantity P i =1 φ i describedbelow equation 7.53. The vacuum expectation value of equation 7.54 can be writtenin the form: h Φ i = 12 v v (7.68)This vacuum value breaks the global SU(2) L × SU(2) R down to a single SU(2) sym-metry denoted SU(2) L + R , with the action h Φ i → L h Φ i L † for L ∈ SU(2) L + R leavingequation 7.68 invariant. This is equivalent to the SO(3) ⊂ SO(4) symmetry actingon the four components φ i when taking the values of an arbitrary fixed Euclidean160-vector, such as ( φ , φ , φ , φ ) = (0 , , v,
0) in equation 7.68. The global SU(2) L + R symmetry itself is broken for hypercharge coupling g ′ = 0, which involves gaugingthe U(1) Y ⊂ SU(2) R subgroup via the σ action of equation 7.67, which is both thehypercharge generator itself and also the third component of the SU(2) R action.For the Standard Model in the limit g ′ → W αµ gauge fields transformas a triplet under the unbroken SU(2) L + R global symmetry, and hence the massesgained from electroweak symmetry breaking are identical, with M W ± = M Z (as canbe seen from equations 7.60 and 7.62 for g ′ → g ′ theunbroken U(1) Q symmetry corresponding to the massless photon determines a weakmixing angle θ W with cos θ W = g g + g ′ which also determines the mass ratio of theheavy gauge bosons at tree level according to equation 7.63. This relation ρ = 1 isprotected from radiative corrections by the approximate SU(2) L + R symmetry, whichis hence named ‘custodial symmetry’ [46, 45].Masses for all three generations of fermions are included in the Standard ModelLagrangian by appending gauge invariant terms with Yukawa couplings to the Higgsfield: L Y = − Γ iju ¯ q iL ǫφ ∗ u jR − Γ ijd ¯ q iL φ d jR − Γ ije ¯ l iL φ e jR + h.c. (7.69)(where ‘h.c.’ is the Hermitian conjugate of all the preceding terms). Here the Yukawacouplings Γ u , Γ d and Γ e are 3 × i, j = { , , } and hence, for example, u iR ≡ { u R , c R , t R } denotes thethree generations of u -type right-handed quarks. When the Higgs field acquires thevacuum value h φ i as expressed with the gauge choice of equation 7.54 the fermionstates acquire Dirac mass terms via the Yukawa couplings: L M = − M iju ¯ u iL u jR − M ijd ¯ d iL d jR − M ije ¯ e iL e jR + h.c.where M iju,d,e = Γ iju,d,e v √ (7.70)are the three fermion mass matrices. Physical particle states may be identified bydiagonalising each M ij matrix using independent unitary transformations applied toeach left and right-handed fermion set via 3 × A ij , such as for: u L → u ′ L = A † u L u L (7.71) u R → u ′ R = A † u R u R (7.72)Hence u ′ L ≡ { u ′ L , c ′ L , t ′ L } and u ′ R are mass eigenstate fields with the masses of the three u -type quarks read off from the diagonal elements of: M ′ u = A † u L M u A u R = m u m c
00 0 m t (7.73)with L M = − m u ¯ u ′ L u ′ R − m c ¯ c ′ L c ′ R − m t ¯ t ′ L t ′ R + h.c. (7.74)as the Lagrangian Dirac mass terms for the u -type quarks (with u ′ , c ′ and t ′ heredenoting the individual first, second and third generation u -type quarks). The u -quark itself hence has mass m u = Y u v √ with the Yukawa coupling Y u = Γ ′ u extracted161rom the diagonalised basis. From equation 7.60 the u -quark mass can be related tothe W ± gauge boson mass as: m u = √ g Y u M W with Y f = g m f √ M W (7.75)where Y f is the Yukawa coupling for each fermion f to the Higgs field φ , includingthe similar cases for the d -type quarks and charged leptons as following also fromequation 7.70. (The neutrino mass may be treated differently and may not involve aYukawa coupling, [42] chapter 7). The couplings Y f are typically small since m f ≪ M W except for the case of the top quark – with the mass m t observed to be approximatelythe sum of M W and M Z . All of the Yukawa couplings are added by hand in order tomatch the empirically determined fermion masses.In the physical mass eigenstate basis there is no Yukawa mixing between gen-erations, as can be seen in equation 7.74 in comparison to equation 7.70 where inthe latter expression the quark states coupling to the weak SU(2) L gauge fields aregenerally composed of a linear combination of the physical quark states. The weakSU(2) L doublets in the quark sector may be written as (cid:0) u ˜ d (cid:1) L , (cid:0) c ˜ s (cid:1) L and (cid:0) t ˜ b (cid:1) L , with theinter-generation mixing expressed purely in terms of the d -type quark states: ˜ d ˜ s ˜ b = V CKM d ′ s ′ b ′ (7.76)Here the weak states ˜ d, ˜ s, ˜ b are related to the physical states d ′ , s ′ , b ′ via the 3 × V CKM = A † u L A d L . With fiverelative global phase transformations between the six quarks ( u, d, c, s, t, b ) only fourof the nine parameters of the unitary matrix V CKM are physical. These four parametersdescribe three real mixing angles between the three generations and one complex phasewhich gives rise to CP violating phenomena. Together with the six quark masses atotal of ten physical parameters (contributing just over half of the 18 Standard Modelparameters listed in table 15.2) may hence be deduced from the Lagrangian for thequark sector after the above field redefinitions. (Again, the description of neutrinomixing in the leptonic sector is a little different, [42] chapter 7).The weak interaction terms for the quarks with the charged gauge bosons W ± may be described by the Lagrangian: L qW = − g √ u i γ µ (1 − γ ) ˜ d i W − µ + h.c (7.77)with the implied sum for i = 1 , , W + µ term). Expressing the d -type quarks as a linear combinationof the mass states the above Lagrangian can be written in terms of the six physical162uarks as: L qW = − g √ (cid:16) ¯ u ¯ c ¯ t (cid:17) γ µ (1 − γ ) V CKM d ′ s ′ b ′ W − µ + h.c (7.78)In these equations the operator P L = (1 − γ ) of equation 7.11 has been put inby hand to project out the left-handed components of the Dirac spinors, describingmaximal parity violation for the charged weak current. This CKM mixing originatesfrom the mismatch between the Yukawa and weak interactions in the Standard ModelLagrangian, with the corresponding mass and weak quark eigenstates for the u and d -type quarks related via unitary transformations such as equations 7.71 and 7.72. On theother hand the neutral currents are flavour-diagonal and such terms are unchanged bythe unitary transformations relating the mass and weak states, that is with A † u L A u L = and so on. Hence there are no flavour changing neutral currents coupled to the Z µ or A µ fields, and only the W ± fields mediate mixing between the generations. While the action of E on h O studied in chapter 6 describes a symmetry of time it isalso of course desirable that the mathematical structures arising in the present theoryshould bear a close resemblance to the symmetries and structures experimentally iden-tified in particle physics. This data is summarised in the Standard Model, as reviewedin the previous two sections, which describes the non-gravitational interactions betweenfundamental particles in terms of the gauge symmetry group SU(3) c × SU(2) L × U(1) Y .Hence in this section we make a preliminary assessment of the suitability of the Liegroup E , both generally and as constructed in chapter 6, as a unifying symmetry.It is well known that the three subgroup components of the Standard Modelgauge symmetry are related to the series of normed division algebras, as introducedhere in section 2.1 in the context of forms of temporal flow and discussed further insection 6.2. Indeed, U(1) is isomorphic to the complex numbers C of unit magnitudeunder multiplication, while SU(2) is similarly isomorphic to the quaternions H of unitmagnitude, and SU(3) is the subgroup of G , the automorphism group of the octonions O , that leaves invariant a given imaginary octonion element. The aesthetic appeal andelegance of such observations have led a number of authors to speculate on a directconnection between the existence of these unique mathematical objects and the natureof the physical structure of the world (see for example [47, 48, 49, 50]). However, whileidentifying a relationship between the mathematical properties of the division algebrasand features of the Standard Model of particle physics much of this work is lacking inany underlying conceptual motivation for the importance of such mathematical objectsin nature.Since the octonion algebra features significantly in the present paper, in theaction of the group E ≡ SL(3 , O ) on the space h O , the references cited above suggest areasonable likelihood of identifying some relation between the structures of the present163heory and those of the Standard Model. Such a correspondence will be described in thefollowing chapter. In the present theory we have both a clear conceptual understandingof the source of these algebras through the symmetry of the flow of time and in turna well defined constraint on the introduction of these algebraic structures into theequations of physics through the relation L ( v ) = 1 and its symmetries.Also in the present theory, as well as aiming to account for the internal gaugeinteractions of the Standard Model through the higher-dimensional structures, gravita-tion is included on the base manifold M with a subspace h C ⊂ h O locally identifiedwith the tangent space TM and with the subgroup SL(2 , C ) ⊂ E being the cover-ing group of the external Lorentz group. As described towards the end of section 3.4general relativity can be presented in the form of a gauge theory with a local Lorentzsymmetry constructed in terms of the components of both a Lorentz Lie algebra-valuedconnection A abµ ( x ) and a tetrad field e aµ ( x ). In terms of the covering group it can inturn be considered to be an SL(2 , C ) gauge theory with an sl(2 , C )-valued connection,which can accommodate a description of both vector and spinor objects in spacetime.While the dynamics of such an SL(2 , C ) ‘gauge theory’ of gravitation [51, 52] aredifferent to those of a standard Yang-Mills gauge theory, as also described in section 3.4,an extension for internal gauge symmetries might be more readily achieved with sucha theory of gravity. (Considering gravity as a gauge theory contrasts with the Kaluza-Klein approach reviewed in chapter 4 for which an internal gauge theory derives fromgeneral relativity with extra spatial dimensions.) Indeed an SL(2 , C ) × U(1) theory ofgravitation and electromagnetism can be obtained by introducing an additional e iα ( x ) phase factor element for the group U(1). This can be achieved by augmenting theset of symmetry actions S ∈ SL(2 , C ) with det( S ) = 1 to include also the actions U = e iα ( x ) ∈ U(1). The mapping of equation 7.31, now incorporating also the U(1)action h → U h U † , then remains one that preserves the value of det( h ) and leavesthe metric g µν ( x ) on M invariant, as for the original SL(2 , C ) action. Further, the U(1)action in fact leaves each of the four components of h invariant and hence effectivelyacts as an ‘internal symmetry’, as described also at the end of section 6.3. Withinthe set of E symmetry actions on h O the action of S \ q in equation 6.43, particularlyon a type 1 h C ⊂ h O subspace, is most reminiscent of the above U(1) symmetryaction on h C and this property is suggestive for the choice of the U(1) Q action forthe electromagnetic gauge symmetry in the present theory.By further augmenting the internal degrees of freedom such unification schemeswhich begin with an SL(2 , C ) theory of gravity can be extended to an SL(2 , C ) × G theory where G may be the gauge symmetry group for the internal forces as identifiedexperimentally in the Standard Model, that is SU(3) c × SU(2) L × U(1) Y . For sucha model there remains the task of introducing states which transform as fermionsunder the external SL(2 , C ) symmetry and under the appropriate representations ofthe internal symmetry as summarised in equation 7.36. However such an approach,with the appropriate interpretation of the gauge groups and their empirically motivatedrepresentations, only serves to describe gravity together with internal field interactionsin a more unified framework.In the present theory, however, the unification group ˆ G = E includes the ex-ternal spacetime symmetry central to general relativity in the form of the subgroupSL(2 , C ) ⊂ E . It is then through the distinctive role of this subgroup, in the identifi-164ation of the necessary perceptual background for the world, that the larger symmetryis broken down to local gauge groups with representations on the broken fragments ofthe space h O . The local gauge groups themselves will be initially identified as the‘stability’ group leaving the space of vectors v ∈ TM , via equation 7.30 equivalentto h ∈ h C ⊂ h O , invariant, generalising from the above case of the e iα ( x ) ∈ U(1)action on h C .Having at hand the real form of E acting on h O , as described in the previouschapter, a detailed study of this symmetry breaking over TM is possible. Initially,however, in this section the symmetry breaking patterns for E and the question ofwhether this group is large enough to actually contain both SL(2 , C ) and SU(3) × SU(2) × U(1) will be addressed at the level of the complex
Lie algebras, in order togain an overview, before returning to the specific real forms of these algebras in thefollowing chapter.One of the main motivations for studying the complexified forms of real Liealgebras in general is the existence of a concise classification scheme. Indeed, everycomplex simple Lie algebra belongs to one of just four sets of classical algebra types,which include the complex forms of the rotation algebras so( p, q ), or is otherwise identi-fied with one of the five exceptional cases, which include L (E ). A further motivation isthat each complex simple Lie algebra has a one-to-one correspondence with a ‘Dynkindiagram’, with semi-simple Lie algebras likewise corresponding to disconnected Dynkindiagrams. The analysis of such diagrams gives a good deal of guidance towards thepossible symmetry breaking patterns for a complex Lie algebra and its real forms asencountered in the context of a theoretical model for physical phenomena.Firstly, we briefly review the relationship between Lie algebras and their repre-sentations. In general, each complex simple Lie algebra, as exemplified by the Dynkindiagrams shown later in this section, and taking its place amongst the systematic clas-sification of such algebras, may be associated with several real forms, with each realalgebra in turn associated with one of more Lie group, and finally each Lie group pos-sesses an unlimited number of representations. This situation is depicted in figure 7.1.Figure 7.1: Any given complex Lie algebra L C , which has a unique Dynkin diagram,is in general associated with a multiplicity of real algebra forms L R , groups G andrepresentations R .While Dynkin analysis at the level of L C is described in this section, in thispaper we generally deal with the structures of L R and G , with notation such as so( p, q )used for a real Lie algebra and SO( p, q ) for the related Lie group, with the distinctionbeing otherwise understood from the context. As an example of the chain of relationsin figure 7.1 the case for L C = so(10) with links through to the R = representation,of particular interest here and featuring for example in equation 8.3 in the opening ofthe following chapter, is described in table 7.1.165 C L R G R
O(1,9)SO(1,9) so(10) SO + (1,9) so(10) րց → → so(1,9) րց → → Spin + (1,9) րց → → so(2,8) Spin(1,9) ... Pin(1,9) ...Table 7.1: The complex Lie algebra so(10), called D in Cartan’s notation, with cor-responding real algebra forms L R , groups G and representations R (labelled by theirdimension), as an example of the general case depicted in figure 7.1, with a particularchain of forms discussed in the text highlighted by the horizontal arrows.In developing a theoretical model the initial motivation often begins from theleft-hand side of figure 7.1, by identifying a complex Lie algebra which exhibits anappropriate symmetry breaking pattern to account for the gauge groups of the Stan-dard Model as described in previous section; and then the task remains to identify theappropriate representations for particle states such as those of equation 7.36. In thispaper such an approach also serves as a useful guide, as we describe in this section.However, here our starting point is rather more anchored in the right-hand side offigure 7.1 since the mathematical form L ( v ) = 1 strongly motivates the possible rep-resentations, with the set of real numbers composing the vector v already belongingto a representation space transforming under the relevant symmetries of L ( v ) = 1.As a preliminary observation we note that given our use of the R = repre-sentation of the particular group G = E − , this uniquely leads back via the real Liealgebra L R = L (E − ) to the complex Lie algebra L C = L (E ) as we step from rightto left through figure 7.1. The structure of symmetry breaking feeding down from thecomplex Lie algebra is largely preserved in terms of semi-simplicity of the algebra andgroup and in terms of the reducibility of the algebra and group representations. Hencewe here consider the Dynkin diagrams for the relevant complex Lie algebras and thesignificant Lie subalgebras involved.The ‘rank’ of a Lie algebra is the dimension of the Cartan subalgebra, com-posed of a maximal subset of mutually commuting generators, which is unique up toautomorphisms of the Lie algebra. For a rank- n Lie algebra there are n ‘simple roots’in the dual root space which is constructed out of the eigenvalues in the adjoint rep-resentation of the algebra in the Cartan-Weyl basis. The properties of a rank- n Liealgebra can be described in terms of geometric relations between these simple rootsin the Euclidean R n root-space and encoded in the topology relating the n nodes ofthe corresponding Dynkin diagram, such as those depicted in figure 7.2 for the rank-6 L (E ), rank-5 so(10), rank-4 su(5) and rank-2 Lorentz Lie algebras. For example, theDynkin diagram for the Lorentz algebra consists of two disconnected nodes, mean-ing that the corresponding two simple roots are at 90 in root space, whereas nodes166igure 7.2: The four Dynkin diagrams for the (a) L (E ), (b) so(10), (c) su(5) and(d) Lorentz or sl(2 , C ) Lie algebras.connected by a single line denote an angle of 120 . At the level of the complexifiedLie algebra L C the Lorentz algebra has the semi-simple composition su(2) ⊕ su(2), asdescribed earlier in equations 7.21–7.23, which in this case is not respected by thecorresponding real form L R = so + (1 ,
3) of the Lorentz Lie algebra which is simple.An explicit basis for the Cartan subalgebra for the real form of L (E ) of importancein this paper was given in equation 6.56 as represented by vector fields on the space T h O . Regular subalgebras, that is those respecting the Cartan-Weyl decompositionof the complex Lie algebra, may be readily obtained from the Dynkin diagrams. Amaximal subgroup G ′ ⊂ G is one for which there is no intermediate G ′′ such that G ′ ⊂ G ′′ ⊂ G as a series of proper subgroups, with a similar definition for the correspondingmaximal subalgebra. A regular maximal subalgebra can be obtained from a Dynkindiagram by the prescription of removing one node and including an extra U(1) factor,which also means that the algebra obtained is not ‘semi-simple’. For example figure 7.3shows a possible symmetry breaking pattern for the su(5) algebra for the well-knowncase [43], as alluded to in the previous section, in which the Standard Model localgauge group is obtained.Figure 7.3: Removing a node from the Dynkin diagram for the Lie algebra of the groupSU(5) reveals a breaking to SU(3) × SU(2) × U(1), which motivates the use of SU(5)in unified theories.Similarly from figure 7.2 it can be seen that SO(10) contains SU(5) as asubgroup, by removing either of two appropriate end nodes. Hence the full Stan-dard Model gauge group can be obtained by first breaking SO(10) to SU(5) andthen breaking SU(5) as described in figure 7.3. Hence the 45-dimensional groupSL(2 , O ) ≡ Spin + (1 ,
9) constructed in section 6.3 as the double cover of SO + (1 , automorphism group of O is reduced to the subgroup SU(3) ⊂ G if a complexsubspace, for example with the imaginary unit l ∈ O , is fixed, as also alluded to nearthe opening of this section. Similarly the subgroup SU(3) ⊂ G ⊂ SO + (1 ,
9) may beobtained through the selection of a preferred subspace h C ⊂ h O , since this choice alsofixes an imaginary unit of h O . Here the mechanism for such a selection is provided bythe nature of perception on the base manifold M with the vector space TM ≡ h C andh C ⊂ h O through the identification of an external SO + (1 , ⊂ SO + (1 ,
9) symmetry.167owever breaking the rank-2 Lorentz group out of the rank-5 so(10) clearly does notleave sufficient symmetry to describe the full rank-4 Standard Model gauge group.It was also shown in section 6.4 how 3 copies of SL(2 , O ), described with atotal of (3 ×
45) = 135 generator actions, lock tightly together as an independentbasis set of 78 generators, summarised in table 6.3, for the E action on X ∈ h O preserving det( X ). This space hence describes a highly symmetric form of temporalflow L ( v ) = 1 motivating the study of this exceptional Lie group.As well as composing a rich symmetry of a multi-dimensional form of L ( v ) = 1,additional motivation for the use of E indeed comes from the fact that this Lie group iswell known as a good candidate for the unifying symmetry group in models describinga unification of the non-gravitational fundamental forces of nature. Further, unlike thetwo larger exceptional Lie groups, E and E , the group E has complex representationsand these are needed to describe the observed multiplets of states in particle physics ofequation 7.36 which are not left-right symmetric. From figure 7.2 it can be seen thatE contains SO(10) and hence in turn SU(5) and finally also the Standard Model gaugesymmetry, with the chain of subgroups: E ⊃ SO(10) ⊃ SU(5) ⊃ SU(3) × SU(2) × U(1).The potential of E as a unifying group has been known since the early history of theStandard Model of particle physics even as it was still taking shape in the 1970s (seefor example [53]) and continues today (see also, for example [54] pp.302–308).In this case the higher rank of E over that of SU(5), with 2 additional Dynkinnodes, suggests that in principle the physical phenomena of the rank-2 Lorentz trans-formations might be described alongside the rank-4 Standard Model gauge groupwithin the full the rank-6 symmetry group E . However it is not possible to breakE into the combined Lorentz and Standard Model algebras by the Dynkin analy-sis prescribed above. While it can be shown that E contains subgroups such asSU(3) × SU(2) × U(1) × SU(3), for example by removing the central node in fig-ure 7.2(a), a similar decomposition but with a rank-2 SU(3) replaced by the rank-2Lorentz group is not possible. An alternative prescription for obtaining semi-simpleregular maximal subalgebras via an intermediate ‘extended’ Dynkin diagram does nothelp this situation. However, to some extent this Dynkin analysis oriented within theCartan-Weyl basis for complex forms of the Lie algebras represents a ball-park pictureand is not tailored to fit the fine details for a real form of E represented within thecontext of a specific theory.To study these details not only is the real form of the group action needed butalso an understanding of how the dynamics arises, and the means by which a symmetrysubgroup of L ( v ) = 1 might be associated with gauge field interactions, in order toaccount for the phenomena observed in the laboratory. In particular the structure ofthe symmetry breaking itself, involving the extended spacetime manifold M , will needto be considered more explicitly. In the meantime, the observation that the Lorentzgroup and Standard Model gauge groups almost fit together at the level of this static Dynkin diagram analysis is an encouraging feature.In principle then, the possibility of identifying features of the full gauge symme-try group for the strong and electroweak particle interactions for the theory presentedhere based on the E symmetry of L ( v ) = 1 is worth pursuing, as we explore inthe following chapter. It is further noted that the representation in table 7.1, thatis the Majorana-Weyl spinor introduced in section 5.4 and described in the following168ection, as exemplified by the Spin + (1 ,
9) spinor θ of equations 6.26 and 8.2, possessesa branching pattern under the SU(3) × SU(2) × U(1) ⊂ SO(10) subgroup into repre-sentation multiplets corresponding to the 15 particle types of a complete generation ofStandard Model fermions of equation 7.36 (plus a right-handed neutrino). However a different approach will be followed here, involving both the incorporation of the exter-nal Lorentz symmetry within Spin + (1 ,
9) as well as the extension to the E symmetry.Indeed we begin in the opening section of the following chapter by identifying objectswhich transform as fermions under the external symmetry.We also note here the possible significance of the three possible embeddings ofan h O subspace, as represented by the components X ∈ h O in equations 6.32, 6.34and 6.35, within the space h O , with equivalent symmetry transformation properties,and in particular three copies of the Spin + (1 ,
9) spinor θ representation. These threeembeddings are related by the matrix T of equation 6.33, as described in section 6.4,and in terms the octonion triality isomorphism as discussed alongside equations 6.49and 6.50, relating to the rich symmetry of this form of L ( v ) = 1. This is suggestivesince we shall have to account for three generations of fermion families, related throughthe CKM matrix of equation 7.76 in the case of the quarks, which might here be relatedthrough the full set of E symmetry transformations. On the other hand only one embedding of h C ⊂ h O will be associated with the local tangent space TM in thesymmetry breaking, potentially lifting the degeneracy between the three generationsof fermions in the present theory.Again, while the connection between some of these algebraic structures and theStandard Model is well known, here there is an underlying motivation for the origin ofthese mathematical forms in a physical theory based on the symmetries of L ( v ) = 1representing a multi-dimensional form of temporal flow.Considering then the demands from both ends of figure 7.1 at the same time,with the choice of L C guided by general features of the Standard Model and the space R identified under a highly symmetric form of L ( v ) = 1, we naturally converge uponthe group E acting on the representation space h O , such that the matrix determi-nant is invariant, as being of particular interest. Indeed this motivated the detailedstudy in chapter 6 based on references [37, 38, 39, 40, 41]. Further, the identification ofthe Lorentz subgroup of E acting upon the subspace h C representing 4-dimensionalspacetime explicitly provides the symmetry breaking mechanism through which thebroken internal subgroups of the larger symmetry may be realised as the local gaugegroups. The symmetry breaking was pictured in figure 5.1 for the provisional modelwith an SO + (1 ,
9) symmetry acting on the form L ( v ) = 1. That case for a 10-dimensional spacetime symmetry, now described by Spin + (1 ,
9) acting on h O , consti-tutes a significant intermediate stage between the full 27-dimensional form of temporalflow and the external 4-dimensional spacetime structure.In order to analyse the physical content of this theory it will be necessaryto dissect the anatomy of the explicit real form of E constructed in chapter 6 inthe context of symmetry breaking over the extended M manifold. In the followingchapter we first study the action of the external Lorentz symmetry on the full set ofh O components, building on the analysis of equation 7.35 presented at the end ofsection 7.1, and then assess how the properties of the internal symmetry, surviving thesymmetry breaking, compare with the Standard Model.169 hapter 8 E Symmetry Breaking O Having at hand a complete mathematical description of the E symmetry action fromchapter 6, preserving the determinant on the space h O as a form of L ( v ) = 1, thephysical significance of various subgroup actions can be considered locally with re-spect to the spacetime manifold M . In particular a distinguished set of symmetrytransformations will act on the components of v ∈ h O lying in the local spacetimetangent space TM . These transformations form the subgroup SL(2 , C ), the doublecover of the Lorentz group, which is identified then as the external symmetry group.This spacetime symmetry is central to general relativity, while in the flat spacetimelimit these Lorentz transformations form a global symmetry on M as for the theoryof special relativity. With the flow of time expanded into the 27-dimensional space of3 × O there are 23 extra dimensions beyond thoseneeded to locate events taking place in our 4-dimensional spacetime world. The ex-plicit action of the external Lorentz symmetry on all components of the space h O willbe described this section, based on the real form of E as constructed in chapter 6.The form of h O matrices transforming under the type 1 SL(2 , C ) and SL(2 , O )subgroups of E , with the structure described in equation 6.29, is compatible with theisomorphism of vector spaces ([1] p.30):h O ∼ = R ⊕ h O ⊕ O (8.1) X θ (cid:16) θ † (cid:17) n → ( n, X, θ ) (8.2) E → ( + + ) Spin + (1 , (8.3)The three parts of this decomposition are respectively the scalar, vector and spinor170epresentations of the 10-dimensional spacetime symmetry group SO + (1 , + (1 , ≡ SL(2 , O ). A spinor representation with both Ma-jorana and Weyl properties is only possible for d = (2 , mod 8) spacetime dimensions,as is the case for SO + (1 , θ corresponds to the Majorana-Weyl spinorrepresentation, denoted , which can be described by 16 real numbers owing to thereality condition for Majorana spinors (in general a Majorana spinor ψ is one which isequal to its ‘charge conjugate’ ψ c , this reality condition is also possible in 4-dimensionalspacetime).As described in [1] the decomposition of equations 8.1–8.3 gives a representa-tion of Spin + (1,9) as linear transformations of h O which do not preserve the Jordanalgebra but do, importantly for the present considerations, preserve the determinant ofh O , as presented explicitly in equation 6.31 of section 6.4. The relationship betweenthe complex Lie algebra L C = so(10), its real forms, the group Spin + (1 ,
9) and its rep-resentations was presented explicitly in table 7.1. Similarly as for so(10) in the Dynkinanalysis of section 7.3 we can consider the above decomposition as a mathematicallyintermediary stage in studying the Lorentz subgroup, Spin + (1 , ≡ SL(2 , C ), in E .While the 27-dimensional irreducible representation of E decomposes as a re-ducible representation of Spin + (1 , v = ( v , v , v , v ) ≡ h in the upperleft-hand 2 × × O , as was the casefor h C ⊂ h C in equation 7.35. The relation det( X ) = 1 with X ∈ h O is preservedunder operations of SL(2 , C ) representing the Lorentz group upon this space as: X → S
00 1 h h + a (6) ch + a (6) h bc b n S †
00 1 (8.4)with h = v + v , h = v − v lh = v + v l, h = v − v (8.5)with S ∈ SL(2 , C ), and with ‘1’ describing the identity transformation in the triv-ial 1-dimensional representation of this group, acting upon the components of X ofequation 6.26. This action preserves the value of det( h ) = h , as it is simply thetransformation of equation 7.31, as well as leaving det( X ) = 1 invariant. In equa-tion 8.4 a (6) denotes the 6-dimensional imaginary part of a ∈ h O of equation 6.26,that is excluding the real a = v and imaginary a l = v l components of a ∈ O whichare associated with the external 4-vector v ∈ TM .The four components of the projected v ( x ) ⊂ v ( x ), forming a tangent vectorin TM locally on the spacetime manifold M , transform as the components of a Lorentz4-vector. These components are embedded within the space h O via the 2 × h ∈ h C . While in section 7.1 { , i } denoted the base units for the space C , for171xample for σ in equation 7.14 as used in equation 7.30 (and also in section 6.3, forexample equation 6.19), here the preferred subspace C ⊂ O basis is taken to be { , l } for v , as indicated in equation 8.5, in conformity with the conventions of sections 6.4and 6.5, and in particular equation 6.58, and as employed in the following section.Since the SL(2 , C ) actions, based on this { , l } complex subspace are embedded in the‘type 1’ location of equation 6.32 this group will be denoted SL(2 , C ) .The full set of actions of the real form of E on the space h O was constructed insection 6.4. With the group action of SL(2 , C ) on h O in equation 8.4 embedded withinthe type 1 group action of SL(2 , O ) on the same space as displayed in equation 6.29we can write: SO + (1 , ≡ SL(2 , C ) ⊂ SL(2 , O ) ⊂ SL(3 , O ) ≡ E (8.6)where the first ‘ ≡ ’ strictly applies at the Lie algebra level. This shows explicitly howthe action of the Lorentz group may be embedded within the higher symmetry groupE acting on the space h O . The direct physical interpretation of the former symmetryin the shape of the perceptual background of the spacetime manifold M provides adirect source for the breakdown of the latter symmetry.The six Lorentz group generators as a subset of the 78 E generators werelisted in equation 6.57 of section 6.5. They can be read off from the full E Liealgebra table [38] and seen to satisfy the SO + (1 ,
3) algebra which is reproduced herein table 8.1. [ • , • ] ˙ R zl ˙ R xz ˙ R xl ˙ B tx ˙ B tl ˙ B tz ˙ R zl − ˙ R xl ˙ R xz − ˙ B tz ˙ B tl ˙ R xz ˙ R xl − ˙ R zl ˙ B tz − ˙ B tx ˙ R xl − ˙ R xz ˙ R zl − ˙ B tl ˙ B tx B tx − ˙ B tz ˙ B tl R xl − ˙ R xz ˙ B tl ˙ B tz − ˙ B tx − ˙ R xl R zl ˙ B tz − ˙ B tl ˙ B tx R xz − ˙ R zl Lie algebra table in [38]). The Lie algebra struc-ture for the set of Lorentz generators of equation 6.57, with bracket composition[ ˙ R zl , ˙ R xz ] = − ˙ R xl etc. The type superscripts ‘1’ are omitted in the table entries,which are all generators of SL(2 , C ) . (Each entry is equivalent to that for the cor-responding 6 × { J , − J , J , − K , K , − K } with the Liebracket of equations 7.17–7.19, via the correspondence of equation 8.9).The corresponding 2 × q = l in table 6.1 of section 6.3. Since each of theseactions involves the composition of matrix elements from a single complex subspace,with base units { , l } , and with each of a, b, c ∈ O (or p, m, n ∈ R ) as elements of h O appearing in separate product terms, the symmetry transformations are equivalent tothose based on H subalgebras and are hence associative. Consistent with the discussion172n the paragraphs following equation 6.35 this means that the symmetry group andcorresponding Lie algebra can be represented in terms of the transformation matricesthemselves. A matrix representation for the Lorentz Lie algebra is therefore providedby defining ˙ M = ∂∂α M (cid:12)(cid:12)(cid:12) α =0 for the corresponding six matrix actions in table 6.1 (herepresented in a different order), that is:˙ M zl = − l − l , ˙ M xz = − , ˙ M xl = − l
00 + l , (8.7)˙ M tx = + , ˙ M tl = l − l , ˙ M tz = + − , (8.8)where the latter three are the boost generators as can be identified by the time com-ponent ‘ t ’ label in the subscript. Expressing the three Pauli matrices as σ = (cid:0) (cid:1) , σ = (cid:0) − ll (cid:1) , σ = (cid:0) − (cid:1) , that is equation 7.14 with i replaced by the imaginary unit l ,the six elements of this Lorentz Lie algebra can be written as: ˙ M zl ˙ M xz ˙ M xl = − l σ + l σ − l σ ∼ + J − J + J , ˙ M tx ˙ M tl ˙ M tz = + σ − σ + σ ∼ − K + K − K (8.9)The associations with the Lorentz rotation { J a } and boost { K a } generators ofequation 7.16 are such that with { J , K , K } → {− J , − K , − K } the Lie algebra ofequations 7.17–7.19 matches that of the commutators in table 8.1. Hence in the contextof the SL(2 , C ) action in equation 8.4 these sign conventions, { ˙ R, ˙ B } ∼ ±{ J, K } ,in equation 8.9 should be noted, which at the group level simply corresponds to asign flip for a subset of the six real parameters { r a , b a } in equation 7.20, and hencein turn will relate to the definition of left and right-handed spinors. As describedin the discussion following table 6.5 in section 6.5 here the key orientation for suchconventions is provided by the E Lie algebra table of reference [38] from which table 8.1is extracted. Ultimately a different set of L (E ) sign conventions may be preferred, inalignment with the physical application.We next address the action of the external Lorentz symmetry on a generalelement X ∈ h O , including the full set of 16 real components of θ = (cid:0) c ¯ b (cid:1) ∈ O , that isthe 16-dimensional Majorana-Weyl spinor under SL(2 , O ) , composed of the octonionentries c and ¯ b , as introduced in equation 6.26 and described after equations 8.1–8.3.The two-sided SL(2 , C ) action on h O in equation 8.4 only transforms the realdiagonal entries h and h together with the h = a + a l and h = a − a l components of a ∈ O . The six components of a (6) ∈ Im( a ) remain invariant as may bededuced from the form of the six SL(2 , C ) generators in table 6.6 or from equation 6.54for the case q = l . (This is equivalent to the invariance of v under SO + (1 ,
3) for themodel of figure 5.1). Of the additional 17 components in h O the real diagonal n entryis also invariant, as is clear from equation 8.4, while all
16 components of b, c ∈ O transform non-trivially under the one-sided SL(2 , C ) action.173he spinor θ l = (cid:0) c ¯ b (cid:1) l ∈ C will denote the { , l } components of c and ¯ b in θ , that is (cid:0) c ¯ b (cid:1) ∈ O restricted to the { , l } complex subspace. By comparison withequation 7.35 this object transforms as a left-handed Weyl spinor ψ L = θ l under theSL(2 , C ) action in equation 8.4. Consistent with the above comments on the signconventions for the Lorentz generators here we take this S ∈ SL(2 , C ) action on θ l to define the left-handed spinor representation, guided but not constrained by thestandard definitions of section 7.1.Due to the anticommuting property, for example in equation 7.1, Clifford alge-bras are also related to the division algebras ([1] section 2.3), with analogous rotationalproperties as alluded to following equation 7.4. Indeed it can be shown, for example,that C (0 ,
2) = H for the Clifford algebra associated with the 2-dimensional vector space R , , while C (1 ,
3) = H (2), that is the Clifford algebra for 4-dimensional spacetime isisomorphic to the algebra of 2 × C ( p, q ) is in all cases an associative algebra there are no such isomorphismsinvolving the octonion algebra).As a representation of C (1 ,
3) the algebra H (2) acts, by matrix multiplication,on the spinor space H rather than the usual Dirac spinor space C . We consider firstthe quaternionic spinor as a subspace of the octonionic spinor θ = (cid:0) c ¯ b (cid:1) with base units { , l, i, il } : θ H = c ¯ b H = c + c l + c i + c ilb − b l − b i − b il ∈ H ⊂ O (8.10)Upon restriction to the subset of 2 × , C ) ⊂ SL(2 , H ) ⊂ H (2)(which is isomorphic to the group Spin + (1 ,
3) as identified within the Clifford algebra C (1 ,
3) = H (2)), with base units { , l } , the spinor space θ H decomposes into two parts: θ l = c + c lb − b l and θ i = c il + c i − b il − b i (8.11)which transform independently. As was described for equation 6.14 the group ac-tions are considered as active transformations. Further, under the left action by theimaginary unit l the components of c transform as:( c + c l ) → l ( c + c l ) = ( − c + c l )( c il + c i ) → l ( c il + c i ) = ( − c il + c i ) (8.12)and hence the ( c , c ) components of θ i transform under left multiplication by l in anidentical manner to the respective components ( c , c ) of θ l , which is also trivially truefor multiplication by the real unit 1. This observation applies also to the ¯ b componentsof θ l and θ i , while the structure of the 2 × , C ) applies inthe same way on both of these objects. In fact the transformations of θ l and θ i inequation 8.11 are identical both for the generators of SL(2 , C ) and for the finite groupactions, as can be readily seen by explicit calculation. For example applying theLorentz symmetry rotation matrix M zl ( α ) from table 6.1 to θ l and θ i results in the174espective transformations: R zl ( α ) θ l = cos α − l sin α − l sin α cos α c + c lb − b l = (cos α c − sin α b ) + ( − sin α b + cos α c ) l (cos α b + sin α c ) + ( − sin α c − cos α b ) l R zl ( α ) θ i = cos α − l sin α − l sin α cos α c il + c i − b il − b i = (cos α c − sin α b ) il + (sin α b + cos α c ) i ( − cos α b + sin α c ) il + ( − sin α c − cos α b ) i Here it can be seen that the four real coefficients { c , c , b , − b } of the spinor θ l map onto the components of R zl ( α ) θ l in precisely the same way that the coefficients { c , c , − b , − b } of θ i map onto the components of R zl ( α ) θ i . A similar observationapplies for θ l and θ i under the remaining five Lorentz symmetry actions. Hence as wellas the original left-handed Weyl spinor θ l the components of θ i also transform exactlyas a left-handed spinor of SL(2 , C ) . This representation of SL(2 , C ) on the two left-handed spinors θ l and θ i of equation 8.11 in H contrasts with the representationconstructed in equations 7.15 and 7.29 on the left and right-handed spinors ψ L and ψ R in C .Considering the further two quaternionic subspaces with base units { , l, j, jl } and { , l, k, kl } it can be seen that the original full octonionic spinor θ = (cid:0) c ¯ b (cid:1) , with 16real components, reduces to a total of four left-handed Weyl spinors under the actionof SL(2 , C ) , augmenting the set in equation 8.11 to: θ l = c + c lb − b l , θ i = c il + c i − b il − b i , θ j = c jl + c j − b jl − b j , θ k = c kl + c k − b kl − b k (8.13)There is an equivalent decomposition for a corresponding set of conjugate spinors in θ † = (¯ c b ) under the right action of S † on h O as implied in equation 8.4. Thisset of four Weyl spinors in equation 8.13 will be important for interpreting furthersymmetries, internal to the action of SL(2 , C ) on h O , in the following section.In this section we have described how the decomposition of the representa-tion of E under the subgroup Spin + (1 ,
9) of equation 8.3 further reduces under thesubgroup SL(2 , C ) as summarised in table 8.2.This may be compared with the SL(2 , C ) action on the subspace h C ⊂ h O as described in equation 7.35 for which the nine real components of h C transform asone 4-vector h , one Weyl spinor ψ L and one scalar n (closely relating to v , θ l and n respectively in table 8.2). The six extra scalars and three extra spinors in table 8.2result from the additional 27 − O . In both cases eachWeyl spinor, as for the space C , has four real parameters.175pin + (1 ,
9) SL(2 , C ) Components scalar (0,0) scalar n vector ( , ) vector6 × (0 ,
0) scalars v a (6) spinor 4 × ( ,
0) spinors θ l,i,j,k Table 8.2: The further decomposition of the ( + + ) representation ofSpin + (1 , ⊂ E of equation 8.3 under the subgroup of external 4-dimensional space-time symmetry SL(2 , C ) ⊂ Spin + (1 ,
9) actions of equation 8.4, and the correspondingcomponents of h O transformed.The spinor components of θ l,i,j,k represent ‘internal’ dimensions of the spaceh O in the sense that, unlike v ∈ TM , they are not tangent to the external spacetime M , but they do transform in a non-trivial manner, as spinors, under the externalSL(2 , C ) symmetry, and in this sense they are not purely internal objects. Thisfeature for the cubic form of temporal flow L ( v ) = 1 is hence distinct from that seenfor a quadratic form with a spacetime symmetry. For example for the 10-dimensionalspacetime form considered in section 5.1 the external SO + (1 ,
3) symmetry acts on theexternal v ⊂ v components only , as pictured in figure 5.1(b) and applies also forthe corresponding gauge field in equation 5.51, as for the external symmetry of anyhigher-dimensional spacetime structure. For the present theory based on temporalprogression, here taking a cubic form, of particular interest in the following sectionwill be the nature of the internal symmetry transformations on the four SL(2 , C ) spinors from the final line of table 8.2. (3) c × U (1) Q Symmetry
Physically the SL(2 , C ) symmetry studied in section 8.1 is considered ‘external’ as it isthe two-to-one cover of the Lorentz group which in the full theory acts on the tangentspace TM of the extended 4-dimensional spacetime manifold. This structure is centralto the theory of general relativity and gravitation, as described in sections 3.3, 3.4 and5.3. On the other hand the ‘internal’ symmetry will consist of further subgroups of E ,which will be central to the structure of local gauge theories and the Standard Modelof particle physics as reviewed in the previous chapter.At the end of the previous section the branching of the representationof Spin + (1 ,
9) into a set of four Weyl spinors under the external Lorentz subgroupSL(2 , C ) was described, as listed in table 8.2. Independently it is also known that thesame Spin + (1 ,
9) Majorana-Weyl 16-dimensional representation branches into a set ofmultiplets describing the 15 states of one generation of Standard Model quarks andleptons, as listed in equation 7.36, together with a right-handed neutrino, all expresseduniformly in terms of left-handed fields, under the internal subgroup SU(3) c × SU(2) L × U(1) Y , as noted in section 7.3, but it is not the approach we follow here. In this176ection we consider the internal symmetry derived from the subgroup Stab( TM ) ⊂ E ,defined below, and its relation to the set of four Weyl spinors derived from the externalsymmetry SL(2 , C ) ⊂ Spin + (1 , , C ) symmetry, of theremaining (78 −
6) = 72 generators of E those which leave all tangent space vectors v ∈ TM untouched may literally be considered to constitute an internal symmetry,surviving the symmetry breaking, and are expected to be significant for the physicsof local gauge theories. While leaving TM invariant these internal symmetries willin general have non-trivial actions on the remaining, ‘extra dimensions’ within v ,through which we may seek to identify a relation with the phenomena of physicalparticle interactions as observed in the laboratory and described by the StandardModel. Here then, as a preliminary definition, and in contrast to the external symmetry,the internal symmetry will be obtained from the set of all E actions on h O which leavethe four components for any v = ( v , v , v , v ) in equation 8.5 (that is, h ∈ h C of equation 8.4) invariant. These components, including v ≡ a associated with theimaginary unit l of a ∈ O , can also be expressed in the combination ( p, m, a , a ) withrespect to the parametrisation of equation 6.1. The corresponding symmetry groupis complementary to the actions of SL(2 , C ) and will be denoted Stab( TM ) as thestability group of all vectors v ∈ TM . By inspection from tables 6.6 and 6.7, forthe 78 elements in the preferred basis for the Lie algebra of E defined on the space T h O , the group Stab( TM ) is generated by the 31 elements listed in table 8.3. Inparticular we shall be looking to identify closed subgroups within Stab( TM ) for whicheach generator is independent of SL(2 , C ) in terms of Lie bracket composition.Category 1 and 2: Boosts and Rotations R xz − ˙ B tx ) , ( ˙ R xz + ˙ B tx ) 2( ˙ R zq + ˙ B tq ) , ( ˙ R zq − ˙ B tq ) 14Category 3: Transverse Rotations˙ A q , ˙ G l , ˙ S l
9( ˙ G q + 2 ˙ S q ) q = { i, j, k, kl, jl, il } TM ). Thesubscript q denotes any of the seven imaginary octonion units { i, j, k, kl, jl, il, l } unlessstated otherwise.The 16 vector fields on T h O generating the Category 1 and 2 elements ofStab( TM ) are written out explicitly in equations 8.14 and 8.15 in which the invariantaction on the 4-dimensional subspace h C ⊂ h O is clear. (In fact they leave all 10177omponents of h O ⊂ h O invariant, see also [41] equations 4.12(27) and 4.13(28)).˙ R xz − ˙ B tx = − ¯ a − m − a − m − b x , ˙ R xz + ˙ B tx = p ap ¯ a c x (8.14)˙ R zq + ˙ B tq = aq mq − qa − mq − b q , ˙ R zq − ˙ B tq = pq aq − pq − q ¯ a c q (8.15)Of the 15 transverse rotations in table 8.3 the first 9 are basis vectors whichexplicitly leave the components of h invariant, while for each of the remaining six( ˙ G q + 2 ˙ S q ) actions the non-zero ˙ a l contributions in table 6.7 cancel.Although the category 1 and 2 transformations of type 1 as originally com-posed on the 10-dimensional space h O each act as a simple rotation or boost in a2-dimensional plane the effect on the components of the spinor θ is less straightfor-ward in the full h O action, as was seen for the case of the external symmetry SL(2 , C ) in the previous section. This is also seen for the type 2 and 3 internal transformations ofequations 8.14 and 8.15. Hence these actions, together with the 15 internal transverserotations, stir up the θ components in non-trivial ways.Of particular interest is the SU(3) subgroup introduced below equation 6.42 anddiscussed shortly after figure 7.3 (as described in [38] pp.115 and 136, following [55]).This SU(3) is defined in terms of the transverse rotations acting on the octonion space O alone as the subgroup SU(3) ⊂ G of the octonion automorphism group that leavesone imaginary unit, here l , invariant. The corresponding Lie algebra su(3) is describedby the set of 8 generators { ˙ A q , ˙ G l } which, as transformations of E on the full spaceh O , act on each of the octonion elements a, b, c ∈ O in the same way leaving invariantthe complex { , l } subspaces, and as elements of table 8.3 identified within stab( TM )may be provisionally associated with the colour su(3) c of the Standard Model. Thisalgebra is also independent of SL(2 , C ) in terms of the Lie bracket composition, thatis [ X, Y ] = 0 for all X ∈ sl(2 , C ) and Y ∈ su(3) c , and hence we have the semi-simplesubgroup: SL(2 , C ) × SU(3) c ⊂ E (8.16)The Lie algebra composition of the { ˙ A q , ˙ G l } ∈ su(3) c elements from the E commu-tation table in [38] is reproduced here in table 8.4. The Lie algebra in table 8.4 isisomorphic to the su(3) Lie algebra represented by the eight 3 × i accompanying the λ α matrices in the algebra isomorphism listed in table 8.6. The factors of i belong to the same complex algebra C used in the components of the Gell-Mann matrices themselves,but are independent of the octonion algebra elements on the left-hand side. (That isthe isomorphism is between the basis { ˙ A q , ˙ G l } and the anti-Hermitian matrices ∼ iλ α • , • ] ˙ A i ˙ A j ˙ A k ˙ A kl ˙ A jl ˙ A il ˙ A l ˙ G l ˙ A i A k − ˙ A j − ˙ A jl ˙ A kl ˙ A l − ˙ G l − ˙ A il A il ˙ A j − ˙ A k A i − ˙ A il − ˙ A l − ˙ G l ˙ A kl ˙ A jl A jl ˙ A k ˙ A j − ˙ A i − A l − ˙ A il ˙ A jl A kl A kl ˙ A jl ˙ A il A l − ˙ A i − ˙ A j − A k A jl − ˙ A kl ˙ A l + ˙ G l ˙ A il ˙ A i − ˙ A k − ˙ A j − A j ˙ A il − ˙ A l + ˙ G l − ˙ A kl − ˙ A jl ˙ A j ˙ A k A i − A i ˙ A l ˙ A il − ˙ A jl − A kl A k ˙ A j − ˙ A i G l − A il − A jl A j A i Lie algebra table in [38]). The Lie algebra structurefor the SU(3) c generators { ˙ A q , ˙ G l } , with bracket composition [ ˙ A i , ˙ A j ] = ˙ A k etc. λ = λ = − i i λ = − λ = λ = − i i λ = λ = − i i λ = √ − with [ λ α , λ β ] = if αβγ λ γ and f = 2, f = f = √ f = − f = f = f = f = − f = 1Table 8.5: The set of eight complex Hermitian Gell-Mann matrices of su(3), withrepresentatives of the completely antisymmetric structure constants f αβγ which arenon-zero. 179ather than directly with the Hermitian Gell-Mann matrices. This is analogous tothe relation between the external SL(2 , C ) generators and the conventional Lorentzalgebra in equation 8.9, where factors of i would also appear if the J a were defined asHermitian rather than anti-Hermitian in equation 7.16).˙ A k ∼ − iλ ˙ A kl ∼ − iλ ˙ A l ∼ iλ ˙ A i ∼ − iλ ˙ A il ∼ iλ ˙ A jl ∼ − iλ ˙ A j ∼ − iλ ˙ G l ∼ − i √ λ Table 8.6: The isomorphism between the su(3) c ⊂ E Lie algebra basis { ˙ A q , ˙ G l } andthe eight Gell-Mann matrices λ α ([38] p.137, table 4.5).The 3 × u ∈ C corresponding, in the context of an SU(3) c gauge theory, to the interactions be-tween ‘red’, ‘blue’ and ‘green’ quark states encountered in quantum chromodynamics.Similarly the { ˙ A q , ˙ G l } algebra elements, as transformations on the space h O mix thecomponents of the Spin + (1 ,
9) spinor θ = (cid:0) c ¯ b (cid:1) ∈ O . For example the tangent vectorfield ˙ A i on the (cid:0) c ¯ b (cid:1) components of h O , obtained from table 6.7, are:˙ A i : ˙ c ˙¯ b = ˙ c + ˙ c l, + ˙ c il + ˙ c i, + ˙ c jl + ˙ c j, + ˙ c kl + ˙ c k ˙ b − ˙ b l, − ˙ b il − ˙ b i, − ˙ b jl − ˙ b j, − ˙ b kl − ˙ b k = l, +0 il + 0 i, − c jl − c j, + c kl + c k − l, − il − i, + b jl + b j, − b kl − b k (8.17)The components here have been ordered to match those of the four left-handedWeyl spinors ( θ l , θ i , θ j , θ k ) of equation 8.13. The fact that each real component of c transforms in the same way as the corresponding component of b is expected sinceSU(3) c acts on each of a, b, c ∈ O in precisely the same way. However, it is also notedthat the action ˙ A i in equation 8.17 respects the 4-way spinor decomposition, with forexample ˙ c and ˙ c of ˙ θ j taking the respective values of − c and − c from the spinor θ k .This apparently non-trivial observation applies to all eight SU(3) c generators, whichhence represent a mixing of the four Weyl spinors, as a structure maintained withinthe mixing of the eight real components of the octonion elements.The extraction of the components of a spinor θ into a matrix of real numberswill be denoted by [ θ ]. For example, from equation 8.13 the spinor θ i can be mappedto the 2 × θ i ] = (cid:0) c c − b − b (cid:1) (with components ordered to matchthose of the spinor θ l under SL(2 , C ) transformations, as described for equations 8.11–8.13). With this notation and the Lorentz spinor definitions in equation 8.13 the above180quation 8.17 can be expressed as:˙ A i : [ ˙ θ ] = ( [ ˙ θ l ] , [ ˙ θ i ] , [ ˙ θ j ] , [ ˙ θ k ] )= ( 0 , , − [ θ k ] , [ θ j ] ) (8.18)where 0 represents the 2 × c generator ˙ A i mixes the external SL(2 , C ) spinors θ j and θ k identifiedin the previous section. The tangent vectors of all eight generators { ˙ A q , ˙ G l } of SU(3) c on the spinor space θ ∈ O are listed in table 8.7 alongside the actions of the Gell-Mann matrices, using the correspondence in table 8.6, on the vectors u ∈ C . On theleft-hand side the elements { ˙ A q , ˙ G l } are already expressed as tangent vectors, while onthe right-hand side the tangents are obtained by matrix multiplication of the λ α into u ∈ C . ([ ˙ θ l ] , [ ˙ θ i ] , [ ˙ θ j ] , [ ˙ θ k ]) ( ˙ u , ˙ u , ˙ u )˙ A i = ( 0 , , − [ θ k ] , [ θ j ]) ∼ λ ⇒ ( u , , u )˙ A il = ( 0 , , [ lθ k ] , [ lθ j ]) ∼ λ ⇒ ( − iu , , iu )˙ A j = ( 0 , [ θ k ] , , − [ θ i ]) ∼ λ ⇒ ( 0 , − iu , iu )˙ A jl = ( 0 , − [ lθ k ] , , − [ lθ i ]) ∼ λ ⇒ ( 0 , u , u )˙ A k = ( 0 , − [ θ j ] , [ θ i ] , ) ∼ λ ⇒ ( u , u , A kl = ( 0 , [ lθ j ] , [ lθ i ] , ) ∼ λ ⇒ ( − iu , iu , A l = ( 0 , [ lθ i ] , − [ lθ j ] , ) ∼ λ ⇒ ( u , − u , G l = ( 0 , [ lθ i ] , [ lθ j ] , − lθ k ]) ∼ λ ⇒ √ ( u , u , − u )Table 8.7: The tangent vector generators for the SU(3) representations on O and C .The column vectors of C are displayed as a row vectors for convenience in the table.In table 8.7 a term such as [ lθ i ] denotes multiplying the spinor θ i on the left by l before extracting the coefficients of lθ i with the Im( O ) units ordered as in equation 8.13.This notation is used to isolate the mixing effect on the real number coefficients, withcare for the joint effects of the division algebra composition as well as matrix algebracomposition. For the case of u ∈ C , the two real degrees of freedom for each of the u , u , u belong to the same complex space C (with base units { , i } ) but occupy different components of the 1 × C . For the case of θ ∈ O the four real degrees of freedom for each of the θ l , θ i , θ j , θ k belong to a different O (with base units { , l } , { il, i } , { jl, j } , { kl, k } respectively)but occupy the same components of the 1 × O .Hence, as seen in table 8.7, the six transformations ˙ A q ( q = l ) mix the com-ponents of the three Weyl spinors θ i , θ j , θ k in a similar manner that the Gell-Mann181atrices λ α ( α = 3 ,
8) mix the three C components u , u , u , with the correspon-dence between the objects of each representation space depending on the form of theisomorphism in table 8.6, which is arbitrary up to the automorphism group of su(3).In both cases there are two remaining diagonal generators as listed at the bottom of ta-ble 8.7. (The physics here is determined by the { ˙ A q , ˙ G l } transformations as generatorsof SU(3) c rather than the particular choice of correspondence with the λ α matrices,as was similarly the case for the external SL(2 , C ) action of the previous section asdescribed after equation 8.9). In the case of the full set of { ˙ A q , ˙ G l } acting on thecomponents of h O there is a copy of the same set of mixing transformations withinthe components of the Hermitian conjugate spinor θ † = (¯ c b ) of equation 6.26 whichalso transforms under the internal SU(3) c symmetry (similarly as described for theSL(2 , C ) spinors below equation 8.13).In conclusion the internal SU(3) c symmetry action in the left-hand column oftable 8.7 dovetails neatly with the external SL(2 , C ) spinor structure of equation 8.13.The mixing action of SU(3) c in table 8.7 takes a form summarised as: θ = ( θ l , θ i , θ j , θ k | {z } SU(3) c action ) (8.19)which implies that as a gauge theory the SU(3) c internal symmetry will mediate in-teractions between the Weyl spinors θ i , θ j , θ k , transforming under the fundamentalrepresentation, which in turn will hence be identified with the three colour degreesof freedom of the quark states. On the other hand the invariance of θ l , transform-ing under the trivial representation of SU(3) c , suggests that these components shouldbe associated with the leptonic sector of the Standard Model (with the subscript l originating from the { , l } base units for θ l also then serving as a mnemonic for itsleptonic character). Further aspects of the Standard Model might then be expected tobe uncovered by exploring further aspects of the internal symmetry group within E ,which will occupy the remainder of this chapter.In particular the Standard Model Abelian gauge group U(1) Q , underlyingMaxwell’s equations and the phenomena of electromagnetism, might also be soughtas an internal symmetry within Stab( TM ). Of the 31 generators for the internal sym-metry group Stab( TM ) listed in table 8.3 there is a (31 −
8) = 23-dimensional setwhich as a vector space is independent of the internal SU(3) c generators. Of these 23there are 3 sets each of 6 elements:( ˙ R zq + ˙ B tq ) , ( ˙ R zq − ˙ B tq ) , ( ˙ G q + 2 ˙ S q ) (8.20)with q = l , totalling 18 elements each of which fails to commute with some of the in-ternal SU(3) c generators in the set { ˙ A q , ˙ G l } . As a cross-check this observation appearsto hold for any linear combination of elements selected from the 18 in equation 8.20,by further inspection of the E Lie algebra table [38]. Hence none of the 18 elements inequation 8.20 can belong to a group which may be appended to the subgroup decompo-sition SL(2 , C ) × SU(3) c in equation 8.16 (in fact the first 12 elements in equation 8.20also fail to commute with SL(2 , C ) ). This then leaves a set of only (31 − −
18) = 5internal generators which in terms of Lie algebra composition, and not only as a vectorspace, is independent of su(3) c . These are the elements:( ˙ R xz − ˙ B tx ) , ( ˙ R xz + ˙ B tx ) , ( ˙ R zl + ˙ B tl ) , ( ˙ R zl − ˙ B tl ) , ˙ S l (8.21)182ndeed, each of the nine individual component parts listed within equation 8.21 com-mute with all eight elements of the internal su(3) c basis set. However the first 4elements in equation 8.21 each fail to commute with the external SL(2 , C ) generators.This leaves ˙ S l as the only E Lie algebra generator of Stab( TM ) which is independentof both SL(2 , C ) and SU(3) c . Hence of the many possible U(1) ⊂ E subgroups theone generated by ˙ S l is identified as the most suitable candidate for the internal U(1) Q gauge symmetry of electromagnetism.Moreover the generator ˙ S l is also closely associated with the diagonal sym-metry action S \ l , described by equation 6.43, and leaves the 4-dimensional spacetimecomponents in h C invariant as a residual of the SL(2 , O ) action on h O as describedat the end of section 6.3. A similar internal U(1) symmetry associated with electro-magnetism has been considered for the SL(2 , C ) × U(1) gauge theories as discussed inthe opening paragraphs of section 7.3. While as elements of the vector space T h O we have ˙ S l = ˙ S \ l , as discussed following equation 6.45, the group actions S (1) l ( α ) and S \ (1) l ( α ) diverge at O( α ) and in any case, although suggestive, this argument aloneis insufficient in itself to associate the U(1) Q symmetry with S l out of many possibleU(1) ⊂ E subgroups. Here the main case for this association is the observation thatthe U(1) subgroup S l uniquely both belongs to Stab( TM ) and at the Lie algebra levelis independent of the SL(2 , C ) × SU(3) c subgroup of equation 8.16.Hence here the internal U(1) generated by ˙ S l is a natural candidate to considerfor the U(1) Q component of the Standard Model gauge symmetry group. From table 6.7it can be seen that the generator ˙ S l impacts on all 8 real components of both c and¯ b of θ ∈ O . In fact, and in comparison with equation 8.17, the tangent vector ˙ S l onthe spinor components θ = (cid:0) c ¯ b (cid:1) is given explicitly by:˙ S l : ˙ c ˙¯ b = ˙ c + ˙ c l, + ˙ c il + ˙ c i, + ˙ c jl + ˙ c j, + ˙ c kl + ˙ c k ˙ b − ˙ b l, − ˙ b il − ˙ b i, − ˙ b jl − ˙ b j, − ˙ b kl − ˙ b k = − c + c l, + c il − c i, + c jl − c j, + c kl − c k b + b l, − b il + b i, − b jl + b j, − b kl + b k (8.22)with [ ˙ θ ] = (cid:18) + 32 [ lθ l ] , −
12 [ lθ i ] , −
12 [ lθ j ] , −
12 [ lθ k ] (cid:19) (8.23)which may be compared with the su(3) c action on θ in equation 8.18 and table 8.7. Herethe two components of c ∈ O within each of the four Weyl spinors are mixed, and sim-ilarly for the corresponding pair of ¯ b ∈ O components, with no mixing of components between different spinors. This is consistent with the nature of the electromagneticinteraction which does not transform between different fermion types.A further observation from equation 8.23 regards the factor of found for the θ l spinor in contrast to the factors of aligned with the three remaining spinors θ i , θ j , θ k .Hence, with ˙ S l provisionally associated with electromagnetism and by comparison withequation 8.19, the apparent ‘electromagnetic charge’ assigned to the leptonic sector is three times larger than that assigned to the quark sector. Associating θ i , θ j , θ k with thethree colour states of a d -quark this observation in principle accounts for the ‘fractionalcharge’ of magnitude as theoretically ascribed and empirically confirmed for d -quark183tates relative to the electron charge. Based on this observation we introduce thenotation: ˙ S –– al = 23 ˙ S al (8.24)(for a = 1 , ,
3) such that the above charge values and are normalised to 1 and under ˙ S –– l , representing the generator of U(1) Q , for ease of comparison with theStandard Model convention for which the electron charge is −
1. The ‘bar’ through˙ S –– l is a mnemonic symbol for this normalisation of fractional charges relative to the e − charge. (The corresponding normalisation for components of the group action S –– l ,which is not needed here, would need to take into account the nested composition ofequation 6.39. This group normalisation would hence be different for the single actionof S \ l of equation 6.43).Hence the subgroup in equation 8.16 may be augmented to:SL(2 , C ) × SU(3) c × U(1) Q ⊂ E (8.25)with the internal group SU(3) c × U(1) Q generated by { ˙ A q , ˙ G l , ˙ S –– l } ∈ stab( TM ). Theaction of this larger internal symmetry on the four SL(2 , C ) spinors also augmentsequation 8.19 as: θ = ( θ l , θ i , θ j , θ k | {z } )SU(3) c : (8.26)U(1) Q : +1 − − − With the generator ˙ S –– l hence associated with electromagnetic charge it is in-structive to consider this action on the full set of h O components. From table 6.7the diagonal components of ˙ S –– l are trivial, with ˙ p = ˙ m = ˙ n = 0, while action on theremaining components a, b, c ∈ O , via equation 8.24, may be summarised as:˙ S –– l = ˙ a ˙ b ˙ c = l a ,l + l a (6) − l b ,l − l b (6)+1 l c ,l − l c (6) (8.27)where a ,l ≡ ( a + a l ) and a (6) ≡ ( a il + a i + a jl + a j + a kl + a k ), with similarexpressions for b and c , following the component order of the spinors in equation 8.13.The same definition of a (6) is implied in equation 8.4. By comparison with the abovediscussion leading to equation 8.26 the expression for ˙ S –– l in equation 8.27 incorporates‘charges’ of 0 and for the ˙ a components, that is we have: a = ( a ,l , a il,i , a jl,j , a kl,k | {z } )SU(3) c : (8.28)U(1) Q : 0 + + + where the SU(3) c action on a ∈ O is identical to that on the octonion components of θ = (cid:0) c ¯ b (cid:1) in equation 8.26. While physical lepton states are invariant under SU(3) c and184re hence associated with the Weyl spinor θ l in equation 8.26, the neutrino states arealso invariant under the U(1) Q of electromagnetism, that is with zero charge, and areprovisionally associated with the a ,l components in equations 8.27 and 8.28; while a setof u -quarks with fractional charges is similarly associated with the a (6) components.However, unlike θ = (cid:0) c ¯ b (cid:1) the a ∈ h O component does not correspond to aset of SL(2 , C ) Weyl spinors, as can be seen from table 8.2. Further, the ‘neutrino’components a ,l = a + a l = v + v l have already apparently been accounted for aspart of the external vector v ∈ TM on the base manifold, as described in equations 8.4and 8.5. These features clearly require further investigation.While in the Standard Model the e − lepton charge is − d -quarkcharge is − , with positive charges for their antimatter counterparts, the conventionand interpretation of the ± -signs of equations 8.23 and 8.26 will depend upon theconventions used and the identification of particle and antiparticle states as relatingto the spacetime dynamics of the theory. As for GUT theories in which particle andantiparticle states may coexist within the same SU(5) multiplet [43], see sections 7.2and 7.3, the apparently opposite charges in equation 8.23 may relate, for example, to acombination of ‘antimatter’ electrons and ‘matter’ d -quarks in the components of h O (which may in turn ultimately relate to the nature of the asymmetry between matterand antimatter in the universe).Within the above caveats, aligned with the charges of 1 and for the electronand d -quark Weyl spinors of equation 8.26 the respective U(1) Q charges of 0 and inequation 8.28 correlate with charges of (cid:0) − (cid:1) for the (cid:0) νe (cid:1) lepton doublet and (cid:0) +2 / − / (cid:1) forthe (cid:0) ud (cid:1) quark doublet of the Standard Model. In addition the states associated witheach left-handed doublet of charges interact via the exchange of W ± gauge bosons inthe Standard Model. Hence it remains to be understood how interactions within eachof these doublets may be mediated via an SU(2) L symmetry, and how such ν -leptonand u -quark components of a ∈ O ⊂ h O gain a Weyl spinor structure under theexternal SL(2 , C ) action.While the empirical charge structure of the Standard Model fermions is inprinciple accounted for by a U(1) Q symmetry associated with the generator ˙ S –– l ofequation 8.27, further elaboration of this theory is required in order to further re-construct the pattern of particle multiplets listed in equation 7.36. Guided by theStandard Model it will be necessary to understand the origin of weak interactions inorder to address these details. Hence in the following section we investigate the possi-ble identification of an SU(2) L gauge symmetry within the structure of the broken E action on h O in the present theory. (2) Transformations and SU (3) s Symmetry
Within the set of 31 internal basis elements in table 8.3 it is possible to identify anumber of SU(2) subgroups, for example generated by the three elements ˙ G q + 2 ˙ S q with q = { i, j, k } or a different triplet of imaginary units (excluding l ) belonging toa common line in figure 6.1 and hence generating a quaternion subalgebra. While185ndependent of { ˙ A q , ˙ G l } as a vector space none of these su(2) generator sets is inde-pendent of the su(3) c algebra in terms of the Lie bracket (that is with [ X, Y ] = 0 forall X ∈ su(3) c , Y ∈ su(2)), as discussed after equation 8.20.It is an open question whether all possible internal symmetry subgroups shouldhave physical significance. In the above case the generator ˙ G q + 2 ˙ S q for q = i, j and k mixes the components of θ l with those of θ i , θ j and θ k respectively, hence mixingbetween ‘leptons’ and ‘quarks’, and would apparently correspond to ‘new physics’ withrespect to the Standard Model. However this particular SU(2) action does not describea ‘fundamental representation’ on the set four Weyl spinors, as was the case for SU(3) c on the left-hand side of table 8.7 or for U(1) Q in equation 8.23.In any case here we attempt to identify an SU(2) symmetry which, as for thecase of the ˙ S –– l generator identified for equation 8.25 for an internal U(1) Q symmetry, isindependent of the internal SU(3) c . The other four internal generators in equation 8.21form a trivial algebra with zero Lie bracket for all products – although non-zero com-mutators are obtained if ˙ S l is included (with [( ˙ R xz − ˙ B tx ) , ˙ S l ] = ( ˙ R zl + ˙ B tl ) forexample) but this is still insufficient structure to form an su(2) algebra. It is alsothe case that none of these four elements commute with SL(2 , C ) and in fact noneof the 31 − TM ) in table 8.3 commute with thesubgroup SL(2 , C ) × SU(3) c of equation 8.16, as implied in the discussion followingequation 8.21. Further, it is to be expected from the Dynkin analysis described in sec-tion 7.3 that in fact there is no possibility of identifying an SU(2) subgroup of E whichis independent of both an external SL(2 , C ) and an internal SU(3) × U(1) symmetrygroup. However, although the full internal gauge symmetry group of the StandardModel reads SU(3) c × SU(2) L × U(1) Y there are a number of features of weak inter-actions associated with SU(2) L , as observed in high energy physics experiments andwritten into the Standard Model, which qualitatively differ from the strong and elec-tromagnetic interactions associated with SU(3) c and U(1) Q respectively. If an internalSU(2) were to be found at this stage, at the level of symmetry groups and their rep-resentations, in a similar manner as for the internal SU(3) c × U(1) Q in the previoussection, it seems unlikely that the kind of distinctive properties observed for the weakinteractions could arise purely in the dynamics of the full theory. The differences inempirical properties between the strong and electromagnetic interactions themselvesoriginate largely out of the differences between the non-Abelian SU(3) and AbelianU(1) symmetries at the group and representation level, with many more interactionspossible in the quantum theory for the former case. However while the non-Abeliangroup SU(2) is mathematically intermediate in size between SU(3) and U(1) the phys-ically observed features associated with the gauge group SU(2) are of a quite differentnature, as described in section 7.2 and summarised in the following paragraph.Firstly the weak interactions violate parity symmetry, prompting the subscript‘ L ’ for the left-handed character of this chiral SU(2) L gauge theory. Secondly, theStandard Model SU(2) L is closely association with a U(1) Y gauge symmetry, withU(1) Q surviving the electroweak symmetry breaking, which is in turn associated withthe Lorentz scalar Higgs field transforming as an SU(2) L doublet and providing themechanism by which three gauge bosons, the W ± and Z , gain a non-zero mass.Thirdly, the weak interactions mix particle states from the three distinct generations186f fermions, as described by the CKM matrix.Here we initially focus upon the simple fact that weak SU(2) L transformationsact on fermion doublets of the form (cid:0) νe (cid:1) and (cid:0) ud (cid:1) , which have been associated with the (cid:0) aθ (cid:1) components of h O for the present theory. This was described at the end of theprevious section where it was noted that the U(1) Q electromagnetic charges associatedwith the ˙ S –– l action on θ = (cid:0) c ¯ b (cid:1) and the a component are respectively aligned with thecharges of the ( e -lepton, d -quark) and ( ν -lepton, u -quarks) particle states.The type 1 SL(2 , C ) action on the four Weyl spinors of equation 8.13 is com-plemented by SL(2 , C ) and SL(2 , C ) transformations of type 2 and 3, all involvingquaternion algebra composition with l ∈ O being the only imaginary octonion unitappearing in the transformation matrices. Two SU(2)s are immediately identifiablein terms of the rotation subgroups of the type 2 and type 3 Lorentz groups, as de-noted by SU(2) , generated by the set { ˙ R zl , ˙ R xz , ˙ R xl } , and SU(2) , as generated by { ˙ R zl , ˙ R xz , ˙ R xl } . Neither SU(2) ⊂ SL(2 , C ) nor SU(2) ⊂ SL(2 , C ) is independentof SL(2 , C ) within the E Lie algebra, with for example [ ˙ R xz , ˙ R xz ] = ˙ R xz = 0, andneither of them forms a subgroup of Stab( TM ), and hence they do not appear toform an internal symmetry by the original definition which led to table 8.3. Howeverowing to the properties described below in exploring further the structure of thesetransformations the groups SU(2) , are found to be of some interest in relation to thestructure of electroweak theory.By reference to equations 6.32, 6.34 and 6.35 of section 6.4, and with the spinorcomponents θ a = (cid:0) θ θ (cid:1) ∈ O represented by (cid:0) c ¯ b (cid:1) , (cid:0) a ¯ c (cid:1) and (cid:0) b ¯ a (cid:1) for the type a = 1 , M ( a ) ∈ SL(2 , C ) a action, with eachset generated by equations 8.7 and 8.8, are of the form: M (1) ! c ¯ b ! , M (2) ! a ¯ c ! , M (3) ! b ¯ a ! (8.29)with an equivalent right composition θ † M † = ( M θ ) † associated with each action above,as seen in the example of the full type 1 embedding of equation 6.29. In all caseshowever the group action is by left translation, that is with group representations R ( g ) R ( g ) = R ( g g ) as discussed in section 6.2 after equation 6.14, and involveselements of the non-commutative quaternion algebra.The type 1 action of SL(2 , C ) decomposes the space θ = (cid:0) c ¯ b (cid:1) ∈ O into thefour Weyl spinors of equation 8.13. The transformations SL(2 , C ) , of type 2 and 3,with complementary transformation matrices also based on the units { , l } , similarlyrespect the octonion decomposition aligned to the four base unit sets: { , l } , { il, i } , { jl, j } , { kl, k } (8.30)based on the same quarternion subalgebras, now for all three of a, b, c ∈ O . Hence thesubgroups SU(2) , ⊂ E describe transformations between the components of equa-tion 8.26 and those of equation 8.28 respecting the alignment of the four componentpieces, and hence acting independently on the corresponding doublets of leptonic andquark states as appropriate for weak interactions. With respect to the embedding of a, b, c ∈ O as components of h O in equation 6.1, the spinor representation mixing187ctions of SL(2 , C ) , , can also be displayed graphically as: ¯ a ca ¯ b ¯ c b ✛ ✲✛ ✲ ............................ ✛ ✲ ✻❄✻❄ ✻❄ ............................ with ✛ ✲ SL(2 , C ) ✛ ✲ SL(2 , C ) ......... ✛ ✲ SL(2 , C ) (8.31)This again shows how the (cid:0) c ¯ b (cid:1) spinor components under SL(2 , C ) are replacedby (cid:0) a ¯ c (cid:1) and (cid:0) b ¯ a (cid:1) spinors under SL(2 , C ) and SL(2 , C ) respectively, depending on thealignment of the θ a = (cid:0) θ θ (cid:1) components in equations 6.32–6.35. It is the observation thatthe SL(2 , C ) , actions relate the θ = (cid:0) c ¯ b (cid:1) ∈ O components with the a ∈ O componentin equations 8.29 and 8.31, while respecting the four-way octonion decomposition ofequation 8.30, that suggests that these transformations might be closely related to theweak interactions.In section 8.1 the Weyl spinors θ i , θ j , θ k were identified alongside θ l in equa-tion 8.13 originating from the one-sided action of SL(2 , C ) ⊂ SL(2 , H ) ⊂ H (2) on (cid:0) c ¯ b (cid:1) H ∈ H . The quark spinors θ i , θ j , θ k are formed out of subspaces of H with quar-ternion base units { il, i } , { jl, j } , { kl, k } ∈ O respectively, with the actions on theseobjects by matrices composed of the base units { , l } , completing the 3 sets of H sub-algebras of the octonions, involving in particular the quaternionic left multiplication by l as demonstrated in equation 8.12. Although the full set of H (2) matrix actions are notinvolved this asymmetric one-sided action is apparently incomplete in terms of the setof possible actions of the non-commutative quaternion algebra on these components.This observation might in principle relate to a possible mechanism for theorigin of chirality in SU(2) interactions in the Standard Model. This situation can becontrasted with left-right symmetric gauge theories with the internal symmetry groupSU(2) L × SU(2) R × U(1) formulated in terms of fields defined over the quaternion algebra(see for example [56] and the references therein). A mechanism is then required throughwhich the symmetry in these parity conserving models is broken to SU(2) L × U(1) tomatch the observed parity violating phenomena of weak interactions.Here since the set of external
Lorentz transformations of SL(2 , C ) act asym-metrically on the left on θ ∈ O and on the subspaces of Weyl spinors θ i , θ j , θ k wemay expect to identify a set of actions on these spinor components algebraically com-posed from the right , which have a complementary effect owing to the non-commutingproperty of the H algebra, potentially forming a distinct internal symmetry, at leastwith regards to the quark states represented by these three Weyl spinors. Similarly,in the present context, for actions involving multiplications by elements belonging toSL(2 , H ) a ⊂ H (2) some chiral behaviour might be expected to arise in this theory asthe type a = 2 , , C ) ⊂ SL(2 , H ) . Further, although only one fermiongeneration has been considered explicitly, the structure of equation 8.31 is suggestive interms of the need to ultimately account for the CKM mixing between three generationsof fermions. 188owever while these possibilities provided some of the initial motivation forstudying the actions of the groups SU(2) , the mechanism for the above physicalphenomena will require further developments. The source of parity violation in thepresent theory will be described in section 9.2, having explicitly constructed both leftand right-handed Weyl spinors by extending the form of temporal flow beyond theaction of E on h O . As will be described in section 9.3 a further expansion to ayet higher-dimensional flow of time may be required in order to account for threegenerations of fermions and the phenomena of CKM mixing.Here the main motivation for studying the SU(2) , ⊂ E subgroups is thestructure of the action on the doublet components of h O as described for equa-tions 8.29–8.31 above in relation to the weak interaction transformations for doubletsof fermions in the Standard Model. In this subsection we hence further explore thisgroup structure before focusing on a pattern of symmetry breaking that closely paral-lels the properties of electroweak symmetry breaking in the remainder of this section.In particular the subgroup SU(3) c × SU(2) × U(1) ⊂ E , provisionally considered asan ‘internal symmetry’ (where U(1) is the type 2 equivalent of U(1) = U(1) Q iden-tified in the previous section), is analogous to the Standard Model gauge symmetrySU(3) c × SU(2) L × U(1) Y ; with the impingement of the action SU(2) × U(1) on theexternal spacetime components of h C ⊂ h O breaking this symmetry down to U(1) Q .This will be described in the following subsection and constitutes a ‘mock electroweaktheory’. We will then ultimately need to address how to combine these structures withthe external SL(2 , C ) symmetry, which within the E structure is not independent ofthe SU(2) , actions.In fact with SU(2) a ⊂ SL(2 , C ) a for a = 1 , , a action described by ˙ S –– al for a = 1 , ,
3, as introducedin equation 8.24, within the full E action on the space h O . The SU(3) c action oftable 8.7, corresponding to the set of eight generators { ˙ A q , ˙ G l } , not only transforms a, b, c ∈ O ⊂ h O in precisely the same way (table 6.7), acting on the components of a (6) as a triplet and a ,l as a singlet (in the notation of equation 8.27), but is alsoindependent of both the SL(2 , C ) , , and S , , l actions in terms of the E algebra Liebracket. This means that the SL(2 , C ) a and S al actions may effectively be stripped outand considered independently of the SU(3) c action. This may aid the identificationof an internal SU(2) L × U(1) Y symmetry, and its relation with the external Lorentzsymmetry SL(2 , C ) , bearing in mind that the former is expected to be ‘broken’ to theU(1) Q symmetry associated with ˙ S –– l .The nine generators of the combined type a = 1 , a forma closed subalgebra of E , which is eight dimensional due to the linear dependence ofthe ˙ R axl generators as displayed in equation 6.46. This subalgebra is in fact an su(3), alinearly independent basis for which can be described by the eight rotation generators([38] p.128): su(3) s ≡ { ˙ R xl , ˙ R xl , ˙ R xz , ˙ R xz , ˙ R xz , ˙ R zl , ˙ R zl , ˙ R zl } (8.32)These generate a group denoted SU(3) s (where ‘ s ’ denotes the ‘standard’ representationor embedding of this group in E [38]). As implied above within E the subgroup SU(3) s is independent of the colour subgroup SU(3) c , as generated by the eight elementsof table 8.4, with the Lie bracket composition of any element of equation 8.32 withany element of { ˙ A q , ˙ G l } being zero. The generators of SU(3) c are explicitly ‘type189ndependent’, in that there are no type labels on any of the eight generators { ˙ A q , ˙ G l } ([38] p.128), none of which distinguish between the three types. The subgroup SU(3) s is also ‘type independent’, in that all three types play an equivalent role, however theindividual generators do carry type labels as for example in equation 8.32.The group product SU(3) s × SU(3) c ⊂ E is a rank-4 subgroup of the completerank-6 symmetry group E . In fact SU(3) s ⊂ SL(3 , C ) s where SL(3 , C ) s is the 16-dimensional rank-4 group generated by the type 1 , , { , l } complex subspace. Taking into account equation 6.51 we have ([38] p.128):sl(3 , C ) s ≡ su(3) s ∪ { ˙ B tz , ˙ B tz , ˙ B tx , ˙ B tx , ˙ B tx , ˙ B tl , ˙ B tl , ˙ B tl } (8.33)In fact sl(3 , C ) s is the closed subalgebra formed collectively out of the three types ofLorentz generators sl(2 , C ) a for a = 1 , ,
3, with group actions as pictured in equa-tion 8.31, which also act on the complex subspace h C ⊂ h O formed with base units { , l } with for example the type 1 action of equation 7.35.At the level of complex Lie algebras L C we have the semi-simple decompositionsl(3 , C ) ≡ su(3) × su(3), and hence the rank-6 subgroup obtained for this real form ofE : SL(3 , C ) s × SU(3) c ⊂ E (8.34)is closely related to an SU(3) × SU(3) × SU(3) ⊂ E decomposition, which may bereadily obtained by the analysis described in section 7.3 via the extension of the Dynkindiagram for the complex E Lie algebra of figure 7.2(a). In the present theory it is theSL(2 , C ) ⊂ SL(3 , C ) s ⊂ E Lorentz symmetry of external spacetime that breaks thefull E symmetry.In fact E also contains the following rank-6 subgroup (listed as one of a numberof possible decompositions from a mathematical point of view in [38] p.187) whichaugments equation 8.25:SL(2 , C ) × U(1) Q × D(1) B × SU(3) c ⊂ E (8.35)with SL(3 , C ) s broken to SL(2 , C ) × U(1) Q × D(1) B , and where U(1) Q is generated by˙ S –– l = ˙ S l with ˙ S l = ( − ˙ R xl − R xl ), via equations 8.24 and 6.47, and D(1) B is generatedby ( ˙ B tz + 2 ˙ B tz ). This latter generator is presented explicitly in equation 13.5 togetherwith a possible physical interpretation of the D(1) B subgroup in the context of thepresent theory as described in section 13.2. Further contained within this symmetrybreaking pattern is the choice of SU(2) × U(1) Q ⊂ SU(3) s ⊂ SL(3 , C ) s with theidentification of U(1) Q = U(1) , which, in relation to the three possible type a =1 , , a × U(1) a ⊂ SU(3) s will be seen to be closely related to thephenomena of electroweak symmetry breaking in the Standard Model.Before describing this connection we note that within the context of the presenttheory in principle it may be possible to mutually constrain the values of the gauge fieldcouplings associated with a range of internal subgroups in terms of the normalisationof the underlying simple E Lie algebra as expressed by the Killing form. The Killingmetric K αβ = c ρασ c σβρ in terms of the algebra structure constants c αβγ was introducedin the discussion leading to equation 4.1. Using this expression the components K αβ ofthe complete Killing form for the 78 generators of E in the preferred basis of table 6.3190an in principle be determined directly from the rows of the E Lie algebra table in [38].For example for the su(3) c ≡ { ˙ A i , ˙ G l } generators of the colour symmetry described inthe previous section we find: K ( ˙ A i , ˙ A i ) = − , K ( ˙ A l , ˙ A l ) = − , K ( ˙ G l , ˙ G l ) = − c can be compared with those for the SU(2) × U(1) generators identified in SU(3) s , as adopted in a ‘mock electroweak theory’, andin principle used to mutually normalise all coupling constants, including α s = g s π forthe strong interactions, under the unifying simple group E , for comparison with therelative couplings adopted for Standard Model gauge group SU(3) c × SU(2) L × U(1) Y .With a view towards studying such a mock electroweak theory here we analyse theKilling form for the generators relevant to su(3) s , and calculate from the rows of theE Lie algebra table in [38]: K ( ˙ R xl , ˙ R xl ) = − , K ( ˙ R xl , ˙ S l ) = 0 , K ( ˙ S l , ˙ S l ) = − K ( ˙ R zl , ˙ R zl ) = − , K ( ˙ R zl , ˙ R zl ) = − , K ( ˙ R xz , ˙ R xz ) = − R axl and ˙ S bl ( a, b ∈ { , , } ) with:˙ R xl = − ˙ R xl − ˙ S l ⇒ K ( ˙ R xl , ˙ R xl ) = − S l = + ˙ R xl − ˙ S l ⇒ K ( ˙ S l , ˙ S l ) = − R xl = − ˙ R xl + ˙ S l ⇒ K ( ˙ R xl , ˙ R xl ) = − S l = − ˙ R xl − ˙ S l ⇒ K ( ˙ S l , ˙ S l ) = − { ˙ R axl , √ ˙ S al } , for either a = 1 , K ( ˙ R xl , ˙ S l ) = 0 while K ( ˙ R xl , ˙ S l ) = +36indicating that the Killing form is not diagonal in the latter basis.Alternatively, restricting the computation of K αβ = c ρασ c σβρ to the SU(3) s subalgebra all elements of the corresponding 8 × K for the subalgebrabasis of equation 8.32 are determined, with for example: K ( ˙ R xl , ˙ R xl ) = − , K ( ˙ R xl , ˙ R xl ) = − , K ( ˙ R xl , ˙ R xl ) = + 32where the latter element is the only non-zero off-diagonal entry of the symmetric Killingform. Hence we replace the basis element ˙ R xl in equation 8.32 with √ ˙ S l such thatthe SU(3) s basis:su(3) s ≡ { ˙ R xl , √ S l , ˙ R xz , ˙ R xz , ˙ R xz , ˙ R zl , ˙ R zl , ˙ R zl } (8.36)191as normalised Killing metric K = − ), where is the 8 × c in table 8.6 a correspondence may be found between theSU(3) s generators of equation 8.36 and the representation of su(3) in terms of Gell-Mann λ matrices, as described here in table 8.8.˙ R zl ∼ iλ ˙ R xz ∼ iλ ˙ R xl ∼ iλ ˙ R zl ∼ − iλ ˙ R xz ∼ iλ ˙ R zl ∼ iλ ˙ R xz ∼ iλ √ ˙ S l ∼ iλ Table 8.8: The isomorphism between the su(3) s ⊂ E Lie algebra basis of equation 8.36and the eight Gell-Mann matrices of table 8.5.The choices of basis elements { ˙ R axl , √ ˙ S al } for type a = 2 or 3 in place of a = 1 correspond to two further possible correlates of the basis matrices { λ , λ } inthe Gell-Mann representation of su(3). These three possibilities correspond to threeclosely related ways to embed the subgroup SU(2) × U(1) in SU(3). Here we firststudy SU(2) ⊂ SU(3) s and the corresponding set of generators { ˙ R zl , ˙ R xz , ˙ R x } . Theseare the type 2 versions of the three actions of equation 8.7 which, as described inequation 8.9, are respectively associated with the three Pauli matrices σ = (cid:0) (cid:1) , σ = (cid:0) − ll (cid:1) , σ = (cid:0) − (cid:1) , within factors of ± l .In the Standard Model electroweak theory the su(2) L Lie algebra-valued con-nection 1-form W ( x ) = W α ( x ) τ α , with W α ( x ) = W αµ ( x )d x µ , τ α = σ α from equa-tion 7.37 and α = 1 , ,
3, is parametrised by the three gauge fields W αµ ( x ). Thecharged gauge boson fields W ± µ ( x ) are associated with complex linear combinations ofthe SU(2) L generators σ ± = ( σ ± iσ ) as was described in equations 7.57 and 7.58.Guided by this construction based on SU(2) L generators, here in the complex algebrafor SU(2) ⊂ E we define: ˙Σ (2) ± := ˙ R zl ± i ˙ R xz (8.37)Here the imaginary unit i ∈ C in the complexification of the E Lie algebra commuteswith the elements of T h O , which are based on an independent octonion algebra O .This is the standard notion of a complexified Lie algebra L C ≡ L R + iL R , as for exampledescribed for figure 7.1, applied here to L R as the real E Lie algebra represented inthe space of vector fields in T h O .The generator ˙ S –– l was associated with the internal symmetry U(1) Q and elec-tromagnetic charge in the previous section. As a rotation the corresponding grouptransformation S –– l takes the form of unitary 3 × i ˙ S –– l to be an Hermitian generator in thecomplexified E algebra real eigenvalues may be obtained under the adjoint represen-tation. In particular, reading off the corresponding entries in the Lie algebra table192n [38] for the complex element of equation 8.37 it is found that:[ ˙ S l , ( ˙ R zl + i ˙ R xz )] = ˙ R xz − i ˙ R zl = − i ( ˙ R zl + i ˙ R xz ) (8.38)hence [ i ˙ S –– l , ( ˙ R zl + i ˙ R xz )] = +( ˙ R zl + i ˙ R xz )and [ i ˙ S –– l , ˙Σ (2) ± ] = ± ˙Σ (2) ± (8.39)with real charge eigenvalues ±
1. Hence the generators ˙Σ (2) ± of equation 8.37 areassociated with the same magnitude of U(1) Q charge under ˙ S –– l as was found for theelectron in the leptonic components θ l ⊂ h O as described in equations 8.22–8.26.Since such factors of as seen in equation 8.38 are relatively sparse in the E Liealgebra table [38], with none appearing for example here in table 6.4, this seems tobe a non-trivial correspondence of ˙ S –– l charges. In the su(3) s basis of equation 8.36the generators ˙Σ (2) ± are in fact two of the eigenvectors of elements of the Cartansubalgebra, which in turn has a basis { ˙ R xl , √ ˙ S l } , under the adjoint representationin the complex su(3) s algebra. Indeed we find also:[ i ˙ R xl , ˙Σ (2) ± ] = ± ˙Σ (2) ± (8.40)More generally for a Lie algebra of rank- n the elements of the Cartan subalgebra { H i } , i = 1 . . . n , are mutually commuting and any element, or linear combinationof elements, in { H i } generates a U(1) symmetry. In any representation of the Liealgebra the eigenvalues, or ‘weights’, of such a U(1) generator can be considered as‘charges’. In the present case the U(1) Q generator ˙ S –– l , which also belongs to the E Cartan subalgebra as can be seen from equation 6.56, is associated with electromagneticcharge. In the Cartan-Weyl basis of a complex Lie algebra the eigenvectors E α of ele-ments of the Cartan subalgebra { H i } in the adjoint representation have real eigenvalues α i : [ H i , E α ] = α i E α (8.41)with [ E α , E − α ] = ( K ij α j ) H i (8.42)(where K ij are components of the Killing metric restricted to the Cartan subalge-bra). The eigenvalues, or ‘weights’, α i of the adjoint representation are also called‘roots’, the full set of which under { H i } is central to the classification of complex Liealgebras, as alluded to in section 7.3 alongside figure 7.2. Since the elements of theLie algebra form the vector space span( H i , E α ) upon which the adjoint representationacts, the dimension of this representation is equal to the dimension of the Lie algebraitself. Generally in a given representation r of a Lie algebra on a vector space V witheigenvectors | v i and weights λ i , that is with: H ( r ) i | v i = λ i | v i then: H ( r ) i ( E ( r ) α | v i ) = E ( r ) α H ( r ) i | v i + [ H ( r ) i , E ( r ) α ] | v i = ( λ i + α i )( E ( r ) α | v i ) (8.43)using equation 8.41. That is, the E ( r ) α act as ‘raising’ operators (while the E ( r ) − α act as‘lowering’ operators) on the eigenstates in the representation.Hence by comparison of equation 8.39 with equation 8.41 above the complexlinear combinations ˙Σ (2) ± of equation 8.37 as eigenvectors ˙ S –– l under the 78-dimensional193djoint representation of E indeed have charges of ± Q which acts on the e -lepton and d -quark states identified in the θ = (cid:0) c ¯ b (cid:1) components of the 27-dimensional representation of the E symmetry on the spaceh O . Further, according to equation 8.43, the ˙Σ (2) ± actions are expected to transformstates in the h O representation with a change of ± ⊂ SL(2 , C ) these raising and lowering operations areassociated with the θ = (cid:0) a ¯ c (cid:1) components of h O as shown explicitly in equations 8.29and 8.31. In this subsection we have focussed precisely upon this doublet action ofthe SU(2) symmetry which appears to be closely related to transformations withinthe lepton (cid:0) νe (cid:1) and quark (cid:0) ud (cid:1) doublets as mediated by the W ± gauge bosons in theStandard Model.In the Cartan-Weyl basis generally the Lie bracket [ E α , E − α ] describes anelement of the Cartan subalgebra, as can be seen from equation 8.42. From the E Liealgebra table in [38] we find:[ ˙Σ (2)+ , ˙Σ (2) − ] = [( ˙ R zl + i ˙ R xz ) , ( ˙ R zl − i ˙ R xz )] = − i ˙ S l − i ˙ R xl (8.44)which is indeed in the Cartan subalgebra of equation 6.56, for the complexified E Lie algebra, and also for the complex su(3) s subalgebra. This is consistent with theidentification of the ˙ R xl ‘charges’ for ˙Σ (2) ± in equation 8.40. However it the S –– l actionthat has been associated with the internal symmetry U(1) Q in the previous sectionand in turn the eigenvalues of ˙ S –– l associated with physical electromagnetic charges. Itis the latter charges of ± (2) ± which will be provisionally associatedwith the W ± µ ( x ) charged gauge fields in the mock electroweak theory.In quantum field theory the creation and annihilation operators associated with real fields do not describe charged particles, rather conserved charges are associatedwith complex fields, or complex linear combinations of real fields, as will be describedin section 10.3. A complex scalar field Y ( x ) has charge q under a U(1) symmetry ifit transforms as Y → e iqα Y , with α ∈ R and e iqα ∈ U(1), with q also labelling theirreducible representation of U(1). The derivative of this transformation at α = 0 canbe written as ∂ Y /∂α = ˙ Y = + q ( i Y ). This has the same form as equation 8.23, whichvia equation 8.24 implies the U(1) Q action ˙ S –– l on the field components θ l ( x ) reads[ ˙ θ l ] = +1[ lθ l ], with the complex imaginary unit l and charge represented by the realeigenvalue q = +1.As for the electron and d -quarks charges identified in the components of h O it remains to be seen how the charges for gauge bosons derived from generators suchas ˙Σ (2) ± relate to the likelihood of physical processes such as observed in high energyphysics experiments for the present theory. This will be discussed in section 11.2in comparison with standard QFT for which the charges are placed by hand intoLagrangian terms, leading to calculations of transition amplitudes and cross-sections.The phenomenon of running coupling, as described in section 11.3, will ultimatelyalso need to be considered for any comparison between theoretical couplings derivedfrom a normalised Killing form for a simple Lie algebra and the couplings measuredempirically in the laboratory. As well as accounting for quantisation a full dynamicaltheory will also be required, incorporating for example self-interactions for non-Abeliangauge fields, as explored in relation to Kaluza-Klein theories here in chapters 4 and 5.194 .3.2 SU (2) × U (1) Mixing Angle
The four type 1 actions { ˙ R zl , ˙ R xz , ˙ R xl } , ˙ S –– l generate the group SU(2) × U(1) Q . HereSU(2) , generated by { ˙ R zl , ˙ R xz , ˙ R xl } , is the rotation subgroup of the external Lorentztransformations, as studied in section 8.1, which commutes with the internal symmetryU(1) Q , underlying Maxwell’s electromagnetic field, generated by ˙ S –– l as identified in sec-tion 8.2. For the case of the corresponding set of four type 2 actions { ˙ R zl , ˙ R xz , ˙ R xl } , ˙ S –– l , a similar structure can be identified for SU(2) × U(1) . In a similar way that ˙ S –– l com-mutes with su(2) , and indeed with the Lorentz group sl(2 , C ) , it is also the case that˙ S –– l commutes with su(2) and hence with ˙Σ (2) ± of equation 8.37:[ ˙ S –– l , ˙Σ (2) ± ] = 0 (8.45)This commutator is consistent with those in equations 8.39 and 8.40 given the lineardependence obtained from equations 6.48 and 8.24:˙ S –– l = ˙ R xl − ˙ S –– l (8.46)While the generator ˙ S –– l is associated with electric charge Q the generator ˙ R xl is asso-ciated with T , the third component of SU(2) . The linear dependencies in the E Liealgebra of equations 6.47 and 6.48 also imply the relation: − ˙ S –– l = ˙ R xl + ˙ S –– l (8.47)which is closely reminiscent of the relation: Q = T + Y (8.48)of equation 7.38, within the choice of sign conventions. This suggests associating ˙ S –– l with Y as a candidate for the generator of the hypercharge symmetry U(1) Y ∼ U(1) which commutes with SU(2) , as generated by { ˙ R zl , ˙ R xz , ˙ R xl } , and as provisionallyassociated with SU(2) L for a mock electroweak theory in the previous subsection. Thegenerator ˙ S –– l may also be expressed as the linear combination of type 1 elements inequation 8.46, which lies in the Cartan subalgebra of the E Lie algebra. Hence the‘weights’ of ˙ S –– l may indeed be considered as ‘charges’, which are termed hyperchargesfor the corresponding U(1) Y symmetry.More generally opening up consideration of the three SL(2 , C ) a actions in theprevious subsection also motivates an examination of the U(1) charge structure asso-ciated with ˙ S –– al for all three types a = 1 , ,
3. To understand the relationships betweenthese charges all three generators ˙ S –– al , for a = 1 , ,
3, from table 6.7 with a factor of × from equation 8.24, are explicitly written out in terms of T h O components in equa-tion 8.49. Each entry of the form ( x, y ) represents the factors of l which multiply thecomponents of h O algebraically from the left side – where x is the ‘leptonic part’, thatis on the real and l components, while y is the ‘quark part’, that is on the remainingsix imaginary units of each a, b, c ∈ O . The components for ˙ S –– l in equation 8.49 containthe same information as equation 8.27 rearranged into the 3 × T h O . Itcan be seen here that ˙ S –– l + ˙ S –– l + ˙ S –– l = 0, for each component a, b and c , consistent withequation 6.42. Also shown are the corresponding components of ˙ R xl as obtained from195able 6.6, which can be seen to be consistent with equation 8.47.˙ S –– l ˙ S –– l ˙ S –– l (0 , + ) (1 , − )(0 , + ) (1 , − )( − , − ) ( − , − ) ( − , − ) ( − , − )(1 , − ) (0 , + )(1 , − ) (0 , + ) (1 , − ) (0 , + )( − , − ) ( − , − )(0 , + ) (1 , − ) ˙ R xl (+ , − ) ( − , + )( − , − ) ( − , , + ) (+1 , all as l ¯ a ca ¯ b ¯ c b ∈ T h O (8.49)Hence the ˙ S –– l ‘hypercharge values’ of ( − , − ) on the ¯ a and c componentsin equation 8.49 match the Standard Model hypercharge values of Y ( l L ) = − and Y ( q L ) = + for the left-handed doublets of leptons and quarks respectively of equa-tion 7.36 (up to a sign convention, which again will ultimately depend on the definitionof particle and antiparticle states in spacetime). As can be seen in equation 8.31 anddescribed in the previous subsection these components (¯ a c ) are also linked by theSU(2) ⊂ SL(2 , C ) actions and corresponding ˙Σ (2) ± operators provisionally associ-ated with the W ± µ ( x ) charged gauge fields. Although some of these observations arenaturally mutually correlated, the (1 , , − ) L and (3 , , ) L pieces of equation 7.36 arehence closely associated respectively with the (cid:0) a ,l θ l (cid:1) and (cid:0) a (6) θ i,j,k (cid:1) components of h O inequations 8.26 and 8.28.While right-handed fermion states remain to be identified, the hyperchargesof the right-handed fermion singlets in equation 7.36 are also closely correlated withthe ˙ S –– al charges in equation 8.49. This is expected since Q = Y for these cases andthe electric charge Q is well described by ˙ S –– l . Also in the top row of ˙ S –– l the values(1 , − ), (0 , + ) have the same magnitude as the Y values for the right-handed singlets e R , d R , ν R and u R respectively, although these components do not correspond tothe correct electromagnetic charges Q under ˙ S –– l for those respective fermion states.However these observations do suggest opening up consideration of the (correlated)charges for all three ˙ S –– al generators. Indeed as well as ˙ S –– l the generator ˙ S –– l should alsorelate to hypercharge as SU(2) × U(1) also forms a possible mock SU(2) L × U(1) Y action with the following linear dependence also found within the E algebra:˙ S –– l = ˙ R xl − ˙ S –– l (8.50)This equation is the type 3 version of equation 8.47. Further linear relations include˙ S –– l = − R xl + ˙ S –– l and ˙ S –– l = 2 ˙ R xl + ˙ S –– l which combine non-commuting type 2 and3 actions on the right-hand side. Such equations of linear dependence, relating thegenerators ˙ S –– al and ˙ R bxl for a, b = { , , } , are fixed by the structure of the E Liealgebra and closely resemble equation 7.38 which is constructed to relate the electriccharge Q , third component of weak isospin T and hypercharge Y in the Standard196odel. However a fuller understanding of this structure in the present theory willrequire the identification of right-handed fermion states, and in particular such stateswith T = 0. The origin of both left and right-handed states, together with theirmutual relation will be considered explicitly in section 9.2, while in the meantime wefurther consider the structure of the mock electroweak theory within the E framework.In particular, moving away from the a static analysis of the E symmetrybreaking pattern to a more dynamic perspective, we next study the structure of anSU(2) × U(1) gauge theory based on the symmetry generators { ˙ R zl , ˙ R xz , ˙ R xl , ˙ S –– l } .These act on the doublet components of the type 2 spinor θ = (cid:0) a ¯ c (cid:1) in h O . Restrictedto the complex subspace C ⊂ O with { , l } basis units the components θ l = (cid:0) a ¯ c (cid:1) l provi-sionally represents the lepton doublet (cid:0) νe (cid:1) . Since these components do not correspondto complete SL(2 , C ) Weyl spinors for either the neutrino or the electron part thisSU(2) × U(1) symmetry is clearly not directly equivalent to the SU(2) L × U(1) Y symmetry of electroweak theory. However the components of θ l do transform underthe internal symmetry SU(3) c × U(1) Q appropriately to represent such a lepton dou-blet, as described in section 8.2, and hence the SU(2) × U(1) symmetry serves as auseful intermediate model, considered as a mock electroweak theory. The equations ofmotion for the corresponding field θ l ( x ) in spacetime M will then involve the gaugecovariant derivative (essentially as described in section 3.1): D µ θ l ( x ) = ∂ µ θ l ( x ) + ˜ g ˜ W αµ ( x ) ˙ R (2) α ( θ l ) + ˜ g ′ ˜ B µ ( x ) 12 ˙ S –– l ( θ l ) (8.51)where α = 1 , , R (2) α ≡ { ˙ R zl , ˙ R xz , ˙ R xl } , and for example ˙ R zl ( θ l ) denotes the θ l components of ˙ R zl . The couplings ˜ g, ˜ g ′ and the gauge fields ˜ W αµ ( x ) , ˜ B µ ( x ) associatedwith the SU(2) × U(1) gauge symmetry are introduced by analogy with the StandardModel case in equation 7.40 and hence similar notation is adopted. However it isimportant to contrast the corresponding gauge coupling terms implied in equations 7.40and 8.51 with for example respectively: D µ l L ∼ i g ′ B µ ( x ) Y l L ) l L and D µ θ l ∼ ˜ g ′ ˜ B µ ( x ) ˙ S –– l θ l ) (8.52)In the former case, apart from the conventional factor of ‘ i ’ there are four factors: thecoupling g ′ , the gauge field B µ ( x ), the hypercharge generator Y ( l L ) = − (cid:0) (cid:1) andthe lepton doublet l L = (cid:0) νe (cid:1) L . In the latter case there are only three factors: with thecoupling ˜ g ′ and gauge field ˜ B µ ( x ) having a similar role as for the first case, while thethird part ˙ S – l ( θ l ) corresponds to the action of the hypercharge generator represented directly on the θ l = (cid:0) a ¯ c (cid:1) l components of h O , which is equivalent to the combination Y ( l L ) l L for the standard case.In principle in the second case the coupling ˜ g ′ may be absorbed into the gaugefield ˜ B µ ( x ) since we are here dealing with the pure covariant derivatives, as originallyexpressed in equation 2.38 of subsection 2.2.3 for the gauge field A µ ( x ) without anycoupling constant. Adopting couplings such as ˜ g ′ = 1 is also compatible with theconstruction of a direct relationship between the curvature for the external linearconnection and that for the internal gauge connection as described in section 5.1 andequation 5.20 (with a factor such as χ = 1 in principle determined by the geometricstructure), in comparison with Kaluza-Klein theory. A similar observation applies for197he coupling ˜ g associated with the gauge field ˜ W αµ ( x ) in equation 8.51. Ultimatelyboth ˜ g and ˜ g ′ will be absorbed into the relevant gauge fields and effectively set equalto one. In turn the ‘charges’ of individual states will depend upon the representationwhich is already determined directly by the values of ˙ S – l ( θ l ) in the second expression ofequation 8.52, which are closely analogous to the case for the electromagnetic chargesobtained from ˙ S –– l ( θ ) in equations 8.23 and 8.24 and further discussed towards theend of the previous subsection. More generally this will require a suitable mutualnormalisation of the generators ˙ R (2) α and ˙ S –– l based on the Killing form of the full E Lie algebra, as also described in the previous subsection, to relate the charges for thevarious subgroups of the internal gauge symmetry. With the gauge groups representeddirectly on the space T h O this structure parallels that employed for Kaluza-Kleintheory based on homogeneous fibres as described in section 4.3.For now considering ˜ g and ˜ g ′ as free parameters in equation 8.51 allows acloser comparison with the structure of the electroweak theory in the Standard Modelfor which the couplings g and g ′ are independent. However here neither the SU(2) generated by { ˙ R zl , ˙ R xz , ˙ R xl } nor the U(1) generated by ˙ S –– l are internal symmetriesin the sense of table 8.3, that is within Stab( TM ), with each of these four generatorsimpacting upon the components of the type 1 subspace h C ⊂ h O , which representcomponents of the external spacetime TM , as can be seen explicitly from the formof these four generators in tables 6.6 and 6.7. The breaking of the full E symmetryaction on h O in this identification of the type 1 subspace h C with the local tangentspace of the external spacetime hence includes the breaking of the SU(2) × U(1) ⊂ E subgroup.The covariant derivative applied to the θ l components in equation 8.51 canbe applied to the components of h O more generally and written out explicitly usingtables 6.6 and 6.7. In particular we find that applied to the type 1 embedding of the2 × X ∈ h O ⊂ h O and h ∈ h C ⊂ h O this covariantderivative reads respectively: D µ X = ∂ µ X + g ˜ W µ ( cl ) + ˜ g ˜ W µ ( c ) + ˜ g ˜ W µ ( ¯ al ) + ˜ g ′ ˜ B µ ˙ S –– l (¯ a )˜ g ˜ W µ ( − l ¯ c ) + ˜ g ˜ W µ ( ¯ c ) + ˜ g ˜ W µ ( − la ) + ˜ g ′ ˜ B µ ˙ S –– l ( a ) ˜ g ˜ W µ ( b l ) + ˜ g ˜ W µ ( b x ) + 0 + 0 (8.53) D µ h = ∂ µ h + ˜ g (cid:16) ˜ W µ ( c l − c ) + ˜ W µ ( c + c l ) + ˜ W µ ( a l + a ) (cid:17) + ˜ g ′ ˜ B µ ( − a l − a ) ˜ g (cid:16) ˜ W µ ( − c l − c ) + ˜ W µ ( c − c l ) + ˜ W µ ( − a l + a ) (cid:17) + ˜ g ′ ˜ B µ ( a l − a ) ˜ g ˜ W µ ( b ) + ˜ g ˜ W µ ( b ) (8.54)where the second equation shows that indeed each of the four gauge fields ˜ W αµ ( x ),˜ B µ ( x ) has non-zero impact on the { , l } components of X ∈ h O , that is on the 4-dimensional vector h ∈ h C of equations 8.4 and 8.5, unlike the case of equations 8.14and 8.15 for example, and hence are not associated with a purely internal symmetryin the sense of Stab( TM ). However an orthogonal linear combination of gauge fields198ay be taken with: ˜ B µ = cos θ M ˜ A µ − sin θ M ˜ Z µ ˜ W µ = sin θ M ˜ A µ + cos θ M ˜ Z µ (8.55)by analogy with equations 7.43 and 7.44, where θ M (with subscript M denoting‘mock mixing angle’ of type 2) plays a similar role to the weak mixing angle θ W . Thecorresponding contribution from the ˜ B µ ( x ) and ˜ W µ ( x ) fields to the ¯ a ,l components inthe top-right element of equation 8.54 is then: D µ ¯ a ,l = . . . +˜ g θ M ˜ A µ ( a l + a )+ ˜ g ′ θ M ˜ A µ ( − a l − a )+ ˜ g θ M ˜ Z µ ( a l + a ) − ˜ g ′ θ M ˜ Z µ ( − a l − a )(8.56)Hence the gauge field ˜ A µ ( x ) represents a purely internal field, with no actionon the external h ∈ h C components, provided:˜ g sin θ M = ˜ g ′ cos θ M that is: tan θ M = ˜ g ′ ˜ g (8.57)This relation is closely analogous to equation 7.46 for electroweak theory in the Stan-dard Model. However here in the case of equation 8.57 neither a Lagrangian for-malism, using for example equation 7.41, nor a Higgs field is required to break theSU(2) × U(1) symmetry down to a U(1) symmetry associated with the gauge field˜ A µ ( x ). Considering more generally the ˜ A µ ( x ) field part of the covariant derivative D µ of equation 8.51, via equation 8.55, on all of the components of X ∈ h O , with forexample ˙ R xl ≡ ˙ R xl ( X ), we have: D µ X ( x ) = ∂ µ X ( x ) + ˜ g sin θ M ˜ A µ ( x ) ˙ R xl + ˜ g ′ cos θ M ˜ A µ ( x ) 12 ˙ S –– l = ∂ µ X ( x ) + ˜ g sin θ M ˜ A µ ( x ) ˙ R xl + ˜ g tan θ M cos θ M ˜ A µ ( x ) 12 ˙ S –– l = ∂ µ X ( x ) + ˜ g sin θ M ˜ A µ ( x ) ( ˙ R xl + 12 ˙ S –– l )= ∂ µ X ( x ) + ˜ g sin θ M ˜ A µ ( x ) ( − ˙ S –– l ) (8.58)where the final line is fixed by the linear dependence of equation 8.47 for the generatorsof the E Lie algebra. The gauge field ˜ A µ ( x ) is hence associated with ˙ S –– l which as anelement of stab( TM ) has been identified as the generator of the internal gauge sym-metry U(1) Q of electromagnetism in the previous section (see the discussion followingequation 8.21). The zero charge of the ν -lepton, associated with the a ,l componentsin equation 8.28, is here taken to be entirely equivalent to the fact that the action ˙ S –– l does not impinge on the { , l } components of a ∈ O ⊂ h O . The apparently ambiguousnature of these a ,l components, which have been associated both with the neutrinostate and with part of the vector space h C ≡ TM on the external spacetime, will beresolved in section 9.2. 199he lines of equation 8.58 are closely analogous to those of equation 7.47 fromelectroweak theory, with ˜ A µ ( x ) ∼ A µ ( x ) and − ˙ S –– l ∼ Q . The apparent electromagneticcoupling ˜ e may be identified directly in equation 8.58 as:˜ e = ˜ g sin θ M (8.59)which is also analogous to equation 7.48 in the Standard Model.We next employ a basis for the E algebra with a normalised Killing form withcomponents proportional to the unit 78 ×
78 matrix. In this case it will be possibleto determine the value of the mixing angle θ M in the breaking of the SU(2) × U(1) symmetry to U(1) Q . For the su(3) s subalgebra such a normalised basis is provided byequation 8.36 with three possible choices of { ˙ R axl , √ ˙ S al } , for type a = 1 , ,
3, for thefirst two elements. The covariant derivative of equation 8.51 may be rewritten in thenormalised Killing form basis, with √ ˙ S al = √ ˙ S –– al (via equation 8.24) and with thecouplings ˜ g and ˜ g ′ absorbed into the gauge fields as: D µ θ l ( x ) = ∂ µ θ l ( x ) + ˜ W αµ ( x ) ˙ R (2) α ( θ l ) + ˜ B µ ( x ) √
32 ˙ S –– l ( θ l ) (8.60)where again α = 1 , , R (2) α ≡ { ˙ R zl , ˙ R xz , ˙ R xl } . The gauge fields { ˜ W µ ( x ) , ˜ B µ ( x ) } aligned with the generators { ˙ R xl , √ ˙ S –– l } may be expressed in a new basis with gaugefields { ˜ Z µ ( x ) , ˜ A µ ( x ) } aligned with the generators { ˙ R xl , √ ˙ S –– l } . In this basis the ‘in-ternal’ gauge field ˜ Z µ ( x ) is associated with ˙ R xl which as a generator of SL(2 , C ) , asoriginally listed in equation 6.57, is in fact a purely external action! However here weare dealing with a mock electroweak theory for which some inappropriate features maybe observed, as was the case for the ambiguity of the a ,l components noted above. Inany case the electromagnetic gauge field ˜ A µ ( x ) associated with √ ˙ S –– l in the new basisdoes represent a purely internal action. Transferring to the new basis we have:˜ W µ ˙ R xl + ˜ B µ √
32 ˙ S –– l ⇒ ˜ Z µ ˙ R xl + ˜ A µ √
32 ˙ S –– l hence: sin θ M ˜ A µ ˙ R xl + cos θ M ˜ A µ √
32 ˙ S –– l = ˜ A µ √
32 ˙ S –– l sin θ M ˜ A µ ( −
12 ˙ R xl −
34 ˙ S –– l ) + cos θ M ˜ A µ √
32 ( ˙ R xl −
12 ˙ S –– l ) = ˜ A µ √
32 ˙ S –– l (8.61)where in the second line the orthogonal transformation of equation 8.55 has beenapplied to the left-hand side and only the ˜ A µ field part has been retained on bothsides. Equations 6.47 and 6.48, together with equation 8.24, have been used for thebottom line. By equating the basis vector ˙ R xl and ˙ S –– l parts separately in this final lineabove it can be deduced that:sin θ M = − √
32 and cos θ M = −
12 (8.62)and hence: sin θ M = 34 with θ M = 240 (8.63)200s the mixing angle. Performing a similar analysis for the type 3 case of SU(2) × U(1) breaking to U(1) Q leads to a similar result, except with sin θ M = + √ and θ M = 120 .Setting ˜ g = 1 the magnitudes of the coupling constants to substitute into equations 8.51and the bottom line of equation 8.58 in order to match the normalised expressions ofequations 8.60 and the right-hand side of equation 8.61 are, relative to ˜ g :˜ g : ˜ g ′ = √ g : ˜ e = √
32 ˜ g (8.64)These values are consistently obtained from equations 8.57 and 8.59 by substitutingin the value of θ M from equations 8.62 and 8.63, with a similar observation applyingfor the type 3 case.This analysis is useful for comparison with the Standard Model for which thegauge groups SU(2) L and U(1) Y are not obtained from a single unifying group andhence the respective gauge couplings g and g ′ of equation 7.40 are independent. Indeedequation 8.60 above may be compared with the form of the covariant derivative of aleft-handed doublet of leptons in the Standard Model, which from equations 7.37 and7.40 can be written as: D µ = ∂ µ + igW αµ ( x ) 12 σ α − ig ′ B µ ( x ) 12 σ (8.65)In this case the third component of weak isospin T = σ and hypercharge Y = − σ combine to form the charge operator Q = (cid:0) − (cid:1) via equation 7.38 for the leptondoublet. In this particular case for equation 7.47 we have the weak mixing combination: ig W µ σ − ig ′ B µ σ ⇒ ie A µ Q which may be directly compared with:˜ W µ ˙ R (2) xl + ˜ B µ √
32 ˙ S –– l ⇒ ˜ A µ √
32 ˙ S –– l from the top line of equation 8.61. In the former case the set of 2 × { σ , σ , σ , σ } , as well as forming a basis for elements of the vector spaceh C ⊂ C (2), forms a basis for the Lie algebra SU(2) L × U(1) Y with the normalisa-tion convention tr( τ α τ β ) = δ αβ , here including τ = σ with α, β = 0 . . .
3. Thecouplings g and g ′ are introduced in this basis. For the empirically measured casethe electroweak mixing angle is determined to be sin θ W ≃ .
23 at the energy scaleof M Z [44], with corresponding electroweak couplings from equations 7.46 and 7.48approximately in the proportions: g : g ′ ≃ . g : e ≃ . g (8.66)Given the unit electron charge for the leptonic component θ l in equation 8.26the action of the generator ˙ S –– l is analogous to that of the unit 2 × σ of equa-tion 7.14. Similarly the normalisation of the type 1 actions { ˙ R zl , ˙ R xz , ˙ R xl } , as seen inequations 8.7 and 8.9, parallels the set of 2 × { τ , τ , τ } = { σ , σ , σ } .Transferring this analysis to the type 2 case, the set of generators { ˙ R zl , ˙ R xz , ˙ R xl , ˙ S –– l } also parallels the set of matrix actions { σ , σ , σ , σ } .201ince this is the generator normalisation used initially in equation 8.51 it maynaively be expected that the couplings obtained for the SU(2) × U(1) symmetrybreaking via the constraint of the E algebra Killing form in equation 8.64 may be di-rectly compared with the corresponding values for SU(2) L × U(1) Y electroweak theoryobtained empirically as displayed in equation 8.66. The significant differences in thesevalues hinges on the differing values for the calculated sin θ M = of equation 8.63and the empirical sin θ W ≃ .
23. However, as emphasised earlier in this subsection theSU(2) symmetry does not act on SL(2 , C ) Weyl spinors in the appropriate way to de-scribe weak interactions, and here we are dealing with a provisional ‘mock electroweaktheory’, which nevertheless exhibits some of the features associated with correspondingstructures of the Standard Model such as the identification of a mixing angle itself.It is also noted that in the mock theory the calculated value of sin θ M = effectively corresponds to a ‘unification scale’ whereas the empirical value of sin θ W ≃ .
23 is determined at the practical energy scale of M Z ∼ GeV. In standard quantumfield theory the phenomena of ‘running coupling’ for an Abelian compared with a non-Abelian gauge theory implies that the ratio g ′ : g increases with the energy scale aswill be described in section 11.3 and depicted in figure 11.10. Hence the need of aquantisation scheme for the present theory, as alluded to at the end of the previoussubsection and as proposed in chapter 11, with the consequence of running coupling,may be one factor leading to the large calculated mixing angle for the present theory.This observation would apply even if the gauge group SU(2) L × U(1) Y were to becorrectly identified in the theory.In any case in this subsection it has been demonstrated how the relativecouplings of the internal gauge groups may in principle be related through unifica-tion within the simple Lie group E . Finally here we consider how a more realisticelectroweak theory might be constructed within this framework. In the above wehave assumed a symmetry breaking pattern of SU(2) × U(1) → U(1) Q , whereasthese subgroups are actually embedded in a larger symmetry breaking structure withSU(3) s → U(1) Q . That is, instead of equation 8.60 we might rather begin with thegauge covariant derivative: D µ θ l ( x ) = ∂ µ θ l ( x ) + W αµ ( x ) ˙ R α ( θ l )where now α = 1 . . . s generators in equa-tion 8.36. Here all three embeddings of SU(2) a × U(1) a ⊂ SU(3) s , for type a = 1 , , ⊂ SL(2 , C ) as the rotation subgroupof the Lorentz group acting on external spacetime breaking the symmetry.While this symmetry breaking structure requires further study the fact that themock electroweak symmetry SU(2) a × U(1) a may be embedded in SU(3) s in two ways,of type a = 2 or a = 3, while the symmetry SU(2) × U(1) has only one embedding, oftype a = 1, may be of some significance. Within su(3) s the three U(1) a generators arelinearly dependent by equation 6.42, while by equation 6.46 only the ˙ R axl part of thethree SU(2) a generators are linearly dependent. These observations offer a hint thatfor an internal SU(2) a combining types a = 2 and 3 the ratio of the effective coupling˜ g to that for the effective U(1) Q coupling ˜ e maybe somewhat larger than that for thetype a = 2 case alone which led the final expression of equation 8.64, once the lineardependencies of the generators are taken into account, which may result in a closercorrespondence with the Standard Model case in equation 8.66.202ype 2 gauge fields:˜ W (2) ± µ ( x ) = ˜ W (2)1 µ ( x ) ∓ i ˜ W (2)2 µ ( x ) (8.67)may be associated with the type 2 generators ˙Σ (2) ± of equation 8.37 in the complexSU(2) subalgebra (by comparison with equations 7.57 and 7.58 for the StandardModel, although neglecting possible factors of √ or here). As described aboveand as can be seen from table 6.6 the relevant generators ˙ R zl and ˙ R xz mix the a component of h O with the c component only. The fact that in the mock theory theSU(2) × U(1) symmetry acts on the θ = (cid:0) a ¯ c (cid:1) ⊂ h O components, and not physicalfermion doublets, is one reason not to expect the calculated mixing angle to matchthe empirical case. That is, while the type 2 symmetry SU(2) × U(1) has some ofthe properties associated with the electroweak symmetry SU(2) L × U(1) Y , the SU(2) transformations do not relate the a ∈ h O component to both components of θ = (cid:0) c ¯ b (cid:1) .On the other hand type 3 gauge fields ˜ W (3) ± µ ( x ) = ˜ W (3)1 µ ( x ) ± i ˜ W (3)2 µ ( x ) maybe associated with similar generators in the complex SU(2) subalgebra:˙Σ (3) ± = ˙ R zl ∓ i ˙ R xz (8.68)The ± signs are chosen such that the generators ˙Σ (3) ± , as for ˙Σ (2) ± in equation 8.39,carry charges of ±
1, that is: [ i ˙ S –– l , ˙Σ (3) ± ] = ± ˙Σ (3) ± (8.69)Adding to the discussion towards the end of the previous subsection, together ˙Σ (2) ± and ˙Σ (3) ± describe four of the six eigenvectors of the Cartan subalgebra in the Cartan-Weyl basis for the adjoint representation of the complexified su(3) s algebra. The fullset of six eigenvectors are sometimes denoted U ± , V ± and T ± in the su(3) root spacediagram (as for example in the context of the SU(3) flavour symmetry between u , d and s -type quarks).For the case of equation 8.68, as can also be seen from table 6.6, the generators˙ R zl and ˙ R xz mix the a component of h O with the b component only, that is withinthe θ = (cid:0) b ¯ a (cid:1) ⊂ h O components as shown in equation 8.31. Hence it appears thatphysical charged gauge boson fields W ± µ ( x ) must indeed be related to both type 2 ˙Σ (2) ± and type 3 ˙Σ (3) ± operators to act on fermion doublets. Ultimately the interactions ofthe physical W ± particle states will need to be appropriately oriented with respect tophysical fermion states. The latter will in turn require a possible SL(2 , C ) Weyl spinorinterpretation of the a ∈ h O components, which have been provisionally associatedwith neutrino and u -quark states according the internal SU(3) c × U(1) Q transformationsof equation 8.28.The possible means of identifying Weyl spinor states for the ν -lepton and u -quarks within the a ∈ O ∈ h O components will be addressed in section 9.1. Theidentification of both left and right-handed Weyl spinors together with the Dirac rep-resentation of the external SL(2 , C ) symmetry will then be described in section 9.2.Finally the possibility of identifying three generations of fermions and the phenomenaof CKM mixing will be outlined in section 9.3. All of the above features may need tocome together in order to fully identify the physical SU(2) L symmetry together with203tandard phenomena of electroweak theory within the context of the present theory.In the meantime in the following subsection we study further suggestive features ofthe SU(2) × U(1) mock electroweak theory based within the E framework, and inparticular concerning the source of finite mass for the both the gauge bosons and thefermion states. Then we shall briefly consider further possible SU(2) ⊂ E subgroupsas candidate components of an electroweak symmetry, before extending beyond E inthe following chapter. The empirical weakness of the weak interaction relative to electromagnetic phenomenaowes not to the value of the coupling g , in equation 7.40 for example, which is aroundtwice the value of e , equation 8.66, but to the large values for the masses of the W ± and Z gauge bosons. Although in this chapter we are dealing primarily at the level of theLie algebra structure, together with the simple dynamic expressions introduced in theprevious subsection, it will be considered here how mass terms for particle states mayoriginate in the symmetry breaking structure, not only for the massive gauge bosonsbut also for leptons and quarks in the full theory. Here ˜ W ± and ˜ Z gauge bosonswill be provisionally associated with the appropriate fields of the SU(2) × U(1) mockelectroweak theory, and hence we first look in more detail at the field ˜ Z µ ( x ).The gauge field ˜ Z µ ( x ) appearing in the top line of equation 8.61 was iden-tified along with ˜ A µ ( x ) as aligned to the choice of basis elements { ˙ R xl , √ ˙ S –– l } . Asdescribed earlier the apparent association of the ‘internal’ field ˜ Z µ ( x ) with the ‘exter-nal’ generator ˙ R xl is one of a number of significant caveats associated with the mockelectroweak theory. Through the orthogonal transformation of equation 8.55; that iswith ˜ Z µ = cos θ M ˜ W µ − sin θ M ˜ B µ , in analogy with electroweak theory and equa-tion 7.43, as for the ‘photon’ field ˜ A µ ( x ), the field ˜ Z µ ( x ) is associated with a linearcombination of the generators ˙ R xl and ˙ S –– l . Since in the E Lie algebra [ ˙ S –– l , ˙ R xl ] = 0and [ ˙ S –– l , ˙ S –– l ] = 0 any such linear combination of ˙ R xl and ˙ S –– l has zero electromagneticcharge. Hence the ˜ Z gauge boson and ˜ γ photon, associated with the fields ˜ Z µ ( x )and ˜ A µ ( x ) respectively, are neutral, unlike the case of the charged ˜ W ± gauge bosonsassociated with fields ˜ W (2) ± µ ( x ) of equation 8.67 corresponding to the generators ˙Σ (2) ± of equations 8.37 and 8.39. This is also the case when such linear combinations areextended to include the type 3 form of these generators since also [ ˙ S –– l , ˙ R xl ] = 0 and[ ˙ S –– l , ˙ S –– l ] = 0.From the type 2 ˜ W µ and ˜ B µ terms in equation 8.53 it can be seen that thetransformations associated with the fields ˜ Z µ ( x ) and ˜ A µ ( x ) mix the components of a ∈ O within h O . This is unlike the more involved transformations associated with thefields ˜ W (2)1 µ ( x ) and ˜ W (2)2 µ ( x ), as can also can be seen in table 6.6 for the correspondinggenerators ˙ R zl and ˙ R xz which mix components of h O ⊂ h O with those not in h O .Isolating the interaction of the ˜ Z µ field with the a and θ = (cid:0) c ¯ b (cid:1) components separatelymay allow a determination of the coupling of the ˜ Z to the lepton pairs as well asquark pairs, which might be directly compared with the electromagnetic coupling ofthe photon to the same components as summarised in equations 8.23, 8.26 and 8.28.This may be more straightforward than for interactions involving the ˜ W ± gauge204osons as here not only is the interaction restricted to single components but also onegeneration of fermion states may suffice since there are no flavour changing neutralcurrents in the Standard Model, as described at the end of section 7.2. Hence a moredetailed study of interactions for the field ˜ Z µ ( x ) in comparison with the field ˜ A µ ( x )may prove enlightening in comparison with the relevant properties of the StandardModel described in section 7.2 and in particular with respect to the determination ofthe relative couplings.This may involve linear combinations of type 2 and type 3 actions on θ = (cid:0) c ¯ b (cid:1) with generators of the weak neutral field ˜ Z µ ( x ) being complementary to the generator˙ S –– l of the electromagnetic field ˜ A µ ( x ) with respect to the full SU(3) s ⊂ E symme-try, as considered towards the end of the previous subsection, with the ˜ Z µ ( x ) fieldassociated with a different linear combination of charge neutral SU(3) s generators.Ultimately however for comparison with weak neutral interactions described in theStandard Model via equation 7.49 both left-handed and right-handed fermions willneed to be identified, such that T = 0 for right-handed states, and this itself willrequire an extension beyond the study of E on the space h O . Such an extension willalso be required to identify the physical SU(2) L × U(1) Y symmetry, independent of theexternal SL(2 , C ) generators, and fully account for both W ± and Z interactions.In the meantime here we consider broader features of the mock electroweaktheory as described in the previous two subsections, and in particular how masses mayarise for gauge bosons through the impingement of the SU(2) × U(1) ⊂ E symmetryon the 4-dimensional subspace h C ⊂ h O associated with the tangent space TM ofthe external spacetime. Having in mind comparisons with the Standard Model wereturn to the convention of equation 8.51 with coupling parameters ˜ g and ˜ g ′ in placeof employing generators normalised according to the E Killing form.In equation 8.56 the coupling of the ˜ A µ ( x ) and ˜ Z µ ( x ) fields to the ¯ a ,l = a − a l components of h C ⊂ h O was extracted. The constraint tan θ M = ˜ g ′ / ˜ g was derivedin equation 8.57 in order for the ˜ A µ ( x ) contribution to vanish. With this constraint theimpingement of the field ˜ Z µ ( x ) on the ¯ a ,l subcomponent part of h C from equation 8.56can be written: D µ ¯ a ,l = . . . + ˜ g θ M ˜ Z µ ( a l + a ) − ˜ g ′ θ M ˜ Z µ ( − a l − a )= 12 (˜ g cos θ M + ˜ g ′ sin θ M ) ˜ Z µ ( a l + a )= (˜ g cos θ M + ˜ g tan θ M sin θ M ) ˜ Z µ
12 ( a l + a )= ˜ g cos θ M ˜ Z µ
12 ( a l + a ) (8.70)This compares with the impingement of the type 2 fields ˜ W (2) ± µ ( x ) of equation 8.67,as composed of ˜ W (2)1 µ ( x ) and ˜ W (2)2 µ ( x ), on the same off-diagonal elements of h C inequation 8.54, which is proportional to ˜ g/
2. Hence the coupling of the correspondinggauge fields to the subcomponent ¯ a ,l of h C ⊂ h O is in the following ratio:˜ Z µ : ˜ W ± µ : ˜ A µ ˜ g cos θ M : ˜ g : 0 (8.71)205his suggests, given the Standard Model expression for M W in equation 7.60and its relation to M Z in equation 7.62, that the interaction with components of h C originating here in equation 8.54 is closely related to the masses of the gauge bosons inthe present theory (within the caveats that the impingement on all four components ofh C may need to be addressed and factors of 2 or √ φ .Mass terms such as for equations 7.60 and 7.62, arising in the Standard ModelLagrangian, are quadratic in the gauge boson fields due to the quadratic composi-tion ( D µ φ ) † D µ φ constructed for the Lorentz invariant initial Lagrangian L H in equa-tion 7.51. In the present theory the composition of the gauge fields with the compo-nents of h O in expressions such as D µ L ( v ) = 0 has a different structure, linear inthe gauge fields. Here the concept and nature of particle ‘mass’ is yet to be identified,and will require an understanding of quantisation and physical particle states as willbe described in chapter 11. However the impingement of the ‘internal’ SU(2) , sym-metry upon the components of the external spacetime tangent space TM is expectedto correlate closely with the phenomenology of the massive W ± and Z , involving thekinematic properties of these gauge boson states in spacetime, and hence accountingfor the short-range nature of the weak interaction. If gauge boson masses may beobtained through these interactions this raises the question of how further elements ofthe present theory might correspond to the Higgs sector of the Standard Model.In the present theory the Lorentz SO + (1 ,
3) symmetry acts on the form L ( v ) =( v ) − ( v ) − ( v ) − ( v ) = h of equation 5.46 with the components of v ∈ h C embedded in h O under the SL(2 , C ) ⊂ E subgroup action. In a suitable choice offrame a Lorentz 4-vector can be expressed as v = ( v , , ,
0) which in turn can bewritten as h = v v (8.72)that is with the three components v = v = v = 0 in equations 8.4 and 8.5. This 4-vector is invariant under the SU(2) ⊂ SL(2 , C ) transformations h → S h S † , whichpreserve the form of equation 8.72, with the type 1 rotations S ∈ SU(2) ⊂ SL(2 , C ) generated by { ˙ R zl , ˙ R xz , ˙ R xl } ; that is the subset of Lorentz generators in equation 6.57leaving the t ≡ v component in equation 6.58 fixed.As described in section 7.2 for the Higgs sector an SU(2) custodial symmetryoriginates as a subgroup of the SO(4) symmetry of the form of the potential V ( φ ) = f ( φ + φ + φ + φ ) implicit in equation 7.53. The vacuum value of the StandardModel Higgs field can be expressed in terms of the bi-doublet Φ of equation 7.64 inthe form of equation 7.68, that is with h Φ i = (cid:0) v v (cid:1) , which is invariant under thetransformations h Φ i → L h Φ i L † with L ∈ SU(2) L + R , where SU(2) L + R is the custodialsymmetry, highlighting the close similarity to the symmetry of a given Lorentz 4-vector h ∈ h C such as that in equation 8.72 in the present theory.Rather than a Higgs complex doublet field φ and Lagrangian L H with ‘acciden-tal’ global SO(4) symmetry (for g ′ → v with anexternal SO + (1 ,
3) symmetry for the form L ( v ) = h which is ‘spontaneously broken’206y the non-zero particular projected value of v ∈ TM . Here the value v = 0 inequation 8.72 is simply the magnitude of the Lorentz 4-vector v , projected out of thecomponents of v ∈ h O , onto the tangent space TM ; without the need for a ‘Mex-ican hat potential’ such as equation 7.53 to provide the mechanism for ‘spontaneoussymmetry breaking’ and induce a non-zero ‘vacuum value’ for the field. This choice ofvacuum value for v ∈ TM is in addition to the E symmetry breaking through thenecessary choice of a Lorentz subgroup SL(2 , C ) ⊂ E associated with the externalspacetime and as explored in the earlier sections of this chapter.Here the Lorentz symmetry itself, expressed with h = v · σ as h → S h S † in equation 7.30 and 7.31, is reduced to the choice of S ∈ SU(2) ⊂ SL(2 , C ) for a particular 4-vector v ∈ TM , as a close analogy to the custodial symmetrySU(2) L + R ⊂ SU(2) L × SU(2) R for the Higgs case as described following equation 7.68.However the present theory may need to be developed beyond the model based onthe SU(2) × U(1) symmetry towards a more standard SU(2) L × U(1) Y electroweaktheory before a more precise correlate of the ‘custodial symmetry’ might be identified.In the Standard Model electroweak theory three of the four SU(2) L × U(1) Y generators are spontaneously broken since they change the vacuum expectation valueof the Higgs field ( h φ i in equation 7.54 or h Φ i in equation 7.68), while maintaining theminimum of the Higgs potential ( V ( φ ) in equation 7.53 or V (Φ) in equation 7.65). Thethree degrees of freedom of the Higgs field associated with the broken generators giverise to the mass terms for the W ± and Z gauge bosons in the Lagrangian. Fluctuationsaround the vacuum v + H ( x ) in the fourth degree of freedom are associated with a massterm for the Higgs scalar particle, which is also proportional to the vacuum value of v ≃
246 GeV, as described shortly after equation 7.62. The unbroken U(1) Q generator Q leaves both h φ i and V ( φ ) invariant, with the vacuum carrying zero electric charge,as described just after equation 7.54.In the present theory, while the v ∈ h C vacuum value of equation 8.72 is alsoinvariant under the U(1) Q symmetry, in fact with the S –– l action leaving all componentsof h C ⊂ h O unchanged, the SU(2) × U(1) generators associated with the ˜ W ± µ and˜ Z µ fields mix the v components in h O such that | v | is not invariant, unlike the casefor V ( φ ) in the standard electroweak theory as described above. While here we aredealing with a mock theory the possibility of associating three of the four degrees offreedom for δ v ( x ) (in particular for the spatial components { v ( x ) , v ( x ) , v ( x ) } ) withlongitudinal components for the gauge bosons and hence masses for the ˜ W ± and ˜ Z particles may assist in the identification of the physical SU(2) L × U(1) Y electroweaktheory within the present framework. In any case here fluctuations in the Lorentz scalarmagnitude | v | (closely related to variation in the remaining temporal component v + ˜ H ( x )) will be associated with the Higgs field and corresponding massive bosonparticle state. That is, ˜ H ( x ) ∼ δ | v ( x ) | in the present theory is provisionally correlatedwith the scalar field H ( x ) in the Standard Model.In the full dynamical quantum theory it will of course be necessary to explainhow the phenomenology of the Standard Model Lorentz scalar Higgs field and particlestate, as observed in the laboratory, may be derived in detail from the components ofthe fundamental 4-vector field v , which will be referred to in this context as a ‘vector-Higgs’. This structure brings to mind other models for which there is no fundamentalHiggs scalar field, with the latter for example composed out of fermion states. Hence207ere we briefly review some of the properties of technicolor models ([57, 58], see also[46]) for comparison and contrast with the present theory.For QCD with two flavours Q L,R = (cid:0) ud (cid:1) L,R in the massless fermion limit themanifold of vacuum states, that is the 2 × h Q L Q R i 6 = 0, breaks theglobal symmetry of the Lagrangian resulting in three Goldstone bosons correspondingto the three pion states π ± and π . Coupling the quarks to SU(2) L × U(1) Y thisgauge symmetry is broken by the vacuum since the SU(2) only couples to the left-handed fermions, while a U(1) Q gauge symmetry is preserved. The symmetry breakinggenerates masses for the corresponding W ± and Z gauge bosons which are, however,too small compared to the empirical values since the pion decay constant f π is onlyaround 93 MeV.Motivated by these observations and difficulties associated with a fundamentalscalar Higgs in the Standard Model, a new strongly interacting sector of fermionscalled ‘techniquarks’ is postulated which couple to a new ‘technicolor’ gauge symmetrySU( N ) tc . The techniquarks T L,R = (cid:0) UD (cid:1) L,R also transform under SU(2) L × U(1) Y butare singlets under the standard colour symmetry SU(3) c . Scalar combinations of T and T condense in the vacuum owing to the new strong technicolor interaction. Asfor the QCD case above the vacuum state is termed a ‘condensate’ by analogy withphenomena in condensed matter physics, and in particular the formation of BCS pairsof electrons in superconductivity.For such a model the technipion decay constant may be taken to be F Π ≃
246 GeV to replicate the masses of the W ± and Z as previously obtained with ascalar Higgs sector. The Standard Model relation M W /M Z = cos θ W of equation 7.62is also reproduced. The techniquark Lagrangian includes kinetic terms of the form: L tc ∼ T L,R γ µ ( ∂ µ + ig N G tc µ + igW µ + ig ′ B µ ) T L,R (8.73)with technicolor gauge field G tc µ and coupling g N as well as the SU(2) L × U(1) Y gauge fields and couplings. In the quantum field theory masses for the W ± and Z gauge bosons are generated by the corrections introduced into the corresponding gaugeboson propagators through the interaction terms in equation 8.73, with massless tech-nipions effectively appearing as the longitudinal components of massive W ± and Z bosons. The low energy behaviour can be described by an effective phenomenologicalLagrangian for the vacuum expectation value Ψ( x ) = h T L T R i 6 = 0 with: L Ψ ∼ F tr( D µ Ψ † D µ Ψ) (8.74)For the two techniquark model the scalar Ψ( x ) is a 2 × x ).Masses for ordinary quarks and leptons are introduced by replacing the scalarHiggs in the Standard Model Yukawa terms of equation 7.69 by techniquark bilinearsresulting in 4-fermion interactions with quartic terms such as the scalar: L ∼ Q L ( T L T R ) Q R (8.75)Here Q L,R are ordinary quarks which gain mass when the techniquarks form a conden-sate h T L T R i 6 = 0. A suitable variety of quartic interactions and coupling parameters208re needed to reproduce the empirical values for the standard quarks and leptons.Higher-order Lagrangian terms such as 6-fermion interactions may also be considered.As a theory of electroweak symmetry breaking without a fundamental scalarHiggs the above technicolor model has some resemblance with the present theory. Thestructure of the 2 × h T T i as composed out of fermions indeed baressome resemblence to the spinor decomposition of the vector h = χχ † + φφ † of equa-tion 7.32. However the latter expression merely represents the algebraic substructure within the components of h without the need of a new technicolor interaction withgauge group SU( N ) tc to condense fermions into a single object. Hence for the presenttheory the association of the scalar Higgs with the scalar magnitude | v | of the ‘vector-Higgs’ v ≡ h is analogous to technicolor models with a scalar condensate composedof a new set of fermions, in that in both cases the need to postulate a fundamentalscalar Higgs field is avoided. The absence of any observation of states belonging to atechnihadron spectrum rules out a number of technicolor models.Unlike the case of the Standard Model for the present theory the mass for the W ± and Z states is expected to arise from terms in D µ L ( v ) = 0 which are linear inthe gauge fields, as suggested in part by the relative couplings listed in equation 8.71and described earlier in this subsection. These gauge field interaction terms are in factsimilar in structure to those of the Lagrangian of equation 8.73 and hence the originof the gauge boson masses in the present theory also resembles the correspondingstructure of the technicolor model. At a suitably low energy scale the present theorymight also be compatible with an effective Lagrangian term quadratic in the gaugefields similar to equation 8.74 for the technicolor case.An origin for the masses of the fermion states in the present theory is alsorequired, as a correlate of the ‘Yukawa interactions’ introduced in the Standard ModelLagrangian. As described above for the W ± and Z gauge bosons, mass terms forthe fermions might also be expected to involve a form of coupling with the externalcomponents v ∈ h C ⊂ h O , which have been shown to exhibit properties analogousto those of the Standard Model Higgs field. The expression of the full form of L ( v ) =1 as the determinant of X ∈ h O matrices, as written out in equation 6.28, includesthe terms p | b | , m | c | and n | a | . With the projected components on TM related under L ( v ) = h in equation 5.46 and adopting the v ≡ h components of equation 8.72with v = h embedded within h O as for equations 8.4 and 8.5 the determinant canbe written as: det( X ) = h n − h | b | − h | c | − n | a | + 2Re(¯ a ¯ b ¯ c ) = 1 (8.76)The b, c internal components of h O hence have a multiplicative ‘coupling’ with thevacuum value h from the components of v in the form h ( b ¯ b + c ¯ c ). These are analogousto the Yukawa coupling terms between the Higgs field and fermion fields in StandardModel Lagrangian of equation 7.69. This suggests that the fermion masses may beproportional to h , in a similar way that they are proportional to the Higgs field vacuumvalue v , equation 7.70, in the Standard Model.Since L ( v ) = 1 is invariant under the transformations of E , and hencealso under the external and internal subgroups, in the dynamics of the theory theactual mass terms may correspond to gauge invariant expressions as for the case ofthe Lagrangian approach. Possible quartic or higher-order terms within a higher-dimensional form of L ( v ) = 1 as a source of mass for the standard fermions, considered209owards the end of section 9.2, are also analogous to the technicolor Lagrangian termsof the form of equation 8.75, at least with regard to their non-standard quartic nature.As for the case of the massive gauge bosons, for which quadratic mass termsdo not arise in the basic elements of the present theory as discussed above, ultimatelycomparison between this theory and the Standard Model should be made at the levelof empirical phenomena rather than a Lagrangian, which in any case is absent in thepresent theory. In addition to the field dynamics the role of mass in the calculations ofquantum field theory and its relation to ‘renormalisation’ and physical particle statesas studied in high energy physics experiments will need to be understood, as will bediscussed in chapter 11.Although only one generation of fermions has so far been considered in relationto the components of h O , in the discussion following equation 8.31 in subsection 8.3.1it was hinted that for the full theory the existence of three generations of physicalfermion states might ultimately be correlated with the existence of three types ofSL(2 , O ) ⊂ E subgroup action, as introduced in equations 6.32–6.35. From this per-spective, given the asymmetric structure of the three terms hb ¯ b , hc ¯ c and na ¯ a withrespect to h in equation 8.76, and the need for renormalisation in the full theory, itis possible that the physical mass eigenstates of empirically studied particles will notbe aligned neatly with the type a = 1 , θ a components of h O . Insteadthe choice of the external v ∈ h C ⊂ h O may be skewed relative to the three gen-erations of physical fermions, which may each then be related to v via a continuous (defined in [38] p.127, as alluded to here after equation 6.35) rather than discrete typetransformation, leading to the spectrum of masses observed for the leptons and quarks.In the Standard Model the phenomena of CKM mixing in the quark sectorrelates to a mismatch between weak interaction and mass eigenstates as was reviewedin section 7.2. In this section we have established a correlation between the weakinteraction and the subgroups SU(2) and SU(2) of E in the context of the presenttheory. If the mass states of three generations of quarks are skewed into the componentsof h O via a continuous type transformations as described above this contrasts with thecharged weak interaction of the ˜ W ± gauge bosons associated with the SU(2) , actionswhich constitute a discrete type complement to the external SL(2 , C ) symmetry. Thisstructure hence provides a possible basis for the mismatch between weak and masseigenstates responsible for the CKM mixing between three generations of quarks, witha similar structure accounting for neutrino oscillations in the leptonic sector.As described earlier the transformations for the symmetry group SU(3) c ⊂ E ,generated by { ˙ A q , ˙ G l } , act on each of the three a, b, c ∈ O components of h O in ex-actly the same way, in manner that is independent of both discrete and continuous typetransformations (as contrasted with the SU(3) s actions after equation 8.32). In thepresent context this symmetry of the SU(3) c action on the three octonion componentsin h O , together with its independence from the SL(2 , C ) , , and S , , l transforma-tions, is likely to be physically relevant for the observation of three generations offermions, at least for the quark content and the corresponding phenomena of CKMmixing between generations with each of the three generations of quarks subject to anidentical coupling to the SU(3) c strong interaction gauge bosons.However while the existence of three generations of fermions may ultimatelybe correlated with the three types of embedding of the θ , , components in h O ,210s presumed for the discussion above, a somewhat larger space will be required toexplicitly house all of the degrees of freedom, as we shall explore in the followingchapter. The expansion of the form of temporal flow L ( v ) = 1 will be accompaniedby a corresponding expansion of the group of symmetry transformations, opening upthe possibility of identifying an internal SU(2) L × U(1) Y symmetry matching all theproperties of the Standard Model.Finally in this section we consider further possible candidates for the StandardModel SU(2) L gauge symmetry in terms of generators confined to the E Lie algebrain the present theory. We return to { ˙ R azl , ˙ R axz , ˙ R axl , ˙ B atx , ˙ B atl , ˙ B atz } as the three sets of sixgenerators for SL(2 , C ) a for each of a = 1 , , a = 1 set was listed in equation 6.57)with the Lorentz Lie algebra of table 8.1 satisfied in all three cases, as considered insubsection 8.3.1. In particular we look more generally to construct explicit SU(2)subgroups out of the collection of 12 generators of this form with a = 2 or 3. Theseform a subset of the 16 generators for the sl(3 , C ) s subalgebra described in equation 8.33and presented explicitly within table 6.6, taking q = l , including the elements ˙ R , xl and˙ B tz which do not belong to the preferred 78-dimensional basis for E .As for the case of the six generators of the Lorentz algebra, listed equation 6.57and table 8.1, in the complexified Lie algebra the SL(2 , C ) subalgebra of type 2 is alsoisomorphic to SU(2) × SU(2). The generators of these two SU(2)s may be denoted A a and B b , in correspondence with equations 7.21–7.23 and 8.7–8.9 (within the choice ofsign conventions as noted for the latter equations), with: { A , A , A } = {
12 ( ˙ R zl + i ˙ B tx ) ,
12 ( ˙ R xz + i ˙ B tl ) ,
12 ( ˙ R xl + i ˙ B tz ) } and { B , B , B } = {
12 ( ˙ R zl − i ˙ B tx ) ,
12 ( ˙ R xz − i ˙ B tl ) ,
12 ( ˙ R xl − i ˙ B tz ) } such that: [ i ˙ S –– l , ( A ± iA )] = ± ( A ± iA )and [ i ˙ S –– l , ( B ± iB )] = ± ( B ± iB )with the latter two expressions hence describing charge eigenstates. Such eigenstatesmight in principle be correlated with charged gauge bosons ˜ W ± as described in theprevious two subsections. A similar analysis follows for the SL(2 , C ) subalgebra oftype 3. In addition to this by using the full set of 12 generators for both SL(2 , C ) andSL(2 , C ) two further SU(2)s can be identified in the complexified algebra in this casewith A a and B b composed as: A = 1 √ R zl + ˙ R zl + i ˙ B tx + i ˙ B tx ) B = 1 √ R zl + ˙ R zl − i ˙ B tx − i ˙ B tx ) A = 1 √ R xz + ˙ R xz + i ˙ B tl + i ˙ B tl ) B = 1 √ R xz + ˙ R xz − i ˙ B tl − i ˙ B tl ) A = ( ˙ R xl + ˙ R xl + i ˙ B tz + i ˙ B tz ) B = ( ˙ R xl + ˙ R xl − i ˙ B tz − i ˙ B tz )(8.77)However in this case none of the linear combinations ( A ± iA ) or ( B ± iB ) isa charge eigenstate of i ˙ S –– l under the adjoint representation in the complexified E algebra. In any case in order to identify a candidate for the SU(2) L gauge symmetryof the Standard Model a real SU(2) subalgebra of the real form of E is required. Such211 compact real form of SU(2) can be obtained from a combination of the type 2 and 3rotation generators with: { J , J , J } = {√
2( ˙ R zl + ˙ R zl ) , √
2( ˙ R xz + ˙ R xz ) ,
2( ˙ R xl + ˙ R xl ) } However again here a complex linear combination of J and J fails to form a chargeeigenstate under i ˙ S –– l . It can also be noted that the third generator J is in fact equal to − R xl , by equation 6.46, which is a generator of the type 1 SU(2) rotation subgroupand hence not even independent of the external Lorentz symmetry SL(2 , C ) in termsof the vector space of generators. A similar observation applies to A and B inequation 8.77, and indeed was also noted for the gauge field ˜ Z µ ( x ) associated with ˙ R xl for the mock SU(2) × U(1) theory before equation 8.61. These observations are notsurprising since the Dynkin analysis for the Lie algebra of E in section 7.3 suggests thatit is not possible to append any SU(2) subgroup alongside an SL(2 , C ) × SU(3) × U(1) ⊂ E decomposition, as recalled near the opening of subsection 8.3.1.However, of the possible SU(2) structures examined within the E algebra,which in some sense are complementary to the type 1 Lorentz subgroup SL(2 , C ) , thesubgroups SU(2) and SU(2) are the most promising in terms of properties resembling the SU(2) L gauge symmetry of the Standard Model, as has been described in thissection. These observations supplement the identification of the subgroup SL(2 , C ) × SU(3) c × U(1) Q ⊂ E in equation 8.25 which exhibits properties correlating closely withfeatures of the Standard Model as described in sections 8.1 and 8.2. These observationsalso helped motivate the detailed study of the subgroup SU(2) × U(1) in this sectionin an attempt to account for aspects of electroweak theory within the scope of the E action on the form of L ( v ) = 1 in the present theory.While a number of features of this mock electroweak theory resemble those ofthe Standard Model the lack of a complete match, together with the knowledge that thefull Standard Model external and internal symmetry cannot be accommodated withinE , now motivates the consideration of a higher-dimensional form of temporal flow,with a higher degree of symmetry, with the goal of incorporating the physical SU(2) L gauge symmetry. The aim will be to retain the significant traits of electroweak theoryas identified in this section, within the breaking of the E symmetry of L ( v ) = 1 overthe external spacetime M , in developing a higher-dimensional expression. As a furtherfeature in reconstructing the full details of the Standard Model it will be necessary toexplain how a set of Weyl spinors might be obtained from the a ∈ O ⊂ h O componentslisted in equation 8.28 for the ν -lepton and u -quark states. This will be the topic ofsection 9.1. In section 9.2 an explicit higher-dimensional form of L ( v ) = 1 will bepresented resulting in the identification of both left and right-handed Weyl spinors.Finally, bearing in mind the need to incorporate three generations of fermions, thepossibility of a further expansion will be described in section 9.3, with the featuresof the Standard Model so far identified in the context of the present theory thensummarised. 212 hapter 9 Further Dimensions O and Further Weyl Spinors In aiming towards the identification of a physical SU(2) L symmetry acting on doubletsof SL(2 , C ) Weyl spinors in the present theory we first recall how the symmetry E acting on the space h O relates to lower-dimensional forms of temporal flow expressedas L ( v ) = 1. In particular the E symmetry of the cubic form det( X ) = L ( v ) = 1,with X ∈ h O , may be contrasted with the case of taking the full symmetry of L ( v ) = 1to be the group Spin + (1 ,
9) acting on the space h O , intermediate in size between h C and h O , such that the quadratic form det( X ), with X = (cid:0) p ¯ aa m (cid:1) ∈ h O , is preserved asthe full form of temporal flow. With Spin + (1 ,
9) being the double cover of SO + (1 , L ( v ) = 1 this is essentially the model described in section 5.1 asdepicted in figure 5.1. In this case there is an SL(2 , C ) ⊂ Spin + (1 ,
9) subgroup, basedon the choice of an imaginary octonion unit such as q = i for equation 6.19, which actsas the external symmetry of 4-dimensional spacetime upon the subspace h C ⊂ h O with the two-sided action of equation 6.22 similarly as for the 10-dimensional case. Thisbreaks the set of 45 generators of SL(2 , O ) ≡ Spin + (1 ,
9) acting on the space h O to aninternal Stab ( TM ) set of symmetry operations which here consist purely of transverserotations amongst the remaining six imaginary units of the a ∈ O component of h O .This is again sufficient to contain SU(3) c × U(1) Q as an internal symmetry group.Indeed it can be seen from the Dynkin diagram of figure 7.2(b), by removing thecentral node with the most connections, that the Lie algebra so(10) has a breakingpattern to sl(2 , C ) × su(3) × u(1).However in the present theory we are not restricted to the consideration ofextra spatial dimensions, which might lead to the study of such a Spin + (1 ,
9) symme-try of 10-dimensional spacetime. Here we are dealing with a higher-dimensional formof temporal flow, allowing the structure of the above paragraph to be augmented tothe group E acting as the symmetry of a cubic form on the 27-dimensional spaceh O . This larger structure incorporates three interlocking Spin + (1 ,
9) actions, withassociated representations on three spinor spaces θ a ∈ O , for a = 1 , θ = (cid:0) c ¯ b (cid:1) and a within h O transform under the internalSU(3) c × U(1) Q ⊂ Stab( TM ) symmetry as a generation of leptons and quarks, withthe appropriate fractional charges, as summarised in equations 8.26 and 8.28 of sec-tion 8.2. The three-way embedding of SL(2 , O ) ⊂ SL(3 , O ) is analogous to the empiricalobservation of three generations of leptons and quarks, although it remains to be seenwhether these features do actually correlate.Further, in augmenting the 2 × O , upon which the symme-try of 10-dimensional spacetime may be represented, to the 3 × O ,with the structure of a temporal symmetry, Weyl spinor states are identified in the θ components of the additional column of this matrix, as listed in equation 8.13, underthe external SL(2 , C ) symmetry. This is analogous to the motivation of the originalKaluza-Klein theories [11, 12] in which the 4 × g µν describing the gravitationalfield is augmented to the case of a 5-dimensional spacetime such that the four compo-nents A µ = g µ in the extra column of the 5 × ω αa included in the metric, as describedfor equation 4.5 in section 4.1). In the present theory we identify fermion states inthe extra temporal components, rather than gauge bosons in the additional metriccomponents.For the case of E acting on L ( v ) = 1 not only is the above set of fourWeyl spinors under the external SL(2 , C ) symmetry identified in the θ components,as described in section 8.1, but they are also seen to be aligned with the internalSU(3) c × U(1) Q transformation properties of the electron and a triplet of d -quarkstates, as described in section 8.2. In the present theory rather than generalising to ahigher-dimensional spacetime here the augmentation is applied to a multi-dimensionalform of temporal flow, the concept of which is further compared and contrasted withthe Kaluza-Klein approach in chapter 5. These temporal structures are not restrictedto a quadratic, or even cubic, form and in principle may be extended to a homogeneouspolynomial form of arbitrary order.In the previous section we assessed the possibility of identifying the structureof electroweak theory in the breaking of the E symmetry on the components of h O .For example the gauge field components corresponding to ˜ W ± and ˜ Z gauge bosons,identified in a mock electroweak theory based on the SU(2) × U(1) ⊂ E symmetry,impinge on the external h C ≡ TM components as described in equation 8.54, whichled to equation 8.71, providing a possible mechanism for identifying gauge boson massterms analogous to the standard Higgs sector. In addition the physical W ± states ofthe Standard Model act as charge raising and lowering transformations in interactionswith left-handed doublets of leptons and quarks, that is (cid:0) νe (cid:1) L and (cid:0) ud (cid:1) L respectivelyfor the first generation of fermions as listed in equation 7.36, where each component { ν, e, u, d } is a left-handed Weyl spinor under the external SL(2 , C ) symmetry.One aim of the present theory has been to derive the spectrum of particle statesof the Standard Model, and in particular the above doublets of left-handed fermions,from the components of h O under the broken E action. Towards this end the actionof the ˜ W ± µ ( x ) fields associated with both the SU(2) and SU(2) transformations on the e -lepton and d -quark states, which have already been associated with the componentsof θ = (cid:0) c ¯ b (cid:1) through equation 8.26, should serve as a useful guide. As can be seen214rom equations 8.29 and 8.31 the SU(2) and SU(2) actions mix the components of θ = (cid:0) c ¯ b (cid:1) ∈ O with the a ∈ O component. Further, the subcomponents of a transformin the same way as those of b and c under the internal SU(3) c symmetry, containing acolour singlet and a colour triplet, and the corresponding elements of a have the correctelectromagnetic charges of 0 and under the U(1) Q generator ˙ S –– l to described the ν -lepton and u -quark respectively as can be seen in equations 8.27 and 8.28 and reviewedabove. However, as described in table 8.2 under the external Lorentz transformationsof SL(2 , C ) the ‘leptonic’ components of a transform as part of the vector v whilethe ‘quark’ components a (6) are scalars; and hence these components appear to beunsuitable to describe fermion states. In this section we focus on the possible meansof constructing these further required spinors.To see how such spinors may potentially arise and account for the ν -lepton and u -quark states the 10 real components of X ∈ h O , embedded in the type 1 locationof h O as depicted in equations 6.26, 6.29 and 6.32, may be provisionally composed interms of the 16 real components of a new object θ X = (cid:0) ¯ rs (cid:1) ∈ O , with r, s ∈ O and X = θ X θ X † (9.1)Hence the vector X is considered to be the square of the spinor θ X , in the form as orig-inally presented in equation 6.20. The compatibility relationship between the vectorand spinor actions for the octonion case in equation 6.23 also applies here since the2 × M ∈ SL(2 , O ) are required to have this property: M X M † = M ( θ X θ X † ) M † = ( M θ X )( M θ X ) † (9.2)In particular this shows that under the Lorentz transformations via M = S ∈ SL(2 , C ) the components of θ X = (cid:0) ¯ rs (cid:1) decompose into a set of four Weyl spinors, as is the casefor θ = (cid:0) c ¯ b (cid:1) under the same transformations as described in equations 8.10–8.13.This hence shows how in principle further left-handed Weyl spinors may be indeedbe identified within the E action on L ( v ) = 1 by opening up further dimensionsthrough the decomposition of equation 9.1.In terms of the real p, m and octonion a, r, s components equation 9.1 can bewritten in more detail as: X = p ¯ aa m = θ X θ X † = ¯ rs (cid:16) r ¯ s (cid:17) = ¯ rr ¯ r ¯ ssr s ¯ s (9.3)that is with p = | r | , m = | s | and a = sr (and as may be compared with equation 6.20).The fact that there are 16 real components of θ X given the original 10 real componentsof X is compatible with the underlying conceptual motivation of the present theoryfor which an n -dimensional form of temporal flow, such as L ( v ) = 1, is derived giventhe original 1-dimensional progression of time, and represents what is essentially a further generalisation and extension of this idea to a still higher-dimensional structure.This structure provides a means to identify a set of Weyl spinors from the externalSL(2 , C ) Lorentz action on X which might in principle be associated with physicalparticle states. 215e shall consider how these new spinors identified within the components of θ X may correlate with the first generation ν -lepton and u -quark states, in a similarway that the e -lepton and d -quark states were identified within the components of θ according to equation 8.26. These fermion states corresponding to SL(2 , C ) Weylspinors will be required to be mutually oriented within the components of θ X and θ with respect to ˜ W ± gauge bosons which mix the corresponding leptonic or quarkstates, raising or lowering the electromagnetic charge of the fermion state by one unit.This SU(2) mixing of Weyl spinors between the components of θ X and θ should beanalogous to the SU(3) c mixing of the θ i,j,k Weyl spinors within the θ components asdescribed on the left-hand side of table 8.7.The close relationship between the 10-dimensional vector X ∈ h O and 10-dimensional vectors of the form θθ † ∈ h O is also exhibited by the 10-dimensionalLorentz inner product in the final term of equation 6.27 for det( X ) with X ∈ h O . The10-dimensional type 1 Lorentz transformations SL(2 , O ) leave both terms of det( X ),that is both det( X ) n and 2 X · ( θ θ † ), invariant. However the subgroups SU(2) , ⊂ E mix the components of these two terms, as well as mixing components between X and θ , and it is these properties which might be studied in order to describe for examplea u ↔ d -quark interaction in terms of θ X ↔ θ components.Substituting the θ X = (cid:0) ¯ rs (cid:1) components r and s in place of p, m and a fromequation 9.3 into the expression for det( X ) in equation 6.28 leads directly to:det( X ) = | r | | s | n − | r | | b | − | s | | c | − n | sr | + 2Re(¯ r ¯ s ¯ b ¯ c ) (9.4)Since for any r, s in the division algebra O we have | s || r | = | sr | the first and fourthterms above cancel, leaving a quartic expression in r, s, b, c ∈ O . Hence in principleequation 9.4 describes a homogeneous form L ( v ) = 1 with 32 dimensions, namely thereal components of { r, s, b, c } ∈ O , with a symmetry group deriving from the action ofE on L ( v ) = 1.However, one significant difference between any elements X ∈ h O and θ ∈ O with θθ † ∈ h O is that in the former case det( X ) ∈ R may take arbitrary realvalues while in the latter case we necessarily have det( θθ † ) = 0, as was described inequation 6.21. This can be seen here from the right-hand side of equation 9.3 forwhich det( θ X θ X † ) = | r | | s | − | r | | s | = 0, for any θ X = (cid:0) ¯ rs (cid:1) , and accounts for thecancellation of the two quintic terms in equation 9.4. It also clearly implies that thedecomposition of X as suggested in equations 9.1 and 9.3 is not possible for the generalcase if det( X ) = 0.This apparent incompatibility may be remedied by further generalising equa-tions 9.1 and 9.3 by introducing an additional spinor φ X = (cid:0) ¯ r ′ s ′ (cid:1) ∈ O , with identicaltransformation properties as the original θ X in equation 9.2, such that: X = θ X θ X † + φ X φ X † (9.5)with M X M † = M ( θ X θ X † ) M † + M ( φ X φ X † ) M † = ( M θ X )( M θ X ) † + ( M φ X )( M φ X ) † (9.6)This introduces a further 16 real parameters in φ X which transform as a further setof four Weyl spinors under the Lorentz actions with M = S ∈ SL(2 , C ) . The valueof det( θ X θ X † + φ X φ X † ) ∈ R may now be compatible with the determinant of X in the216eneral case. This is analogous to the case of the 4-dimensional Lorentz vector de-composition in equation 7.32, with the spinor substructure of the vector X potentiallyproviding a source of microscopic physical structure, as suggested after equation 7.32for the 4-vector field v ( x ). The form of X in equation 6.1 and det( X ) in equation 6.27when substituting in equation 9.5 become: X = | r | + | r ′ | ¯ r ¯ s + ¯ r ′ ¯ s ′ csr + s ′ r ′ | s | + | s ′ | ¯ b ¯ c b n ∈ h O (9.7)det( X ) = det( θ X θ X † + φ X φ X † ) n + 2( θ X θ X † + φ X φ X † ) · ( θ θ † ) (9.8)Here the first part of the expression for det( X ) contains quintic terms, which nowdo not cancel in general as they did in equation 9.4, while the second part containsfurther quartic terms. This expression hence represents an inhomogeneous polynomialform, and hence deviates from the form of L ( v ) = 1 in equation 2.9 of section 2.1on incorporating a further higher-dimensional dissolving of specific components, suchas those of X above. The potential physical consequences of such a mathematicalpossibility in relation to the original homogeneous form of L ( v ) = 1 requires furtherclarification. While such inhomogeneous expressions may be explored upon examina-tion they appear indeed inconsistent with the underlying conceptual basis employedin deriving equation 2.9, as relating to infinitesimal intervals of temporal flow δs . Onthe other hand expressions such as equation 9.8 may represent an intermediate steptowards the derivation of a higher-dimensional homogeneous form, such as a purelyquintic expression for L ( v ) = 1, as will be proposed hypothetically in section 9.3 inthe light also of the physical motivation described below.The need to generalise from equation 9.1 motivated the introduction of a combi-nation of spinors, θ X and φ X , in equation 9.5. This latter expression X = θ X θ X † + φ X φ X † ,with θ X and φ X having identical transformation properties under the external and inter-nal symmetry actions, suggests that for example the first generation u -quark and thesecond generation c -quark states might be accommodated in the θ X and φ X componentsrespectively.As described in the previous section the SU(2) , ⊂ SL(2 , C ) , actions indi-cated in equation 8.31 transform the octonion components ( a ) ↔ ( b, c ) in a seeminglyasymmetric way. Hence, with a = sr from equation 9.3, a combination of ˙Σ (2) ± and˙Σ (3) ± , of equations 8.37 and 8.68, appear to be needed in order to transform any ofthe four Weyl spinors located in the components of θ = (cid:0) c ¯ b (cid:1) into the doublet partnerlocated within the corresponding components of θ X = (cid:0) ¯ rs (cid:1) .Together the observations of the above two paragraphs suggest the possibilityof a Cabibbo-like mixing between the first two generations of quarks. Empirically thegauge action SU(2) L mixes the quark states ( u ) ↔ ( d cos θ c + s sin θ c ), where s denotesthe strange quark and θ c is the Cabibbo angle which may be generalised to the fullCKM matrix for three generations, as described towards the end of section 7.2. Inthe Standard Model the coupling between the first and third quark generations is verysmall. In the present theory quark exchanges via the ˜ W ± bosons associated with theSU(2) , symmetry may open up a full set of possible states associated for example with217 = θ X θ X † + φ X φ X † + ψ X ψ X † , further augmenting equation 9.5 and lifting the degeneracyto a complete set of three generations interacting via a CKM-like mixing, although asimilar degeneracy will also need to be identified relating to the θ components. Inprinciple this offers a possible means of accommodating three generations of fermionsinto the theory which may not relate directly to the existence of the three types ofSL(2 , O ) embedded within SL(3 , O ) as described in equations 6.32–6.35.The full study of these phenomena, as discussed in the previous subsection,will require the identification of each of the physical mass states, as observed in thelaboratory, in relation to the components of θ = (cid:0) c ¯ b (cid:1) and θ X = (cid:0) ¯ rs (cid:1) for example. Aroundequation 8.76 it was proposed that fermion masses will relate to the degree of couplingwith the scalar magnitude | v | = h , and in particular the ‘vacuum value’ of h in theprojection of v ∈ TM , by analogy with Higgs phenomena in the Standard Model.Since terms containing both the a ∈ O and the v = h components of X ∈ h O do notappear in det( X ) in equation 8.76 an explicit higher-dimensional form of L ( v ) = 1,such as introduced in the following section, may be needed for further study of possiblemass terms. On the other hand composition with the scalar field n ( x ), such as for the | a | term in equation 8.76, might also provide a source of fermion mass terms.With the possible generalisation of equation 9.5 and the above weak interac-tions in mind it is also necessary to determine the internal SU(3) c × U(1) Q symmetrytransformations of the θ X components, with similar transformations implied for φ X .The SO(8) ⊂ SL(3 , O ) subgroup can be generated by the composition of 3 × M (1) of type 1 in the form of equation 6.49 based on the 2 × M = (cid:0) q
00 ¯ q (cid:1) .Acting via the conjugation X → M XM † on X in equation 9.3 the component a ∈ O transforms under the vector representation of SO(8) while the θ X components s ∈ O and r ∈ O transform individually via the spinor and dual spinor representations ofSO(8) as can be seen via equation 9.2.These three 8-dimensional representations are mutually related through trialitymaps described in the opening of section 6.1 and around equation 6.50 – the trialitystructure in the present mathematical context is also described in more detail in ([38]pp.77–80 and 120–126). Further, elements of the G ⊂ SO(8) octonion automorphismsubgroup transform the vector, spinor and dual spinor, here represented by the octo-nions a, s and r respectively, in precisely the same way via symmetric, left and rightmultiplication by the same sequence of octonions. This property of termed ‘strongtriality’ in ([38] p.123).Hence under the colour gauge symmetry SU(3) c ⊂ G ⊂ SO(8) each of theoctonion components of θ X = (cid:0) ¯ rs (cid:1) transform in the same way as the component a , andhence also in the same way as the octonion components of θ = (cid:0) c ¯ b (cid:1) . This means thatthe four Weyl spinors, obtained from the reduction of θ X under the external SL(2 , C ) action, transform as a leptonic singlet and quark triplet under the internal SU(3) c , justas is the case for θ as summarised in equation 8.19. There then remains the questionof how the U(1) Q charges for the θ X Weyl spinors compare to those for the θ Weylspinors deduced for equation 8.26.The electromagnetic U(1) Q generator ˙ S –– l is contained in the group SO(7) ⊂ SO(8) ⊂ SL(2 , O ) , but unlike the SU(3) c generators { ˙ A q , ˙ G l } it is not containedwithin the subgroup G ⊂ SO(7). Although ˙ S \ l = ˙ S l the simpler unnested singlegroup action S \ l of equation 6.43 is not used here since it is not constructed as a218compatible’ action in the sense of equation 6.23 or 9.2. We hence employ S l as amember of the preferred basis incorporated into the E Lie algebra composition asdiscussed following equation 6.43.The group action of S l ( α ) on the components of X = (cid:0) p ¯ aa m (cid:1) and θ X = (cid:0) ¯ rs (cid:1) may bedetermined from table 6.2 and equation 6.39 together with table 6.1 and equation 6.24as the type 1 nested compositions: X → R il,i ( α ) ◦ R jl,j ( α ) ◦ R kl,k ( α ) X (cid:18) p ¯ aa m (cid:19) → M il,i ( M il,i ( M jl,j ( M jl,j ( M kl,k ( M kl,k (cid:18) p ¯ aa m (cid:19) M † kl,k ) M † kl,k ) M † jl,j ) M † jl,j ) M † il,i ) M † il,i θ X → R il,i ( α ) ◦ R jl,j ( α ) ◦ R kl,k ( α ) θ X (cid:18) ¯ rs (cid:19) → M il,i ( M il,i ( M jl,j ( M jl,j ( M kl,k ( M kl,k (cid:18) ¯ rs (cid:19) ))))) (9.9)where the expression for the θ X transformation is a consequence of equation 9.1 or 9.5together with the compatibility of the S l ( α ) action as defined in equation 6.23 or 9.2.The following notation for the 6 nested actions, with factors of ± N ( ≡ − ( il cos α i sin α (cid:0) il (cid:0) ( jl cos α j sin α (cid:0) jl (cid:0) ( kl cos α k sin α (cid:0) kl ) N † ≡ − kl (cid:1) ( kl cos α k sin α (cid:1) jl (cid:1) ( jl cos α j sin α (cid:1) il (cid:1) ( il cos α i sin α a, s and r then transform under S l ( α ) in the manner (as may be compared with equation 6.50): a → N ( a ) N † s → N ( s )))))¯ r → N (¯ r )))))hence r → ((((( r ) N † (9.10)While here we have symmetric, left and right multiplication on a , s and r respectively by the same sequence of octonions, as described by ‘ N (’ and ‘) N † ’, thesethree actions are not mutually related by triality since they do not describe the sametransformation on O . Indeed since the action S l ( α ) is not part of the G ⊂ SO(8)subgroup it does not exhibit the property of ‘strong triality’, but rather participatesin the SO(8) triality structure collectively when further generators are considered.Taken at face value equations 9.10 imply that the electric charge identified with˙ S –– l for the ¯ r, s components here hence differs from that for the a component. Thisis an undesirable feature which means that the (cid:0) / (cid:1) charge structure observed forthe component parts of a under ˙ S –– l in equations 8.27 and 8.28, as sought for ν -leptonand u -quark fermion states, has apparently been lost for the set of SL(2 , C ) Weyl219pinors in θ X = (cid:0) ¯ rs (cid:1) . In fact, as expected from the compatibility of the S l group action,the U(1) Q transformations of the spinor θ X = (cid:0) ¯ rs (cid:1) are identical to those of the spinor θ = (cid:0) c ¯ b (cid:1) and hence both spinors possess the same ˙ S –– l charge values of 1 and , asdescribed for equation 8.26, which have been associated with the e -lepton and d -quarkstates. Indeed, as implied in the discussion around equation 9.2, the components of θ X transform in precisely the same way as those of θ under the action of the subgroupSL(2 , C ) × SU(3) c × U(1) Q ⊂ SL(2 , O ) owing to the compatibility requirement of all M ∈ SL(2 , O ) group transformations.A possible solution would be to maintain the same S l ( α ) action on X whileredefining the transformation properties of θ X under the U(1) Q subgroup. That is, with X → θ X θ X † in equation 9.1 or 9.5, on expanding the 10-dimensional space for X tothe 16-dimensional space for θ X there is a degree of redundancy in the transformationproperties of θ X under U(1) Q provided X transforms in the same way. While equa-tion 9.9 represents the simplest assumption for the action of S l ( α ) on the componentsof θ X , based on the notion of compatibility in equation 9.2, in principle there may befurther choices such as: X → N ( X ) N † = N ( θ X θ X † ) N † with θ X → N ( θ X ) N † (9.11)rather than θ X → N ( θ X ), although with care needed to take into account the non-associative properties of octonion composition. On employing equation 9.11 in placeof equation 9.10 the action of the U(1) Q symmetry S l ( α ) on a, s, r ∈ O would beexpressed uniformly as: a → N ( a ) N † s → N ( s ) N † (9.12) r → N ( r ) N † Hence in this case the (cid:0) / (cid:1) charge structure of a under ˙ S –– l would also applyto the r, s ∈ O components of θ X . These U(1) Q charges for θ X are here contrived by inserting further nested N ( † )6 actions for θ X ( † ) in the appropriate places for equa-tion 9.11. However the introduction of the 16 real component object θ X = (cid:0) ¯ rs (cid:1) itself,obtained from the 10 real component object X in equation 9.1, is contrived and definedin order to construct a possible set of doublet partners for the θ = (cid:0) c ¯ b (cid:1) componentsunder the action of ˜ W ± charge raising and lowering operators. With physical statesassociated with definite representations under the Lorentz symmetry of 4-dimensionalspacetime M , such as the above Weyl spinors within θ and θ X , the purpose here isto demonstrate the mathematical possibility of recovering weak interactions betweenfermions, such as those mediated via ˜ W ± gauge bosons, from within the present theory.A mathematical justification for transformations such as those of equation 9.12 mightultimately be sought within a natural higher-dimensional homogeneous form L ( v ) = 1.Hence θ X = (cid:0) ¯ rs (cid:1) provisionally describes a possible mathematical constructionwhich possesses the appropriate transformation properties under the external symme-try SL(2 , C ) and internal symmetry SU(3) c × U(1) Q to represent the neutral ν -leptons220nd charge- u -quarks. These latter states are related to the charge-1 e -leptons andcharge- d -quarks of θ = (cid:0) c ¯ b (cid:1) via ˜ W ± interactions. The relation between the fermionstates and the unit charge raising and lowering action of the ˜ W ± oriented with respectto the θ X components serves to mutually motivate and aid the determination of boththe Weyl spinor states and the internal gauge symmetry. From this point of view, with X ∈ h O composed in the form of equation 9.5, the ˜ W ± and ˜ Z may derive from aweak SU(2) × U(1) action which is less directly related to the SU(2) , × U(1) , ⊂ E subgroups than suggested in the previous section.To conclude the above discussion, while equation 9.5 describes a possible wayto include the required further Weyl spinor states there are several questions whichremain to be resolved – these include the means by which equation 9.8 might beincorporated into a higher-dimensional homogeneous form of L ( v ) = 1 with a largersymmetry group incorporating an appropriate electroweak SU(2) L × U(1) Y subgroupaction and the means by which the electromagnetic charges for the ν -lepton and u -quark states may be retained from the ˙ S –– l action on the components of the original a ∈ O component of h O . Further, while here we are working at the level of thebasic group and representation structure, a leading question in the full theory will beto understand the nature of physical particle states in general, and in particular forthe gauge bosons, three generations of fermions and also a Higgs state as empiricallyobserved.The above identification of fermions by opening up the 10-dimensional vector X ∈ h O according to equation 9.5 can similarly be applied to the 4-dimensionalLorentz vector h ∈ h C , for the subspace h C ⊂ h O , according to the decompositionof equation 7.32. This latter Weyl spinor substructure of the vector h = χχ † + φφ † in terms of the spinors χ, φ has some analogy with composite Higgs and technicolormodels in which fermion states are combined in scalar condensates in the vacuum,hence replacing the fundamental scalar Higgs of the Standard Model, as reviewed insubsection 8.3.3. Here opening up the h ∈ h C components to form spinors in thisway incorporates the a ,l components of equations 8.27 and 8.28, leaving the set ofSL(2 , C ) Lorentz scalars in a (6) which transform under the internal SU(3) c × U(1) Q symmetry as a colour triplet of u -quarks. In principle each of the three scalars a il,i , a jl,j and a kl,k of equation 8.28 might be composed in terms of a suitable scalar product ofWeyl spinors of the form χ † χ with the aim of describing the fermion nature of u -quarksunder the external symmetry.These spinor decompositions involve C or H subalgebras of a ∈ O . Hence,in comparison with equation 9.11, an extra intermediate factor of the form e − lβ e lβ might be inserted for the decomposed vector h → χχ † in augmenting the S –– l ( α )action, to ensure the charge neutrality of the candidate neutrino states, avoiding anycomplication due to the non-associative nature of the octonions. With the possibilityof a similar insertion for the u -quark states in principle the U(1) Q charges under S –– l of (cid:0) / (cid:1) , as originally found for the (cid:0) a ,l a (6) (cid:1) components in equation 8.28, might bemaintained under the spinor decomposition of these components which might henceindeed be associated with (cid:0) νu (cid:1) fermion states. Again, while such a structure might bemathematically contrived as a proof of principle, ultimately the aim will be to accountfor the external and internal symmetry properties of all Standard Model fermion statesin a natural manner in the components of a higher-dimensional homogeneous form of221 ( v ) = 1.In keeping the (cid:0) νu (cid:1) particle type interpretations aligned with the (cid:0) a ,l a (6) (cid:1) com-ponents in this way, based on the spinor decomposition of h ∈ h C (unlike the casefor the collective decomposition X → θ X θ X † ∈ h O as originally considered in equa-tion 9.1), also suggests that exchanges with the corresponding doublet partners (cid:0) ed (cid:1) in the components of θ might be mediated by ˜ W ± states closely associated with theSU(2) , ⊂ E actions as described in section 8.3 for the mock electroweak theory.Based on the { , l } base units these SU(2) , actions preserve the 4-way decompo-sition of octonion components as listed in equation 8.30 for the transformations be-tween the a, b, c ∈ O components of h O of the kind described in equations 8.29 and8.31. This is consistent with an electroweak SU(2) gauge symmetry action on indepen-dent lepton and quark doublets as accommodated respectively within the { , l } and( { il, i } , { jl, j } , { kl, k } ) components of both a and θ = (cid:0) c ¯ b (cid:1) .In particular the neutrino state is associated with the a ,l components, whichalso form part of the 4-vector h ∈ h C as projected onto the external spacetime TM .In the present theory the scalar degree of freedom | h | = p det( h ), or an alternativescalar combination of the spinor components χ, φ in the implicit substructure of h described above, will provide a candidate for the origin of the observed Higgs particle asdescribed in subsection 8.3.3. This apparent inconsistency with the degrees of freedomof h ∈ h C seemingly required to play a double role as the correlate of both theneutrino and the Higgs will be resolved in the following section.The approach of the present theory is to gently coax the known properties of theStandard Model out of the symmetry breaking structures of forms of L ( v ) = 1 over thebase manifold M , with an awareness of the known empirical features while being con-scious of not contriving them ultimately for the complete theory. However the possibil-ity of contriving an augmented structure based on the components of a ∈ h O under theE action transforming as a ν -lepton and u -quark under SL(2 , C ) × SU(3) c × U(1) Q ⊂ E , as described in this section, and with the further possibility of incorporating a sec-ond and third generation through additional spinors such as described for equation 9.5,is at least consistent with the possibility that a higher-dimensional form of L ( v ) = 1,for example with an E symmetry as will be considered in section 9.3, might naturallycontain these structures. Similarly the empirical properties of left and right-handedspinors, as we recap below, will contribute to the motivation for the study of an E symmetry of a higher-dimensional form of time in the following section.In the Standard Model Lagrangian each fermion kinetic term, such as equa-tion 7.39, or interaction term, such as equation 7.77, contains either left or right-handedfermion states, while the Yukawa or Dirac mass terms combine opposite chiralities, asfor example in equations 7.69 or 7.74. In all cases the operators P L = (1 − γ ) or P R = (1 + γ ), of equations 7.11 and 7.12, may be used to project out the respectiveleft or right-handed chirality states from a 4-component Dirac spinor.The factors of P L = (1 − γ ) which appear in all fermion terms involvingthe SU(2) L gauge symmetry, such as in equations 7.77 and 7.78, are placed in theLagrangian by hand in order to replicate the parity violating phenomena observedempirically for the weak interaction. This parity violation is maximal for the caseof interactions via W ± gauge bosons but non-maximal for Z interactions, which isassociated with a linear combination of SU(2) L and U(1) Y generators, equation 7.49, as222escribed in section 7.2. It is this asymmetry in the chiral structure, with different weakisospin transformations for left- and right-handed fields, implying that no fermion statetransforms under the complex conjugate representation of that of any other fermion,which necessitates the introduction of Yukawa couplings to the Higgs field, as forequation 7.69, in order to include fermion mass terms in the Lagrangian.In the present theory we have described how the components of θ = (cid:0) c ¯ b (cid:1) formthe set of four left-handed Weyl spinors of equation 8.13 under the external Lorentzsymmetry SL(2 , C ) . In this section a similar decomposition has also been identified forthe components of X = (cid:0) p ¯ aa m (cid:1) , for example via the spinor θ X = (cid:0) ¯ rs (cid:1) ∈ O as introducedin equation 9.1. Hence the projection operator P L = (1 − γ ) has not been introducedsince only left -handed Weyl spinors under SL(2 , C ) have so far been considered. Therethen remains the question of how right -handed Weyl spinor counterparts may identifiedwithin this framework, and related to the above left-handed components in a single4-component Dirac spinor ψ ( x ) to describe, for example, a physical electron state.Here we began with the cubic form det( X ) = 1, with X ∈ h O , as a 27-dimensional expression of temporal flow L ( v ) = 1. Determinant preserving E trans-formations were then considered on this space, with for example X → MX M † for the3 × × M representing the matrix M with each entry replaced by its octonion conjugate, as described in equation 6.7,transformations of the form: X → M X M † (9.13)clearly also leave the value of det( X ) invariant. With E a symmetry of the form oftime L ( v ) = 1 the two possible representations and are equally valid whileonly one of them has been used so far.Correspondingly the set of six actions with M , for M ∈
SL(2 , C ) , providesan alternative choice for the type 1 Lorentz transformations acting on the vector com-ponents v ∈ h C ⊂ h O , with h → S h S † in place of equation 7.31. In turn thesetransformations are represented on θ as a set of four right -handed Weyl spinors. Thegroup SL(2 , C ), as for the full group E , has complex representations, and the actionsof S ∈ SL(2 , C ) and S describe distinct sets of transformations of θ , as explained insection 7.1. This means that the left and right-handed transformations are not equiva-lent to each other but are instead mutually related as described in equations 7.26 and7.27. In the context of the present theory both S ∈ SL(2 , C ) and S act as symmetrytransformations leaving the form L ( v ) = 1 invariant, and hence both the left and right Weyl spinor compositions of θ should in principle play a role.The apparent asymmetry in the choice of the E or representation toexpress L ( v ) = 1, with a corresponding choice of left or right-handed representationsof SL(2 , C ) ⊂ E , the need to clearly identify both left- and right-handed fermions ψ L and ψ R , in particular with reference to an SU(2) L gauge symmetry, and the existenceof a homogeneous quartic form as a candidate for a higher-dimensional temporal flowin the form L ( v ) = 1 all point to consideration of the group E as a symmetry oftime, as will be described in the following section.223 .2 E Symmetry and the Freudenthal Triple System
The introduction of further dimensions in the previous section and the observation ofthe quartic expression of equation 9.4, with the extension to equation 9.8 includingquintic terms also, suggests the possibility of a higher-dimensional expression for theflow of time generalising beyond the cubic form L ( v ) = 1 described in chapter 6. Ahigher-dimensional homogeneous polynomial form is desired, in conformity with thederivation of equation 2.9 in chapter 2. While the determinant preserving symmetry ofthe space X ∈ h O describes the lowest-dimensional non-trivial representation of E thesmallest non-trivial representation of the exceptional Lie group E is 56-dimensionaland may be constructed in terms the elements x of the Freudenthal triple system F (h O ) ([59, 60, 61], [1] p.48).In the above references and related publications these mathematical structuresare applied in two very different contexts – namely the classification of black hole solu-tions in string theory and the entanglement of qubits in quantum information theory– with a correspondence between these applications identified through the mathemat-ical forms they share. Neither application is relevant for the present discussion. Whilemuch of the literature describes a more general algebraic framework or particular casesinvolving for example the ‘split octonions’ O s or takes an underlying field of integers Z ,here we are interested in the octonion O case over an underlying field of real numbers R as we summarise in the following.In order to describe the Freudenthal triple system F (h O ) it is useful to firstintroduce further definitions regarding the exceptional Jordan algebra h O itself. The structure group Str(h O ) leaves the cubic norm det( X ) of equations 6.27 and 6.28invariant up to a real scalar factor, that is:Str(h O ) = { g ∈ GL(h O ) | det( σ g ( X )) = λ ( g ) det( X ) , ∀X ∈ h O } (9.14)with λ ∈ R depending only on g . The norm preserving subgroup with λ = 1 isidentified as the reduced structure group Str (h O ) ≡ SL(3 , O ). This latter symmetrycorresponds to the 27-dimensional representation of E − as described in detail inchapter 6.A trace bilinear map may be defined for any elements X , Y ∈ h O of the Jordanalgebra, mapping h O × h O → R with:( X , Y ) = tr( X ◦ Y ) (9.15)where on the right-hand side the ◦ denotes the Jordan algebra product of equation 6.2.An adjoint s ∗ for any transformation s ( X ) with s ∈ E may be defined with respectto the above trace bilinear form such that:( s ( X ) , Y ) = ( X , s ∗ ( Y )) ∀X , Y ∈ h O (9.16)Along with the Jordan product there is a second natural composition for the elementsof h O which is called the Freudenthal product and may be defined by:
X ∧ Y = X ◦ Y − (cid:0) tr( X ) Y + tr( Y ) X (cid:1) + 12 (cid:0) tr( X )tr( Y ) − tr( X ◦ Y ) (cid:1) ∈ h O (9.17)224or any X ∈ h O a quadratic adjoint map h O → h O can be defined in terms of theFreudenthal product as: X ♯ = X ∧ X (9.18)or explicitly: X ♯ = X − tr( X ) X + 12 [tr( X ) − tr( X )] (9.19)This ‘sharp’ operation satisfies the relations ( X ♯ ) ♯ = det( X ) X and X ◦ X ♯ = det( X ) .The linearisation of the quadratic adjoint is written as: X × Y = ( X + Y ) ♯ − X ♯ − Y ♯ (9.20) ≡ X ∧ Y (9.21)For the elements of h O the quadratic adjoint is in fact the classical adjoint, that isthe transposed cofactors of X ∈ h O which, for the components of X presented inequation 6.1 or 9.25 below, can be written explicitly as the 3 × X ♯ = mn − | b | cb − n ¯ a ¯ a ¯ b − mc ¯ b ¯ c − na pn − | c | ac − p ¯ bba − m ¯ c ¯ c ¯ a − pb pm − | a | ∈ h O (9.22)The vector space F (h O ) has 56 real components and may be introduced ac-cording to Freudenthal’s construction with the vector space composition (which maybe compared with the further decomposition of equation 8.1): F (h O ) ∼ = h O ⊕ h O ⊕ R ⊕ R (9.23)Correspondingly elements x ∈ F , with F = F (h O ) , are generally written in the formof a ‘2 × x = α XY β , with X , Y ∈ h O , α, β ∈ R (9.24)and X = p ¯ a ca m ¯ b ¯ c b n , Y = P ¯ A CA M ¯ B ¯ C B N (9.25)here with the real
P, M, N and octonion
A, B, C components of Y distinguished fromthe lower case counterpart components of X . A non-degenerate bilinear antisymmetricquadratic form mapping F × F → R may be defined on this space which acts on x = (cid:0) α XY β (cid:1) , y = (cid:0) γ WZ δ (cid:1) ∈ F as: { x, y } = αδ − βγ + ( X , Z ) − ( Y , W ) (9.26)Of more significance for the present theory there is also a homogeneous quartic norm q : F → R defined on the components of x ∈ F as follows: q ( x ) = − αβ − ( X , Y )] − α det( X ) + β det( Y ) − ( X ♯ , Y ♯ )] (9.27)225here all the necessary definitions contained within this expression are inherited fromthose for the Jordan algebra h O as described above. The quadratic and quartic formsof equations 9.26 and 9.27 may be used in turn to define a trilinear mapping of thespace F × F × F → F , which is the triple product by which the ‘Freudenthal triplesystem’ gains its name. When written out explicitly in terms of the real and octonioncomponents of equations 9.24 and 9.25 there are a large number of quartic terms in q ( x ). In fact, via equations 9.15 and 9.22, and cross-checking with ([60] equation 9.51),we have for equation 9.27: q ( x ) = − h αβ − pP − mM − nN − (cid:0) h a, A i + h b, B i + h c, C i (cid:1) i − h βP M N + αpmn − pP mM − pP nN − nN mM + ( pm − βN ) | A | + ( P M − αn ) | a | + ( mn − βP ) | B | + ( M N − αp ) | b | + ( np − βM ) | C | + ( N P − αm ) | c | + 2 β Re( ¯ A ¯ B ¯ C ) + 2 α Re(¯ a ¯ b ¯ c ) − | a | | A | − | b | | B | − | c | | C | − ( cb − n ¯ a )( ¯ B ¯ C − N A ) − ( CB − N ¯ A )(¯ b ¯ c − na ) − ( ac − p ¯ b )( ¯ C ¯ A − P B ) − ( AC − P ¯ B )(¯ c ¯ a − pb ) − ( ba − m ¯ c )( ¯ A ¯ B − M C ) − ( BA − M ¯ C )(¯ a ¯ b − mc ) i (9.28)where the inner product h a, A i = ( a ¯ A + A ¯ a ), which has the property h a, A i = h ¯ a, ¯ A i ,was defined in equation 6.10. Equations 9.27 and 9.28 for the quartic form q ( x ) are theanalogue of equations 6.27 and 6.28 respectively for the cubic form det( X ). Clearlythere are many more terms for the above quartic from in equation 9.28 as an extensionfrom the cubic form of equation 6.28.The group Inv( F ) of all invertible transformations σ in F preserving the abovequartic norm with q ( σ ( x )) = q ( x ), as well as the bilinear form of equation 9.26 with { σ ( x ) , σ ( y ) } = { x, y } , is also denoted Aut( F ) since it in turn forms the automorphismgroup of the trilinear product defined for the Freudenthal triple system. This group isfound to be the non-compact real form E − of the exceptional Lie group E . Hence,in particular, under this symmetry group the invariance of the quartic form q ( x ), asa homogeneous polynomial, describes a possible 56-dimensional form of temporal flowwhich may be denoted L ( v ) = 1. The possible physical implications of this form andthe accompanying E symmetry will be assessed in the remainder of this section andsummarised in the following one.The symmetry of the cubic form L ( v ) = 1, in the form of det( X ) or det( Y ),is contained within this structure as can be seen from equation 9.27. In fact theelements X and Y , with 54 real components in total, may be considered to representa ‘complexification’ of the space h O , with both the 27-dimensional representation ofE and its complex conjugate contained within the E action on q ( x ). Including theactions of the subgroup E ⊂ E on the elements x → s ( x ) ∈ F the transformationsof full symmetry E ≡ Inv( F ) may be categorised in terms of four sets. With s ∈ E ,226 ∈ R and C, D ∈ h O these are [59, 60, 61]: T ( s ) : α XY β → α s ( X ) s ∗− ( Y ) β (9.29) λ : α XY β → λ − α λ X λ − Y λ β (9.30) φ ( C ) : α XY β → α + ( Y , C ) + ( X , C ♯ ) + β det( C ) X + βC Y + X × C + βC ♯ β (9.31) ψ ( D ) : α XY β → α X + Y × D + αD ♯ Y + αD β + ( X , D ) + ( Y , D ♯ ) + α det( D ) (9.32)where s ∗ is the adjoint of s ∈ E as defined in equation 9.16. The set of actions s ∗− in equation 9.29 is equivalent to the complex conjugate of the representation definedby the set of actions s ∈ E on h O . Under the subgroup E − ⊂ E − the space F decomposes into the reducible representation ([60] equations 9.45 and 9.46): E → ( + + + ) E (9.33)compatible with the structure of equation 9.23 (and can be compared with the furtherreduction under Spin + (1 ,
9) in equation 8.3). The 78 actions of E combined with thesingle dilation action λ of equation 9.30 applied to an h O ⊂ F subspace together formthe 79-dimensional group Str(h O ) as defined in equation 9.14. The 27 independentactions of φ ( C ) together with the further 27 for ψ ( D ) in equations 9.31 and 9.32 furtheraugment the E symmetry to complete the full (78 + 1 + 27 + 27) = 133-dimensionalexceptional Lie group E . (Building up the symmetry structure this way is analogousto augmenting the F algebra by the D B maps in equation 6.4 to complete the fullE symmetry.) In addition to the continuous actions of equations 9.29–9.32 a discretesymmetry τ := φ ( − ) ψ ( ) φ ( − ) such that: τ : α XY β → − β −YX α (9.34)with τ ( x ) = − x , may also be defined. Since ψ ( C ) = τ φ ( − C ) τ − the set of actions φ and ψ are conjugate with respect to τ . Between equations 9.29 and 9.34 the furtherrelationship τ T ( s ) = T ( s ∗− ) τ is also found.At the Lie algebra level the actions ˙ s ∈ L (E ) may be divided into the 14elements of L (G ), denoted D G , and the action of the 64 tracefree octonion matrices x , denoted D S in equation 6.5. The latter set further divides into the 26 boosts withHermitian x and 38 rotations with anti-Hermitian x . Such decompositions were alsodiscussed in the opening three paragraphs of section 6.5. The subgroup G itself mayalso be obtained through sequences of nested rotations as described in section 6.4. The227 ual representation of L (E ) may be obtained by defining the action ˙ s ′ ( X ) for each˙ s ∈ L (E ) such that ([62] equation 4):( ˙ s ( X ) , Y ) = − ( X , ˙ s ′ ( Y )) ∀X , Y ∈ h O (9.35)which may be contrasted with equation 9.16 at the group level. For the L (E ) maps X → x X + X x † the dual transformations correspond to: x ′ = x for rotations (9.36) x ′ = − x for boosts (9.37)that is with x ′ = − x † in general ([62] equation 5). It also follows that ˙ s ′ = ˙ s for the L (G ) actions derived from transverse rotations. Applied to the subalgebra L (E ) ⊂ L (E ) acting on the elements of the Freudenthal triple system these 78 generatorsform the first of the four sets of E actions at the Lie algebra level (correspondingto equations 9.29–9.32 at the group level) which may be listed as the infinitesimaltransformations of x = (cid:0) α XY β (cid:1) ([62] section 2): T ( ˙ s ) : s ( X )˙ s ′ ( Y ) 0 (9.38)˙ λ : − ˙ λα ˙ λ X− ˙ λ Y ˙ λβ (9.39) φ ( ˙ C ) : ( Y , ˙ C ) β ˙ C X × ˙ C (9.40) ψ ( ˙ D ) : Y × ˙ Dα ˙ D ( X , ˙ D ) (9.41)with ˙ s ∈ L (E ), ˙ λ ∈ R and ˙ C, ˙ D ∈ h O . Higher-order terms such as C ♯ = C ∧ C andthe cubic norm det( C ) appear for the finite group actions of equations 9.31 and 9.32.Having extended beyond the L (E ) subalgebra to the full L (E ) we next focuson the generators of the 4-dimensional spacetime Lorentz subgroup SL(2 , C ) ⊂ E ⊂ E of type 1 as studied in section 8.1. As for all E transformations for the actions ofthe Lorentz subalgebra sl(2 , C ) ⊂ L (E ) the dual transformations ˙ s ′ in equation 9.38have identical rotation generators to ˙ s while the boosts are reversed, by equations 9.36and 9.37. As was described for equations 7.24 and 7.25 of section 7.1 reversing the signof the boosts, there parametrised by b a , is precisely the operation which interchangesbetween the L and R representations of SL(2 , C ).Hence while the components of θ l in θ within X ∈ h O , defined in equa-tion 8.11, transform as a left -handed Weyl spinor under SL(2 , C ) the correspondingcomponents of θ L = (cid:0) C + C lB − B l (cid:1) within the θ component of Y ∈ h O , extracted fromequation 9.25, transform as a right -handed Weyl spinor under the same SL(2 , C ) ⊂ E ⊂ E action. (The subscript ‘ L ’ on θ L denotes both the use of the imaginary unit228 and the identification of the ‘leptonic’ components of θ = (cid:0) C ¯ B (cid:1) in Y , as will be seenbelow. In general the superscript ‘1’ is not appended to components such as θ l and θ L since they are unambiguously extracted from ‘type 1’ θ components, while a su-perscript is included for the ‘type 2’ or ‘type 3’ case as for θ l in equation 8.51 forexample). Considered as an action of 2 × S ∈ SL(2 , C ) on the 2-componentWeyl spinors θ l and θ L , extracted from the corresponding θ components of X and Y respectively, and using equation 7.27, the action of equation 9.29 may be summarisedas: θ l θ L → S S †− θ l θ L (9.42)This is precisely the Lorentz transformation of a 4-component Dirac spinor ψ as de-scribed in equations 7.15 and 7.29. Alternatively the above expression could be ob-tained directly at the group level from equation 9.29 using the definition of the adjoint s ∗ in equation 9.16 applied directly to the SL(2 , C ) ⊂ E group transformations.As explained in section 8.1 the components of θ within X ∈ h O under theaction of SL(2 , C ) actually decompose into a set of four left-handed Weyl spinors { θ l , θ i , θ j , θ k } as listed in equation 8.13. Hence equation 9.29 contains both the origi-nal representation of SL(2 , C ) on X , which contains the set of four left-handed Weylspinors in the θ components, simultaneously with an equivalent of the complex conju-gate representation on Y , which hence contains a corresponding set of four right-handedWeyl spinors, which may be denoted { θ L , θ I , θ J , θ K } ⊂ Y . Correspondingly a set offour 4-component Dirac spinors have hence been identified with: ψ = ψ L ψ R = θ l θ L , θ i θ I , θ j θ J or θ k θ K (9.43)with ψ = ψ L ψ R → S S †− ψ L ψ R (9.44)under S ∈ SL(2 , C ) ⊂ E ⊂ E transformations in each case.The above analysis applied to the θ components of X and Y similarly appliesfor the left-handed SL(2 , C ) Weyl spinors contained within θ X under the decompositionof equation 9.1 or 9.5. In this case a corresponding set of four right-handed spinors arefound in the components of θ Y obtained in turn under a decomposition which may bedenoted Y = θ Y θ Y † for the h O ⊂ h O components of Y . A similar observation appliesfor the alternative spinor decomposition of Y beginning with the h C ⊂ h O subspaceas described towards the latter part of the previous section.The internal SU(3) c × U(1) Q symmetry, described in section 8.2, is composedas a subgroup of E purely out of the subset of rotations. Hence, by the discussionaround equation 9.36 above, these actions are identical on the components of X and Y in equation 9.29. Hence in turn the SU(3) c action on the components of X , includingupon the θ components as detailed in table 8.7 and summarised together with theU(1) Q action in equation 8.26, is identical for the corresponding components of Y , andthe corresponding U(1) Q charges for the respective subcomponents of equation 9.25229re also the same. Hence the ψ L and ψ R components carry matching SU(3) c × U(1) Q transformation properties for the set of four Dirac spinors in equation 9.43 (justifyingthe identification of both θ l and θ L as leptonic components). Similarly the SU(2) , × U(1) , ⊂ E rotations, for the mock electroweak theory described in section 8.3, alsoact on the X and Y components of x ∈ F (h O ) in the same way.While the total number of dimensions has been increased from 27 to 56 itremains the case that only a single set of 4 dimensions will describe the externalspacetime. This can be chosen as an h C ⊂ h O subset of components v ⊂ X ,under an SL(2 , C ) ⊂ E action, or as an h C ⊂ h O subset of components v ⊂ Y ,transforming under the complex conjugate representation, but not both . Here we choose v ≡ h ∈ h C as embedded within the Y = (cid:0) P ¯ AA M (cid:1) ∈ h O components of Y inequation 9.25 to represent external spacetime, with Lorentz transformations hencedescribed by: h → h ′ = S †− h S − (9.45)rather than equation 7.31, under the action of S ∈ SL(2 , C ) ⊂ E . The complexsubspace with base units { , l } still underlies both the SL(2 , C ) subgroup and thesubspace for the vectors h ∈ h C . These h components of Y will also now be takento form the ‘vector-Higgs’ correlated with the phenomena of the Standard Model Higgssector and Yukawa couplings, as was described for the original case of L ( v ) = 1 insubsection 8.3.3. Here for the case of L ( v ) = 1 this now implies that none of the27 components of X ∈ h O ⊂ F (h O ) are identified with components of the externalspacetime vectors v ∈ TM .In particular this means that in addition to the d -quark and charged leptoncomponents of left-handed Weyl spinors in θ ⊂ X , potentially both u -quark and neutral lepton left-handed Weyl spinors might be identified in the X components of X as described in the previous section. The a ∈ O component of X has the cor-rect (0 , ) charge structure to describe ( ν -lepton, u -quark) particle states, as seen inequations 8.27 and 8.28, and is now free to accommodate both states. However whilethe corresponding imaginary A (6) components of Y also have an ˙ S –– l charge of , the A ,l = ( A + A l ) part of A ∈ O in Y is occupied by the above components h ∈ h C ,representing the vector-Higgs and external spacetime, as depicted in equation 9.46. α X ∼ θ X θ X † θ θ † n X Y ∼ θ Y θ Y † θ θ † N Y β ∼ ‘ ν L ’‘ u L ’ e L d L X v ≡ h ‘ u R ’ e R d R Y (9.46)This provisionally provides an explanation for the existence of the left-handedneutrino ν L while the corresponding right-handed state ν R is prohibited, at least atthe level of the basic symmetry structures, as a feature of the breakdown of left-right symmetry through the identification of external spacetime in the breaking of230he full symmetry of L ( v ) = 1. This observation is accompanied by the caveatconcerning the Weyl spinor composition of the components of X ⊂ X and Y ⊂ Y .With this in mind, and hence with quote marks placed on the ν L , u L and u R states,the relation between the component structure for elements of x ∈ F (h O ), in the formof equations 9.24 and 9.25, and the first generation of Standard Model fermions issummarised in equation 9.46.As described in the previous section, in order to obtain left-handed Weyl spinorsin the components of X = (cid:0) p ¯ aa m (cid:1) a further decomposition is required, as for examplein equation 9.1 or 9.5; with a similar decomposition of Y = (cid:0) P ¯ AA M (cid:1) , as for example Y = θ Y θ Y † with θ Y = (cid:0) ¯ RS (cid:1) ∈ O , also required to obtain the corresponding right-handed spinors within the components of Y ⊂ Y . With a = sr in equation 9.3 or a = sr + s ′ r ′ in equation 9.7 for the a ∈ O component of X , and similarly with A = SR for example for the A ∈ O component of Y , this decomposition is relatedto the octonion property of triality for SO(8) transformations, as described near theopening of section 6.1 and around equation 6.50. In fact, with the E ‘rotations’acting in the same way on the X and Y components by equation 9.36 and followingthe discussion before equation 9.9 in the previous section, the triality symmetry impliesthat each of a, s, r, A, S, R ∈ O transform in precisely the same way under the actionof any SU(3) c ⊂ SO(8) transformation.The Lorentz spinor structure under the external SL(2 , C ) symmetry may alsobe obtained under an alternative decomposition of X and Y based on the h C ⊂ h O subspaces, as for example in equation 7.32, as also described in the previous section.This possibility may also be relate to the structure of the technicolor models reviewedsubsection 8.3.3. With the external 4-vector h ∈ h C accommodated within the Y components and left-handed neutrino ν L to be accommodated in the X componentsultimately a different decomposition of the X and Y components may be involved inconsistently accounting for the corresponding empirically observed phenomena. Thesephenomena require the correct matching of the internal SU(3) c × U(1) Q action to theobserved fermion multiplets of equation 7.36. Indeed, as also described in the previoussection, some care is needed in order to maintain the ˙ S –– l charge structure correlatingwith ν -lepton and u -quark states in the spinor decomposition. Ideally a yet higher-dimensional form of L ( v ) = 1 may prove the best guide for uncovering this structurein a mathematically natural manner.While further components are needed to unfold the full spinor structure, underthe enlargement of the symmetry group from E to E on the temporal form L ( v ) = 1we next consider the possible identification of an internal SU(2) L action within the E symmetry structure. The Dynkin diagram for the rank-7 Lie algebra E is comparedwith that for the rank-6 Lie algebra E in figure 9.1.Figure 9.1: The Dynkin diagrams for the (a) L (E ), (b) L (E ) and (c) L (E ) Liealgebras, which may be contrasted with those for the subalgebras listed in figure 7.2.231nlike the case for E , the Lie algebra E does contain a rank-6 subgroup cor-responding to the combined external Lorentz symmetry and internal gauge symmetryof the Standard Model, that is:SL(2 , C ) × SU(3) × SU(2) × U(1) ⊂ E (9.47)The description of the internal symmetry, defined in section 8.2 as the stabilitygroup of the external h C ≡ TM spacetime components and adapted here with respectto the external components of h ⊂ Y , will be augmented beyond the 31 E generatorsof table 8.3 to a complete set for Stab ( TM ) ⊂ E . These will include for example theactions ψ ( ˙ D ) of equation 9.41 for which ˙ D ( h ) = 0 as well as any linear combinationof the four sets of E generators in equations 9.38–9.41 which sum to zero on thefour projected components of h ⊂ Y in equation 9.46. An SU(2) ⊂ Stab ( TM ) ⊂ E subgroup, independent of the SL(2 , C ) × SU(3) c symmetry, acting upon the left-handed spinors of X and, together with the identification of a further U(1) action,completing the E decomposition of equation 9.47 may be considered as a candidatefor the SU(2) L × U(1) Y gauge symmetry of the Standard Model. Indeed such anSU(2) ⊂ Stab ( TM ) ⊂ E , having not been identified within the E generators, beinginternal to the components h ⊂ Y whilst acting freely on X and hence constructedasymmetrically in terms of φ ( ˙ C ) and ψ ( ˙ D ), would be expected to have an asymmetricaction on the left and right-handed spinors identified in equation 9.46.Empirically it is the gauge bosons of an SU(2) L gauge symmetry which mediateinteractions within doublets of quarks (cid:0) ud (cid:1) L and leptons (cid:0) νe (cid:1) L . Hence the identificationof such an internal symmetry within the present theory may be a valuable guide tothe full identification of left-handed u -quark and ν -lepton states in equation 9.46 giventhat we have already identified left-handed d -quark and e -lepton states within the θ components of X . With a different action on the X and Y components of x = (cid:0) α XY β (cid:1) inprinciple the identification of such an SU(2) L ⊂ E gauge symmetry is free to act onthe left-handed doublets derived for example from the components of (cid:0) θ X θ (cid:1) L identifiedwithin X , without impinging upon the external spacetime components of Y . Thishence provides a free channel for charged weak transitions within the leptonic (cid:0) νe (cid:1) L and quark (cid:0) ud (cid:1) L doublets which may be extracted from equation 9.46. The analysis ofsuch an SU(2) L action relating to W ± gauge boson interactions, consistent with theappropriate SU(3) c × U(1) Q transformations and charges for the left-handed states, mayalso clarify the structure of left-handed spinors themselves within the X components.More generally, guided by standard electroweak theory, the identification of the ν -lepton and u -quark left-handed spinors in the components of X will be mutually relatedto a determination of the composition of an internal SU(2) L × U(1) Y ⊂ E symmetryaction itself.Towards this end, and in contrast with the opening of section 8.2, an internalsymmetry might be defined as any group G consistent with the subgroup decomposi-tion SL(2 , C ) × G ⊂ E for which the set of SL(2 , C ) spinors transform under thetrivial or fundamental representations of G . That is, while the external SL(2 , C ) sym-metry partitions the components of L (ˆ v ) = 1 into irreducible pieces, including thespinors θ l,i,j,k of equation 8.13 and table 8.2 each composed of four real components,the internal symmetry G respects this partitioning in treating the Weyl spinors asindividual components of a representation of G . This definition excludes for example232he SU(2) generated by ˙ G q + 2 ˙ S q for q = i, j and k which, as described in the openingof subsection 8.3.1, does not transform the spinors θ l,i,j,k as a fundamental represen-tation, but does still include the internal SU(3) c × U(1) Q symmetry as identified insection 8.2, with the actions on the spinors described in table 8.7 and equation 8.23as summarised, via equation 8.24, in equation 8.26. The question then regards theuniqueness of this SU(3) c × U(1) Q action or the existence of further internal symmetrygroups which possess a similarly tidy action on the spinors.At the same time the action of SU(2) L ⊂ E might still be expected to be closelyrelated to the subgroups SU(2) , × U(1) , ⊂ E acting on the components of X , sincethe latter have desirable properties in relation to electroweak theory as described inthe ‘mock electroweak theory’ of section 8.3. These include the ˙ S –– l charges for the˜ W ± and ˜ Z gauge bosons, for example for ˙Σ (2) ± in equation 8.39, and the similarityof the linear dependencies for the corresponding E generators, as seen for example inequation 8.47, to the structure of equation 8.48 for the Standard Model. In attemptingto fit an SU(2) L × U(1) Y symmetry into the E analysis the generator ˙ S –– l was also foundto provide the correct hypercharges for the left-handed fermion states in X as describedfollowing equation 8.49. An SU(2) L × U(1) Y ⊂ E symmetry action will differ for the X and Y components of x ∈ F (h O ), with for example presumably Q = Y requiredfor the right-handed spinors in Y as singlets of SU(2) L .A quantitative test of the E symmetry breaking structure might be foundin a calculation of the electroweak mixing angle θ W , following a similar derivationthat led to sin θ M = in equation 8.63 for the mock electroweak theory within theE structure. As described in section 8.2 the relative coupling of the U(1) Q gaugesymmetry to the fermions, in terms of the fractional charges of the quarks, alreadymatches the observed values. The relative value of the internal SU(3) c coupling tothe spinor components of equation 9.46, in comparison with the electroweak couplings,with respect to a normalised L (E ) Killing form could also in principle be calculated.The explicit structure of the E symmetry actions on the cubic form of X ∈ h O , obtained by generalisation of Lorentz transformations on quadratic forms [38,39, 40, 41], could ideally be further generalised to obtain the structure of the E symmetry actions on the quartic form of x ∈ F (h O ). This would involve an additional133 −
78 = 1 + 27 + 27 = 55 generators from equations 9.39–9.41 now expressed astangent vectors to the 56-dimensional space of F (h O ). In principle the applicationof equation 6.55 for the full set of E actions on x ∈ F (h O ) could in turn be usedto determine the full 133 × L (E ) table, building upon the 78 × L (E ) table in[38]. Another approach to such a construction might be based on the identification of L (E − ) with the Lie algebra of the symplectic group Sp(6 , O ) as described in [62].An SU(2) L action might then be sought using the new generators, either solelyor in combination with the original 78 E generators, acting on a set of left-handedWeyl spinors, via a spinor decomposition of X , identified within the components of X in equation 9.46, as guided by the nature of electroweak interactions for the fermions.Since the SL(2 , C ) Lorentz symmetry and SU(3) c × U(1) Q internal symmetry havealready been identified in sections 8.1 and 8.2 within the E actions of equation 9.38it may be possible to use this as a starting point to more directly construct an SU(2) L symmetry with appropriate properties out of the further generators listed in equa-tions 9.38–9.41. That is in seeking a particular SU(2) L ⊂ E action as represented on233he components of X in equation 9.46 the full 133 ×
133 Lie algebra table for E maynot be required.On the other hand the study of the complete algebra, and the subalgebras itcontains, may be necessary to both identify the actions SU(2) L × U(1) Y correspondingto electroweak theory and to determine the weak mixing angle sin θ W for the presenttheory. Even in this case a ‘quantisation’ of the theory to describe the phenomenaof ‘running coupling’ may be necessary in order to make comparison with the valueof sin θ W ≃ .
23 as empirically determined at the energy scale of M Z , as alluded toshortly after equation 8.66 in subsection 8.3.2. This full picture may also be needed toinclude the SU(3) c interactions in this comparison, given the differing behaviour of therunning coupling associated with each of the three components of SU(3) c × SU(2) L × U(1) Y in the Standard Model as sketched in figure 11.10.In constructing an SU(2) L ⊂ E action with an appropriate action on thecomponents of X in equation 9.46, including upon the four θ l,i,j,k left-handed Weylspinors, as part of an SL(2 , C ) × SU(3) c × SU(2) L × U(1) Y subgroup decomposition,as an exemplification of equation 9.47, the SU(2) L action might also be found to actnon-trivially on the Y components of equation 9.46 and in particular impact upon theexternal h ∈ h C ≡ TM components. This is analogous to the D(1) B ⊂ E actionin the decomposition of equation 8.35 which, although independent of SL(2 , C ) in theLie algebra, with the generator of equation 13.5, clearly impacts upon the externalspacetime components.In the present theory it is proposed that some of the differing properties ofthe internal gauge interactions associated with SU(2) L compared with SU(3) c × U(1) Q arise since the latter forms a subgroup of Stab( TM ) ⊂ E , and even of Stab ( TM ) ⊂ SL(2 , O ) considered as a subgroup of a 10-dimensional spacetime symmetry as de-scribed in the opening of section 9.1, while the former is only to be identified as asubgroup of E , acting on a quartic form of temporal flow, such that SU(2) L is not asubgroup of Stab ( TM ). This structure is further proposed to be closely related to thephenomena of electroweak symmetry breaking in the Standard Model, based on thestudy of the mock electroweak theory described for the SU(2) , × U(1) , subgroupsof E , acting on a cubic form, as described in section 8.3.These features of a higher-dimensional temporal form L (ˆ v ) = 1 of cubic orhigher polynomial order are distinct from those of a quadratic spacetime form. Forthe model considered in section 5.1 with the quadratic form L ( v ) = 1, representinga 10-dimensional form of time which can also be interpreted as a higher-dimensionalspacetime structure, the external SO + (1 ,
3) and internal SO(6) components of thebroken full SO + (1 ,
9) symmetry act independently on the external v and internal v components of temporal flow v , respectively, as depicted in figure 5.1(b). On ex-tension to the cubic form of time L ( v ) = 1 the external SO + (1 ,
3) symmetry wasfound to also act on the extra ‘internal’ dimensions of θ = (cid:0) c ¯ b (cid:1) , identifying a set offour Weyl spinors, as described in section 8.1 and contrasted with the 10-dimensionalspacetime case at the end of that section. Here we make the complementary obser-vation that a component of the internal symmetry G , in the subgroup decompositionSL(2 , C ) × G ⊂ ˆ G with ˆ G = E or E , can itself act on the projected external v ∈ TM spacetime components. This possibility, for a cubic or higher form of temporal flow, isproposed to underlie the origin of mass for the corresponding gauge bosons.234ith significant physical properties deriving from the combined action of theexternal and internal symmetry on both the external and internal temporal componentsthe present theory deviates significantly from models based on a higher-dimensionalspacetime. In particular these observations mark a departure from the resemblancewith Kaluza-Klein theories, as reviewed in chapter 4 and incorporated into the geo-metric structures of the present theory in section 5.1, which may assist in the aim ofderiving a relation between the external and internal geometry, in the form of equa-tion 5.20, in the context of the present theory alone.Similarly as for the proposed SU(2) L × U(1) Y ⊂ E subgroup the SU(2) , × U(1) , ⊂ E actions are not contained within Stab( TM ) ⊂ E . However in section 8.3the impingement of these actions on the h ∈ h C components of X were seen to beanalogous in structure to the Higgs mechanism of electroweak symmetry breakingand the origin of the masses for the W ± and Z gauge bosons. The object h , nowaccommodated in the Y components in equation 9.46, together with the properties ofits components, is now considered as the ‘vector-Higgs’, providing the source for theempirically observed Higgs phenomena.The corresponding components ‘ h ’ ⊂ X ⊂ X , in the complementary h C ⊂ h O ⊂ h O subspace of x ∈ F (h O ) can be opened up by a spinor decomposition,as described in the previous section, to account for the ν L fermion state. On theother hand while the vector-Higgs h ⊂ Y ⊂ Y could be interpreted to be composedof spinors, in the form of equation 7.32 and by analogy with technicolor models forexample, the physical expression of these components is directly in terms of a tangentvector v ∈ TM in the external 4-dimensional spacetime. Hence in particular a right-handed neutrino ν R cannot be accommodated in the Y components.The two kinds of interaction for the SU(2) L × U(1) Y gauge fields on the X and Y components of equation 9.46 contain analogous structures to the Standard ModelLagrangian terms respectively for the weak interactions of left-handed fermions, suchas equations 7.39 and 7.40, and weak coupling to the Higgs field, such as equations 7.51and 7.52 – with the same mixing angle θ W applying in both cases as described afterequation 7.62. For the present theory the SU(2) L × U(1) Y symmetry breaks to U(1) Q as generated by ˙ S –– l ∈ L (E ) ⊂ L (E ) which acts upon X and Y in the same way. Inthe case of the X components the S –– l ≡ U(1) Q action misses the ν L components ofequation 9.46 accounting for the charge neutrality of the neutrino, which is described inthe Standard Model in terms of equations 7.39–7.46. In the case of the Y componentsthe S –– l ≡ U(1) Q action misses the h components of equation 9.46 here potentiallyaccounting for the massless nature of the photon, as suggested for the gauge field˜ A µ ( x ) for the mock electroweak theory in equation 8.71, and as constructed for theStandard Model in equation 7.61.These two different aspects of electroweak theory may hence here be describedtogether in terms of the broken E action on the X and Y components of L ( v ) = q ( x ) = 1 in equation 9.46. While the SU(2) L × U(1) Y ⊂ E action may differ on the X and Y components, involving the asymmetric actions of equations 9.31 and 9.32, aunique mixing angle θ W and surviving U(1) Q symmetry with the same action on X and Y should result from the symmetry breaking over the external h ⊂ Y components. Inprinciple a complementary SU(2) R × U(1) ′ Y ⊂ E subgroup might also be identified, byreversing the contributions from equations 9.31 and 9.32, however such a symmetry,235cting on right-handed doublets of quarks (cid:0) ud (cid:1) R , may be heavily suppressed due to alarger impact on the external h ⊂ Y components.In the quantum theory the propagators for the gauge fields will attain a finitemass through interaction with the external h components. Naturally a ‘quantisation’scheme and particle concept will be needed in order to assess the properties of theparticle content of the theory (as will be developed in chapter 11), with all physicalparticle states transforming under well-defined representations of both the externaland internal symmetry.In identifying an SU(2) L × U(1) Y ⊂ E symmetry it will be desirable to main-tain the features of the electroweak theory studied in section 8.3, in particular with adegree of impingement on the h ⊂ Y components accounting for the correspondinggauge boson masses. This is counter to the provisional assumption in the opening ofsection 8.2 that an internal symmetry should belong to the stability group Stab( TM )of the external spacetime components h ∈ h C ≡ TM . Rather here in a decomposi-tion such as equation 9.47 the emphasis is upon defining internal subgroups throughthe structure of their well-defined representations on the external SL(2 , C ) spinors.The internal SU(3) c × U(1) Q symmetry may also be motivated in this way, asa component of the E decomposition with well-defined representations on the spinorcomponents, as seen in equation 8.26 for example. In this case the fact that it also happens that SU(3) c × U(1) Q ⊂ Stab ( TM ) is responsible for the fact that the gaugebosons associated with QCD and QED happen to be massless. It may also be thecase that an internal SU(2) L × U(1) Y ⊂ E symmetry might also impinge on anyof the scalars α, β, n and N of equation 9.46, all of which are invariant under theSU(3) c × U(1) Q action. The possible physical consequences of these scalar componentsremains to be seen, whether in terms of masses for the gauge bosons and fermions orother effects (as will be considered in chapter 13). It also remains to be seen whetherthe action of an SU(2) L × U(1) Y ⊂ E on the h ⊂ Y components may be more closelyanalogous to the action of the Standard Model group SU(2) L × U(1) Y on the Higgscomplex doublet φ than was the case for the mock electroweak theory.In the Standard Model fermion masses are introduced through Yukawa cou-plings to the Higgs field, as described in the Lagrangian of equation 7.69. In the presenttheory there is neither a fundamental scalar Higgs field nor an explicit Lagrangian,however amongst the long list of quartic terms in the expression for q ( x ) ≡ L ( v ) = 1in equation 9.28 the top line includes the terms: q ( x ) ∼ ( αβ − ph − mh − nN ) ( h b, B i + h c, C i ) (9.48)with the ‘vacuum value’ P = M = v = h substituted in using equation 8.72 appliedto the ‘vector-Higgs’ h ⊂ Y ⊂ Y ∈ h O components of x ∈ F (h O ). Terms of thisform contain both left-handed (cid:0) c ¯ b (cid:1) ⊂ X and right-handed (cid:0) C ¯ B (cid:1) ⊂ Y components and inthis sense are reminiscent of the Standard Model Lagrangian mass terms deriving fromequation 7.69. The terms of equation 9.48, in potentially contributing to the fermionmasses in the full theory, supersede the cubic terms such as h ( b ¯ b + c ¯ c ) obtained fromthe form det( X ) ≡ L ( v ) = 1 as described for equation 8.76. In both cases the fullset of terms transform under the broken symmetry of L (ˆ v ) = 1 rather than under thesymmetry of a Lagrangian. The introduction of non-standard mass terms, as extractedfrom equation 9.28, is not unprecedented as can be seen by comparison with the quarticterm of equation 8.75 for the technicolor model described in subsection 8.3.3.236 key part of developing the present theory will be the identification of theempirically observed properties of the neutrino sector. Of particular interest will beto identify a description of neutrino oscillations, and contrast that structure with theCKM mixing in the quark sector. These structures are also expected to relate closelyto the identification of fermion masses. The low value of the left-handed neutrinomass may correlate in this theory with the lack of a right-handed counterpart in thecomponents of Y . Again with reference to equations 9.24, 9.25 and 9.46, and basedon the structure of equation 8.28, the neutrino is associated with the a ,l componentsof X . As possible contributions to the particle masses, in addition to equation 9.48above, equation 9.28 contains the quartic terms: q ( x ) ∼ P M | a | + M N | b | + N P | c | = h | a | + hN ( | b | + | c | ) (9.49)where the second line again follows on substituting the vacuum value P = M = v = h .Hence, as well as relating to the lack of right-handed A ,l neutrino components in Y ,the low mass of the neutrino in comparison with the electron might depend upon themagnitude of h in comparison with that of the scalar field N ( x ). In any case the quartic‘mass terms’ for the neutrino state do differ from those for the charged lepton, and ina somewhat more complicated manner than suggested by the terms of equation 9.49alone. Although the u -quarks have both left and right-handed components there arealso differences between such quartic terms for the u and d quark states identified inequation 9.46; in this case required to account for the smaller empirical mass differencewith m u m d ∼ . u and d quark masses may in fact be relatedto the attainment of a stable vacuum value for L ( v ) = h , as will be discussed insection 13.2, and which may also correlate with the low mass for the neutrino.With the ambiguity over the possible mathematical ways in which to decomposethe components of X, Y ∈ h O into a set of spinors, as described in the previous section,ultimately an understanding of the nature of the empirically observed particle statesas originating out of the present theory may be required in order to motivate a natural physical choice for such a decomposition. The full identification of scalar, spinor andgauge boson particle states will require consideration of a means of ‘quantisation’ forthe present theory, and we pursue that direction in the following two chapters.The scheme in equation 9.46 accounts for one family of quarks and leptonswith the appropriate transformations under the internal SU(3) c × U(1) Q symmetryand external SL(2 , C ) symmetry, within the above caveat for the u -quark and ν -lepton fermion states. In addition the particle states yet to be identified include thesecond and third generation of fermions, as related through CKM mixing in the caseof the quark states, and their relation to the massive gauge bosons associated withelectroweak theory in the Standard Model. In the following section we speculate onthe possible nature of a yet higher-dimensional form of temporal flow in principlecapable of accommodating these phenomena.237 .3 E Symmetry and the Standard Model
The extension from E acting on L ( v ) = 1 to E acting on L ( v ) = 1 can beconsidered as a continuation of the progression to higher-dimensional forms of temporalflow which began with the SL(2 , C ) Lorentz symmetry of the quadratic form L ( v ) = 1on 4-dimensional spacetime. This progression, the first stages of which were alsodescribed in the opening of section 9.1, is summarised here in table 9.1.form dimensions space symmetry L ( v ) = 1 quadratic 4 spacetime v ∈ h C SL(2 , C ) 6 L ( X ) = 1 quadratic 10 spacetime X ∈ h O SL(2 , O ) 45 L ( X ) = 1 cubic 27 temporal X ∈ h O E − L ( x ) = 1 quartic 56 temporal x ∈ F (h O ) E − L ( v ) = 1 has a symmetrybreaking pattern to E − ⊂ E − with the representations of equation 9.33 asexhibited by the structure of equation 9.29. This is analogous to the further breakingpattern of E to SL(2 , O ) ≡ Spin + (1 , , O ) symmetry of 10-dimensional spacetime is an intermediate stage on theway down to the Lorentz SL(2 , C ) symmetry which further decomposes the represen-tation space into a Lorentz 4-vector, Weyl spinors and Lorentz scalars, as describedin table 8.2 and now applied to both X and Y ∈ h O , with the external Lorentz 4-vector v ∈ TM accommodated within the Y components in equation 9.46, where twofurther Lorentz scalar components α and β are also identified.Apart from the three additional scalars, N , α and β in equation 9.46, theincrease in dimension from 27 to 56 does not contain any redundancy in terms ofcomparison with the structures of the Standard Model. Most of the additional 29dimensions are interpreted as an augmentation from 2-component Weyl spinors to4-component Dirac spinors, together with a separation in the identification of the left-handed neutrino state and the external spacetime h C ≡ TM components.At the level of the Dynkin diagrams of figures 9.1(a) and (b) the E algebramarks a minimal extension from E , but one which together with the representa-tion as described in the previous section may be sufficient to account for much of thestructure of Standard Model symmetries and particle spectrum with little further aug-mentation. This additional augmentation is needed to account for the identification ofthe u -quark and ν -lepton spinor components which will require a further decomposi-tion of the X ⊂ X and Y ⊂ Y components, with for example X = θ X θ X † + φ X φ X † + . . . as described in equations 9.1 and 9.5 of section 9.1. The features of these two meansof augmenting the form of temporal flow to a higher dimension, as described in thetwo previous sections, will need to be combined in the complete theory. One possible238eans of achieving this will be described in the present section.In terms of the dimension of the underlying space, as listed for the sequence offorms L ( v ) = 1 in table 9.1, we first note that a further expansion from 56 to ∼
80 realcomponents would be sufficient incorporate Weyl spinors for the ν L , u L and u R statesof equations 9.46. This is deduced by observing that a ∈ O of equation 8.28 has 8 realcomponents while a set of four Weyl spinors requires a total of 16 real components,or alternatively by noting that the decomposition of the form X = θ X θ X † involves anaugmentation from 10 to 16 real components. With a complete generation of StandardModel fermions then accounted for the second and third generations might also bedirectly incorporated under a further augmentation from 80 to ∼
240 real components.Given the progression to larger symmetry groups summarised in table 9.1 froma mathematical point of view it is also natural to consider whether the Lie groupE , as the largest exceptional Lie group, represented on a quintic homogeneous form L ( v ) = 1, may mark one further and final possible step in this sequence. (While werefer to such a hypothetic ‘quintic’ form, essentially an order greater than quartic isimplied). With the smallest non-trivial representation of E being 248-dimensional,this possibility is particularly worth consideration in light of the observations of theprevious paragraph. In a similar way that extending the symmetry from E to E led to the incorporation of right-handed as well as left-handed fermion states, ideallya further extension to E would subsume both the E symmetry of the structure inequation 9.46 and explicitly incorporate also the u -quark and ν − lepton spinor statesand a full three generations of fermions all under a higher-dimensional form of L ( v ) = 1with an E symmetry.The smallest non-trivial representation of E is the which can be expressedas the symmetry of the cubic form L ( v ) = 1, while for E the representation, againthe lowest-dimensional non-trivial representation, can be expressed as the symmetryof the quartic form L ( v ) = 1. However the representation for E is expressed interms of the adjoint representation on the 248-dimensional E Lie algebra itself, withno clear interpretation in terms of a symmetry of a form of temporal flow L ( v ) = 1.Indeed the Lie algebra E can be essentially introduced in terms of its action on itself,and constructed in purely algebraic terms which may involve the octonions [1], withthe absence of any geometric motivation or application which might be related to aform L ( v ) = 1.The group E can be defined as the symmetry group of a 57-dimensional mani-fold based on F (h O )+ R , known as the ‘extended Freudenthal triple system’ equippedwith extra geometric structure [61], with E ⊂ E now identified as a subgroup. Thesemathematical objects have some connection with the structures of M-theory [60, 61],and indeed E features heavily in some branches of theoretical physics as for examplein E × E heterotic string theory. There is also some debate in the literature concern-ing whether or not the structure of the E Lie algebra alone is large enough to fullydescribe the Standard Model together with gravity (see for example [63]).For the present theory with an E symmetry acting on a hypothetical formof temporal flow, which may be denoted L ( v ) = 1, three generations of fermionstogether with a vector-Higgs could be accommodated within the 248 temporal com-ponents, as outlined above, while the external Lorentz group and internal gauge sym-metries could all in principle be identified within the E group actions. Without the239eed to employ a supersymmetry there are no SUSY states or set of ‘mirror’ particlesof any kind, although it is of course possible that new physics might be predicted withconsequences that might be tested. However in the present theory the primary guidingprinciples are driven by conceptual ideas, rather than taking a fundamental motivationfrom a notion of mathematical elegance. Hence here it is the possible forms of tempo-ral flow L ( v ) = 1, together with their symmetries, which lead the development of thetheory, and this may or may not involve the Lie group E . The progression towardshigher dimensions in table 9.1 does however strongly hint towards consideration of E ,hence allowing this one lead from the perspective of mathematical beauty, we considerthe possible marriage of this ‘aesthetically pleasing mathematics’ with the underlyingconceptual form of the present theory.The fact that the smallest non-trivial representation of the 248-dimensionalE Lie algebra is expressed as the adjoint representation does not itself preclude thepossibility that a representation may also be identified in terms of the symmetriesof a quintic form L ( v ) = 1. By comparison for example the smallest non-trivialfaithful representation of SO(3) is the adjoint representation on the 3-dimensional so(3)Lie algebra, but in this case there is also a fundamental representation preserving themagnitude of vectors v ∈ R in a 3-dimensional Euclidean space. Indeed the SO(3)symmetry of the 3-dimensional form L ( v ) = 1 of equation 2.14 was the example ofa symmetry of a multi-dimensional form of temporal flow with which we began insection 2.2. In this case the two representations of SO(3) are closely related since thebilinear Killing form on the elements of the so(3) Lie algebra has the same structureas the quadratic scalar product in the space R as used in forming the magnitude | v | .A quintic form underlying L ( v ) = 1, invariant under an E symmetry action,may not be as closely related to the adjoint representation and the L (E ) algebrastructure, unless such a quintic form (or more generally a homogeneous polynomialform of order greater than four) might be related to the bilinear Killing form on L (E )in some way. Further, given the progression from the cubic polynomial form det( X )of equation 6.28 as an expression of L ( v ) = 1 with an E symmetry to the termsof the quartic form q ( x ) of equation 9.28 underlying the form L ( v ) = 1 with anE symmetry, a possible quintic form for L ( v ) = 1 with an E symmetry maybe a considerably more complicated mathematical object still. Hence it is perhapsconceivable that such a structure has not been identified through purely algebraicmeans, even over fifty years after the corresponding E and E structures were firstrealised. On the other hand if such a mathematical structure does exist, namely aquintic form L ( v ) = 1 with an E symmetry, then as for the other forms of table 9.1it would naturally apply for the present theory, based on multi-dimensional forms oftemporal flow, and further physical consequences would be expected to be uncoveredin this further progression.In reference [64], as an example of a more geometrical approach, all of theclassical Lie groups are accounted for as isometry groups of bilinear or sesquilinearforms and the first four exceptional Lie groups, G , F , E and E , are described asisometry groups constructed for cubic or quartic forms, but with E essentially absentfrom the discussion. More generally little reference has been identified in the literaturein which a 248-dimensional representation of E is described in terms of an action on aquintic, or any other homogeneous polynomial, form. However in [65, 66] a polynomial240f degree eight which is invariant as a 248-dimensional representation of the compactreal form of E is described, and is closely related to an invariant polynomial forthe real form E . For the present theory it is then an open question whether anoctic form with an E symmetry might contain the quartic form with E symmetry.Such a natural extension consistent with the form of temporal flow L ( v ) = 1 mayalso be required to have a symmetry described by a non-compact real form of E inorder that temporal causality may be respected for physical structures identified onthe base manifold M , with a local SO + (1 , ⊂ E symmetry, as will be discussed insection 13.3.Considering the possible real forms of E more generally, a suitable candidatewould be E − since the following maximal subgroups involving the exceptional Liegroups are well known (see for example [67]):E − × SU(1 , ⊂ E − E − × SO(1 , ⊂ E − (9.50)This suggests the employment of the chain of non-compact real forms E − → E − → E − as symmetry groups for the corresponding forms of the sequence L ( v ) = 1 → L ( v ) = 1 → L ( v ) = 1, where the first two stages have been de-scribed here in chapter 6 and section 9.2 respectively, while the third form remainshypothetical. As for the structure of the first two stages it seems likely that a con-struction of the final form in this progression will involve the algebraic structure of theoctonions in a significant way.At the level of the complex Lie algebras the corresponding Dynkin diagrams forE , E and E are displayed alongside each other in figure 9.1. The Lie group generatedby the rank-8 L (E ) algebra is large enough to contain a rank-8 decomposition of theform: SL(2 , C ) × SU(3) × SU(2) × SU(2) × U(1) × U(1) ⊂ E (9.51)as can be shown by straightforward analysis of the Dynkin diagrams involved. Hencewhile the degrees of freedom of the components of v , as an extension from v ≡ x ∈ F (h O ) of equation 9.46, are sufficient to contain a full three generations ofStandard Model fermions and a vector-Higgs, the E symmetry group is comfortablylarge enough to describe the external Lorentz symmetry together with the internalSU(3) c × SU(2) L × U(1) Y gauge group.While the higher-dimensional extensions of section 9.1 were contrived , for exam-ple via equation 9.5 leading to the inhomogeneous expression of equation 9.8, in orderto describe the further necessary spinors and generations for the Standard Model,ideally these structures will be found to arise naturally within a homogeneous form L ( v ) = 1 under an E symmetry broken over an external 4-dimensional spacetime M . This natural structure should include a full set of SU(3) c × U(1) Q transformationsand charges aligned with the SL(2 , C ) spinors, completing the structure identifiedwithin the E action in equations 8.26 and 8.28, and in particular supplying a mathe-matical justification for the electromagnetic charges of the ν -lepton and u -quark spinorstates through a U(1) Q action which might be related to the form of equation 9.12 forexample. 241owards the end of the previous section an SU(2) R × U(1) ′ Y subgroup wasconsidered as a possible complementary alternative to SU(2) L × U(1) Y within a de-composition of E in the form of equation 9.47, however the E breaking structure ofequation 9.51 can in principle accommodate both subgroups together. In general adecomposition of E in the form of equation 9.51, arising from the symmetry break-ing through a choice of SL(2 , C ) on the external spacetime TM , will contain internalsymmetry groups acting on the set of spinors which do not belong to the stability groupStab ( TM ) ⊂ E . This may include for example an ‘SU(2) R ’, acting asymmetricallyon the X and Y components of the v ⊂ v subspace in equation 9.46 or other gaugegroups with a significant impingement on the vector-Higgs v ≡ h ∈ h C ≡ TM com-ponents and hence associated with very massive gauge bosons, which are hence as yetunobserved as are the corresponding gauge interactions. An internal SU(2) L × U(1) Y isanticipated which is also broken through a degree of impingement on the vector-Higgscomponents, resulting in the empirically observed massive W ± and Z gauge bosonsand associated electroweak phenomena, as a progression from the ‘mock electroweaktheory’ described in section 8.3.Given the projection of v ∈ TM with an external SL(2 , C ) symmetry andthe set of fermions identified in the residual v components transforming under theinternal SU(3) c × U(1) Q symmetry, the further internal SU(2) L symmetry action willbe sought as acting on doublets of quark (cid:0) ud (cid:1) L and lepton (cid:0) νe (cid:1) L left-handed Weyl spinors,and not only for the first but also the second and third generation of Standard Modelfermions. Masses for the fermions are anticipated to arise through interaction withthe vector-Higgs as expressed in the quintic terms of L ( v ) = 1 as an extension fromthe quartic terms such as those of equations 9.48 and 9.49 for the L ( v ) = 1 case. Amisalignment between the SU(2) L weak doublet states and the mass eigenstates for thequark sector is expected to give rise to the phenomena of CKM mixing, as describedfor the Standard Model towards the end of section 7.2, with a related considerationleading to the phenomena of neutrino oscillations in the lepton sector.It would be possible to attempt to embed the structures of the Standard Model,as alluded to above, into the components of a quintic form L ( v ) = 1 with an E symmetry if the latter structure was already known and described in the literature.This would continue the approach adopted for the E symmetry of L ( v ) = 1 andE on L ( v ) = 1, as based on the corresponding mathematical structures originallydiscovered in the 1950s [34] and 1960s [35] respectively, for which the consequences ofsymmetry breaking over M have been studied here in chapter 8 and section 9.2.Alternatively the mathematical structure of E acting on a quintic form un-derlying L ( v ) = 1, if it exists, might itself be constructed through its application inthe present theory as a form of temporal flow based on a knowledge of the empiricalproperties of the Standard Model. That is, continuing the progression of table 9.1through the Standard Model structure identified in the components of F (h O ) underthe broken E symmetry in equation 9.46, and using the need to identify spinor com-ponents for the ν -lepton and u -quarks, together with three generations of fermionsoriented under an SU(2) L action and relating to CKM mixing, all in a structural cor-respondence with the Standard Model, might lead to the identification of a suitableunderlying 248-dimensional space. The study of this mathematical structure, incorpo-rating the subspaces of h O and F (h O ) under the subgroups E and E respectively,242ay lead to the identification of an E symmetry represented on the form L ( v ) = 1,which might then be rigorously studied as an objective mathematical entity in its ownright. Such an interplay between the development of physical theories and mathemat-ical structures has a long history of stimulating mutually beneficial progress, as forthe parallel development of gauge theories and the structure fibre bundles reviewed inchapter 3.Essentially this is the approach we have set out to follow in section 9.1 inattempting to open up further components to account for further spinors and furthergenerations through augmentation such as that in equation 9.5. The aim is then to combine that form of extension with the augmentation to the action of E on F (h O )described in section 9.2 in seeking an E action on a 248-dimensional space such thata homogeneous quintic, or higher order, norm L ( v ) = 1 is invariant. As well asaiming to incorporate the essential structure of the Standard Model, in progressing inthis way it is also possible that new features will appear for E acting on L ( v ) = 1as the full form of temporal flow.For the subgroup action of SL(2 , O ) ⊂ E ⊂ E on the components of F (h O )in equation 9.46 the subspace elements X ∈ h O and Y ∈ h O transform as 10-dimensional spacetime vectors, and need to be opened up to identify a spinor substruc-ture as discussed in section 9.1. However under the corresponding SL(2 , O ) ⊂ E sub-group on the hypothetical form L ( v ) = 1 the object θ (denoted θ in equation 8.2)might be directly identified along with further Majorana-Weyl spinors, including θ X and φ X of equation 9.5, all as naturally occurring representations within the enlargedstructure and without any direct vector representations in the components of v .Further, as for the four-way decomposition of θ in equation 8.13 the Majorana-Weylspinor θ X , for example, will decompose into a set of four left-handed Weyl spinors { θ Xl , θ Xi , θ Xj , θ Xk } under the Lorentz subgroup SL(2 , C ) ⊂ SL(2 , O ) ; with an inter-nal SU(2) L ⊂ E symmetry action sought on doublets of the SL(2 , C ) Weyl spinorsidentified within (cid:0) θ X θ (cid:1) .In this case in place of decomposing a vector into spinor representations, asfor equation 9.5, for the SL(2 , C ) ⊂ E action the need is rather to construct a 4-component vector v to be locally associated with the external tangent space TM .This may be achieved by going the other way and composing together right-handedspinors for example (such as effectively associated with a subset of the Y componentsin equation 9.46), under SL(2 , C ) ⊂ E to form the 4-vector: h = θ Y L ( θ Y L ) † + φ Y L ( φ Y L ) † (9.52)This is essentially equation 7.32, interpreted as composing the right-hand side to formthe left-hand side rather than as a decomposition of the latter. Here the Weyl spinorsare fused together through the projection of the full temporal flow onto the externalspacetime M as an arena for perception in the world, with the local Lorentz symmetryacting on the 4-vectors h ≡ v ∈ TM which also forms the vector-Higgs in thepresent theory. While this structure is analogous to the formation of a scalar Higgs intechnicolor models, as a condensate of a set of proposed techniquarks interacting underan SU( N ) tc gauge symmetry as reviewed in subsection 8.3.3, in the present theoryit is the identification of the geometric form of an external spacetime as an innatefeature of perception, as described in section 2.2, which necessarily draws together243pinor components into a 4-vector composition. This 4-vector h under SL(2 , C ) ⊂ SL(2 , O ) ⊂ E ⊂ E can be seen directly in the components of the various formsof L ( v ) = 1 for the progression in table 9.1, as shown explicitly for example in thecomponents of F (h O ) on the right-hand side of equation 9.46, now considered asintermediate stages on the way to the full form L ( v ) = 1.The fusing of U(1) Q charge neutral Weyl spinors θ Y L and φ Y L (perhaps alsowith a third spinor ψ Y L ) to form the external spacetime vector h ≡ v ∈ TM inequation 9.52, through the requirement of perception on the extended manifold M , isconsistent with absence of physical particle states corresponding to the right-handedneutrino (for all three generations). On the other hand the complementary θ Xl , φ Xl and ψ Xl spinors under SL(2 , C ) ⊂ E remain free as a full set of three generationsof left-handed neutrinos. This analysis is similar to that described for equation 9.46under the E symmetry, with h ∈ TM accommodated in the Y components and the ν L -neutrino derived from the X components, for the first generation only.Although provisional, this discussion for the hypothetical action of E on L ( v ) = 1 describes a possible marriage of a full form of temporal flow, deducedon the basis of mathematical elegance as a further progression from the sequence intable 9.1, together with the basic conceptual framework of the present theory, with aknowledge of the Standard Model structure presiding over the union.Whether or not E will ultimately feature in a significant way for the presenttheory remains to be seen. Here the primary focus is upon homogeneous forms oftemporal flow expressed as L ( v ) = 1, as derived in section 2.1, and if it happens thatE does not form a symmetry group of such an object then it seems unlikely that thislargest exceptional group will play an important role in this theory. However suchhomogeneous forms are known for Lie groups as large as E , as we have describedin this paper. We end this chapter with a summary of the Standard Model features,based on the gauge symmetry SU(3) c × SU(2) L × U(1) Y , which have been identifiedup to this stage within the breaking of known symmetries of L ( v ) = 1 forms throughthe extraction of an external Lorentz symmetry. These are described in relation to theprogression of higher-dimensional forms of temporal flow listed in table 9.1. • L ( v ) = 1: External spacetime Lorentz symmetry SL(2 , C ) acting on v ∈ TM .The Lorentz transformations on 4-dimensional spacetime are subsequently iden-tified with the subgroup SL(2 , C ) ⊂ E within the larger symmetry, as generatedby the basis elements { ˙ B tz , ˙ R xl , ˙ B tx , ˙ B tl , ˙ R xz , ˙ R zl } for sl(2 , C ) of equation 6.57. • L ( X ) = 1: Internal symmetry SU(3) c × U(1) Q actions may be identified inStab ( TM ) ⊂ SL(2 , O ). In the context of the subsequent E action this symme-try is generated by the basis elements { ˙ A q , ˙ G l } + ˙ S –– l acting on the components X = (cid:0) p ¯ aa m (cid:1) ∈ h O ⊂ h O , with the transformations of the a ∈ O components asdescribed in equation 8.28. (This form is closely related to the L ( v ) = 1 modelof figure 5.1). • L ( X ) = 1: The additional θ = (cid:0) c ¯ b (cid:1) ∈ O ⊂ h O components ( θ in equation 6.26)transform under the external SL(2 , C ) as 4 left-handed Weyl spinors θ l , θ i , θ j , θ k as subspaces of O (equation 8.13). These spinors neatly dovetail with the corre-sponding internal SU(3) c × U(1) Q ⊂ Stab( TM ) ⊂ E actions, as deduced from244able 8.7 and equation 8.23 and summarised in equation 8.26, hence identifyinga charged lepton singlet and d -quark triplet.Although the group E , acting on X ∈ h O , is not large enough to contain anadditional internal SU(2) symmetry a number of the more esoteric propertiesof SU(2) L × U(1) Y electroweak theory are reflected in the action of the type2 subgroup SU(2) × U(1) ⊂ E generated by { ˙ R zl , ˙ R xz , ˙ R xl } + ˙ S –– l , and simi-larly for the corresponding type 3 case, which complement the type 1 externalSL(2 , C ) actions, as described in section 8.3. These properties include the dou-blet actions of equations 8.29 and 8.31, a ‘mock electroweak’ symmetry breakingpattern leading to the mixing angle deduced for equation 8.63 and the potentialorigin of gauge boson masses for the broken SU(2) × U(1) generators derivingfrom an impingement on the external spacetime components as described forequation 8.71. Fermion mass terms are similarly considered to arise through in-teractions with the projected external v ∈ TM components under L ( v ) = 1as described for equation 8.76. The vector v ∈ TM itself is considered to consti-tute a ‘vector-Higgs’, with the degree of freedom of the magnitude | v | providesa candidate for the empirically observed scalar Higgs.In addition to the SL(2 , C ) spinors identified from the components of θ a cor-responding set of 4 left-handed Weyl spinors may be identified within the com-ponents of θ X for example, upon introducing the decomposition X = θ X θ X † ofequation 9.1. These further Weyl spinors can be interpreted as the componentsof a neutrino and triplet of u -quarks, although care is needed to maintain thenecessary electromagnetic charges of 0 and as explained around equation 9.12,and further some of these components coincide with the external v ∈ TM ,which has provisionally been associated with the above vector-Higgs. • L ( x ) = 1: Containing now the E and representations, equation 9.33,the external Lorentz symmetry SL(2 , C ) ⊂ E can be taken to act on the v ≡ h ⊂ Y ∈ h O ⊂ F (h O ) components, which continue to both represent externalspacetime and also account for the Higgs sector, as depicted in equation 9.46.The left-handed electron and d -quark Weyl spinors of the L ( X ) = 1 case abovenow have right-handed counterparts, combining in 4-component Dirac spinors,as described in equations 9.43 and 9.44. A left-handed neutrino (along with aset of u L -quark spinors) might now be identified by expanding the X ∈ h O ⊂ h O ⊂ F (h O ) components, while a right-handed counterpart may be excludedby the external h ∈ TM components of Y (while a set of u R -quark spinorsremain), as also indicated in equation 9.46.The internal SU(3) c × U(1) Q ⊂ E symmetry acts on the X and Y componentsof equation 9.46 in the same way. Further internal symmetries may be soughtwhich also act on the set of spinors in the shape of trivial or fundamental repre-sentations. In particular an internal SU(2) L ⊂ E , with an asymmetric action onthe X and Y components, might now be accommodated within the larger group.An explicitly left-right asymmetric coupling to fermion doublets for an internalsymmetry SU(2) L × U(1) Y ⊂ E may be possible for this structure, with furtheranalysis of the E algebra required. The U(1) Q action, surviving the mock elec-troweak symmetry breaking over TM , is identical on the Y and corresponding245 components, accounting for the massless nature of the photon in the first caseand the charge neutrality of the left-handed neutrino in the second case.In augmenting the full symmetry from E to E , and hence embedding theLorentz symmetry in the latter, there is a two-way choice regarding the embeddingof the external spacetime h C ⊂ F (h O ) in either the X or Y components, with thelatter option taken in equation 9.46 as described in the text and alluded to above.The necessary asymmetry in this choice is then ultimately responsible for the left-rightasymmetry observed for physical phenomena, and in particular leads to the parityviolating properties of the weak interaction.The Lie group E does not have complex representations and is hence unsuit-able as a unification group for the purely internal symmetry structure of the StandardModel, as mentioned in section 7.3. However the external Lorentz symmetry does havecomplex representations and including the SL(2 , C ) action within the structure of theE action on F (h O ) in this asymmetric way in turn implies a left-right asymmetryfor the action of the residual internal symmetry. In the present theory this mecha-nism provides the source of parity violating phenomena (rather than such phenomenaarising from the complementary actions of SU(2) , with respect to SU(2) in termsof non-commutative quaternion subalgebras, as had been briefly considered in subsec-tion 8.3.1 shortly after equation 8.31 as guided by [56] for example). In addition to theSU(2) L action the hypercharge symmetry U(1) Y also remains to be specifically iden-tified, although the latter derives from the ˙ S –– , l generators for the mock electroweaktheory as described for equation 8.47 for example.Further, the three possible embeddings of SL(2 , O ) acting on h O and O ac-cording to equations 6.32–6.35 within the structure of the E action on h O may relateto the empirical observation of three generations of fermions. The embedding of achoice of Lorentz symmetry SL(2 , C ) ⊂ E acting on the type 1 subset h C ⊂ h O breaks the discrete three-way symmetry between the type 1 , symmetry on L ( v ) = 1 may need to be further augmented to an E action on L ( v ) = 1, incorporating a spinor expansion of the original componentsin a form such as equation 9.5 with also a third term ψ X ψ X † . The possibility of thisfurther extension to E , which is currently hypothetical, has been the main topic ofthis section.The above observations, through to the E action on F (h O ), currently markthe point of closest approach between the present theory and the empirical worldof elementary particle phenomena recorded in high energy physics experiments. Apossible extension to an E action of L ( v ) = 1 is suggested partly on aestheticmathematical grounds and partly through the known structure of the Standard Modelitself considered in the context of the present theory.Here we have largely only considered a somewhat ‘static’ picture based onthe structures of the forms L ( v ) = 1 and the corresponding symmetry groups, withemphasis on the explicit structure of the Lie group E acting on the space h O . For thiscase in addition to the terms arising from the expansion of equation 8.76 the constant246alue of L ( v ) = 1 will be expressed ‘dynamically’ on an extended spacetime manifold M as the zero covariant derivative D µ L ( v ) = 0. The terms of the latter expressionresulting from the symmetry breaking contain the internal gauge fields Y µ ( x ), as wasdescribed for the L ( v ) = 1 model in equation 5.51 – which includes an interactionbetween the gauge field Y µ and the internal v components. The cubic temporal form L ( v ) = 1 does not have an interpretation as a higher-dimensional spacetime formand in this case, through the terms of D µ L ( v ) = 0, an internal gauge field canalso impinge upon the external v ∈ TM .It is through this impingement that massive gauge bosons are anticipated to arise asdescribed for the mock electroweak theory in subsection 8.3.3, with the field v ( x )termed the vector-Higgs through association with Higgs phenomena. In the full theorythe masses for the W ± and Z gauge bosons of the Standard Model might be identifiedin this manner, while the fermion masses may arise through the composition of fermioncomponents with the vector-Higgs under the full form L (ˆ v ) = 1.Within the expansion of D µ L ( v ) = 0 there are also terms of the form h ( bY µ ¯ b + cY µ ¯ c ), by comparison with equation 8.76, with similar terms deriving from the quarticnorm in the E case, describing a coupling between the gauge field Y µ ( x ) and thefermion components within h O . In this way the internal gauge field Y µ ( x ), takingvalues for example in the SU(3) c Lie algebra, will mix the components of the Weylspinors, such as those of the set { θ i , θ j , θ k } in equation 8.19, creating the possibilityof field interactions. Ultimately the consequences of the mutual couplings of all fieldsin the terms of L (ˆ v ) = 1 and D µ L (ˆ v ) = 0 will need to be assessed for the full form oftemporal flow.The initial dynamical equations for this theory derived from the relation be-tween the geometry of the external spacetime and the curvature of the internal gaugefields, as deduced in section 5.1 and culminating in equation 5.20, as guided by thestructure of Kaluza-Klein theories. Hence the gauge fields, such as A µ ( x ) , W ± µ ( x ) . . . ,in being closely related to the spacetime geometry G µν = f ( A, W, . . . ) in the form ofequation 5.31, seem to take some priority over possible fermion states which may beidentified in turn through the field interactions, as will be described in chapter 11.That is given for example an initial W ± µ ( x ) field in turn fermion fields ψ ( x ) withindoublets such as (cid:0) νe (cid:1) L or (cid:0) ud (cid:1) L will be drawn into relation with the external spacetimegeometry G µν = f ( A, W, ψ, . . . ) from the components of x ∈ F (h O ) via interactionswith a gauge fields as an example of the generalisation described for equation 5.32 insection 5.2. In section 13.1 a direct relation between the spacetime geometry and themagnitude of the vector-Higgs field with G µν = f ( v ) will also be derived, leading toa further and more direct link with fermions through the terms of L (ˆ v ) = 1.An understanding of the empirical consequences of all of the possible fieldinteractions, and the phenomena of high energy physics in general, will require a fulldynamical and quantum expression of the theory. This will include an understandingof how macroscopic ‘mass’ as central to general relativity through the field equation G µν = − κT µν is related to particle ‘mass’ as observed in the laboratory, and anexposition of a unified conceptual basis for describing both gravitational and quantumphenomena more generally. Within this unified framework the concept and the natureof physical elementary particles themselves might be addressed. A quantised theorydynamically expressed on the spacetime manifold M will also be required in order to247educe the empirical particle spectrum as well as to express kinematic quantities suchas the masses of the particle states, for fermions as well as gauge and the Higgs bosons.In quantum field theory (QFT) the particle masses feature in ‘propagators’while charges and coupling constants appear in interaction ‘vertex’ terms. Both of theseobjects are intrinsic to calculations of cross-sections via the transition amplitude M fi ,as will be described in the following chapter. The propagator factors in calculations ofprocess probabilities contain various kinematic quantities with the dimension of mass.For example the Feynman propagator for the scalar Higgs field has a particularly simpleform, i/ ( p − M H ) where p is the 4-momentum, which may provide a guide for the roleof mass terms for the present theory. Here the effective incorporation of finite massinto the propagators for massive gauge bosons is expected to be related to that fortechnicolor models as described between equations 8.73 and 8.76 in subsection 8.3.3.Similarly while equations 8.26 and 8 .
28 describe the correct U(1) Q charge struc-ture for a generation of leptons and quarks, it will need to be understood how these‘charges’ enter into cross-section calculations and hence actually account for the elec-tromagnetic charge structure as observed for particle states in high energy physics(HEP) experiments. The interpretation of such QFT calculations within the contextof the present theory will need to be addressed before the concepts of charge and masscan be fully comprehended here. The nature of field interactions and the concept ofparticles themselves will also need to be addressed in the course of this study, as weexplore in the following two chapters.Rather than beginning with fields or particles which are then postulated to have various properties and forms of interaction, in the present theory we begin essentiallywith a composition of, or coupling between, components of the full form L (ˆ v ) = 1together with the generators of the symmetry transformations. Here ‘masses’ and‘charges’ originate in the terms of the expressions for L (ˆ v ) = 1 and D µ L (ˆ v ) = 0.Only once these mathematical relations are expressed in terms of dynamical equationsover the manifold M , with spacetime geometry G µν ( x ) = f ( Y, ˆ v ) in the notation ofequation 5.32, might particle states themselves be identified as a phenomenon arisingout of the field interactions. In turn the observable characteristics of such particlephenomena might be determined.The particle properties, including masses and mixing parameters, althougharising from the underlying interactions of the fields, are not necessarily expected tobe literally read off directly from the E or E symmetry breaking level. Indeed someparticle characteristics, such as their behaviour under CPT transformations and theidentification of antiparticles necessarily requires a theory expressed in an extendedspacetime. In dealing with the bare F (h O ) components together with the algebraicform of the E symmetry actions it can only be expected to uncover a shadow of thefull variety of Standard Model phenomena at this level. However it is also desirablethat this shadow should possess identifiable features, such as the correct fractionalcharges and a left-right asymmetry, that may plausibly underlie the empirical data.The dynamic aspects of the theory and a quantisation scheme will need to be devel-oped in order to make more rigorous comparisons with the full variety of laboratoryphenomena.In the meantime, a collection of general properties of the Standard Model havealready been identified in the study of the breaking of the E symmetry of L ( v ) = 1248n chapter 8 and E symmetry of L ( v ) = 1 as presented in section 9.2. In particularthe structure of the E symmetry on the components of F (h O ) when broken over TM makes significant contact with the Standard Model, as also summarised in this section,and further inroads may be possible by further exploring this structure. However theaim here is to avoid the possibility of contriving the appearance of Standard Modelproperties, but rather to be primarily guided by the development of the theory itself,albeit very much in the light of known empirical phenomena. A number of features,including the identification of u -quark and ν -lepton spinors and their SU(2) L interac-tions with the d -quarks and e -leptons respectively, the ‘Yukawa couplings’ and originof mass, the structure of three generations of fermions and the mixing between them,remain to be better understood.The progression towards higher-dimensional forms of time listed in table 9.1,together with the need to fill out the empirical picture, hints at the possibility ofuncovering an E symmetry action on a quintic form L ( v ) = 1 as the final ‘Russiandoll’ in the sequence of enveloping symmetries of time, as we have described in thissection. However, as well as extensions to higher dimensions a quantised theory and anunderstanding of physical particle states, as considered in the following two chapters,will be needed to identify further details of the Standard Model from within the presenttheory for a thorough comparison with and testing against the empirical data. Untilthen the extent to which the E stage is sufficient or otherwise to account for theStandard Model will not be completely clear.In this regard the main question concerns the identification of the structureof particle-like states within the theory before returning to further assess the corre-spondence between the present theory and empirical data, and then progress towardsmaking predictions which may be tested. Before comprehending the particle conceptit will be necessary to understand how in the present theory quantum phenomena arisetogether with the mathematical structures of quantum field theory which are intrinsicto calculations of high energy physics processes. Hence in the following chapter webegin by reviewing the standard machinery of QFT as applied for HEP experiments.249 hapter 10 Particle Physics
The concept of particle phenomena as observed in HEP experiments in the context ofthe theory presented in this paper will be examined here and in the following chapter.In this theory the world appears in our experience necessarily within the geometricalconfines imposed in order for it to actually be perceived through the flow of time, withthe geometrical conditions for the perceived 4-dimensional spacetime world projectedout of a general higher-dimensional progression in time. The arbitrary nature inherentin a degenerate set of possible geometric solutions manifests itself as quantum andparticle phenomena – such as observed in the detector apparatus of high energy physics(HEP) experiments, and through which we interact with and experience the world ingeneral. This perspective, introduced in this section, will be described more thoroughlyin the next chapter.The phenomena of particles are observed in the laboratory in the limit of near‘vacuum’ conditions as elementary transitions of the world as recorded in detectorcomponents. Similar phenomena will be manifest more generally in a curved spacetimeassociated with an arbitrary distribution of matter, however in the flat spacetimelimit of the near vacuum, approximating the laboratory environment as consideredhere, these phenomena may be simpler to categorise. The ‘particles’ observed in HEPexperiments are states of matter that arise in this simplifying limit, rather than thefundamental ‘building blocks’ of matter itself.It is the aim of experimental high energy physics – employing huge and techno-logically complex macroscopic physical structures in the form of ‘particle’ accelerators,colliders and detectors coupled with sophisticated computer software and data analysis(see for example [68]) – to detect and analyse the most delicate and minimal transi-tions of the state of the perceived physical world. In this way the nature and propertiesof the elementary particles ascribed to such transitions are empirically determined –for example, the relative degree of interaction between particular gauge boson andquark fields in the case of [68]. In the present theory internal symmetries and fermionstates have been identified at the level of the broken E symmetry action on the multi-250imensional temporal form L ( v ) = 1, as described in section 9.2, and will relate to thecomponents of the corresponding gauge fields and quark or lepton fields respectively,subject to dynamical constraints in extended spacetime. While significant contact hasbeen made with structures of the Standard Model, as summarised in equation 9.46and section 9.3, it will be necessary to identify in detail the mathematical correlate ofHEP phenomena within the present framework in order to establish a closer relationbetween the theoretical and experimental environment and hence further assess thevalidity of the theory.It should be kept in mind that the events recorded in a high energy physicsexperiments are not actively made to happen by physicists, rather the complete exper-imental apparatus is designed and built to passively make highly refined observationsof the course of nature. The most elementary and minute transitions of the physi-cal world, expressed for example in terms of gauge or fermion fields, are isolated andamplified through such experiments as exemplified in figure 10.1. Such a process, or‘event’, may involve ‘jets’ of many final state particles as displayed in figure 10.1 orcould be as simple as that sketched in figure 10.2 in the following section.Figure 10.1: The most delicate changes of the macroscopic state of physical structuressuch as HEP detectors are interpreted in terms of ‘particle tracks’ composed out of aseries of such minimal detectable transitions, here exemplified in an event recorded bythe SLD collaboration [69].All ‘material’ objects, such as particle detectors, are apparently infused withand seemingly ‘composed of’ field transitions. The environment of a HEP experiment issuch that a particular series, or chain, of macroscopic transitions of the apparatus canbe reconstructed, via amplified signals and computer algorithms, as a particle track. Atthe elementary microscopic level the particular components of equation 9.46 involved,as developed so far up to the action of E on L ( v ) = 1 for the present theory, willdetermine the particle type, for example an electron or d -quark, with properties suchas the observed bending of an electron track in a magnetic field or the manifestation ofquarks in hadronic states determined by the coupling to the gauge field components.Similarly, the appearance of a set of such particle tracks, as seen in figure 10.1, is251orrelated through the higher-order interactions with other fields, such as that of the Z gauge boson field, hence making connection with mathematical calculations in thecorresponding theoretical framework.Although the higher-order interactions may be complicated empirically theunique properties of elementary particles, such as the masses of the electron and muonfor example, are independent of the external material environment (for example withthe particle production and detection apparatus made of copper, silicon or other ele-ments) as far as we can observe (excepting cases such as an ‘effective mass’ in a solidstate device for example). These properties are measured to be the same in all the va-riety of experiments that have been set up to induce them, and also as they have beenobserved for a range of particle states in natural events such as cosmic ray showers.This robustness arises presumably since there is a universal ‘vacuum’ limit. Hence,although ordinary matter is complex, we expect to be able to isolate the robust andinvariant quantities that describe the observed particle properties in the appropriatelimit for theoretical calculations.The eventual aim will be to calculate the effects seen in particle physics ex-periments in terms of transitions between the fields to determine the properties of theobserved elementary particles. These include their masses and spins which categorisethe particle transformations under the Poincar´e symmetry of 4-dimensional Minkowskispacetime, assuming an approximately flat base manifold M . An ‘electron’, for ex-ample, will be associated with particular field transformations under both a spinorrepresentation of the global external Lorentz symmetry over M and a particular rep-resentation of the internal symmetry of the local gauge group, with for example unitcharge relative to other particle states under the U(1) Q action of electromagnetism.Part of the defining notion of a particle is its local nature. A particle is an entity,whether in experiment or in theory, which causally connects and relates two spacetimeevents or interactions. In HEP experiments the chain of interactions can be tracedfrom the production of the initial particle beams, through interactions with guidingmagnets and accelerating components, into the interaction region of the collider andout into a spray of detector hits and signals to be recorded and analysed. Knowledgeof the spacetime location of the directly detected interactions allows the reconstructionof kinematic quantities, such as the invariant mass or electric charge, of the particlesascribed to these observations.The ultimate ambition here will be to describe what the ‘in’ and ‘out’ particlestates in HEP experiments actually are , physically understood and mathematicallyexpressed, as well as to account for the process taking place in the spacetime volumeof the interaction region. In the spirit of this theory these phenomena, as for allphysical processes in spacetime, will be ‘enveloped’ by the structure of the spacetimegeometry as related to the other fields through G µν = f ( Y, ˆ v ) of equation 5.32, asdescribed in section 5.2. It will be necessary to understand the precise general form ofthe right-hand side of this expression to address the question of what an elementaryparticle, such as an electron, is . For completeness this question will also include thephysical nature of particle states such as quarks which are not observed to propagatemacroscopically as independent objects in spacetime.In standard field theory an independent flat spacetime background is givenas an arena upon which fields may be arbitrarily added. Gauge invariance of a La-252rangian function composed of the fields is then postulated as a means to introduceinteractions between fields, as described in sections 3.5 and 7.2. This constructionis transferred to the corresponding quantum field theory (QFT) in which the gaugetransformations mix internal components of the field operators such as ˆ φ ( x ). Thefield itself may be quantised by applying canonical commutation relations, by analogywith non-relativistic quantum mechanics, to the infinite degrees of freedom of the field,and particle creation and annihilation operators a † ( p ) and a ( p ) identified, as will bereviewed later in this chapter.In contrast, in the present theory all elementary structures arise out of theinterplay of multi-dimensional forms of the flow and symmetry of time expressed in L ( v ) = 1. The higher-dimensional mathematical form of temporal flow L ( v ) = 1gives rise to the components of fields locally in interaction when perceived in physical4-dimensional extended spacetime M in a manner consistent with the underlyingfundamental ordered one-dimensional flow of time. With the action of E on v ∈ F (h O ) broken over M and the derivative D µ L ( v ) = 0 in turn fragmented throughthis 4-dimensional projection the physical manifestation of a degeneracy of causallylinked exchanges between fields describing multiple solutions under G µν ( x ) will beidentified as the origin of indeterministic interactions . It is these interactions whichgive rise to apparent particle phenomena, such as quarks and leptons, as objects ofstudy in high energy physics experiments.Hence the aim is then to understand how such discrete particle phenomena ariseout of the fundamental elements of the theory, without needing to impose creation andannihilation operators, or using similar ad hoc quantisation techniques, to describethis particle-like behaviour. Rather the mathematical structures of the present theoryare intended to match the physical structure of the world down to the most elementarylevel. Here particles should be derived as a phenomenon arising out of the possibilityof multiple field solutions under G µν ( x ) on M .The principle goal of the following chapter will be to consider how the newtheory describes the phenomena observed in high energy physics experiments, yetwithout the conceptual problems – for example regarding the particle interpretation– of quantum field theory. In particular this essentially means to be able to matchthe cross-section calculations for particle interactions in QFT except with both anunderlying motivation for the nature of probabilities in these processes and a clearerunderstanding of the particle concept itself.Quantum field theory, although incomplete, provides a set of pragmatic toolsand strategies which have achieved great empirical success, and hence much of themathematical machinery is expected to remain of importance. The preliminary andgeneral nature of QFT allows for the successful elements to be extracted for comparisonwith the present theory. It is the agreement between calculations based on scatteringmatrix amplitudes in QFT and cross-sections measured in the laboratory that needsto be accounted for in the context of the present theory, and hence in the remainderof this chapter we review some of the standard textbook material on the structure ofsuch calculations for reference in the following chapter.253 In this chapter we consider how quantum field theory (see for example [10, 70, 71]) isemployed in practice to calculate cross-sections for processes observed in high energyphysics experiments, for example in proton machines such as the LHC, but in particularfor the kind of events detected in electron-positron colliders as depicted in figure 10.1.The cross-section σ ( e + e − → X ) for a particular process quantitatively represents thelikelihood for the production of the final state X . The description of this final statein general combines a particular collection of outgoing particles, or of ‘jets’ containinga spray of particles as for the event in figure 10.1, together with a particular range ofkinematic or geometric characteristics.The aim here will be to present the cross-section for such processes and thenstrip down this expression to identify how the basic structure of QFT is used to cal-culate the probability of such events. In the following chapter we describe how suchcalculations might be reconstructed in the context of the present theory. Given thecross-section σ ( e + e − → X ) the predicted event rate R (for N events per t seconds) issimply: R ≡ dNdt = Lσ (10.1)which also defines the luminosity value L at which the machine is operating whileproducing the events. In practice ‘bunches’ of incoming particles are directed throughthe interaction region of the experiment, with bunches of n − electrons facing oncomingbunches of n + positrons (where the apparent number n ± of particles per bunch canbe closely estimated from the total charge or energy carried by the bunch). Withthe effective two-dimensional overlap, normal to the beam direction, of the opposingbunches given by the area A and the rate of bunch crossings given by the frequency f ,in the laboratory centre-of-mass frame, the luminosity is simply: L = f n + n − A (10.2)If this luminosity L , in units of cm − s − , is known in addition to the cross-section σ ,in units of cm , then the rate of detection of the corresponding events will be R inequation 10.1 multiplied by the total efficiency ε for the experimental apparatus toobserve such events. In practice L itself is measured using the detection rate εR for aprocess for which σ in equation 10.1 is both well-known and sufficiently high to achievea small statistical uncertainty for L . In quantum electrodynamics (QED) the cross-section σ ( e + e − → µ + µ − ), for the process depicted below in figure 10.2 and describedsubsequently, is one of the simplest to calculate and is well known. It was used as areference point for e + e − colliders in the 1970s in order to measure the cross-section forhadronic final state production relative to muon pairs as a function of centre-of-massenergy. The approach taken in this chapter is to begin with observable quantities inHEP experiments, writing down the general expression for the cross-section as below,and then show how this is related to calculations in QFT through computation of the S -matrix. This in turn will lead to consideration of the elementary interaction termsin the Lagrangian and a description of the procedure of calculation aided by Feynman254iagrams and rules. We begin then with the differential cross-section dσ ( e + e − → X )for a general process at an e + e − collider experiment (see for example [70] p.106): dσ = 14 E E | v − v | |M fi | (2 π ) δ (cid:16) X f p f − X i p i (cid:17) Y f d p f (2 π ) E f (10.3)where E , and v , are the energy and 3-velocity of the particles in the two opposingincoming beams, E f and p f are the energy and 3-momentum for each final state particleand p i and p f are the 4-momenta of each initial and final state particle ( i = 1 , f = 1 , . . . , N f ), all in the centre-of-mass frame. A further combinatoric factor may beneeded, for example to account for initial or final state particle spins for an unpolarisedcross-section, as for equation 10.11 below, or a factor of 1 /n ! for a total cross-sectionwith n identical particles in the final state.The only non-kinematic quantity in equation 10.3 is the transition amplitude M fi (where here the subscript f i labels the overall process) which contains the dynam-ics of the transformation between the initial and final particle states. The relativisticstate normalisation of equations 10.17 and 10.18 below will be employed and is consis-tent with the Lorentz invariance of M fi as constructed in the following section. Every-thing to the right of |M fi | in equation 10.3 is the ‘Lorentz invariant phase space’ term d Φ for the final state. The only factor on the right-hand side of equation 10.3 which isnot Lorentz invariant is the initial state flux factor (4 E E | v − v | ) − , however thisterm is invariant under Lorentz boosts along the beam direction. Indeed dσ , on theleft-hand side of this equation, transforms as a two-dimensional cross-sectional area un-der Lorentz transformations. When composed with the luminosity L of equation 10.2in equation 10.1 the event rate R exhibits a simple special relativistic time-dilationeffect under a change of Lorentz frame, as for any physical ‘clock’.The cross-section σ can be considered as the effective cross-sectional area withinscattering range of each particle in the beam, or as the number of scattering events perunit time, per unit volume, per unit flux density of the incoming beams. Indeed theabove cross-section formula can be calculated by considering the interaction to takeplace over a finite time period T in a finite spatial volume V , which contains purelyfree fields in the limits t → ±∞ relative to the interaction time around t = 0. Factorsof T and V cancel in the final result of equation 10.3. Alternatively, a more detailedapproach may be followed in which the incoming states are modelled as wave packetslocalised in space ([70] pp.102–106). In this case the final result for dσ is independentof the shape of the wave packets.For either way of deriving this formula the transition amplitude M fi itself inequation 10.3 is calculated for the idealised case of ‘in’ and ‘out’ plane wave states ofdefinite momentum extending throughout spacetime. The resemblance of these statesto the concept of a particle is somewhat limited due to the absence of localisation,however their use in the determination of M fi , and in turn the cross-section for particleinteractions, may be followed pragmatically.The transition amplitude is determined by the matrix element between theinitial e + e − free field state represented by | p , p i in for t → −∞ and a particular finalstate | q , q . . . q N f i out for t → + ∞ , in the respective ‘in’ and ‘out’ Fock space basesfor the incoming and outgoing particles states. While neither of these two bases aresimply related to a further Fock basis for interacting fields, since they both represent255he free-field case they are isomorphic to each other. This isomorphism is described bythe unitary operator S , connecting the ‘in’ and ‘out’ bases such that | P i in = S | P i out with P denoting any state. Unitarity is required here to conserve probabilities, withthe transition probability being determined by the squared modulus of the amplitude,that is |M fi | , by a basic postulate of quantum theory, as discussed further below.This situation can be expressed in a single ‘interaction picture’ basis I withan initial state | i i I evolving in time from t = −∞ , through interactions as describedby the S -matrix, to be measured in the final state | f i I at t = + ∞ with a probabilitydetermined by the matrix element: S fi = out h f | i i in = out h f | S | i i out = in h f | S | i i in ≡ I h f | S | i i I (10.4)where we subsequently drop the subscripts I since the interaction picture, describedfurther in the following section, will be used throughout. The S -matrix can be written: S = + iT (10.5)where represents the trivial identity operation and iT , with the conventional i = √− S -matrix. It is this latter part iT = S − which is of most interest and its matrix element between the initial andfinal states can be written, with p I = P i p i and p F = P f p f , as: h f | iT | i i = (2 π ) δ ( p F − p I ) i M fi (10.6)which isolates the transition amplitude M fi . Expressions for i M fi will later be as-sociated with Feynman diagrams which in turn may be obtained directly from theLagrangian for the field theory. Hence the transition amplitude is identified from thematrix element in equation 10.6 by factoring out an ever-present total 4-momentumconserving delta function. Such delta functions arise as a consequence of treating theexternal particles as idealised states of definite momentum.In deriving the expression for the cross-section a factor of |h f | S | i i| is incor-porated which hence contributes two factors of (2 π ) δ ( p F − p I ); one of which maybe interpreted as the spacetime interaction volume V T and cancels with other factorsof V and T in the final result of equation 10.3. In this expression for the differentialcross-section the surviving delta function is included in the Lorentz invariant phasespace d Φ when composed with the final factor of Q f d p f / ((2 π ) E f ).This latter object is a statistical factor representing the density of final statesin ‘small’ regions of phase space between p f and p f + d p f for each outgoing particle.These regions are constrained by the delta function for the total 4-momentum whenintegrating over the final state degrees of freedom of the differential cross-section. Thefactors of 1 /E f arise in the phase space from the relativistic state normalisation ofequation 10.17. The first factor in equation 10.3 arises in a related way and representsthe flux density for the incoming colliding particle beams.The overall expression is such that the cross-section σ essentially representsthe probability of individual particle on particle interactions and is hence correctlynormalised for equations 10.1 and 10.2. Bearing in mind these latter equations together256ith equation 10.3 the total differential event rate can be written: dR = (cid:18) f n + n − A · E E | v − v | (cid:19) · |M fi | · (cid:18) (2 π ) δ ( p F − p I ) Y f d p f (2 π ) E f (cid:19) (10.7)as a composition of three parts. The factor in the first brackets contributes to thelikelihood of events occurring given the properties of the incoming beams from a purelystatistical point of view. In a similar way the Lorentz invariant phase space d Φ in thefinal set of large brackets represents the range of possible outgoing state configurationsas a further natural statistical factor. These two factors hence arise out of considerationof the basic classical laws of probability, essentially with the probability simply beingproportional to the sum of the ‘number of ways’ that something can happen. A furthercombinatoric factor is possible, such as a sum over outgoing particle spin states, asalluded to after equation 10.3. Observations made in the experiment depend also onthe efficiency ε of the detector, as alluded to after equation 10.2. Further, classicalstatistical methods are used to analyse the data to complete the measurements ofphysical quantities with the results presented along with their statistical and systematicuncertainties.The point of this discussion is to highlight the contrast between this list ofclassical probabilistic factors and the middle term |M fi | of equation 10.7 with whichthey are composed and which has rather different characteristics. Historically thisfinal factor originated from non-relativistic quantum mechanics for which the transitionprobability for a state described by the normalised wavefunction Ψ( x , t ) to be measuredin the normalised eigenstate Φ i ( x , t ) is represented by the squared modulus | A | of theamplitude A = h Φ i ( x , t ) | Ψ( x , t ) i , that is the overlap integral A = Z Φ ∗ i ( x , t )Ψ( x , t ) d x (10.8)This construction of a probability is a postulate of quantum theory, apparently quitedifferent to the notion of probability as being a measure of the ‘number of ways’ thatsomething can happen, as encountered in all non-quantum walks of life. This formof quantum probability was itself originally introduced to represent the likelihood forlocating a particle at the spatial position x by the value of | Ψ( x , t ) | , and dates fromthe formative years of quantum theory in the mid 1920s.As an example the production of muon pairs in the process e + e − → µ + µ − , asdepicted in figure 10.2, will be considered. The cross-section formula of equation 10.3simplifies for this case of scattering to a two-particle final state. The δ functionconstrains | p f | in the centre-of-mass frame to the same fixed value for each outgoingparticle and, taking the approximation that all particle masses are sufficiently belowthe centre-of-mass energy √ s and hence can be neglected, the differential cross-sectionreduces to: dσd Ω = |M fi | π s (10.9)where Ω is the solid angle within which the µ − is produced. For the unpolarised process e + e − → µ + µ − there is a further combinatoric factor corresponding to an average over257igure 10.2: A schematic diagram for the transition from an e + e − incoming state toa µ + µ − outgoing state. In the text the purely QED process is considered.the initial electron spin states and sum over the final muon spin states, with |M fi | above then replaced by: 14 X spins |M fi | = e (1 + cos θ ) (10.10)This is for the lowest non-trivial order of perturbation in the QFT, for which theunpolarised differential cross-section is hence given by ([70] pp.8 and 137): dσd Ω = α s (1 + cos θ ) (10.11)with fine structure constant α = e / π ≃ / e is the charge of the electron,conventionally taken to be negative. In equation 10.11 s is the square of the centre-of-mass energy and θ is the polar angle of the final state µ − , as depicted in figure 10.2.In deriving equation 10.11 it is assumed not only that s ≫ m µ − , and hence the leptonmasses are neglected, but also that s is sufficiently below M Z , so that a contributionfrom the weak interaction can also be neglected. In particular this means that thecentre-of-mass energy is assumed to be somewhat lower than that for the experimentin figure 10.1, which operated on the Z resonance. In this case, for a purely QEDprocess, the lowest-order calculation can be associated with the Feynman diagram offigure 10.3 featuring an intermediate ‘virtual photon’.On integrating over the solid angle the total cross-section is found to be: σ ( e + e − → µ + µ − ) = 4 πα s (10.12)This cross-section, based on the leading order process depicted by the Feynmandiagram in figure 10.3 agrees with observations in HEP experiments to within about10%. Most of this discrepancy is accounted for by the next order in perturbationtheory ([70] p.8), with excellent agreement between the data and theory for a morethorough calculation. 258igure 10.3: Feynman diagram for the process e + e − → µ + µ − to lowest order in QEDperturbation theory. In such diagrams the external lines on the left-hand side repre-sent incoming particle states, while those on the right-hand side represent outgoingparticles. (The direction of the arrows on the external lines is explained under ‘item3’ in the discussion of Feynman diagrams in section 10.5, while the causal structure ofthe two vertices will also be discussed later, for example alongside figures 10.5(b) and10.6 in section 10.4.)The cos θ angular dependence in equation 10.11 arises in |M fi | from thespin- property of the initial and final state particles. The actual calculations in-volving interaction processes in QED are made significantly more complicated by thepresence of Lorentz spinor and vector fields, with the derivation of the right-handside of equation 10.10 for example being non-trivial. Since we are here interested inthe probability interpretation of the transition amplitude M fi in the following sec-tion we consider in detail a simpler, but closely analogous, model based on interactingscalar fields in order to extract the essential mathematical structure that is used in thecalculation of such probabilities in a more transparent manner. For the remainder of this chapter we consider a scalar model for an interacting fieldtheory with three scalar fields, including one real field ˆ φ ( x ) and two complex fieldsˆ X ( x ) and ˆ Y ( x ), that is with a total of five real field components, with the quanta ofthe complex fields being interpreted as charged particles. (The analogy with the HEPprocess described in the previous section being constructed here may be briefly sum-marised by comparing the Feynman diagrams in figures 10.3 and 10.4 for the respectivelowest-order calculations). Both real and complex free fields can be expressed in termsof a corresponding annihilation and creation operator expansion which for the fieldsˆ φ ( x ) and ˆ X ( x ) can be written as:ˆ φ ( x ) = Z d p (2 π ) p ω p (cid:16) a ( p ) e − ip · x + a † ( p ) e + ip · x (cid:17) (10.13)ˆ X ( x ) = Z d p (2 π ) p ω p (cid:16) b X ( p ) e − ip · x + d †X ( p ) e + ip · x (cid:17) (10.14)ˆ X † ( x ) = Z d p (2 π ) p ω p (cid:16) d X ( p ) e − ip · x + b †X ( p ) e + ip · x (cid:17) (10.15)259ith p = ω p = + p p + m in all three expressions. The mass m for each field will beassociated with the corresponding particle states which are identified in the following.In QFT the Fourier field coefficients such as a ( p ) and a † ( p ) in equation 10.13 are takento be linear operators acting on the Fock space of particle states. The ‘quantisation’of the free field is completed by imposing commutation relations on these operators:[ a ( p ) , a † ( p ′ )] = (2 π ) δ ( p − p ′ )[ a ( p ) , a ( p ′ )] = 0 , [ a † ( p ) , a † ( p ′ )] = 0 (10.16)By imposing these relations, largely by analogy with the quantum mechanical simpleharmonic oscillator, the spectrum of states possesses a ladder structure with a † ( p )interpreted as creating a particle of momentum p and a ( p ) annihilating such a state.Hence in turn the e ± ip · x Fourier modes of equation 10.13 are associated with particlequanta of mass m the creation or annihilation of which are attributed to the free scalarfield ˆ φ ( x ). This structure marks an attempt to achieve direct contact with the conceptof particles by modelling their discrete nature, although the associated Fourier modesare clearly not localised in space. With the vacuum represented by the state | i in theFock space the annihilation operator acts as a ( p ) | i = 0 while a single particle state | p i is created as: | p i = p ω p a † ( p ) | i (10.17)such that, given the vacuum normalisation h | i = 1, we have: h p | q i = 2 ω p (2 π ) δ ( p − q ) (10.18)which is Lorentz invariant, justifying the choice of normalisation factor employed inequation 10.17.Analogous relations to equations 10.16 hold for each pair of operators, namely b X ( p ) , b †X ( p ) and d X ( p ) , d †X ( p ), for the ˆ X ( x ) field of equations 10.14 and 10.15. Thesetwo pairs of operators, with the corresponding two sets of commutators, are interpretedas generating two types of particle states, with b †X ( p ) and b X ( p ) respectively creatingand annihilating X − particles, and similarly with d †X ( p ) and d X ( p ) for X + antiparti-cles. (The U(1) charges associated with the particles and antiparticles are actually +1and − e < Y ( x ) and its conjugate ˆ Y † ( x ) can be similarly expanded in termsof corresponding creation and annihilation operators by direct analogy with equa-tions 10.14 and 10.15 and ˆ Y ± particle states similarly described. The normalisationof these single particle states follows the convention of equation 10.17 and hence wedefine the creation operators:ˆ B †X ( p ) = p ω p b †X ( p ) with ˆ B †X ( p ) | i = | p X − i (10.19)ˆ D †X ( p ) = p ω p d †X ( p ) with ˆ D †X ( p ) | i = | p X + i (10.20)ˆ B †Y ( p ) = p ω p b †Y ( p ) with ˆ B †Y ( p ) | i = | p Y − i (10.21)ˆ D †Y ( p ) = p ω p d †Y ( p ) with ˆ D †Y ( p ) | i = | p Y + i (10.22)260ith corresponding conjugate annihilation operators. These may be considered as sub-components of the operator fields ˆ X ( x ) and ˆ Y ( x ), as for the operator in equation 10.17with respect to the field ˆ φ ( x ). We stress that here in sections 10.2–10.5 we are describ-ing the standard constructions of a quantum field theory (as described in more muchdetail in [10, 70, 71] for example) and in the following chapter we shall need to describehow the corresponding elements arise in the context of the new theory presented inthis paper.The Lagrangian for the model under consideration here consists of three freefield parts, each of which is essentially a Klein-Gordon Lagrangian, for the fields ˆ φ ( x ),ˆ X ( x ) and ˆ Y ( x ), with mass parameters m φ , m X and m Y respectively, together with aninteraction part L int consisting of polynomial functions of the fields: L = L φ + L X + L Y + L int with L φ = ∂ µ ˆ φ ∂ µ ˆ φ − m φ ˆ φ L X = ∂ µ ˆ X † ∂ µ ˆ X − m X ˆ X † ˆ XL Y = ∂ µ ˆ Y † ∂ µ ˆ Y − m Y ˆ Y † ˆ Y and L int = − g ˆ φ ˆ X † ˆ X − g ˆ φ ˆ Y † ˆ Y (10.23)where g is the interaction coupling constant. Since the Lagrangian must be a realfunction a complex field appears in each term symmetrically with its conjugate field;for example L X contains the mass term − m X ˆ X † ˆ X . It is the invariance of this totalLagrangian under the global U(1) symmetry with ˆ
X → e iα ˆ X and ˆ X † → e − iα ˆ X † (andsimilarly for the complex ˆ Y ( x ) field) that implies a conserved U(1) charge as describedabove, consistent with Noether’s theorem as briefly reviewed alongside equation 3.100in section 3.5.The simple QFT model described here is not a gauge theory and in equa-tion 10.23 the interaction terms are added by hand. By contrast in QED or scalarelectrodynamics the coupling of the charged fields to the electromagnetic field A µ ( x )is induced by the requirement of a local U(1) gauge invariance of the Lagrangian, asalso described in section 3.5 and exemplified in the final term of equation 3.96, al-though an arbitrary coupling constant can still be employed. In the Standard Modelnon-Abelian gauge theories are also incorporated through such expressions as for ex-ample in equations 7.39 and 7.40 of section 7.2. In all cases such Lagrangian termsimply interactions since the fields mutually influence one another in equations derivedfrom the principle of extremal action. Here with the additional interaction terms ofequation 10.23 the Euler-Lagrange equations of motion from equation 3.89, derivedby varying ˆ φ ( x ), ˆ X † ( x ), ˆ X ( x ), ˆ Y † ( x ) and ˆ Y ( x ) respectively as five independent fieldssubject to the constraint δ R L d x = 0 (in a flat spacetime) are non-linear in the fields:( (cid:3) + m φ ) ˆ φ ( x ) = − g ˆ X † ˆ X − g ˆ Y † ˆ Y (10.24)( (cid:3) + m X ) ˆ X ( x ) = − g ˆ φ ˆ X and with ˆ X → ˆ X † (10.25)( (cid:3) + m Y ) ˆ Y ( x ) = − g ˆ φ ˆ Y and with ˆ Y → ˆ Y † (10.26)and impossible to solve exactly. Neglecting the L int terms in equation 10.23, that isin the limit for the coupling g →
0, each of equations 10.24–10.26 reduces to the free261lein-Gordon equation for which fields of the form in equations 10.13–10.15 provideexact general solutions.Equations 10.24–10.26 correspond to the ‘Heisenberg picture’ in which all ofthe time dependence is ascribed to the operator fields, while for the ‘Schr¨odingerpicture’ the time dependence would apply purely to the states. In all cases in quantumtheory the time evolution is determined by the Hamiltonian operator H which maybe expressed as the sum of a free field part H and in interaction part H int . In the‘interaction picture’ the time dependence of all operators is determined by H only,with the corresponding evolution of free operator fields such as ˆ φ ( x ) then readilyhandled (as for equation 10.13 as a solution of equation 10.24 with g = 0) while H int governs the evolution of the states. In the interaction picture the aim is to expressthe transition amplitude, and hence the scattering probability, purely in terms of freefields. (This structure will be significant for making a link with the conceptual pictureof the present theory, as will be discussed in ‘item 3)’ of section 11.2 for example.)For the model QFT under consideration here the evolution of the states isclosely related to the interaction terms of equation 10.23. Indeed if there are no timederivatives in the Lagrangian density L int the interaction Hamiltonian H int can bewritten simply as: H int = Z d x H int = − Z d x L int (10.27)In the interaction picture the initial state | i i evolves according to the unitary operator U into the state | Ψ( t ) i ≡ U ( t, −∞ ) | i i at time t , with the equation of motion: i ddt | Ψ( t ) i = H int ( t ) | Ψ( t ) i (10.28)with the Hamiltonian H int ( t ) defining the time evolution. The scattering amplitude isobtained from the overlap of the state | Ψ( t ) i evolved to t = + ∞ with the given finalstate | f i , that is the matrix element: S fi = h f | U (+ ∞ , −∞ ) | i i (10.29)In the interaction picture the ‘initial value problem’ for U ( t, −∞ ) is posed by the initialcondition U ( −∞ , −∞ ) = together with the equation of motion obtained directlyfrom equation 10.28: i ddt U ( t, −∞ ) = H int ( t ) U ( t, −∞ ) (10.30)As an Hermitian operator the Hamiltonian H acts as the infinitesimal generatorof a one-parameter unitary group. This unitary symmetry is employed in QFT to modelthe conservation of probability in scattering processes. A solution to equation 10.30,which might naively be expected to take the form U ( t, −∞ ) ∼ e − itH int , when takinginto account the time dependence and operator action can be obtained by iteration(and checked by direct substitution into equation 10.30) and then restructured usingthe time-ordered product T of operators. Considering the evolution for any time262nterval from t to t it is found that: U ( t, t ) = + ( − i ) Z tt dt H int ( t ) + ( − i ) Z tt dt Z t t dt H int ( t ) H int ( t )+ ( − i ) Z tt dt Z t t dt Z t t dt H int ( t ) H int ( t ) H int ( t ) + . . . (10.31)= − i Z tt dt T [ H int ( t )] − Z tt dt Z tt dt T [ H int ( t ) H int ( t )]+ i Z tt dt Z tt dt Z tt dt T [ H int ( t ) H int ( t ) H int ( t )] + . . . (10.32)= ∞ X n =0 ( − i ) n n ! Z tt dt Z tt dt . . . Z tt dt n T [ H int ( t ) H int ( t ) . . . H int ( t n )] (10.33)= T [exp( − i Z tt dt ′ H int ( t ′ ))] (10.34)The factor of appears on the right-hand side in equation 10.32 since theextended integral does everything that is needed for the corresponding term in equa-tion 10.31 twice. This generalises to the factor of 1 /n ! in equation 10.33 for the corre-sponding combinatorial over-counting for the higher-order terms. The final expressionabove is a useful shorthand notation for equation 10.33. The S -matrix, as introducedin equation 10.4, is then defined, on taking t = −∞ and t = + ∞ , as the unitaryoperator: S = U (+ ∞ , −∞ ) = T e − i R + ∞−∞ dt H int ( t ) (10.35)which appeared in equation 10.29 for the transition amplitude for a particular process.Hence the S -matrix contains the information needed to calculate the probability ofscattering from one plane wave state to another. In the interaction picture the basisfor the external plane wave states is expressed in terms of the same sets of annihilationand creation operators which provide the coefficients of the Fourier expansions of thefields ˆ φ ( x ), ˆ X ( x ) and ˆ Y ( x ), of equations 10.13–10.15 for example, through which inturn H int and hence the S -matrix is expressed in equation 10.35, containing all theinformation about the interaction.For the case of L int = 0 the interaction Hamiltonian is zero and trivially S = .In the general case with L int = 0 equations 10.29 and 10.35 together describe a time-ordered chain of field operations between the initial and final states. This time orderingis explicit in equation 10.31 owing to the temporal limits for each integral and theorder of the interaction Hamiltonian operators in the integrand. Essentially the S -matrix represents everything that can happen at all intermediate times between theinitial and final free states according to the L int terms. As will be described below,when the calculation is restructured with a more symmetric set of temporal limits foreach integral in equation 10.32 the time ordering with T ensures causal relations aremaintained through this chain, with Hamiltonian field operators acting in the correctsequence with intermediate field states first being created before being annihilated.263iven this iterative solution for U ( t, t ) described in equations 10.31–10.34,for ( t, t ) = (+ ∞ , −∞ ), the assumption of perturbation theory is that the first fewterms provide a good approximation to the exact full expression. This may be possibleif the magnitude of the first few terms in equation 10.29 decreases (or if there arecancellations between large terms) with increasing order n , as defined in equation 10.33,which will generally be the case if the coupling constant, such as g in equation 10.23or α in equation 10.11 for the case of QED, is sufficiently small. Even in this casemany terms will lead to divergent integrals in QFT which will need to be accountedfor by renormalisation. However, even this does not imply that the expression for U (+ ∞ , −∞ ), and in turn S fi , will converge for large n . Nevertheless the first fewterms of perturbation theory do lead to calculations that have a well-defined meaningin that they generate quantities that can be compared with experiment, as is the casefor muon pair production in the Standard Model as described towards the end of theprevious section.By analogy with the real process e + e − → µ + µ − here for the scalar field modelwe consider the scattering process X + X − → Y + Y − , that is, using equations 10.19–10.22, between:the initial state X + X − : | i i = ˆ D †X ( p ) ˆ B †X ( p ) | i at t = −∞ and final state Y + Y − : | f i = ˆ D †Y ( q ) ˆ B †Y ( q ) | i at t = + ∞ Hence for the process X + X − → Y + Y − under consideration in the scalar fieldmodel the transition amplitude of equation 10.29, via equation 10.35, can be written: S fi = h | ˆ B Y ( q ) ˆ D Y ( q ) T [exp (cid:0) − i Z + ∞−∞ dt H int ( t ) (cid:1) ] ˆ D †X ( p ) ˆ B †X ( p ) | i (10.36)where H int is expressed in terms of a polynomial in the interaction picture operatorfields ˆ φ ( x ), ˆ X ( x ) and ˆ Y ( x ). These are free-fields evolving simply under H and can beexpanded in terms of creation and annihilation operators, that is by substituting thefree fields of equations 10.13–10.15 (as well as for ˆ Y ( x ) and ˆ Y † ( x )) into equations 10.23and 10.27 in turn, hence linking the initial and final states in equation 10.36. Thegeneral problem then in the interaction picture is to evaluate terms of the form: Z dt , dt . . . dt n T [ H int ( t ) H int ( t ) . . . H int ( t n )] (10.37)between the external particle Fock states. This calculation can be somewhat simplifiedby noting that these terms, together with the initial and final state creation operatorsin equation 10.36, are sandwiched between vacuum states which have the property a ( p ) | i = 0 and h | a † ( p ) = 0 for an arbitrary annihilation operator a ( p ) and its conju-gate. Hence the goal is to use the commutation relations for such operators, for exam-ple equation 10.16, to extract the residual non-zero terms from equation 10.36. Thisis achieved by decomposing the time-ordered product into a combination of normal-ordered terms and contractions , which takes a simple form for the product of two fieldvalues: T ( ˆ φ ( x ) ˆ φ ( y )) = : ˆ φ ( x ) ˆ φ ( y ) : + p –—– q ˆ φ ( x ) ˆ φ ( y ) (10.38)264ere the final term is the contraction which can be defined as the difference betweenthe time-ordered product and the normal-ordered product of the field values. Thenormal-ordered product, denoted by the colon braces : ˆ F :, is defined such that allannihilation operators are placed to the right of all creation operators in each term,and hence h | : ˆ F : | i = 0, that is the vacuum expectation value (v.e.v.) for thenormal-ordered product of any collection ˆ F of fields is zero. The contracted productin equation 10.38 is a scalar multiple of the identity operator , as can be shown byconsidering the case for x > y and for x < y . For example: p –—– q ˆ φ ( x ) ˆ φ ( y ) = T ( ˆ φ ( x ) ˆ φ ( y )) − : ˆ φ ( x ) ˆ φ ( y ) : which for the x > y case:= Z d p (2 π ) d q (2 π ) p ω p p ω q n(cid:0) a ( p ) e − ip · x + a † ( p ) e + ip · x (cid:1)(cid:0) a ( q ) e − iq · y + a † ( q ) e + iq · y (cid:1) − (cid:0) a ( p ) a ( q ) e − ip · x e − iq · y + a † ( q ) a ( p ) e − ip · x e + iq · y + a † ( p ) a ( q ) e + ip · x e − iq · y + a † ( p ) a † ( q ) e + ip · x e + iq · y (cid:1)o = Z d p (2 π ) d q (2 π ) p ω p p ω q (cid:16) a ( p ) a † ( q ) e − ip · x e + iq · y − a † ( q ) a ( p ) e − ip · x e + iq · y (cid:17) = Z d p (2 π ) d q (2 π ) p ω p p ω q [ a ( p ) , a † ( q )] e − ip · x e + iq · y = Z d p (2 π ) d q (2 π ) p ω p p ω q (2 π ) δ ( p − q ) e − ip · x e + iq · y = Z d p (2 π ) ω p e − ip · ( x − y ) (for x > y ) (10.39)which is a scalar quantity, and with e − ip · ( x − y ) replaced by e + ip · ( x − y ) in the concludingline found for the case x < y . Hence taking the v.e.v. of equation 10.38, with thenormalisation h | i = 1, shows that: h | T ( ˆ φ ( x ) ˆ φ ( y )) | i = p –—– q ˆ φ ( x ) ˆ φ ( y ) (10.40)which is an object also known as the ‘Feynman propagator’ for the field ˆ φ ( x ). Thecomplete contractions for the fields of equations 10.13–10.15 can be written as: p –—– q ˆ φ ( x ) ˆ φ ( y ) = i Z d k (2 π ) e − ik · ( x − y ) k − m φ + iε (10.41) p –—– q ˆ X ( x ) ˆ X † ( y ) = i Z d k (2 π ) e − ik · ( x − y ) k − m X + iε (10.42)as will be explained in the following section, see for example equation 10.71, wherethe role of ε will also be described. While the above functions are identical the lattercase can be interpreted as representing X − particle propagation for x > y and X + antiparticle propagation for x < y , since in the latter case the antiparticle creationoperator d †X ( p ) of equation 10.14 acts first at the earlier time x .265he generalisation of equation 10.38 for higher-order compositions of fields, inparticular for those occurring in equation 10.37, is given by Wick’s theorem. Thisexpresses the T -product as a sum of terms involving permutations of normal-orderedproducts composed with contracted field pairs. Many of these terms vanish whentaking the v.e.v. due to their normal-ordered part, leaving residual terms expressibleas a product of pair-wise contractions, that is Feynman propagators.However, the terms in the Wick expansion of the T -ordered product in equa-tion 10.36 do not act on the vacuum directly due to the operators for the initial andfinal states and hence it is necessary to consider also the more trivial contractions suchas (by substituting in for example equations 10.14 and 10.19): p –—– q ˆ X ( x ) ˆ B †X ( p ) = h | ˆ X ( x ) ˆ B †X ( p ) | i = h | Z d q (2 π ) p ω q (cid:0) b X ( q ) e − iq · x + d †X ( q ) e + iq · x (cid:1) p ω p b †X ( p ) | i = h | Z d q (2 π ) r ω p ω q e − iq · x (2 π ) δ ( q − p ) | i = h | e − ip · x | i = e − ip · x (10.43) p —–— q ˆ D Y ( q ) ˆ Y ( x ) = h | ˆ D Y ( q ) ˆ Y ( x ) | i = e + iq · x (10.44)which can be interpreted as the position space representation of the one-particle wave-functions for the respective initial and final single particle states. These have a simpleform since there is no time dependence for the operators ˆ B †X ( p ) and ˆ D Y ( q ) and theorder of products in these two expressions is given explicitly with creation operatorsfor initial state particles acting first and those for final state particles acting last intemporal order.For example, substituting R dt H int ( t ) = − R d x L int ( x ) from equation 10.27,with the interaction Lagrangian L int of equation 10.23, into equation 10.36 the lowest-order non-trivial term in the perturbative expansion, corresponding to n = 2 in equa-tion 10.33, for S fi will include a contribution from the expression: S fi | n =2 = − g h | ˆ B Y ( q ) ˆ D Y ( q ) T (cid:16) Z d x ˆ φ ( x ) ˆ X † ( x ) ˆ X ( x ) Z d y ˆ φ ( y ) ˆ Y † ( y ) ˆ Y ( y ) (cid:17) D †X ( p ) B †X ( p ) | i = − g Z d x d y p —–— q ˆ B Y ( q ) ˆ Y † ( y ) p —–— q ˆ D Y ( q ) ˆ Y ( y ) p –—– q ˆ φ ( x ) ˆ φ ( y ) p —–— q X † ( x ) D †X ( p ) p –—– q X ( x ) B †X ( p )(10.45)As an alternative to expressions such as equation 10.45 the operators creatingthe initial and final states, such as B †X ( p ) and the Hermitian conjugate of ˆ D †Y ( q ),can also be expressed in terms of functions of free fields, such as ˆ X ( x ) or ˆ Y ( x ). In thiscase an additional Feynman propagator is introduced for each external particle stateas expressed in the LSZ reduction formula ([70] p.227). In this form each contributionin the expansion of the scattering amplitude is expressed as the Fourier transform ofthe v.e.v. of a T -product of free fields, that is of a Green’s function (or correlation266unction). This full LSZ expression may be needed for example for a consistent treat-ment of ultraviolet divergences in higher perturbative orders. Here we deal essentiallywith the ‘truncated’ Green’s function, describing the internal interactions, in orderto abstract out the general structure needed to calculate the transition amplitude, asrequired to make connection with the present theory in the following chapter.Each non-zero term in the transition amplitude can be represented by a Feyn-man diagram. In practice QFT calculations of such terms begin with the correspond-ing Feynman diagrams as constructed from a small set of rules. For example thelowest-order non-trivial term described in equation 10.45 corresponds to the diagramin figure 10.4.Figure 10.4: Feynman diagram for the process X + X − → Y + Y − to lowest order inperturbation theory in the scalar model; closely analogous to the diagram for the QEDprocess e + e − → µ + µ − shown in figure 10.3 for which the general comments in thecaption apply also here.More generally the essence of the transition amplitude calculation can be dis-tilled out into a collection of Feynman rules and diagrams as will be described insection 10.5 and table 10.1 for the scalar model. These may be obtained either fromthe canonical quantisation route, as described above (taking care to handle fermionstate operator anticommutators correctly in the case of the Standard Model) or thepath integral approach to QFT. Here we are interested in the origin of the Feyn-man rules, which may be written down from the Lagrangian density for a particularmodel, for comparison with the present theory. From this point of view the approachof canonical quantisation will prove to be more illuminating, in particular throughthe intermediate stage of equation 10.31 as will be described in the following chapter.On the other hand the formalism of the path integral, while pragmatically serving asa valuable calculational tool for QFT, seems to provide less in the way of relevantconceptual insight for the present theory.By substituting the contractions in the form of equations 10.41–10.44 theleading-order term of the transition amplitude expressed in equation 10.45 can bewritten out explicitly as (with the integrals covering all terms to the right of the inte-267ral signs): S fi | n =2 = − g Z d x d y e + iq · y e + iq · y i Z d k (2 π ) e − ik · ( x − y ) k − m φ + iε e − ip · x e − ip · x = − g i Z d x d y d k (2 π ) k − m φ + iε e i ( k + q + q ) · y e − i ( k + p + p ) · x = − g i Z d y d k (2 π ) k − m φ + iε e i ( k + q + q ) · y (2 π ) δ ( k + p + p )= − g i Z d y − p − p ) − m φ + iε e i ( q + q − p − p ) · y = − g i ( p + p ) − m φ + iε (2 π ) δ ( q + q − p − p ) (10.46)where the three integrals over d x , d k and d y have been carried out in the third,fourth and fifth lines above respectively. The final expression is relatively simple andexplicitly shows how such terms of the matrix element S fi are functions of the coupling g , the particle masses and the momentum variables. Indeed since HEP experimentsgenerally prepare initial particles in momentum states and measure the final particlesalso in particular momentum states such calculations are simplified by beginning withmomentum space Feynman rules, as will be described in section 10.5. In this case thescattering matrix is calculated in terms of momentum space Green’s functions whichare related to the corresponding position space functions, such as equation 10.41, bya Fourier transform (see also equation 10.72 in the following section).In explicit calculations the final integral over position space always leads to anoverall 4-momentum conserving delta function, as for the bottom line in equation 10.46.This is factored out and not included in the definition of the transition amplitude M fi as was described for equation 10.6, and hence this delta function is also not includedin the Feynman rules for i M fi . Further, in equation 10.45 only complementary halvesof L int from equation 10.23 have been employed under each integral. The reversechoice corresponds to swapping the coordinate labels x and y on the two vertices ofthe Feynman diagram in figure 10.4. Hence the complete expression for S fi | n =2 basedon equations 10.45 and 10.46 will contain a further equivalent contribution with thedummy variables x and y interchanged. More generally an amplitude i M fi will beassociated with each topologically distinct Feynman diagram, with the permutation of n ! ways of associating the n interactions with n vertices for an n th order diagram can-celling the n ! factor in the expansion of equation 10.33. This cancellation is generallyincorporated into the Feynman rules for a quantum field theory, including the case ofthe model QFT considered here as will be described in the opening of section 10.5 (seethe discussion of ‘rule 6’ following table 10.1).With the above observations on mind, and by reference to equations 10.4–10.6, the transition amplitude for this leading-order term can be extracted from equa-tion 10.46 (now including also the x ↔ y case) for the Feynman diagram of figure 10.4268drawn without the explicit x, y labels) as: M fi = − g p + p ) − m φ + iε (10.47)and hence |M fi | = g s (10.48)where for the second equation it has been assumed that s = ( p + p ) ≫ m φ , and also ε has been set to zero as will be explained in the following section. The differential cross-section for X + X − → Y + Y − scattering to lowest non-trivial order is then obtained bysubstituting this transition amplitude into equation 10.9 for this two-particle final stateto find dσd Ω = g π s .The purpose of this section has been to show explicitly how such transitionamplitudes, featuring in the general cross-section and hence event rate formulae ofequations 10.3 and 10.7, are calculated. In the case of muon production the contribu-tion from the lowest-order transition amplitude in equation 10.10 is rather different tothe analogous case for the scalar model in equation 10.48. In the case of e + e − → µ + µ − the coupling e = √ πα is dimensionless, unlike the case for g in the scalar model, and(combined with the kinematic normalisation factors for the Dirac spinor and electro-magnetic fields) this leads to an absence of s in equation 10.10, while for equation 10.48there is no θ dependence since the model deals with scalar fields only. However, under-lying these differences the essential elements of quantum field theory going into thesecalculations are very similar. In the following section we explore further the basicingredients and structure of the transition amplitude in the context of the scalar fieldmodel. Central to the calculation of the amplitude in equation 10.36, via Wick’s theorem forthe general T -ordered product of several fields, is the Feynman propagator. This wasintroduced for the scalar field ˆ φ ( x ) in equations 10.38–10.41 and is generally denotedby the symbol ∆ F (‘delta F’) with a conventional factor of i (or by D F ≡ i ∆ F as for[70]) in the expression: i ∆ F ( x − y ) = h | T ( ˆ φ ( x ) ˆ φ ( y )) | i (10.49)= h | θ ( x − y ) ˆ φ ( x ) ˆ φ ( y ) + θ ( y − x ) ˆ φ ( y ) ˆ φ ( x ) | i (10.50)The θ -function takes the value θ ( t ) = 1 for t > θ ( t ) = 0 for t < θ (0) = less significant since θ ( t ) is generally used under a time integral; seealso the discussion of equation 10.64 below) and explicitly expresses the time orderingof the field product. The Hamiltonian H int is composed of a product of free fieldsin the interaction picture with the scalar field ˆ φ ( x ) having the Fourier expansion ofequation 10.13. The field ˆ φ ( x ) can be constructed as a sum of positive and negativefrequency parts, ˆ φ ( x ) = ˆ φ + ( x ) + ˆ φ − ( x ), with a ( p ) and a † ( p ) operator coefficients269espectively: ˆ φ + ( x ) = Z d p (2 π ) p ω p a ( p ) e − ip · x (10.51)ˆ φ − ( x ) = Z d p (2 π ) p ω p a † ( p ) e + ip · x (10.52)The e − ip · x components are termed ‘positive frequency’ since as wavefunctions theywould represent states of positive energy under the quantum mechanical operator H ≡ ˆ E = i ~ ∂/∂t (as implied for the same operator in equation 11.51 of section 11.4 wegenerally employ natural units with ~ = 1 and c = 1 in this paper). Similarly the e + ip · x modes are termed ‘negative frequency’. Hence decomposing ˆ φ ( x ) into a sumof the positive and negative frequency parts, with ˆ φ + ( x ) | i = 0 and h | ˆ φ − ( x ) = 0,equation 10.50 for the scalar Feynman propagator can be written: i ∆ F ( x − y ) = θ ( x − y ) h | ˆ φ + ( x ) ˆ φ − ( y ) | i + θ ( y − x ) h | ˆ φ + ( y ) ˆ φ − ( x ) | i (10.53)= θ ( x − y ) h | [ ˆ φ + ( x ) , ˆ φ − ( y )] | i + θ ( y − x ) h | [ ˆ φ + ( y ) , ˆ φ − ( x )] | i (10.54)= θ ( x − y ) i ∆ + ( x − y ) + θ ( y − x ) i ∆ + ( y − x ) (10.55)In the final line above the function ∆ + ( x − y ) can be defined in terms of the commutatorof the positive and negative frequency parts of the field and then written out explicitlyusing equations 10.51 and 10.52: i ∆ + ( x − y ) = [ ˆ φ + ( x ) , ˆ φ − ( y )] (10.56)= Z d p (2 π ) p ω p Z d q (2 π ) p ω q [ a ( p ) , a † ( q )] e − ip · x e + iq · y (10.57)= Z d p (2 π ) ω p e − ip · x e + ip · y (10.58)where the constraint on the energy components, such as p = + ω p = + p p + m , isunderstood in these expressions, and equation 10.16 has been used in the final line –which agrees with equation 10.39 for the x > y case as expected. Again here, since∆ + ( x − y ) is simply a function rather than an operator, the vacuum normalisation h | i = 1 has been used to factor out the vacuum states in equation 10.54 above toobtain equation 10.55. Integrals of the form R d p (2 π ) f ( p )2 ω p are Lorentz invariant provided f ( p ) is a general Lorentz invariant function ([70] p.23, equation 2.40), and hence fromequation 10.58 it can be seen that the function ∆ + ( x − y ) is Lorentz invariant. Togetherwith the function: i ∆ − ( x − y ) = [ ˆ φ − ( x ) , ˆ φ + ( y )] = − [ ˆ φ + ( y ) , ˆ φ − ( x )] = − i ∆ + ( y − x ) (10.59)these can be written in the manifestly Lorentz invariant form: i ∆ ± ( x − y ) = ± Z d p (2 π ) e − ip · ( x − y ) θ ( ± p ) 2 πδ ( p − m ) (10.60)270he objects θ and δ are ‘generalised functions’, or ‘distributions’, which typi-cally only make full mathematical sense when composed with regular functions in anintegrand. A representation of the θ -function will be given below. In one dimensionthe Dirac δ -function can be defined by the property: Z dx f ( x ) δ ( x − x ′ ) = f ( x ′ ) (10.61)which is essentially to substitute the value x = x ′ into any function f ( x ). The one-dimensional δ -function can be represented by the following expression, which has thesubsequent properties (while generally in the text denoting four-parameter objects, x and k each represent a single real variable in equations 10.61–10.63): δ ( x − x ′ ) = 12 π Z + ∞−∞ dk e ± ik ( x − x ′ ) (10.62)with Z + ∞−∞ dx δ ( x − x ′ ) = 1 , and Z dx f ( x ) δ ( g ( x )) = X i f ( a i ) | g ′ ( a i ) | with g ( x ) = 0 for x = { a , a . . . } i.e. δ ( g ( x )) ≡ X i δ ( x − a i ) | g ′ ( a i ) | e.g. δ ( x − a ) ≡ a (cid:0) δ ( x − a ) + δ ( x + a ) (cid:1)(cid:12)(cid:12) a ≥ (10.63)The final expression above can be substituted into equation 10.60 and the p integralperformed to show that it is equivalent to the expression for ∆ + ( x − y ) in equation 10.58and to that for ∆ − ( x − y ) via equation 10.59 for the p < + ( x − y ) in equation 10.60 describes the positive energy and‘on-mass-shell’ momentum space overlap integral of the plane waves, or wavefunctions, e − ip · x and ( e − ip · y ) ∗ . In quantum theory the probability for a particle originating atthe spacetime location y to be found at the location x is represented precisely by thisamplitude (which via a Fourier transform is analogous to the wavefunction transitionamplitude of equation 10.8). In quantum field theory the form of this amplitude i ∆ + ( x − y ) = h | φ + ( x ) , φ − ( y ) | i , from equations 10.53 and 10.55, indeed suggeststhe propagation of a particle created at y and annihilated at x . Since the spacetimelocations x and y are arbitrary x may be either later or earlier than y .The Feynman propagator can be expressed either in terms of operators actingon the vacuum state, equations 10.49 and 10.50, or in terms of plane waves as describedin equations 10.55 and 10.58, with the bridge between these forms of ∆ F ( x − y ) providedby the intermediate equations. In either case a temporal ordering is introduced via the θ -functions.For x > y the Feynman propagator is simply ∆ F ( x − y ) = ∆ + ( x − y ), fromequation 10.55, and hence represents the amplitude for a positive energy particle topropagate forward in time from y to x . On the other hand the ‘negative energy’ partin equation 10.60, with p < θ ( − p ) = 1, represents a propagation from x to y inthe x < y part of ∆ F ( x − y ) and in QFT is interpreted as an antiparticle of positive energy carried forward in time from x to y . As described following equation 10.42 for271he complex scalar field case and for x < y the operator ˆ X acts before ˆ X † with d †X creating an antiparticle; while for the real scalar field ˆ φ there is no distinction betweenparticle and antiparticle states. Hence ∆ F ( x − y ) can be consistently interpreted as onlyrepresenting propagation forwards in time. Further, from equation 10.55 ∆ F ( x − y ) isclearly symmetric in x and y , as is the above interpretation.In actual calculations all spacetime location variables, such as { x, y } for thepropagator ∆ F ( x − y ), will appear under an integral, such as the R d x d y in the firstline of equation 10.46, over all spacetime (including regions outside the light cone with( x − y ) <
0) hence showing explicitly how all possible time orderings are includedequally. These integrals essentially represent a Fourier transform to momentum space,allowing for a simplification of the calculations in terms of the momentum space Feyn-man rules as will be presented in the following section.Hence the Feynman propagator ∆ F ( x − y ) combines wave-like functions e ± ip · x and particle-like operators a ( † ) ( p ) of the field ˆ φ ( x ) with structures of causality throughthe θ -functions, for example in equation 10.53; – apparently elements required to de-scribe the dynamics of exchanges between fields in an interacting theory. It is repre-sented pictorially by an internal line in a Feynman diagram such as figure 10.5(b). SuchFigure 10.5: (a) The function ∆ + ( x − y ) represented as the creation, propagationand annihilation of a particle state from y to x in spacetime. (b) The internal lineFeynman propagator between two spacetime points, representing equation 10.55. Notime ordering is implied in either diagram.diagrams do not represent literal particle trajectories but should merely be interpretedas mnemonic symbols for mathematical terms such as ∆ F ( x − y ) which form the basisof perturbative calculations for an interacting QFT. Indeed the form of ∆ F ( x − y )results from the restructuring of the S -matrix calculation of equation 10.31, whichdescribes an explicitly causal chain of operator actions, to the form of equation 10.32with θ -functions implicitly introduced to impose the apparent time ordering requiredfor mathematical consistency with the first equation.Hence with the Feynman propagator ∆ F ( x − y ) employed to aid calculation inthis way there need not be any direct physical interpretation of this object. However,due to the time ordering, the Feynman propagator can be considered to representthe internal part of both ‘processes’ depicted in figure 10.6 below, in which a specifictime direction is indicated. While the latter diagram in particular represents a purely272athematical element of the calculation both of these ‘processes’ are implied in asingle Feynman diagram, such as figure 10.5(b), for which there is no explicit temporaldirection relating the two vertices.Figure 10.6: The two terms in equation 10.53 for the Feynman propagator ∆ F ( x − y )describe respectively the two internal ˆ φ field ‘processes’ depicted here. In (a) an internalparticle state propagates from y to x while in (b) an internal antiparticle propagatesfrom x to y , however with no distinction between particle and antiparticle states for areal scalar field such as ˆ φ . In (b) φ , Y + and Y − particle states are created out of thevacuum at x .The propagator ∆ F ( x − y ) depends only on the 4-vector difference ( x − y ). Thefunctions ∆ ± ( x − y ), and hence also ∆ F ( x − y ), are non-zero outside the light coneregion, ( x − y ) <
0, where they decay exponentially. While the ∆ ± ( x − y ) are Lorentzinvariant the function θ ( x − y ) is not Lorentz invariant for spacelike separationsoutside the light cone. However the combination of both terms in equation 10.55 is Lorentz invariant.The generalised function θ ( t ) itself can be expressed in the Fourier, or integral,representation as: θ ( t ) = lim η → + i π Z + ∞−∞ e − ist s + iη ds (10.64)which as a distribution is differentiable everywhere (unlike the closely related Heaviside function H ( t ) defined with H ( t ) = 1 for t ≥ H ( t ) = 0 for t < dθ ( t ) dt = lim η → + i π Z − is e − ist s + iη ds = 12 π Z e − ist ds = δ ( t ) (10.65)from the representation of the δ -function in equation 10.62. The substitution of the θ -function into equation 10.55 for ∆ F ( x − y ) is aided by first making the change ofintegration variable s → k − ω , with finite real constant ω , in equation 10.64 so that: θ ( t ) = lim η → + i π Z + ∞−∞ e − i ( k − ω ) t k − ω + iη dk = lim η → + i π e + iωt Z + ∞−∞ e − ik t k − ω + iη dk and hence: θ ( t ) e − iωt = lim η → + i π Z + ∞−∞ e − ik t k − ω + iη dk (10.66)273his expression for the θ -function, along with equation 10.58 for the function∆ + ( x − y ), can be substituted into equation 10.55 for the Feynman propagator asfollows:∆ F ( x − y ) = θ ( x − y ) ∆ + ( x − y ) + θ ( y − x ) ∆ + ( y − x ) (10.67)= θ ( x − y ) ( − i ) Z d p (2 π ) ω p e + i p · ( x − y ) e − ip · ( x − y ) + θ ( y − x ) ( − i ) Z d p (2 π ) ω p e + i p · ( y − x ) e − ip · ( y − x ) (cid:12)(cid:12)(cid:12) p =+ ω p =+ √ p + m Since { x, y } are fixed for each value of ∆ F ( x − y ) the θ -function can be moved insidethe d p integral and with p = + ω p , which is constant for each value of the 3-vector p , equation 10.66 above may be substituted into the square brackets below:∆ F ( x − y ) = ( − i ) Z d p (2 π ) ω p e + i p · ( x − y ) h θ ( x − y ) e − iω p · ( x − y ) i +( − i ) Z d p (2 π ) ω p e + i p · ( y − x ) h θ ( y − x ) e − iω p · ( y − x ) i (10.68)= ( − i ) Z d p (2 π ) ω p e + i p · ( x − y ) h lim η → + i π Z e − ik ( x − y ) k − ω p + iη dk i +( − i ) Z d p (2 π ) ω p e + i p · ( y − x ) h lim η → + i π Z e − ik ( y − x ) k − ω p + iη dk i (10.69)Hence the 3-momentum integral has been enlarged to a 4-parameter integral byincluding the full unrestricted range of the k variable associated with the θ -functionintegral. That is while the p component of the 4-vector p is constrained to the value ω p = + p p + m , the free k integration variable is introduced from equation 10.66.In relabelling the 3-momentum p by the 3-vector k the above final expression is seento take the form of an apparent F ( x − y ) = lim η → + Z d k (2 π ) ω k h e − ik · ( x − y ) k − ω k + iη + e + ik · ( x − y ) k − ω k + iη i (10.70)= lim η → + Z d k (2 π ) e − ik · ( x − y ) h ω k (cid:16) k − ω k + iη + 1 − k − ω k + iη (cid:17) i = lim η → + Z d k (2 π ) e − ik · ( x − y ) h ω k − iηω k (( k ) − ω k + 2 iω k η − η ) i = lim ε → + Z d k (2 π ) e − ik · ( x − y ) k − m + iε (10.71)Here the second line is obtained by reversing the sign of all 4 integration variablesin the second term in square brackets in equation 10.70. The third and final linesfollow after some straightforward algebra, with the new limiting parameter ε ≃ +2 ω k η introduced, and with the limit ε → + for the integral understood even if not explicitlystated. Through substituting ω k = k + m (see equation 10.67) into the third line,and with k = ( k ) − k in the final line, k is treated as a Lorentz 4-vector. This is theexpression for the Feynman propagator scalar function quoted in equation 10.41 (with274 factor of i from equation 10.49). This function of the spacetime difference ( x − y )may also be written: ∆ F ( x − y ) = Z d k (2 π ) e − ik · ( x − y ) e ∆ F ( k )with e ∆ F ( k ) = 1 k − m + iε (10.72)being the momentum space representation of the Feynman propagator, obtained asthe coefficients in the Fourier decomposition of the position space function.Unlike the 4-momentum integral expression for ∆ ± ( x − y ) in equation 10.60,for the Feynman propagator in equation 10.71 there is no ‘mass-shell’ condition witha δ ( k − m ) function, and with 4 independent ‘momentum’ variables the Feynmanpropagator represents ‘states’ which are generally ‘off-shell’. This situation motivatesthe term ‘virtual particle’ in referring to the ‘propagating entity’. On the other hand∆ F ( x − y ) is constructed in equation 10.67 out of elements which are on-shell withenergy ω p = + p p + m , with the off-shell interpretation for the full expression arisingthrough the incorporation of the θ -functions.Equation 10.71 follows from the structure of ∆ + ( x − y ), which is found through[ a ( p ) , a † ( p ′ )] commutators appearing for example in the expansion of terms in equa-tion 10.31 between vacuum states to determine a scattering amplitude, together withthe θ -functions, which are deployed when the calculation is reorganised with the timeordering T of equation 10.32. Hence the notion of ‘virtual particle states’ may beconsidered to be a purely mathematical construction arising from this reworking ofthe calculation.In equation 10.64 the θ -function is defined by a contour integration in thecomplex plane. This involves a combination of Cauchy’s theorem and the residuetheorem – respectively for integration contours surrounding a region of the integrandfunction which is regular or containing singularities, together with Jordan’s lemmafor the vanishing of particular e − ist contour integrals depending on the sign of thereal parameter t in the complex s -plane. The result is that θ ( t ) can be expressed inequation 10.64 with the horizontal integration contour C of figure 10.7(a), in whichthe pole in the integrand at s = − iη is also shown.The single pole in the integrand function for θ ( t ) carries over into two polesin the complex plane (since there are two θ -functions in equation 10.68 leading toequation 10.70) for the integrand in equation 10.71 for ∆ F ( x − y ). In this latter equation(which was derived from equation 10.69) it is understood that the k integration shouldbe carried out first following the straight contour C along the real axis in figure 10.7(b).Using Cauchy’s theorem this contour integral can be ‘analytically continued’ by a 90 counterclockwise rotation to the imaginary k axis without encountering any poles.Under this ‘Wick rotation’ to Euclidean 4-space (with k replaced by k = ik to forma Euclidean 4-vector with k ) the parameter η (and hence ε in equation 10.71) may bediscarded.Alternatively equation 10.71 and the real k integration in figure 10.7(b) isequivalent setting ε = 0 and performing the resulting integral:∆ F ( x − y ) = Z C F d k (2 π ) e − ik · ( x − y ) k − m (10.73)275igure 10.7: Integration contours (a) in the complex s -plane for θ ( t ) defined in equa-tion 10.64 and (b) in the complex k -plane for the Feynman propagator ∆ F ( x − y ) inequation 10.71. The single pole in the first case and pair of poles in the second caseare also indicated.following the contour C F with an implied limit of infinitesimal detours below thefirst then above the second pole on the real axis as displayed by the thick line infigure 10.8. Although these expressions are equivalent equation 10.71 is generallyquoted in preference to equation 10.73 since the iε term in the former case serves toexplicitly indicate the side on which the contour avoids the poles.Figure 10.8: The six functions ∆ ± ( x − y ), ∆( x − y ) and ∆ F,R,A ( x − y ) described in thetext can be defined by the integration along six different contours ( C ± , C and C F,R,A respectively) in the complex k -plane for the same integrand function presented inequation 10.73.Maintaining the same integrand while adapting the contour C F employed inequation 10.73 in a total of six different ways leads to the expression of a total ofsix different functions, all related to ∆ F ( x − y ), and each then defined here in arelated mathematical form. However, the primary importance is given to the C F contour and the Feynman propagator in QFT since this object arises prominently in276he calculation of scattering amplitudes. The three contours C F , C R and C A hugthe real axis in figure 10.8 with the integral determined in the limit of vanishinglysmall detours around the poles. However these integrals do include these infinitesimaldetours are not the Cauchy principle values of the integrals which ‘hop over’ the polesin this limit and would then be identical for ‘ C F ’, ‘ C R ’ and ‘ C A ’.The three remaining contours C , C + and C − can be taken anywhere in thecomplex plane, so long as they navigate around the poles with the topology indicated infigure 10.8. These contour integrals in the complex k -plane simply have the values of − πi times the residues enclosed, with a negative sign relative to the residue theoremwhich is based on anticlockwise circulating contours. It is again understood that thiscomplex k integral is performed first in equation 10.73 for the respective contours,before the remaining real R d k , in defining the ∆, ∆ + and ∆ − functions.Here the outer contour C , encompassing both poles in figure 10.8, representsthe Lorentz invariant singular function ∆( x − y ). This function can be introduced inthe discussion of causality relating to field interactions and defined directly in termsof the field commutator: i ∆( x − y ) = [ ˆ φ ( x ) , ˆ φ ( y )] (10.74)= [ ˆ φ + ( x ) , ˆ φ − ( y )] + [ ˆ φ − ( x ) , ˆ φ + ( y )]= i ∆ + ( x − y ) + i ∆ − ( x − y ) (10.75)= Z d k (2 π ) ε ( k ) 2 π δ ( k − m ) e − ik · ( x − y ) (10.76)using equations 10.59 and 10.60 and with ε ( k ) = +1 , , − k > , k = 0 , k < C in figure 10.8 enclosing both poles, which are separately enclosed by C + and C − . With ∆( x − y ) = 0 for ( x − y ) <
0, unlike the case for the individual∆ ± ( x − y ) components, this function represents causality in field interactions throughequation 10.74, in the sense that it implies ˆ φ ( x ) and ˆ φ ( y ) operate independently of eachother outside the light cone. Each of these three functions satisfies the Klein-Gordonequation: ( (cid:3) x + m ) ∆ ( ± ) ( x − y ) = 0 (10.77)where the differential operator (cid:3) x acts on the spacetime variables corresponding to x , and m in the above is understood to be the mass m φ associated with the scalarfield ˆ φ ( x ). In the spatial plane x − y = 0 the function ∆( x − y ) also satisfies thetime derivative equation ∂ ∆( x − y ,
0) = − iδ ( x − y ) which, via equation 10.74, andthe conjugate field ˆ π ( x ) = ∂ ˆ φ ( x ), is consistent with the equal-time field commutationrelation: [ ˆ φ ( x , t ) , ˆ π ( y , t )] = i δ ( x − y ) (10.78)Here we have arrived at this expression by employing the commutation rela-tion [ a ( p ) , a † ( q )] = (2 π ) δ ( p − q ) in order obtain equation 10.76 from equation 10.74via equation 10.57. However the ‘canonical’ commutation relation of equation 10.78may be postulated ahead of equation 10.16 as the field quantisation rule, as a gen-eralisation from the non-relativistic quantum mechanical relation [ˆ x a , ˆ p b ] = i ~ δ ab for a, b = { , , } in the three spatial dimensions.277n contrast to the three C ( ± ) contours for the three ∆ ( ± ) functions in figure 10.8the three remaining contour integrals essentially follow the real k axis, differing onlyin their means of bypassing the two poles as described above. Although figure 10.8provides a neat mathematical way of summarising these six functions it is importantto understand their conceptual meaning and the relationships between them.In particular the two functions ∆ R ( x − y ) and ∆ A ( x − y ) are the ‘retarded’ and‘advanced’ parts of the Lorentz invariant singular function ∆( x − y ), that is:∆ R ( x − y ) = θ ( x − y ) ∆( x − y ) (= 0 for x < y ) (10.79)∆ A ( x − y ) = − θ ( y − x ) ∆( x − y ) (= 0 for x > y ) (10.80)Both of these functions of course vanish outside the light cone since ∆( x − y ) does. Thefunction ∆ R ( x − y ) also vanishes for x < y into the past while ∆ A ( x − y ) vanishes intothe future. In solutions for a classical theory both retarded and advanced waves can beidentified, with the latter then being eliminated on the grounds of causality. Bearingin mind the antiparticle interpretation described earlier in this section, the retardedand advanced functions are of comparable significance in quantum field theory. Thesetwo functions, along with the Feynman propagator ∆ F ( x − y ), are Green’s functionswhich satisfy the inhomogeneous Klein-Gordon equation:( (cid:3) x + m ) ∆ F,R,A ( x − y ) = − δ ( x − y ) (10.81)The conventional factor of i introduced in equation 10.49 is chosen so that sucha factor is absent in the above equation. The choice of detours around the poles forthe contour integration in figure 10.8 reflects different choices of boundary conditionsfor solutions to the differential equation 10.81, such as the vanishing of the functionsinto the past or the future described in equations 10.79 and 10.80. The relation of theFeynman propagator to the retarded and advanced Green’s functions can be seen fromfigure 10.8 to be:∆ F ( x − y ) = ∆ R ( x − y, θ ( k )) + ∆ A ( x − y, θ ( − k )) (10.82)That is, with the θ ( ± k )-functions understood to be attached to the integrand in theright-hand side of equation 10.73, the contour integral for the Feynman propagator∆ F follows the advanced contour C A in the negative frequency k < C R for positive frequency k >
0. Alternatively, on attaching the θ ( ± k ) to the integrand of equation 10.76 which is then substituted into equations 10.79and 10.80 and in turn into equation 10.82 the resulting expression is found to beidentical to equation 10.55, with the latter expressing ∆ F ( x − y ) in terms of the∆ ± ( x − y ) functions.Retarded and advanced propagators are employed in quantum field theory tostudy solutions to the equations of motion. For example, with regard to the scalarmodel of section 10.3, expressions such as:ˆ φ ( x ) = Z d y ∆ R ( x − y ) g ˆ X † ( y ) ˆ X ( y ) (10.83)may be considered. The retarded propagator ∆ R ( x − y ) satisfies equation 10.81, whichapplied to equation 10.83 yields:( (cid:3) x + m ) ˆ φ ( x ) = − g ˆ X † ( x ) ˆ X ( x ) (10.84)278s an equation of motion for the quantum field ˆ φ ( x ) with source term − g ˆ X † ( x ) ˆ X ( x ).This is equation 10.24, for the two fields ˆ φ ( x ) and ˆ X ( x ) of the scalar model, whichin the previous section was derived from the Lagrangian of equation 10.23. Thismethod of obtaining solutions to equations of motion via Green’s functions was origi-nally employed for classical field theories. For the classical case the right-hand side ofequation 10.84 may act as a source of disturbance generating a wave motion for thecorresponding classical field φ ( x ) on the left-hand side, while in the quantum case theright-hand side may act as a source for the production of particles of the quantum fieldˆ φ ( x ). The various systematic procedures involved in calculating a given transition amplitudefor a given interacting quantum field theory can be conveniently summarised in a smallset of rules, which are most simply expressed in the momentum space representation,obtained in turn for the Feynman propagators in their Fourier expansions. The Feyn-man rules associate mathematical elements of the calculation with graphical elementsin a diagram representing a particular contribution to the transition amplitude. Theserules are written down here for the scalar model with the interaction Lagrangian ofequation 10.23 in table 10.1. These rules resemble those for the simpler interacting fieldtheory based on a single scalar field ˆ φ ( x ) with the interaction Lagrangian L int = − λ ˆ φ ([70] p.115), which is often presented as a model QFT.Representing possible terms in the transition amplitude by the possible topolo-gies of graphical diagrams greatly assists the bookkeeping involved in the calculation.While terms in the expansion of S = T e − i R dt H int ( t ) of equation 10.35 can be pic-tured this way the internal lines should not be literally interpreted as representingtrajectories of ‘virtual particles’, indeed there is no reference to location at all in themomentum space Feynman rules. Rather the topology of the diagrams describes thestructure of possible mathematical terms. Here we make some further comments onthese rules 1–6 as listed in table 10.1:1. Each line, whether internal or external, is associated with a particular field type.The direction of an arrow on a line can be used to distinguish a particle from anantiparticle when relevant, as described in ‘item 3’ below. The propagator termis i e ∆ F ( k ) from equation 10.72, where the factor of i follows from the conventionof equation 10.49.2. The coupling g is added by hand in equation 10.23 and hence for the interactionHamiltonian in equation 10.27. The factor of − i originates from equation 10.34and in turn from the evolution equation 10.30.3. The factors in these first three items are multiplied together. The external linescan be labelled with the on-mass-shell 4-momentum k , with an arrow on theline following the momentum transfer (into or out of the terminating vertex)for a particle and in the opposite direction for an antiparticle (with a similarconvention for internal lines), as depicted in figures 10.3 and 10.4.279. For each propagator: ˆ φ or ˆ X , ˆ Y ✲ ik − m + iε
2. For each vertex: s ✟✟✟❍❍❍ ✯❨ − ig
3. For each external line: ˆ φ s or ˆ X , ˆ Y s ✲
14. Impose 4-momentum conservation at each vertex: P a k a = 05. Integrate over each unconstrained loop momentum k : R d k (2 π )
6. Multiply by the symmetry factor: 1Table 10.1: The Feynman rules in momentum space for the scalar model, relatingmathematical terms and instructions to the elements of a Feynman diagram, each ofwhich contributes to a transition amplitude i M fi .4. The momentum conservation for each vertex arises from the R d x over spacetimeassociated with each of n factors of L int ( x ) in the n th order of perturbation, withthe x -dependence in the integrand purely in terms of the form e i ( P a k a ) · x , with the P a k a summing over all lines connected to the vertex. This is seen for examplefor n = 2 in equation 10.46, where all the various factors for the S fi | n =2 term ofequation 10.45 are composed.5. The loop integrals over R d k tend to diverge leading to the need for renormal-isation, as will be discussed below for figure 10.9 and also in section 11.3. Theloop integral includes the full independent range −∞ < k < + ∞ , arising orig-inally from equations 10.66–10.71 as described in the previous section. In otherquantum field theories there may also be a discrete sum over field indices suchas spin.6. This factor is simply the exponential expansion coefficient of equations 10.33 and10.34 multiplied by n ! from the number of ways the dummy integration variables { x, y . . . } can label the n vertices of the Feynman diagram. In other theoriesthere may also be symmetry factors for permutations of identical particles, as forexample in the ˆ φ theory ([70] p.93).Bearing in mind equations 10.5 and 10.6 each Feynman diagram corresponds toa contribution to the S -matrix without the overall factor of (2 π ) δ ( p F − p I ), that is the280ransition amplitude i M fi . Since the amplitude appears as |M fi | in the cross-sectioncalculation of equation 10.3 the overall factor of i is sometimes neglected.The above rules can be applied to the Feynman diagram of figure 10.4, rep-resenting the lowest-order term for the process X + X − → Y + Y − . Reading off theFeynman rules in table 10.1 for this diagram we find directly: i M fi = − g i ( p + p ) − m φ + iε (10.85)This is the same expression for the transition amplitude as obtained in equation 10.47by explicit calculation, as it should be. The Feynman rules, as applied above, stripout the essence of such calculations.We recall here that the transition probability is obtained from the square of theabsolute value of the transition amplitude, by a basic postulate of quantum theory, asdiscussed around equation 10.8. The transition amplitude itself is strictly composed ofall of the terms in the expansion of equation 10.35, of which only the lowest-order non-trivial term for n = 2 has been accounted for in equation 10.85. It is an assumptionof perturbation theory that the subsequent inclusion of terms of higher order into thesum gives a rapidly improving approximation to physical quantities such that very feworders are needed in practice. One aim of the following chapter is to understand howthis procedure works in the context of the theory presented in this paper, but here wefirst explore a next-to-leading order term in the standard QFT approach for the scalarmodel. One of several contributions to the transition amplitude for n = 4 is describedby the Feynman diagram in figure 10.9.Figure 10.9: Feynman diagram for the process X + X − → Y + Y − for a possible higher-order perturbation. At this ‘next-to-leading order’ level an unconstrained internal loopmomentum r appears, depicted here for the ˆ X field.In this case reading off the instructions from table 10.1 ‘rule 5’ is invoked for thefreedom in the internal loop momentum r which is not constrained by the applicationof ‘rule 4’, leading to the amplitude contribution: i M fi = g (cid:16) i ( p + p ) − m φ + iε (cid:17) Z d r (2 π ) ir − m X + iε i ( p + p − r ) − m X + iε (10.86)In QFT such loop momentum integrals are frequently divergent, as is the case here andfor similar terms in the ˆ φ scalar model, giving infinite and hence meaningless answersif taken at face value. This leads to the need for a program of ‘renormalisation’ inorder to extract useful results out of these calculations.281n practice the divergent internal loop integrals are first made finite by intro-ducing a parameter to smooth the integrand or act as a cut-off to the integrationrange, a process known as ‘regularisation’. The theory is then renormalised, essen-tially by calibration against an empirical input, before the regularising parameters areeliminated. The aim is to achieve finite predictive quantities in this way for compari-son with further physical measurements, such as the observation of ‘running coupling’which is a consequence of renormalisation as will be described in section 11.3.In the natural units we are adopting, with ~ = 1 and c = 1, any physicalquantity can be expressed in units of mass, that is with dimension M D , where themass dimension M is reciprocal to that of length and time, that is M ≡ L − ≡ T − .The success of the renormalisation procedure generally depends upon the power ofthe mass dimension D for the coupling parameter itself. Since R L d x represents the‘action’ which is a dimensionless quantity with D = 0, the Lagrangian density L hasdimension D = 4, which is also consistent with equation 10.27 since the Hamiltonian H has dimension D = 1. If the coupling parameter in the interaction Lagrangian has D ≥ D <
0, such asgravitation for which Newton’s constant G N has D = −
2, are non-renormalisable.Hence the renormalisation procedure works for quantum field theories withdimensionless coupling constants, such as QED and the Standard Model in generaland also the scalar model with L int = − λ ˆ φ . For the scalar model considered herewith L int = − g ˆ φ ˆ X † ˆ X − g ˆ φ ˆ Y † ˆ Y the full Lagrangian of equation 10.23 implies that thecoupling g has dimension D = +1, and hence the theory can be renormalised. Such atheory with D > σ has the dimension L the right-hand side of equation 10.3must also have the overall dimension D = −
2, which is also the dimension of the initialstate flux factor in this equation. For a two-particle final state the Lorentz invariantphase space d Φ is dimensionless, implying that the amplitude M fi itself should alsohave D = 0 in this case. This is consistent with the dimensionless coupling e ofQED in equation 10.10 and with the coupling g having D = 1 for the scalar modelin equations 10.48, 10.85 and 10.86. More generally the dimension of the transitionamplitude M fi will depend upon the multiplicity of the final state and the conventionsemployed for initial and final state normalisation, consistent with the composition offactors forming the cross-section having the appropriate net dimension, as is the casefor equation 10.3.Higher-order corrections, as appearing for the internal propagator for the fieldˆ φ of figure 10.4 when dressed as in figure 10.9, will also be important for the externalparticle states. This applies also for calculations in QED and Standard Model QFTcalculations in general. Although the theory begins by describing free field states it isnot possible in the physical world to decouple the electron field from the electromag-netic field (or the ˆ X field from the ˆ φ field in the scalar model) since they are intrinsicelements of a single interacting system.Any parameters, such as masses m φ and m X in the model here, ascribed toa free field will be unphysical and unmeasurable. Instead a finite set of fundamentalphysical parameters can be operationally defined as those quantities which are directlymeasurable in the laboratory. The self-interaction effects for the observed particle282tates are absorbed into these measured parameters, with the Fock space of initialand final states (in the interaction picture basis) assumed to represent precisely theobserved masses and charges of physically produced or detected particles in matrixelement h f | S | i i calculations. These renormalised parameters obey the fundamentalconservation laws of external and internal symmetries in collision processes. The phys-ical renormalised mass is not the same object as the ‘bare’ mass parameter appearingin the Lagrangian of the theory.As well as the obvious necessity to ‘tame the infinities’ for calculations of phys-ical quantities, the finite results obtained must also respect the basic requirement ofprobability conservation, namely that the total probability for something to happenmust always be equal to 1. This fundamental principle translates in quantum theoryinto the unitarity of the S -matrix, with the restrictions of this condition having im-plications for the relationship between physical quantities such as the cross-section σ and the structure of the transition amplitude M fi as will be described here.The unitarity of the S -matrix of equation 10.35, that is the property SS † = S † S = , together with the definition of the operator T = i ( − S ) in equation 10.5,hence with T † = − i ( − S † ), implies that: T T † = T † T = i ( T † − T ) (10.87)and therefore: h f | T T † | i i = i h f | T † | i i − i h f | T | i i (10.88)Inserting a sum over a complete set of intermediate states | m i the left-handside of this expression can be written as: h f | T T † | i i = X m r m Y j = l Z d k j (2 π ) E j h f | T | m ih m | T † | i i (10.89)where r m is the number of particles in each state | m i and d k j (2 π ) E j is the invariant phasespace element for the particle state normalisation adopted, as described in section 10.2and required here for the insertion of the unit operator between T and T † . The twoterms on the right-hand side of equation 10.88 can be written as: i h f | T | i i = i M fi (2 π ) δ ( p F − p I ) (10.90) i h f | T † | i i = i M ∗ if (2 π ) δ ( p F − p I ) (10.91)These are obtained directly from equation 10.6, which can also be applied to the right-hand side of equation 10.89 and hence substituted into equation 10.88 along withequations 10.90 and 10.91 to find: X m M fm (2 π ) δ ( p F − p M ) M ∗ im h (2 π ) δ ( p M − p I ) r m Y j = l Z d k j (2 π ) E j i = ( i M ∗ if − i M fi ) (2 π ) δ ( p F − p I ) (10.92)This is a non-linear relationship between transition amplitudes, with a product on theleft and a sum on the right-hand side, resulting from the unitarity of the S -matrix.Given the second δ -function on the left-hand side the first one δ ( p F − p M ) may be283eplaced by δ ( p F − p I ), which hence cancels with the δ -function on the right-handside. The term in square brackets is simply the Lorentz invariant phase space d Φ,as described for equations 10.3 and 10.7, here for the intermediate states, and henceequation 10.92 can be written simply as: X m (cid:18) M fm M ∗ im Z d Φ (cid:19) = i ( M ∗ if − M fi ) (10.93)Considering a two-particle initial state and setting | f i = | i i , corresponding toelastic forward scattering at a HEP collider with the final state being identical to theinitial state, and by comparison with equation 10.3, the left-hand side above is thenidentical to the total cross-section for the transition from an initial state | i i to any state | m i , up to an initial state flux factor, which again relates to the state normalisation.That is, with | f i = | i i and since |M im | = |M mi | , equation 10.93 becomes: X m (cid:18) |M mi | Z d Φ (cid:19) = 2 Im( M ii ) (10.94) ≡ E E | v − v | σ tot = 2 Im( M ii ) (10.95)where equation 10.3, with an implied integration over the phase space for each finalstate to obtain the total cross-section σ tot , has been substituted in for the left-handside in the second line. (Here Im( M ii ) is of course a real number, as for the standarddefinition of the imaginary part of a complex number, in contrast to the definition ofthe imaginary part of an octonion as described immediately before equation 6.10). Theflux factor can be expressed in terms of the total centre-of-mass energy E T (= √ s )and the momentum of either initial particle in the centre-of-mass frame | p i | (notinghowever that this factor is not fully Lorentz invariant, as described after equation 10.3),such that the total cross-section can finally be written as: σ tot ( i → anything) = Im( M ii )2 E T | p i | (10.96)This relationship, along with its derivation, is a form of the ‘optical theorem’([70] p.231, equation 7.50). It is a consequence of the S -matrix unitarity condition inscattering experiments, which in turn expresses basic properties of the laws of prob-ability, and has further implications for observable quantities. Here it shows howthe total cross-section for the production of any final state is directly related to the imaginary part of the forward scattering amplitude, up to the normalisation factor inequation 10.96. By equations 10.5 and 10.6 the imaginary part of M ii corresponds tothe non-trivial real part of h i | S | i i , with many intermediate processes contributing. Thesignificance of this result in the context of the present paper is that it demonstrates a linear relationship between a cross-section, that is the likelihood of an event occurring,and an amplitude.The generalised optical theorem as expressed in equation 10.93 can also beapplied to the case of a single particle initial state. On again setting | f i = | i i in thiscase an expression for the total decay rate Γ can be identified as:Γ( i → anything) = Im( M ii ) m i (10.97)284here m i is the mass of the initial state particle. For a single particle the tree levelcontribution to M ii is just the propagator e ∆ F ( k ) of equation 10.72. For ε → / ( k − m + iε )) ∼ δ ( k − m ).This observation can be generalised for higher-order perturbations. In fact theapplication of the optical theorem in a quantum field theory can also be demonstratedin terms of Feynman diagrams, where it can also be proved to all orders of perturbationtheory by applying ‘cutting rules’ ([70] pp.232–236, [72] pp.183–196). An exampleobtained by relabelling the Feynman diagram in figure 10.9 to represent an amplitudefor the forward scattering process X + X − → X + X − , with identical incoming andoutgoing particles and momenta, via two ˆ φ field propagators and a ˆ Y field internalloop is shown here in figure 10.10.Figure 10.10: A Feynman diagram for the forward scattering process X + X − → X + X − ,with a ‘cut line’ drawn through the intermediate loop propagators of the ˆ Y field.By careful analysis of the singularities that occur when internal propagators goon-mass-shell under internal loop momenta integrals, twice the imaginary part of theamplitude can be obtained by summing over the ‘cutting’ possibilities (only one forthe diagram in figure 10.10, shown by the vertical dashed line) and replacing the termin the Feynman rule for each propagator that may be simultaneously put on-shell bythe cut as: ik − m + iε → πi δ ( k − m ) (10.98)(with the sign and factors of 2 and i depending on the conventions adopted) beforeperforming the R d r over the loop 4-momentum. Hence the imaginary part of a loopamplitude is obtained by placing the intermediate states on-shell together, as may havebeen expected from the optical theorem itself since the final states for cross-sectionsand decay rates, equations 10.96 and 10.97 respectively, consist of on-shell particles.Each way of placing intermediate states on-shell together, as for figure 10.10, is calleda ‘cut’ after Cutkosky, with the above cutting rules providing a method to computethe imaginary part of a transition amplitude in general.The cutting rules for obtaining the imaginary part of the transition amplitudefor a given Feynman diagram can be derived by summing over sets of replacements ofeach Feynman propagator ∆ F by either ∆ F , ∆ ∗ F , ∆ + or ∆ − in the Feynman rules. Thiscalculational tool involves a sum over permutations of selected vertices which determine285he kind of replacement for each ∆ F (see for example [72] p.186). Indeed it can beseen that replacing ∆ F of equation 10.71 with ∆ ± from equation 10.60 incorporatesthe substitution of equation 10.98 together with the introduction of a factor of θ ( ± k ).This latter factor relates to the time ordering of the corresponding vertices and theresulting interpretation as an apparent particle or antiparticle propagating forwardsin time between the two vertices. While in equation 10.55 or 10.67 the Feynmanpropagator was constructed out of two ∆ ± components, here it is taken apart againand a single on-shell part retained.This on-mass-shell condition is expressed by the δ -function in equation 10.60.On performing the k integral this constraint leads to the form of equation 10.58which in the present context represents the phase space factor for real final stateexternal particles, on-mass-shell and with positive energy, in cross-section or decayrate calculations. The remaining propagators ∆ F , for example for the two internal ˆ φ field lines in figure 10.10 are unchanged, representing their usual (non-physical) aid tocalculation as described in the previous two sections.For example for the Feynman diagram in figure 10.10, by adapting equa-tion 10.86 and applying the substitutions from equation 10.98 to the basic Feynmanrules of table 10.1 we obtain:2 Im (cid:0) M ( X + X − → X + X − ) (cid:1) = g i ( p + p ) − m φ + iε ! Z d r (2 π ) πi δ ( r − m Y ) 2 πi δ (( k − r ) − m Y )(10.99)The latter integral can be more easily performed under the substitution of the original4-momenta q and q , as indicated in figure 10.10, in place of the integral over r , byincluding a 4-momentum constraint in: Z d r (2 π ) ≡ Z d q (2 π ) Z d q (2 π ) (2 π ) δ ( q + q − k )This leads to:2 Im (cid:0) M ( X + X − → X + X − ) (cid:1) = − g i ( p + p ) − m φ + iε ! Z d q (2 π ) Z d q (2 π ) πδ ( q − m Y ) 2 πδ ( q − m Y ) (2 π ) δ ( q + q − k )= − g i ( p + p ) − m φ + iε ! Z d q (2 π ) ω q Z d q (2 π ) ω q (2 π ) δ ( q + q − k )(10.100)by applying equation 10.63 to obtain the bottom line with q , >
0. Here the initial q , part of the R d q , (2 π ) integrals over δ ( q , − m Y ) place the momenta q , on-shell resultingin integrals of the form R d q , (2 π ) ω q , , that is over the relativistic phase space. Together286ith the overall (2 π ) δ ( p F − p I ) for 4-momentum conservation implied in the finaldelta function this identifies the Lorentz invariant phase space factor d Φ for a two-body Y + Y − final state in equation 10.100. The remaining factor, before the first R sign, canbe identified with |M fi | for the scattering amplitude M fi of equation 10.85 for theprocess X + X − → Y + Y − at the level of the Feynman diagram depicted in figure 10.4,and hence (swapping the two sides of equation 10.100): |M ( X + X − → Y + Y − ) | Z d Φ = 2 Im (cid:0) M ( X + X − → X + X − ) (cid:1) (10.101)This equation verifies the optical theorem relation of equation 10.94 for the Y + Y − final state contribution to the total cross-section in X + X − collisions for theFeynman diagram analysis at this level of perturbation theory. A further contributionfor a X + X − final state can be obtained in a very similar manner based on an inter-mediate X + X − loop, in place of the Y + Y − loop, in figure 10.10. The above argumentapplies to arbitrary loop diagrams and this Feynman diagram approach based on thecutting rules can be used to prove the optical theorem to all orders of perturbationtheory ([70] pp.235–236), with care for combinatoric factors and the consistency of theconventions used in general.Our main point here has been to review the relation between a physical cross-section and an expression linear in a component of an amplitude, namely the imaginarypart of the forward scattering amplitude, both in terms of the total cross-section inequation 10.96 and at the level of individual processes as implied in equation 10.101.These expressions relate to the optical theorem and the unitarity constraint whichin turn represents the basic property that the total probability must always equalone. This structure will provide a means to connect calculations of the likelihood ofscattering processes for the present theory with the techniques of quantum field theory,as we shall describe in section 11.2. In the meantime, in the following section, we assessthe nature of basic field interactions in the context of the present theory.287 hapter 11 A Novel Conception of HEPProcesses
In this chapter we consider how the probabilistic nature of quantum phenomena arisesin the context of the present theory, and in particular in the environment of laboratoryexperiments. The main goals will be to relate the calculation of cross-sections, forexample, for the present theory with the corresponding formalism of QFT and toaddress the related question concerning the nature of particle phenomena generally.Here the probability for a particular process will be a measure of the degeneracy offield states describing the mathematical form of a particular 4-dimensional geometry,that is through the symmetry of possible local reinterpretations of fields such as thegauge field Y µ ( x ) or fermion field ψ ( x ) (denoted without ‘hats’, since these are not quantum field operators here) under the same Einstein tensor G µν ( x ). The spacetimegeometry is locally completely insensitive to reinterpretations of the underlying fields,that is exchanges between components of the fields δY ( x ) ↔ δψ ( x ), which leave G µν ( x )locally unchanged, while the geometric contracted Bianchi identity G µν ; µ = 0 remainsglobally valid.The field interactions proceed by a kind of ‘Chinese whispers’ of field indis-tinguishability, as a degenerate mathematical possibility underlying the spacetime ge-ometry. This leads directly to the indeterminate nature characteristic of quantumphenomena. Probabilities, in the form of cross-sections and decay rates, will thenarise in proportion to the sum of the ‘number of ways’ in which such underlying fielddescriptions are possible.We begin however by considering a particular case for which G µν ( x ) is a func-tion of a single internal gauge field. Based on the breaking of the full symmetry of L (ˆ v ) = 1 into external and internal parts over an extended base manifold M a rela-tionship between the external geometry described by G µν ( x ) and internal gauge fields288 µ ( x ) over x ∈ M was developed in chapters 2–5. Structures are identified analogousto those of Kaluza-Klein theory in leading to equation 4.16-4.17, which is conjecturedto arise out of the geometric constraints of the present theory culminating in equa-tion 5.20, for which the practical normalisation convention χ = κ may be adopted. Forthe present theory with an external linear connection Γ ρµν ( x ) and an internal gaugefield A µ ( x ), deriving from an internal Abelian U(1) gauge symmetry, the relation be-tween the external Riemannian curvature and internal gauge curvature is expressedin equation 5.22, via the above connection with classical Kaluza-Klein theory, and asreproduced here: − κ G µν = + F µρ F ρν + 14 g µν F ρσ F ρσ (11.1)Under the Bianchi identity G µν ; µ = 0 this relation implies the source-free ho-mogeneous Maxwell equation 5.30, as explained in section 5.2. Here, within the contextof the present theory the generator of the internal U(1) Q gauge symmetry of electro-magnetism is identified with the element ˙ S –– l within the set of E Lie algebra actionsas described in section 8.2. The electromagnetic gauge field A µ ( x ), associated withthe U(1) Q generator, is in turn identified with the field ˜ A µ ( x ) as described in andfollowing equation 8.58. We next consider the form of the free field A µ ( x ) as a solutionof Maxwell’s equation, which can be written in terms of the gauge field itself as: (cid:3) A µ ( x ) = 0 (11.2)The energy-momentum tensor for the electromagnetic field can be obtaineddirectly through the definition T µν := − κ G µν with the geometry G µν ( x ) determinedin terms of the electromagnetic field tensor F µν according to equation 11.1 above.(A normalisation convention setting κ = 8 πG N = − T µν := G µν ). Here theelectromagnetic gauge field itself is analysed under the assumption of an approximatelyflat spacetime. A real field A µ ( x ) can be expressed in terms of Fourier components,which in terms of trigonometric functions and a single 4-vector k takes the form: A µ ( x ) = A µc ( k ) cos k · x + A µs ( k ) sin k · x (11.3) (cid:20) ≡ A µc cos (cid:16) tan − A µs A µc (cid:17) cos (cid:18) k · x − tan − A µs A µc (cid:19) (cid:21) (11.4)= A µ ( k ) e − ik · x + A µ ∗ ( k ) e + ik · x (11.5) ≡ C ε µr ( k ) A r ( k ) e − ik · x + C ε µr ( k ) A ∗ r ( k ) e + ik · x (11.6)In equation 11.4 the gauge field is expressed in terms of a single cosine function,that is in the form A µ ( x ) = A µ cos( k · x + λ ), with no sum implied over the index µ = 0 , , ,
3, using the trigonometric identity cos α cos β ± sin α sin β = cos( α ∓ β ).Equation 11.5 follows from equation 11.3 with A µ ( k ) = ( A µc ( k ) + iA µs ( k )), and with A µ ( k ) expressed as C ε µr ( k ) A r ( k ) in the final line.Hence for each 4-vector k each of the four vector components of A µ ( x ) canbe associated with the real coefficients A µc ( k ) and A µs ( k ) or the complex coefficients A µ ( k ) and A µ ∗ ( k ), either pair of which can be considered to be independent in termsof possible field interactions as described in the following section. With all physical289henomena being invariant under spacetime translations, and hence with no preferredset of coordinates { x } , and hence with A µs ( k ) = 0 in general, equation 11.4 forms arelatively cumbersome expression for the free field and will not be employed further.The constant coefficient C is introduced with the square root in equation 11.6since each factor of A µ ( x ) appears quadratically in the expression for G µν in equa-tion 11.1. In line with textbook solutions to Maxwell’s equations, and anticipating thequantum field analysis, four polarisation vectors ε µr ( k ) are introduced, representing a4-vector object for each of r = 0 , , , constant basis for the 4-vector A µ ( k ), analogous to the variable tetrad components e µa ( x ) describing four vector fields for a = 0 , , , M , as employed in section 5.3 for example. A standard choice ofbasis is such that the polarisation vectors are real and orthogonal, with respect to theMinkowski metric η µν = diag(1 , − , − , − ε r ( k ) · ε s ( k ) = ε r µ ( k ) ε µs ( k ) = δ rs r = 0 − δ rs r = 1 , , ε µ ( k ) = (1 , , ,
0) (11.8) ε µr ( k ) = (0 , ε r ( k )) r = 1 , , k · ε r ( k ) = 0 r = 1 , ε ( k ) = k / | k | (11.11)The r = 0 case is the scalar, or timelike, polarisation vector, while r = 1 , r = 3 is called the longitudinal polari-sation vector. (With respect to a 3D scalar product with metric δ ij = diag(+1 , +1 , +1)the 3-vector parts of the transverse polarisation vectors satisfy ε r ( k ) · ε s ( k ) = + δ rs for r = 1 , , r = 1 , k · ε r ( k ) = k µ ε µr ( k ) = 0).The full set of four polarisation states ε µr ( k ) for r = 0 , , , A µ ( x ) and suggests the possi-bility of four kinds of photon states corresponding to these four degrees of freedom.However, in the standard theory, the requirement of gauge invariance, which allowssome field excitations to be transformed to zero, together with the massless condition k = 0 for the free electromagnetic field, result in there being only two physical photonstates corresponding to the transverse polarisation states ε µ ( k ) and ε µ ( k ).Maxwell’s equation in the form of equation 11.2 is obtained from equation 5.30under the Lorenz gauge condition ∂ µ A µ = 0, as described for the inhomogeneous caseof equations 3.91 and 3.92 in section 3.5. In turn equation 11.6 forms a solutionof the free field case of equation 11.2 provided that the massless condition k = 0holds. Further the Lorenz gauge condition itself, applied to equation 11.6, requiresthat k µ ε µ = 0 and hence, from equations 11.8–11.11, the transverse polarisation statesare clearly permitted.When substituted into equation 11.1 the gauge field A µ ( x ) of equation 11.6,with A r ( k ) = 0 for either r = 1 or r = 2 only, yields a large number of terms, most290f which are zero due to the conditions k · ε , ( k ) = 0, for these transverse states, and k = 0 (for example no F ρσ F ρσ terms remain) leading to: T µν := − κ G µν = + C k µ k ν (cid:16) | A r | − A r e − ik · x − A ∗ r e +2 ik · x (cid:17) = +2 C k µ k ν (cid:16) | A r | − (cid:0) Re( A r ) cos 2 k · x + Im( A r ) sin 2 k · x (cid:1)(cid:17) = +2 C k µ k ν | A r | (cid:16) k · x + α ) (cid:17) (11.12)The form of the final line, with α ∈ R , follows by a similar argument that led to equa-tion 11.4, in terms of a single real cosine function. Taking the 4-vector k = ( k , , , k ),representing the propagation of the electromagnetic wave in the direction of the x co-ordinate (with transverse polarisation vectors ε µ ( k ) = (0 , , ,
0) and ε µ ( k ) = (0 , , , T µν ( x ) is sketched alongside that for the gauge field A µ ( x ) in figure 11.1 as projected onto the spatial coordinate x on M .Figure 11.1: The energy-momentum tensor T µν ( x ) := − κ G µν ( x ) is modulated by anon-negative cosine function as depicted above, corresponding to an electromagneticvector field A µ ( x ) in the form of a plane wave.Although the field function A µ ( x ) is presented in the Lorenz gauge, the formof the Einstein tensor G µν ( x ) is gauge invariant. Indeed this was one of the conditionsused to derive the relation between the external and internal geometry, as describedfor example in the discussion following equation 5.13 in section 5.1, in leading to equa-tion 11.1 itself. Hence in turn the energy-momentum tensor T µν ( x ), of equation 11.12and figure 11.1, is naturally gauge invariant.As for the electric E i and magnetic B i field components of the gauge invariantelectromagnetic field tensor F in equation 5.23, the geometry of the Einstein tensor G µν ( x ) in equation 11.12 represents an unambiguous physical feature associated withthe electromagnetic wave. Further, as discussed shortly after equation 5.28 the scalarcurvature R associated with any electromagnetic field vanishes, with the Ricci tensor R µν hence identified with the Einstein tensor G µν . Hence equation 11.12 describes a‘wave of Ricci curvature’, which is complementary to the usual notion of a gravitationalwave, with the latter composed of purely Weyl curvature in the Ricci vacuum asdescribed after equation 5.44, also in section 5.2.Indeed under the assumption of an approximately flat spacetime, as employedfor equation 11.14 below, and with R µν = G µν given by equation 11.12 it can be seenfrom equation 5.44 that K ρσµ = 0, that is the source of Weyl curvature vanishes forthis geometry. As implied in the discussion before equation 3.69 the Weyl curvature291anishes for any conformally flat geometry, and hence a metric of the form: g µν ( x ) = (1 + β cos 2 k · x ) η µν (11.13)for a small value of β ∈ R , provides a candidate solution underlying an Einsteintensor in the form of equation 11.12. Indeed, assuming the Levi-Civita connection ofequation 3.53, via equations 3.73 and 3.74 it can be seen that the scalar curvature R vanishes for such a metric if k = 0, and that the resulting G µν ( x ) = 4 βk µ k ν cos 2 k · x ,to first order in β , exhibits a corresponding oscillatory behaviour, although more workis needed to obtain the precise form for a metric underlying the Einstein tensor ofequation 11.12.The physical spacetime curvature described by G µν ( x ) in equation 11.12 is as-sumed to be very small. As described in section 5.2, alongside equation 5.22, while theEinstein tensor is theoretically directly related to the internal gauge field, in the formof equation 11.12 for example, in units appropriate for laboratory measurements theEinstein equation can be written G µν = − κT µν , where the normalisation constant κ isa very small number. Hence while the energy-momentum carried by the electromag-netic wave may be readily detected the distortion of the spacetime geometry away fromMinkowski flatness is extremely small and utterly unobservable via any direct means.In turn the plane wave description of equations 11.3–11.6, modelled on the flat space-time case, can be used to a very good approximation. The divergence of the Einsteintensor in equation 11.12 can be expressed in terms of the energy-momentum tensor inthis approximately flat spacetime limit with T µν ; µ → T µν,µ in Cartesian coordinates, asdescribed in the opening of section 5.2. Consistent with the Bianchi identity G µν ; µ = 0this object can be seen to vanish, due to the condition k = 0, as would be expected: T µν,µ = 2 C k µ k ν | A r | (cid:16) − k µ sin(2 k · x + α ) (cid:17) = 0 (11.14)In the context of the present theory while the polarisation requirement k µ ε µ = 0can again be seen to be a consequence of imposing the Lorenz gauge condition on suchplane wave solutions for the field A µ ( x ), the ‘momentum’ requirement k = 0 is aconsequence of the necessity for the free field solution to satisfy the geometric Bianchiidentity as for equation 11.14. That is as a plane wave the free field is necessarily ‘mass-less’ in order to identify a consistent solution for G µν ( x ) in the form of equation 11.1,that hence might occur in nature. Equivalently the requirement k = 0 could be consid-ered to be a consequence of the homogeneous Maxwell equation 11.2, which itself is adirect consequence of the Bianchi identity G µν ; µ = 0 given equation 11.1, as describedfor equation 5.30. Equation 11.14 also of course directly implies energy-momentumconservation, T µν,µ = 0, for the gauge field described in equation 11.6 as employed inequation 11.12.Recalling that in general relativity the components T µν ( x ) represent the energy-momentum density , the field A µ ( x ) in a spatial volume V carries 4-momentum P µ which may be expressed as: P µ = Z V d x T µ (11.15)In the present theory the energy-momentum is always fundamentally determined by theEinstein tensor G µν ( x ) through the Einstein equation T µν := G µν . As also described292n the opening paragraphs of section 5.2 this is in contrast to the Lagrangian approachfor which an energy-momentum tensor t µν can be defined giving rise to a conserved4-momentum, in the form of equation 11.15, as described in equation 3.102 and thesubsequent discussion of section 3.5.The components P µ are locally four conserved quantities which transformamongst each other covariantly as a 4-vector under Lorentz transformations. Set-ting | A r | = 1, with the real coefficient C taking care of the field normalisation inequation 11.6, and substituting the top line of equation 11.12 into the above expres-sion, taking into account the vanishing of the integral of the e ± ik · x terms for suitablydefined boundary conditions for the volume V , yields: P µ = Z V d x C k k µ = 2 V C k k µ (11.16)Hence by setting the coefficient C = V k the 4-vector k µ in the Fourier componentcan be identified with the 4-momentum P µ of the field in the volume V . Such anobject with field values localised to within the volume V might naively be consideredto represent a ‘particle’, although a less simplistic particle concept that emerges in thepresent theory will be described in section 11.3.Hence in turn C = √ V k is the normalisation required in equation 11.6, giventhe transverse polarisation vectors described in equations 11.7–11.11, taking | A r | = 1and considering the field in the volume V to possess 4-momentum P µ ≡ k µ . Theorigin of this field normalisation factor C here therefore is in the interpretation ofthe Einstein tensor as the energy-momentum, and in particular with G ( x ) as theenergy density in the local reference frame T := − κ G ∼ Ck k ∝ k V relatingthe k component of the Fourier expansion of the A µ ( x ) gauge field directly to thephysical energy P carried by the field (that is, the classical Hamiltonian H ). Thisconstruction is independent of the spatial volume V which, being arbitrary withinthe choice of the boundary conditions, should cancel in all calculations of physicalquantities when interactions are considered, as it does for cross-section calculations inQFT as described in the discussion following equation 10.3.In the present theory the energy of a real field, such as A µ ( x ), is obtaineddirectly by substitution of the field into the appropriate expression for the right-handside of T µν := − κ G µν . Taking a complex-valued expression for the field A µ ( x ) in theform of the first term on the right-hand side of equation 11.6, for example, leads tothe subsequent expression for the energy-momentum tensor: A µ ( x ) = C ε µr ( k ) A r ( k ) e − ik · x (11.17) ⇒ T µν = − C k µ k ν A r e − ik · x (11.18)which, while consistent with T µν,µ = 0, is a complex tensor and hence does not representa real energy-momentum tensor T µν , or a real geometric tensor G µν . In addition here A µ ( x ) is required in any case to be real in order to represent the real components of aU(1) Q Lie algebra-valued vector field, that is a classical macroscopic gauge field.Alternatively, the first term on the right-hand side of equation 11.3, for exam-ple, is real and does, alone, produce a real energy-momentum tensor: A µ ( x ) = C ε µr ( k ) A c r ( k ) cos k · x (11.19) ⇒ T µν = + C k µ k ν A c r sin k · x (11.20)293s a special case of equation 11.12 (with A r = A c r ∈ R ).All three expressions for A µ ( x ) in equations 11.3 (or 11.6), 11.17 and 11.19 alsonecessarily satisfy Maxwell’s equation (cid:3) A µ = 0 since this is implicit in the identity G µν ; µ = 0 when applied to equation 11.1 as was described in equation 5.29 and thesubsequent discussion as reviewed above. In all cases a solution with A µ dependentupon the Fourier mode 4-vector k expressed in the Lorenz gauge requires a polarisationvector with k µ ε µ = 0, and with G µν in the form of equation 11.1 the geometric Bianchiidentity G µν ; µ = 0 implies k = 0.For the standard treatment of a massive vector field with k = m = 0, as forthe case of a massive gauge vector boson, the plane wave expansion in the form of anyof equations 11.3–11.6 can again be employed, and the Lorenz gauge condition againimplies k µ ε µ = 0 for the polarisation vector. In this case however the remaining gaugefreedom, subject to ∂ µ A µ = 0, cannot be used to uncover a cancellation between thescalar and longitudinal components of polarisation. Hence for massive gauge bosonsthere are three possible states, with the longitudinal degree of freedom appended tothe two transverse polarisation states. In this case Maxwell’s equation is replaced byan expression incorporating a mass term:( (cid:3) + m ) A µ = 0 (11.21)In the context of the present theory on substituting the free field in the formof equation 11.6 into the expression G µν = f ( A ) of equation 11.1 the Bianchi identity G µν ; µ = 0 in the form of equation 11.14 is no longer satisfied for this new case with k = 0. This suggests that the direct relationship between the Einstein tensor and agauge field of the form of equation 5.20 and 5.31, as employed for equation 11.1, nolonger holds, but rather a more general expression is to be sought, as suggested bythe form of equation 5.32 in section 5.2. In the present context this latter expression G µν = f ( A, ˆ v ) implicitly incorporates the consequences of interactions between thegauge field A µ ( x ) and components of the temporal flow under the full form L (ˆ v ) = 1.Indeed in subsection 8.3.3 it has been suggested that in the present theory gaugeboson masses arise through an impingement of the corresponding internal symmetryon the external vector h ≡ v ∈ TM of equation 8.72, which forms the componentsof a ‘vector-Higgs’. This argument was constructed in part by analogy with techni-color models, with longitudinal components for massive gauge bosons obtained whenthe propagators are corrected for the field interactions, as described following equa-tion 8.73. In the context of the present theory while equation 11.1 together with theidentity G µν ; µ = 0 implies equation 11.2, the form of G µν = f ( A, ˆ v ) under the sameidentity is expected to be consistent with equation 11.21.The A µ ( x ) gauge field associated with electromagnetism is in fact massless. Inthe present theory in terms of the corresponding U(1) Q ⊂ E internal symmetry thisproperty is attributed to the fact that the U(1) Q generator ˙ S –– l does not impact uponthe external v ∈ TM components of v , as presented for example in equation 8.71.However there are interactions between A µ ( x ) and other temporal components whichsuggest that the free field expansion and equation 11.1 will not represent the fullpicture.Indeed, in the present theory the gauge field A µ ( x ) and the associated internalU(1) Q symmetry are not considered as basic entities in themselves, rather they are294ntroduced since they act on components within the form L (ˆ v ) = 1. These lattercomponents include the Dirac spinors ψ of equations 9.43 and 9.46, as identified in thecomponents of v in the extension to the E symmetry of the full form L ( v ) = 1 insection 9.2, which unlike the v components do transform non-trivially under the U(1) Q action. In principle these temporal components provide a greater freedom for buildingthe spacetime geometry, now with an underlying degeneracy of possible δA ( x ) ↔ δψ ( x )field ‘redescriptions’, always consistent with the Bianchi identity G µν ; µ = 0 for theexternal spacetime.Objects transforming as a 4-vector can be constructed out of Dirac spinors inthe form of ψγ µ ψ via the conjugate field ψ = ψ † γ as introduced for equation 3.96 insection 3.5. As for the standard theory the relationship between ψ and ψ is expectedto relate to the dynamics of fermions and antifermions for physical particle statespropagating in spacetime. For the present theory ‘interactions’ between the vectorfield A µ ( x ) and a fermion field ψ ( x ) take the form of vector field ‘redescriptions’ asprovisionally sketched in figure 11.2.Figure 11.2: Field redescriptions: (a) The same function of spacetime is associatedwith the field ψ ( x ) γ µ ψ ( x ) before time t ∈ T and with the field A µ ( x ) at later times,here in the spatial volume V . (b) The field A µ ( x ) in the spacetime volume V T isredescribed as the field ψ ( x ) γ µ ψ ( x ) from time t ∈ T .While the field function is relabelled A µ ( x ) → ψ ( x ) γ µ ψ ( x ) at time x = t infigure 11.2(b), the function form itself is independent of the choice of t , as is the localgeometric structure G µν = f ( A, ψ ) in spacetime. Figure 11.2(b) does not representthe field A µ ( x ) ‘turning into’ the field ψ ( x ) γ µ ψ ( x ) at time t , rather this possibleredescription is everywhere implicit in A µ ( x ) as a function on M . For example theplane wave described by A µ ( x ) in figure 11.1 might be redescribed in terms of the field ψ ( x ) γ µ ψ ( x ) at any time t with the external form of T µν := G µν remaining unchangedthroughout the spacetime volume V T .This notion of field indistinguishability is closely analogous in spirit to the‘arithmetic indistinguishability’ of the multi-dimensional form L ( v ) = 1 from the orig-inal one-dimensional temporal flow within which the form L ( v ) = 1 is ever implicit,as described in section 2.1. That is one-dimensional time s innately contains pos-sible ‘redescriptions’, such as s = ( x ) + ( x ) + ( x ) , which may potentially beinterpreted as geometric or spatial structures. Further, when projected onto the basemanifold M in the full theory the form L (ˆ v ) = 1 will provide constraints on possible295eld redescriptions, represented by A µ ( x ) ↔ ψ ( x ) γ µ ψ ( x ) here, as will be described forequation 11.33 below for example.Although locally indistinguishable, the possibility of local field redescriptionssuch as depicted in figures 11.2(a) and (b) will lead to globally distinguished and ob-servable phenomena on M . This includes the possible outcomes of a ‘Schr¨odinger’scat’ type experiment, as will be described in section 11.4. This is possible since differentfield descriptions point towards a different set of subsequent possible field redescrip-tions propagating in the broader spacetime environment, always under the constraint G µν ; µ = 0. The relative probability for a specific observable effect will depend directlyupon the degeneracy of the local underlying possible field descriptions, as we shallexplore in the following section.In the full theory the spacetime geometry with metric g µν ( x ) and Einsteintensor G µν ( x ) are continuous and smooth over M and have ‘surveillance’ over theother fields, as described in section 5.2 in the discussion shortly before equation 5.44for example. As well as shaping the equations of motion for macroscopic fields andentities this surveillance will also constrain the form of microscopic field interactionsand exchanges. While the original form of T µν := G µν = f ( A ) is expected to beassociated with photon states in some way, further redescriptions of a form suggestedby the sketches of figure 11.2 will ultimately introduce matter terms T µν primarilyassociated with the spinor ψ field components, to be associated with electron statesfor example.The precise mathematical form of the field redescriptions remains to be fullyunderstood. However this structure, as pictured in figure 11.2, brings to mind Huy-gen’s principle for the description of a field at a later time as propagated from earliertimes and the form of the retarded propagator, closely relating to ∆ R ( x − y ) of equa-tion 10.79 for the scalar case. Here however, rather than fields propagating througha pre-existing spacetime background, the 4-dimensional spacetime M with geometry G µν = f ( A, ψ ) is constructed in terms of the fields. In any case we provisionallyrepresent the structures of figure 11.2(a) and (b) respectively by the mathematicalrelations: A µ ( x ) = Z d y D µνr ( x − y ) ψ ( y ) γ ν ψ ( y ) (11.22) A µ ( x ) = Z d y D µνa ( x − y ) ψ ( y ) γ ν ψ ( y ) (11.23)The role of the functions D µνr ( x − y ) and D µνa ( x − y ) is hence to provide a morerigorous account of the field exchanges depicted graphically and somewhat naivelyin figure 11.2. Here, by analogy with the case of standard electrodynamics, the ‘re-description function’ D µνr ( x − y ) is analogous to the retarded propagator D µνR ( x − y )for a vector field. Such a redescription may also take place ‘into the past’, that is byanalogy with the advanced propagator D µνA ( x − y ), equivalent to the field exchange A µ ( x ) → ψ ( x ) γ µ ψ ( x ) forward in time, as described in terms of D µνa ( x − y ) in equa-tion 11.23 and depicted in figure 11.2(b).Hence we provisionally identify the functions D µνr,a ( x − y ) in equations 11.22and 11.23 with the propagators D µνR,A ( x − y ). The vector retarded propagator forthe massless gauge boson case can be defined in terms of the corresponding scalar296ropagator ∆ R ( x − y ) as: D µνR ( x − y ) = lim m → [ − g µν ∆ R ( x − y )] (11.24)with (cid:3) x D µνR ( x − y ) = g µν δ ( x − y ) (11.25)hence following from equation 10.81, and with a similar construction for the advancedpropagator. In turn equation 11.22 implies: (cid:3) A µ = ψγ µ ψ (11.26)that is Maxwell’s equation F µν ; µ = j ν of equation 3.91 for the inhomogeneous casewith source current j µ = ψγ µ ψ .This relation, deriving from equation 11.22, is incompatible with the classi-cal expression of equation 11.1, which led to (cid:3) A µ = 0. This generalisation fromequation 11.2 with the addition of the source term j µ on the right-hand side is anal-ogous to the extension with a mass term m on the left-hand side in equation 11.21,in both cases arising out of interactions between the gauge field A µ ( x ) and compo-nents of the temporal flow under L (ˆ v ) = 1. In both cases this involves opening upa more general relation between the spacetime geometry and the internal fields, andcorrespondingly more general properties of matter described by the energy-momentumtensor T µν := G µν = f ( Y, ˆ v ), as outlined in the discussion around equation 5.32 in sec-tion 5.2. For the generalisation to G µν = f ( A, ψ ) considered here the evolution of the both fields, A µ ( x ) and ψ ( x ), will in turn be shaped in conformity with the Bianchi iden-tity G µν ; µ = 0 (which led to the source-free Maxwell equation for the electromagneticfield A µ ( x ) alone in equation 11.2).From equations 10.76 and 10.79 the scalar retarded propagator as appearingin equation 11.24 can be written as:∆ R ( x − y ) = − iθ ( x − y ) Z d k (2 π ) ε ( k ) 2 π δ ( k − m ) e − ik · ( x − y ) (11.27)This function contains similar features to those required for D µνr ( x − y ) in equa-tion 11.22, including a θ -function for the temporal ordering of the field redescription,which takes place at time t in figure 11.2(a). Further, this propagator is employedto obtain field solutions in the form of equation 10.83, which also applies for classi-cal fields as described at the end of section 10.4, and which is closely analogous toequation 11.22 for the field redescription above.As used above in deriving equation 11.26 the retarded propagator ∆ R ( x − y ) ofequation 11.27 satisfies equation 10.81. Indeed in QFT the propagators ∆ F,R,A ( x − y )may be introduced as inverse functions for the operator ( (cid:3) x + m ) in equation 10.81,with appropriate boundary conditions, motivated by the search for solutions to differ-ential equations of motion for the fields, such as equation 10.84. These equations ofmotion are themselves derived via the Euler-Lagrange equation 3.89 given an originalpostulated Lagrangian as the starting point, which led for example to equation 10.24(incorporating equation 10.84) for the scalar model. A very similar situation appliesfor the QFT employed for the Standard Model in particle physics.In the standard theory the complete Lagrangian, including the interactionterms, is subject to the Euler-Lagrange equation collectively. For example the com-bined Maxwell and Dirac Lagrangian, given by equation 3.96 for the internal U(1) Q A µ ( x ) and its spacetime derivatives ∂ ν A µ ( x )leads, in the Lorenz gauge, directly to: (cid:3) A µ = ψγ µ ψ =: j µ (11.28)as implied in equations 3.97 and 3.98. In order to arrive at this expression the variationof both the F µν F µν and j µ A µ parts implied in equation 3.96 are mutually related byappearing in the same Lagrangian object under a single Euler-Lagrange equation.By contrast the form of the redescription propagator D µνr ( x − y ) of equa-tion 11.22 is not motivated on the grounds of finding solutions for equations of motionsuch as equation 11.28, but rather in the present theory it is conceptually motivated onthe grounds of a degeneracy of field solutions under the construction of the spacetimegeometry G µν = f ( A, ψ ) over M . In fact here there is no similar direct expressionwith an explicit source term for the microscopic case, as there is in the standard theorywith equation 11.28 above. In the present theory simple differential equations such asequation 11.26 arise as a consequence of the possibility of mutual field redescriptionsat the microscopic level. However, apparent source terms in these expressions mightbe identified which are reminiscent of those seen in the field equations of motion forthe Standard Model. A generalisation of the gauge-fermion field interactions describedin equations 11.22–11.26 for non-Abelian gauge symmetries for comparison with thegeneral case of equations 3.97 and 3.98 in section 3.5 could also be considered.Here, rather than an interaction Lagrangian or Hamiltonian relating the dif-ferent fields as for QFT, the form of temporal flow L (ˆ v ) = 1 places mutual constraintson field values and provides selection rules for possible ‘transitions’ linking possibleinitial, intermediate and final states. Here the field interaction terms appear not in a single Lagrangian function but rather through a range of constraint equations, whichmay be provisionally listed as: L (ˆ v ) = 1; D µ L (ˆ v ) = 0; G µν = f ( Y ); G µν ; µ = 0 (11.29)Of these L (ˆ v ) = 1, as a scalar invariant, is perhaps most closely related to a standardLagrangian, however in being constrained to the fixed scalar value 1 further fieldinteractions are implied in the terms of D µ L (ˆ v ) = 0. The third of these constraints isthe relation between the external geometry and internal degrees of freedom consistingpurely of gauge fields, that is equation 5.20, and relates closely to Kaluza-Klein theoriesas described in section 5.1. Together with G µν ; µ = 0 further geometric structures suchas the Bianchi identity D F = 0 for the internal gauge fields constrain the equations ofmotion.Underlying the more general spacetime geometry G µν = f ( Y, ˆ v ), it is the pos-sibility of gauge-fermion field redescriptions such as expressed in equations 11.22 and11.23 as considered here for the Abelian case, consistent with the selection rules ofequations 11.29, which leads to the identification of the current j µ := ψγ µ ψ in equa-tion 11.26, which is identical in form to equation 11.28. In addition to the vector fieldtransitions described above, spinor field redescriptions may also be considered with forexample: ψ ( x ) = Z d y S r ( x − y ) /A ( y ) ψ ( y ) (11.30)298here /A = γ µ A µ , by analogy with equation 11.22. As for the vector case in figure 11.2this field redescription is also possible for the reverse temporal ordering. The spinorredescription function S r ( x − y ) is here closely related to the spinor retarded propagatorwhich may be expressed as S R ( x − y ) = ( i/∂ x + m )∆ R ( x − y ) in terms of the scalarpropagator of equation 11.27, which satisfies the relation ( i/∂ x − m ) S R ( x − y ) = δ ( x − y )([70] p.63).In a similar way that equation 11.22 led to equation 11.26, that is Maxwell’sequation with a source term, here equation 11.30 leads to the Dirac equation, alsowith a source term, assuming that the properties of the spinor redescription function S r ( x − y ) are similar to the propagator S R ( x − y ). In this case the action of ( i/∂ x − m )on both sides of equation 11.30 results in:( i/∂ − m ) ψ = /Aψ (11.31)This is the Dirac equation that was obtained in section 3.5 via the Dirac Lagrangianin leading to equation 3.99, here with the convention for the gauge covariant derivative D µ = ∂ µ + iA µ .In the context of the present theory the mass m terms in these equations willalso be introduced through field interactions. In the case of equation 11.21 for a mas-sive gauge field the mass arises from the impact of the gauge symmetry upon thecomponents external vector-Higgs v ∈ TM , as recalled in the discussion after equa-tion 11.21, as introduced in the terms of D µ L (ˆ v ) = 0 of equation 11.29. Mass termsfor fermions on the other hand will be incorporated through the constraint of L (ˆ v ) = 1itself in equation 11.29, in the form of Yukawa-like couplings between the fermion com-ponents and the same vector-Higgs, as described for equation 8.76 in subsection 8.3.3in the case of the form L ( v ) = 1 and for equation 9.48 in section 9.2 in the case of theform L ( v ) = 1. Both for gauge bosons and fermions the interaction mass terms willcorrect the form of the corresponding Feynman propagators in the quantum theory.However, in focussing on the gauge-fermion interactions in the following we neglect themass terms and hence equation 11.31 reduces to simply (within a conventional factorof i ): /∂ψ = /Aψ (11.32)In the present theory field redescriptions occur if permitted by the constraintequations 11.29, which effectively provide interaction selection rules. For the caseof an electromagnetic gauge field A µ ( x ) associated with the internal U(1) Q symmetrygenerated by ˙ S –– l ∈ L (E ), as described for example in equations 8.22–8.24 of section 8.2,interactions between the gauge and fermion fields can be identified in the expression D µ L ( v ) = 0, here taking the conserved quantity L ( v ) = 1 as the full form oftemporal flow. This is analogous to the expression for D µ L ( v ) = 0 in equation 5.51,as described towards the end of section 5.4, for the SO + (1 ,
9) model, while here for theE symmetry of the form L ( v ) = 1 of equation 6.28 the expression D µ L ( v ) = 0includes terms of the form: D µ L ( v ) = . . . + pb ( ∂ µ ¯ b + ˙ s f A µ ¯ b ) + m ¯ c ( ∂ µ c + ˙ s f A µ c ) + . . . = 0 (11.33)= . . . + h θ † D µ θ + . . . = 0 (11.34)In the first line ˙ s f carries the ˙ S –– l charges of the corresponding fermion componentsand in the second line the values p = m = v = h via equation 8.72 and the spinor299 = (cid:0) c ¯ b (cid:1) of equation 6.26 have been substituted in. The spinor θ decomposes intothe four Weyl spinors θ l,i,j,k of equation 8.13 under the external SL(2 , C ) symmetry,each of which is augmented to a Dirac spinor ψ of equation 9.43 upon extension to theE symmetry of L ( v ) = 1. The Dirac spinor for the ‘electron’ field for example willconsist of the 4-component object: ψ = c + c lb − b lC + C lB − B l (11.35)in the notation of equation 9.25. Having identified ψ ( x ) in the components of F (h O )its conjugate ψ can also be constructed, and both fields expanded in terms of planewaves with complex coefficients, as was the case for the electromagnetic wave in equa-tion 11.6. The nature of particle and antiparticle states will ultimately need to beaddressed in relation to such field expansions, although here we deal directly with thefields and their mutual exchanges.Hence in generalising from equation 11.34 for the E symmetry case the ex-pression D µ L ( v ) = 0 will contain terms incorporating factors of the form ψ † D µ ψ involving a juxtaposition of gauge and fermion fields, with the latter identified in thecomponents of F (h O ). In the present theory field exchanges in the form of equa-tion 11.30, with the ensuing equations of motion such as equation 11.32, are requiredto be compatible with the constraints such as D µ L ( v ) = 0.The precise means of implementing these constraints remains to be well under-stood, although the terms identified are analogous to the form of those found in theStandard Model Lagrangian. Further, the mutual redescriptions of the field functionsare considered to be discrete, as suggested by the provisional picture of figure 11.2,which is reminiscent of the actions of the creation and annihilation operators in theexpansion of quantum fields which appear through an interaction Lagrangian in ex-pressions such as equation 10.45 in a quantum field theory.Here equations of motion such as equation 11.32, derived from the field re-description of equation 11.30, must be filtered through the selection rules such asequation 11.34, deriving from equations 11.29, with a corresponding range of charges.This is one factor leading to differences in the likelihood of a particular process tooccur. Specifically the relative factors of ˙ s f for different fermion components in equa-tion 11.33 will relate to the relative number of ways in which such a process may bechannelled via equation 11.30, which takes the same form for all such processes, aswill be described further after figure 11.5 in the following section. Hence the factorsof ˙ s f , obtained from the components of ˙ S –– l in equations 8.22–8.24, with | ˙ s f | = 1 and | ˙ s f | = provisionally associated with charged leptons and d -type quarks respectivelyin section 8.2 provide a factor of three in the relative interaction strength betweenthese fermion states and the electromagnetic field, that is with an apparent ‘fractionalcharge’ of for the d -quark relative to the unit electron charge.For both equation 11.33 in the present theory and equation 10.23 in the modelquantum field theory an interaction is mediated since changes is one field influence300nother field through their mutual composition in these expressions, with the constraintof D µ L (ˆ v ) = 0 in the former case and through the Euler-Lagrange equation of motionderived form the total Lagrangian in the latter case. In the present theory equationsof motion with field interactions are induced through consistency with equations 11.29rather than directly as Euler-Lagrange equations of motion from a Lagrangian withinteraction terms.Interactions between gauge and fermion fields arise for the Standard Modelthrough the Lagrangian approach by requiring the invariance of the total Lagrangian L under local internal symmetry transformations, such as with the gauge group U(1) Q in the case of electromagnetism. This implies an equivalence or indistinguishabilitybetween for example a photon and an e + e − pair, with A µ ↔ ψγ µ ψ , or between anelectron and an electron-photon pair, with ψ ↔ /Aψ ; which implies the possibilityof physical interactions between the fields. Similarly in the present theory it is theproperty of invariance of the form L (ˆ v ) = 1 with respect to the internal symmetry,dynamically expressed over M through terms such as those of equation 11.33, thatallows interchanges between gauge and fermion field components corresponding to amultitude of possible solutions for the geometric form G µν = f ( A, ψ ) in 4-dimensionalspacetime.In the present theory it is the possibility of such multiple solutions with couplingbetween the A µ ( x ) and ψ ( x ) fields implied in D µ L (ˆ v ) = 0 terms that leads to theidentification of the current j µ := ψγ µ ψ in equation 11.26. The fields A µ and ψγ µ ψ mutually appear in the field redescriptions of equations 11.22 and 11.23 which are alsosubject to the selection rules implied in D µ L (ˆ v ) = 0 and incorporated into a worldgeometry, with the form of G µν ( x ) generalised from equation 11.1 but always with G µν ; µ = 0 as a further constraining identity.The further constraint G µν = f ( Y ) listed in equations 11.29, referring to the di-rect relation between the external and internal geometry as expressed in equation 5.20,itself will generalise to incorporate gauge-gauge field exchanges for the case of a non-Abelian internal symmetry. That is, for a gauge field Y µ ( x ) associated with a non-Abelian internal gauge symmetry with: − κ G µν = F αµρ F ρνα + 14 g µν F αρσ F ρσα (11.36)and F αµν = ∂ µ Y αν − ∂ ν Y αµ + c αβγ Y βµ A γν (11.37)with the latter from equation 3.38, there will be possible gauge field redescriptionsconsistent with the cubic and quartic terms of G µν = f ( Y ), namely: ∂Y Y Y terms ⇒ Y ↔ Y Y exchanges
Y Y Y Y terms ⇒ Y Y ↔ Y Y exchanges (11.38)Mutual gauge field redescriptions channelled through these constraints will augmentthe form of equation 11.36, similarly as for equations 11.22 and 11.23 and again un-der the identity G µν ; µ = 0, allowing for further possible solutions for the extendedspacetime geometry.In the Standard Model such cubic and quartic gauge field interaction terms fora non-Abelian gauge field similarly appear through terms quadratic in the curvature301ensor F , in this case via a Lagrangian in the form of equation 3.94. In a quantumfield theory for describing particle phenomena, such as for the Standard Model, thereare certain constraints placed on the form of the Lagrangian. In general all possi-ble terms which are allowed by gauge invariance and other symmetries of the theoryshould be included, but there should be no terms involving coupling constants withnegative dimension D , in order to construct a renormalisable theory, as described afterequation 10.86 in section 10.5. For QCD (quantum chromodynamics) in addition toequation 3.94 the Lagrangian term: L = α s π θ F µνα ∗ F αµν with ∗ F αµν = 12 ε µνρσ F α ρσ (11.39)is also admitted. Here α s = g s π is the strong coupling while the index α correspondsto the Lie algebra values and ∗ F αµν is the dual field strength tensor, as originallyintroduced for the electromagnetic field in equation 5.24. The θ -parameter is sometimesconsidered as the 19 th parameter of the Standard Model along with the 18 others (assummarised later in table 15.2 of section 15.2). However this Lagrangian term implies CP violation for strong interactions, contradicting empirical observations, unless the θ -parameter is unnaturally very small. This is the ‘strong CP problem’ in the StandardModel, which indicates that the Lagrangian approach may contain too many terms,leading to effects not seen in nature.In the present theory gauge field interactions have a different origin. Theexpression for G µν in terms of the gauge field strength F αµν as implied in equation 11.29in the form of 11.36 can be rewritten in a form similar to equation 5.27, with a termquadratic in the dual field strength. However, as noted after equation 5.28, there isno term of the form in equation 11.39 and hence the strong CP problem is potentiallysidestepped in this Lagrangian-free theory.Regardless of the nature of the underlying gauge or fermion field content, theobject G µν ( x ), describing the spacetime geometry of M , is a real-valued tensor, whilethe identity G µν ; µ = 0 is a real-valued vector. Similarly the constraints L (ˆ v ) = 1and D µ L (ˆ v ) = 0 are a real-valued scalar and real-valued vector respectively. Thecollection of these objects, as listed in equations 11.29 (with G µν = f ( Y ) interpretedas a local constraint), is analogous to the collection of terms in a single real-valuedscalar Lagrangian, and in the present theory they will also be interrelated throughthe full dynamics. However, as for a real-valued Lagrangian, the components of fieldsunderlying these objects may be mathematically analysed into complex-valued parts,such as the Fourier modes for the electromagnetic field in equations 11.5 and 11.6.More generally, as described in subsection 2.2.3 and equation 2.30, a real-valuedgauge field Y ( x ) on M was originally obtained as the pull-back of the Maurer-Cartan1-form defined on the manifold of an unbroken symmetry group ˆ G . Subsequently inter-nal gauge fields deriving from the symmetry breaking over the base manifold were con-sidered, as appearing in the final term of equation 2.47 for example. Although only thecomplete real field Y ( x ) represents a macroscopic gauge field (as discussed after equa-tion 11.18), the functional form of the gauge field Y αµ ( x ) may be analysed into complexFourier components. Similarly, the 56 real components of a vector v ∈ F (h O ) un-der L ( v ) = 1, including the various fermion subcomponents ψ ( x ) ⊂ v ( x ), whenexpressed as functions over M may be analysed into complex Fourier modes. Further,302n principle such complex e ± ik · x Fourier mode components of the fields Y ( x ) and ψ ( x ),or a hybrid combination, might be composed at the microscopic level to form real expressions for objects such as G µν ( x ) and D µ L ( v ( x )) = 0 over M .While a crucial observation for the present theory is that the spacetime associ-ated with any field propagation is not flat, as pictured in figure 11.1 with G µν = − κT µν for example, here the geometry is assumed to be sufficiently close to flat in order toemploy such a plane wave expansion in essentially Cartesian coordinates, as describedbefore equation 11.14. The field redescription functions, featuring in equations 11.22,11.23 and 11.30 for example, are closely related to the retarded propagator ∆ R ( x − y )of equation 11.27. This latter function itself is expressed as an integral over e − ik · ( x − y ) Fourier modes, suggesting that in turn the exchanges and interactions between thecomponents of fields such as Y ( x ) and ψ ( x ) might also be most conveniently analysedin terms of e ± ik · x Fourier modes, as is the case for the field expansions in quantumfield theory. That is the association of the functions D µνr,a ( x − y ) with the propagators D µνR,A ( x − y ) as provisionally suggested after equations 11.22 and 11.23, as for the asso-ciation of the function S r ( x − y ) in equation 11.30 with the propagator S R ( x − y ), mayinvolve analysis of the corresponding field structures in terms of complex-valued com-ponents. These structures in the present theory will be linked with the cross-sectioncalculations of QFT in the following section.In all cases the mutual field exchanges are required to be consistent with thefull set of constraints of equations 11.29, with the geometric condition G µν ; µ = 0 in4-dimensional spacetime implying 4-momentum conservation through the definition ofenergy-momentum as T µν := G µν . The underlying one-dimensional form of temporalprogression is reflected in the structure of a causal sequence of field redescriptions, asexpressed by the θ -function in ∆ R ( x − y ) of equation 11.27, while the δ -function in thatexpression relates to the appropriate matching of Fourier modes for the general case,for which a finite mass m may result from further field interactions. Each possiblefield redescription itself, for individual Fourier modes such as e − ik · x , may provisionally be associated by analogy with QFT with an element of a Feynman diagram, namely avertex diagram of the kind listed in ‘rule 2’ of table 10.1, as depicted in the examplesof figure 11.3.Figure 11.3: Three Feynman vertex diagrams correlated with the possible field ex-changes (a) A ↔ ψψ , (b) ψ ↔ Aψ and (c) Y ↔ Y Y , as associated with equations 11.23,11.30 (strictly with S r replaced by S a here, since the implied time ordering is from leftto right in these diagrams) and the cubic terms of equation 11.38 respectively; with a4-way gauge field vertex also possible for the quartic terms of the latter equation.The field redescription of equation 11.30, associated with figure 11.3(b), is303irectly suggested by the form of the terms of D µ L ( v ) = 0 in equations 11.33 and11.34 via equation 11.32, although a higher-dimensional full form such as L ( v ) = 1will be needed for more explicit details. More generally the juxtaposition of a gaugefield and quadratic fermion field factor in the terms of D µ L ( v ) = 0 may lead tointeractions between this combination of fields with various spacetime orientations,while sharing the same vertex topology, resulting in the exchange of figure 11.3(a) forexample.Similarly, as well as identifying particular particle states in the components of F (h O ) the distinction between particles and antiparticles, together with their differentdynamic behaviour, will require a full consideration of the fields under the symmetriesof extended 4-dimensional spacetime. The provisional correlation of the combinationof the fermion field ψ and its conjugate ψ with the combination of a fermion and an-tifermion pair, as discussed after equation 11.35, will be dependent upon the temporalorientation of the field components on the extended manifold M .The association between terms of the constraints in equation 11.29 and theform on an interaction Lagrangian, as emphasised by the Feynman vertices of fig-ure 11.3, raises the question of how calculations for quantities such as cross-sections,as measured in HEP experiments, might be determined in the present theory and howsuch calculations might be related to the Feynman rules more generally. This will formthe topic of the following section. Here we make a provisional connection between the calculation of physical quantitiessuch as cross-sections, as described in the previous chapter, and the notion of a de-generacy of field redescriptions underlying the corresponding processes, as introducedin the previous section. Since such calculations in quantum field theory have achievedgreat success in comparison with empirical HEP observations a relation between thepresent theory and the mathematical structures and tools of QFT will be desirable.First we consider as an example a field sequence ψγ µ ψ → ϕγ µ ϕ , where ψ, ϕ ⊂ v ∈ F (h O ) denote fermion components, with the interaction taking place in thespatial volume V over a time period T via an intermediate A µ ( x ) field state. Thissituation is depicted in figure 11.4 which essentially consists of a juxtaposition offigures 11.2(a) and (b) where the initial and final fermion types may differ in general.In this section we consider interactions at the level of such field exchanges. Asalluded to at the end of the previous section the structure of physical particle states inspacetime, including both particle and antiparticle states, is yet to be identified in thistheory. Further, the inclusion of the second and third generation fermions may requirea further extension of the full form L (ˆ v ) = 1, as suggested for example in section 9.3.However a field state such as ψγ µ ψ is provisionally considered to represent fermionpairs such as e + e − or µ + µ − leptons or d ¯ d or t ¯ t quarks for example. Hence the fieldsequence in figure 11.4 mimics a HEP collision process such as e + e − → µ + µ − . Inthe following section the physical nature of the actual incoming and outgoing particlestates observed in HEP phenomena will be considered.In the analogous QFT calculation the initial and final ‘particle’ states are rep-304igure 11.4: The transition from an initial ψγ µ ψ field state to a final ϕγ µ ϕ state via anintermediate description of the field function in terms of a mathematically equivalent A µ ( x ) field state.resented by complex plane waves, that is Fourier modes of the form e ± ik · x , as discussedfor equations 10.43 and 10.44 for example. This is similar to the picture initially con-sidered here in figure 11.4 with the field functions in spacetime expanded in termsof Fourier modes such as those of equation 11.6. However, in the present theory theincoming, interacting and outgoing field states conform everywhere to an expressionof the spacetime geometry described by the real tensor G µν ( x ) = f ( A, ψ, ϕ ).For the case of d discrete intervals of time ∆ t i during which the field exchangesbetween t = 0 and t = T in figure 11.4 may occur the total number of ways N in whichthe overall transition may proceed is simply: N = d X i =1 R ( t ∈ ∆ t i : A → ϕϕ ) i − X j =1 R ( t ∈ ∆ t j : ψψ → A ) (11.40)with R denoting ‘redescription’ such that R (∆ t i : A → ϕϕ ) ≡ A µ ↔ ϕγ µ ϕ is allowed during thetime interval ∆ t i . More generally R ( t ) will take the value 1 if the corresponding fieldexchange is allowed, according to the constraint equations 11.29 as described in theprevious section, and 0 if it is not.For the process with incoming field state ψγ µ ψ the total field function is alreadydistributed everywhere in V from time t = 0 in figure 11.4, and as a function inspacetime it is indistinguishable from that of the outgoing ϕγ µ ϕ field state at t = T .The field redescription applies everywhere in V simultaneously at any time such as t or t since this simply involves a reinterpretation of the same field function, withnothing physically changing in V T . Hence from the point of view of the spacetimegeometry and G µν ( x ) alone it would be possible to link the states ψγ µ ψ and ϕγ µ ϕ directly, without an intermediate A µ ( x ) field description. This is prevented in thepresent theory by the absence of selection rule being provided by constraints such as D µ L (ˆ v ) = 0 which determine whether R ( t ) = 1 or R ( t ) = 0 for a particular fieldredescription.This is closely analogous to the Lagrangian approach in QFT as described forexample for the scalar model where the absence of a coupling term of the form ˆ X † ˆ X ˆ Y † ˆ Y H int ( t ),implies that the collision process X + X − → Y + Y − requires an intermediate φ state asdepicted in the Feynman diagram of figure 10.4. Similarly the lack of a direct electron-muon coupling in the Standard Model Lagrangian leads to consideration of scatteringprocesses via an intermediate photon, such as depicted in figure 10.3, which will beseen to be analogous to figure 11.4 for the present theory.Taking equation 11.40 to the continuum limit, as implied in figure 11.4, ameasure of the total degeneracy D can then be expressed as: D ( T,
0) = Z T dt Z t dt R ( t ) R ( t ) (11.41)The structure of this equation has some similarity to the second-order term in theexpansion of the time evolution operator U ( t, t ) in quantum field theory. In theinteraction picture, with interaction Hamiltonian H int , the operator U satisfies thedifferential equation 10.30, as described in section 10.3, with the iterative solution for U ( t, t ) displayed in equation 10.31.In equation 10.31 the factors of the Hamiltonian operator H int in each termnaturally stand in time order, with the earliest to the right and latest to the left, invirtue of the time integration limits. As explained in section 10.3 this expansion ofthe time evolution operator U ( t, t ) can be written in the familiar more compact formof equations 10.34, via equations 10.32 and 10.33, using the T -product of operatorswhich imposes time ordering over a broadened, and more symmetric, range of timeintegrals. In particular the second-order term in equation 10.31 can be replaced bythat in equation 10.32 since: Z tt dt Z t t dt H int ( t ) H int ( t ) ≡ Z tt dt Z tt dt T (cid:0) H int ( t ) H int ( t ) (cid:1) (11.42)It is the similarity between equation 11.41, as a measure of the degeneracy ornumber of ways in which to describe the field transition sequence ψγ µ ψ → A µ → ϕγ µ ϕ ,and the left-hand side of equation 11.42 that provides a further preliminary entry pointfor the present theory into the workings of QFT. In a similar way that the field exchangeof figure 11.2(b) has been provisionally associated with the Feynman vertex diagramof figure 11.3(a), the A µ ( x ) internal field stage of figure 11.4 might be associated withthe Feynman propagator corresponding to the internal line of the Feynman diagram infigure 10.3 for example, via the relation between equations 11.41 and 11.42 describedabove. Equations 10.31 and 10.32 are matched on a term by term basis and hence theterms of the perturbative expansion of equation 10.34 match those of equation 10.31.In turn the higher-order terms of equation 10.31 can be associated with higher-ordersequences of field redescriptions, such as depicted in figure 11.6 below. In the Feynmanrules for the mathematical elements associated with a Feynman diagram at order n inperturbation theory the factor of 1 /n ! in equation 10.33 cancels against a factor of n !from the possible vertex permutations, as described shortly after equation 10.46 andsummarised for ‘rule 6’ in the opening of section 10.5. Hence in the Feynman rules forthe second-order term correlated with the right-hand side of equation 11.42 the factorof does not appear. 306ia the above associations the field exchange sequence described in figure 11.4is analogous to the Feynman diagram in figure 10.4 for the corresponding scalar modelQFT calculation. While a possible physical interpretation of the Feynman propagator∆ F ( x − y ) in terms of ‘virtual particles’ is conceptually dubious, as discussed in sec-tion 10.4 (for example after equation 10.72), this object is a key part of calculations inQFT and we return to this propagator – which in the scalar field case may be expressedfor the internal field operator ˆ φ ( x ) in canonical QFT by the equation: i ∆ F ( x − y ) = h | T ( ˆ φ ( x ) ˆ φ ( y )) | i (11.43)as we began with equation 10.49 in section 10.4. This object arose when the transitionamplitude calculation was restructured with the time evolution operator U ( ∞ , −∞ ) inthe form of equation 10.32 placed between vacuum states, in particular for the second-order term. This object hence consists of terms implicitly containing time-ordered fieldproducts, such as T ( ˆ φ ( x ) ˆ φ ( y )) in the right-hand side of equation 11.42.The time ordering implies that ∆ F ( x − y ) consists of two parts, associated with θ ( x − y ) and θ ( y − x ), as described in equations 10.50–10.55 and as represented bythe two diagrams in figure 10.6. From the point of view of the concept of field redescrip-tions in the present theory the first diagram, figure 10.6(a), can be physically motivatedas representing the field redescription causal sequence such as ψγ µ ψ → A µ → ϕγ µ ϕ asdepicted in figure 11.4 while the second diagram, figure 10.6(b), represents a figmentof the mathematical restructuring of the calculation, leading in turn to the notion ofintermediate ‘virtual particle’ states.Nevertheless, via the above chain of argument each case of an intermediate A µ ( x ) field state, as depicted in figure 11.4, may be provisionally associated with acorresponding Feynman propagator D µνF ( x − y ). That is, intermediate field redescrip-tions such as that in figure 11.4 may be associated with the ‘virtual particle’ states asrepresented by the internal line in figure 10.4, and in Feynman diagrams in general,considered as a restructuring of a calculation which is here fundamentally based on anunderlying conceptual notion of a degeneracy of field descriptions.Associating a Feynman propagator with each intermediate causal redescription,such as that with the field A µ ( x ) in figure 11.4 as described above, supplements the setof interaction vertices associated with the constraints of equations 11.29, as exemplifiedin figure 11.3. Hence with propagators identified in addition to the vertices theseobjects may be combined to form Feynman diagrams more generally. Beginning fromthe idea that the probability of an observable process is a measure of the number ofways in which it can occur, summing over all possible intermediate field redescriptionsas for example in equation 11.41, the aim is to effectively reproduce a full set ofFeynman rules, for comparison with the Standard Model version of table 10.1, andfurther to use this relation in order to make calculations of empirical quantities suchas cross-sections.Regarding the Feynman vertices the key to understanding how D µ L ( v ) = 0terms, for example, are to be used in place of a Lagrangian here may be found in thecoupling strength, which is put in by hand in the Lagrangian case. In equation 11.33the value of ˙ s f for the leptonic states is 3 times larger than for the quark states, asdetermined in section 8.2 and noted in the previous section. The question then ishow this mathematical factor of 3 corresponds to an empirical factor of 3 in ‘electric307harge’ with an underlying explanation in terms of the degeneracy for the number ofways a process can occur. Consider the processes described by the Feynman diagramsin figures 11.5(a) and (b), either of which may be correlated with, while not literallyrepresenting, the field sequence depicted in figure 11.4 as described above.Figure 11.5: Feynman diagrams for the electromagnetic processes (a) e + e − → µ + µ − and (b) e + e − → d ¯ d , together with a ‘higher-order correction’ via (c) a radiated photonand (d) a gluon exchange between the final state quarks respectively.In the calculation of the degeneracy for a process, as initially described forfigure 11.4, the number of possibilities depends upon the total time T available for theprocess, as can be seen in equation 11.41. For a quantum field theory, the spacetimevolume factor V T for an interaction cancels in cross-section and decay rate calcula-tions, as described in section 10.2 following equation 10.3, essentially since the effectivevalues of V and T in external spacetime are the same for all possible processes. A sim-ilar cancellation might be expected for calculations based on field degeneracies in thepresent theory. On the other hand, unlike the case for the common external dimensionsof the interaction, here for the present theory, the effective ‘charge volume’ C in the internal space dimension varies from process to process, as indicated by the differingvalues of ˙ s f in equation 11.33, and does not cancel in such calculations. In QFT thesethree spaces are closely related, as seen for example in the CPT theorem, while inthe present theory they are mutually related through the structure and symmetriesof the underlying temporal flow in the form L (ˆ v ) = 1. A more precise expression forthe way in which the relative charges channel the relative likelihood for different fieldexchanges, and indeed a fuller understanding of the relation of the present theory tothe Lagrangian approach in general, requires further study, as was also discussed afterequation 11.35.Given the ‘virtual photon’ mediating both processes in figures 11.5(a) and (b)further internal degeneracy, as for example in figure 11.6 below, will be essentially thesame for both cases and not effect the relative rates. That is the branching fractionsor relative cross-sections for competing processes will depend on the differences in thenumber of ways, and this may be dominated by the factors of | ˙ s f | = 1 or | ˙ s f | = associated with the final state vertex in figures 11.5(a) and (b) respectively. Differencesmay also arise due to the mass of the final state particles (upon which the final statephase space depends), relating to further possible field interactions with the compo-308ents of the vector-Higgs h ≡ v ∈ TM , and more generally due to higher-order fieldexchange possibilities, such as those represented in figures 11.5(c) and (d); as will befurther discussed in the following section.In the full theory the possible Feynman diagrams will generalise correspondingto the range of gauge fields and further interactions identified for a full set of internalsymmetries as studied in chapters 8 and 9, and which show a significant resemblance tothe structures of the Standard Model of particle physics. For example, in figure 11.5(d)an SU(3) c gauge field exchange is included. There are eight internal SU(3) c generators,as described in section 8.2 and listed down the left-hand side of table 8.7. Unlike theU(1) Q action in equation 11.33 these mix the components of θ = (cid:0) c ¯ b (cid:1) ∈ O betweendifferent Weyl spinors hence introducing interactions between the corresponding quarkstates. The identification of an SU(2) L ⊂ E (or within a larger symmetry of time suchas E ), also mediating between the external SL(2 , C ) Weyl spinors in F (h O ) (orwithin a higher-dimensional form of time such as L ( v ) = 1) will provide a furtherinternal gauge symmetry action central to an understanding of electroweak theorywithin the present theory.The measure of degeneracy in equation 11.41 can be generalised to higher-order sequences of A µ , ψ, ϕ field exchanges which mirror the general expansion tohigher-order perturbations for QFT in equation 10.31; with the Hamiltonian operator H int ( t ) in the latter case replaced by the ‘redescription parameter’ R ( t ), as determinedby the constraints of equations 11.29, in the former case. The temporal sequence offigure 11.6 provides an example of the ways in which the causal sequence of figure 11.4may be generalised for nested sequences of field indistinguishability to arbitrary highorder.Figure 11.6: The transition from an initial ψγ µ ψ state to a final ϕγ µ ϕ state via anintermediate description of the field function in terms of a sequence mathematicallyequivalent A µ ( x ) → ψ ( x ) γ µ ψ ( x ) → A µ ( x ) field states.The corresponding degeneracy for the chain of field interpretations in fig-ure 11.6, as an augmentation of equation 11.41, is expressed as : D ( T,
0) = Z T dt Z t dt Z t dt Z t dt R ( t : A → ϕϕ ) R ( t : ψψ → A ) R ( t : A → ψψ ) R ( t : ψψ → A ) (11.44)While the sequence of field descriptions pictured in figure 11.4 can be correlated withthe Feynman diagram of figure 10.4, via equations 11.41 and 11.42, the higher-order309rocess of figure 11.6 is similarly analogous to the form of figure 10.9, representing the T -ordered expression for this fourth-order term for the scalar QFT model. A similarcorrespondence may be identified between field sequences for the present theory andFeynman diagrams in QED, as depicted in figure 11.11 in the following section forexample. These figures represent steps in the direction of connecting the structures ofthe present theory with Feynman diagrams and rules more generally.Intuitively the extra sums over the two additional intermediate times, labelled t and t in figure 11.6 and equation 11.44, will lead to a relative ‘infinity’ of new waysin which the overall event may proceed from the initial to the final state. However thedegeneracy measure D for both equations 11.41 and 11.44 is actually finite. On theother hand the intermediate state composed of ψ and ψ between t and t involves twofield contributions simultaneously , each of which may be expanded into Fourier modesindependently with a combined product of the form ∼ e − i ( p + p ) · x which, although thetotal p + p is constrained, leads to a further infinity in the degeneracy of the internal4-momentum. In this case the integral sum over p is unlimited, unlike the situationfor the time integrals, and is expected to be reflected in the divergent momentum loopintegrals, as for example in equation 10.86 for the scalar model, in the correspondencewith QFT calculations. For the present theory, as for QFT, such divergences might beexpected to cancel when observable quantities such as branching ratios are appropri-ately normalised, as will be described in the following section, with such observablesultimately dominated by the charges involved in the final interaction of the sequenceas discussed above.In the present theory the world geometry is necessarily described by the real tensor G µν ( x ) which itself in principle may be composed out of the real or complex components of fields, such as A µ ( x ) and ψ ( x ). Regarding the degeneracy count it-self it is an open question concerning whether there is a unique or optimal way inwhich possible field redescriptions should be counted, consistent with the constraintequations 11.29. This question concerns both the domain of the field functions, as apatchwork of regions in spacetime or in momentum space for example, and also theform of the field functions. Here we are analysing the degeneracy count in terms ofcomplex Fourier modes on the base manifold M . In this sense each e ± ik · x componentis not considered as an independent physical field, rather this decomposition provides a mathematical means of identifying a set of mutually independent field solutions whichmay be summed over.In describing the transitions between fields such as ψ ( x ), A µ ( x ) and ϕ ( x ) itis possible that linear combinations of real sine and cosine expansion terms, ratherthan complex e ± ik · x parts, might be employed to preserve the identity of real , andhence physical, fields under the spacetime geometry G µν ( x ) subject to the constraintequations 11.29 everywhere. For example considering the real Fourier components A µc ( k ) and A µs ( k ) of the field in equation 11.3 to be exchanged independently maintainsa real condition for the field A µ ( x ) which at every stage may compose an intermediate,but physical, gauge field coupled to the fermion fields consistent with D µ L (ˆ v ) = 0.However here we have described field interactions such as A µ ↔ ψγ µ ψ in termsof the indistinguishability of complex Fourier modes of the fields, as expanded for exam-ple in equations 11.5 and 11.6 for the electromagnetic field, in part since this providesa closer link with the framework of QFT. Indeed, as alluded to towards the end of the310revious section, many of the tools involved in QFT, such as the various propagatorsand the δ -function of equation 10.62 and the θ -function of equation 10.64, are conve-niently expressed in terms of complex Fourier modes. Further, complex componentsof gauge fields have already been considered with regard to the charged gauge bosons˜ W (2) ± µ ( x ) of equation 8.67 in section 8.3, by analogy with the standard electroweakgauge fields W ± µ ( x ) of equation 7.57 in section 7.2, which relate to the correspondingphysical interactions with Lorentz spinors. Hence here the field redescriptions will beanalysed in terms complex Fourier modes in the determination of a real measure orcount of the degeneracy of field solutions.As described in the previous section both parts of equation 11.6 are requiredto identify a field state carrying real energy-momentum, which in the present theory isdetermined by the form of the field under T µν := − κ G µν . Hence transitions betweenfields must necessarily link both of the e ± ik · x parts with the external k , as identified through equation 11.16, matched under an everywhere real T µν := − κ G µν energy-momentum tensor, subject to the identity G µν ; µ = 0, and also withthe internal representations of the field components matching under the constraints ofequation 11.29 in general, with the form L (ˆ v ) = 1 broken over the base manifold.With A µ ( k ) = ( A µc ( k ) + iA µs ( k )) ∈ C a general complex number in equa-tion 11.5 transitions in the field A µ ( x ) can be considered to take place treating A µ ( k )and A µ ∗ ( k ) as independent degrees of freedom in terms of possible exchanges withcomplex Fourier modes of the fermion fields. This implies the possibility of intermedi-ate complex fields such as A µ ( x ) and ψ ( x ) while hybrid combinations of these gaugeand fermion fields mutually form under real objects such as G µν ( x ) and D µ L (ˆ v ) = 0.Hence the temporal sequence of redescriptions should be considered indepen-dently for the complex e − ik · x and e + ik · x parts such that, for example, the processesrepresented in figures 11.4 and 11.6 may be generalised for this independence, as de-picted for example in figure 11.7.Figure 11.7: The transition from an initial ψγ µ ψ state to a final ϕγ µ ϕ state generalisedfor an intermediate description of the field function in terms of the complex Fouriermodes e − ik · x and e + ik · x independently in time.With the need to account for both sets of possible sequences as exemplifiedin figure 11.7 the probability P for the overall process ψψ → ϕϕ is proportional to D + × D − , where D + represents the degeneracy of ways via e + ik · x exchanges and D − e − ik · x mode exchanges, each of which has a structure similar to thatin equation 11.41 or 11.44. Alternatively the process probability could be expressedin terms of the degeneracies D c and D s representing the number of field exchangesrelating to the cosine and sine Fourier modes as alluded to above, with for example A µc ( k ) and A µs ( k ) of equation 11.3 independent, and with P ∝ D + D − ≡ D c D s . Inthis case all fields are real-valued and hence can be interpreted as physical entitiesat all times, however here we pursue the equivalent calculation based on the complexdecomposition.Earlier in this section we have described a correlation between the form of adegeneracy count D ( T,
0) and the anatomy of a Feynman diagram, with for examplefigure 11.4 compared with figure 11.5(a) or (b), via the structure of the expansionof the QFT operator U ( t, t ) of equations 10.31–10.34. Here the underlying physical basis of probability calculations is found in the field degeneracies, with the use of T -ordered products in QFT, via the θ -functions, simply implementing a restructuringof the calculations. Hence in turn the representation of the Feynman propagatorin figure 10.6 should not be interpreted as two possible physical processes. On theother hand the fact that the underlying field redescriptions are free to take placeindependently for both the e − ik · x and e + ik · x field components, as depicted for examplefor the process ψγ µ ψ → ϕγ µ ϕ in figure 11.7, does extend the range of possible fieldredescriptions and hence will have physically observable consequences. With both setsof field redescriptions for the e ± ik · x Fourier modes required to link the initial and finalstates the process probability takes the form P ∝ D + D − , and we hence now wish todetermine a correlate for this product in QFT.In figure 11.7 the field states at t = 0 and t = T (and hence also for t → ±∞ )represent real external particle states, that is on-mass-shell particles. This suggeststhat the diagram in figure 11.7 can be ‘unfolded’ to represent an extension of a linear degeneracy count, having the same basic structure as figure 11.4 or 11.6, but with a‘fold line’ denoting on-shell states. The corresponding unfolded diagram is depictedin figure 11.8(a). This field sequence correlates with the structure of the Feynmandiagram of figure 11.8(b), with the fold line mapped to the cut line – for which thepropagators are simultaneously placed on-mass-shell, as originally described for fig-ure 10.10.According to the ‘cutting rules’, as also described in section 10.5, the imaginarypart of the transition amplitude associated with a Feynman diagram is obtained bysumming over the cutting possibilities. These involve adapting the Feynman rules foreach possibility by placing the ‘cut’ virtual states on-mass-shell – and hence open tointerpretation as external particle states – via equation 10.98, which essentially replaceseach corresponding Feynman propagator ∆ F by one of the ∆ ± function componentsdescribed in equations 10.55–10.60. From the unfolding of figure 11.7 the initial andfinal field states in figure 11.8(a) are equivalent, and hence the cutting rules appliedto the corresponding figure 11.8(b) yields the imaginary part of the forward scatteringamplitude for the e + e − initial state | i i , namely in fact 2Im( M ii ), as contributed byplacing the cut line on the intermediate µ + µ − state for this Feynman diagram.The important observation of the present theory is that Im( M ii ) is a real number, and hence might be directly compared with event probabilities with contri-butions of the form P ∝ D + D − based on a count of the ‘number of ways’ in which an312igure 11.8: (a) The unfolding of figure 11.7, with a corresponding ‘fold line’ andreparametrised time intervals. (b) A correlated Feynman diagram for the forwardscattering process e + e − → e + e − , with a ‘cut line’ drawn through the intermediateloop propagators of the muon field.observed process might arise. Adding all possible contributions for all possible finalstates, exemplified by the process in figure 11.7, then correlates, via the generalisa-tion of figure 11.8, with the imaginary part of the forward scattering amplitude forall possible Feynman diagrams for the full perturbative expansion. The resulting realnumber Im( M ii ) is in turn directly related to the total cross-section σ , equation 10.96,via the optical theorem as described in section 10.5. Hence we arrive at a provisionalrelationship between a degeneracy count and a physical observable.As described towards the end of section 10.5 the optical theorem can be provento all orders of perturbation through the analysis of Feynman diagrams. The cutpictured in figure 11.8(b) represents one contribution to Im( M ii ) for this diagram,with a second contribution provided by placing the cut instead through the d ¯ d fermionloop. Hence by the above discussion the determination of Im( M ii ) via the cuttingrules for this diagram correlates with a sum of a D + D − field sequence for both a µ + µ − final state and a d ¯ d final state. Similarly for equation 10.101 the imaginary part ofthe Feynman diagram of figure 10.10 was determined corresponding to opening up a Y + Y − final state, with a further contribution to Im( M ii ) at this order of perturbationobtained by replacing the loop in figure 10.10 with a X + X − state, as described afterequation 10.101.The Feynman diagram with the cut of figure 11.8(b), in placing the µ + µ − pairon-mass-shell and via the optical theorem, contributes to the cross-section σ ( e + e − → µ + µ − ). However the structure of Im( M ii ), in summing over the cuts, generally incor-porates a collection of final states from which individual cross-sections for particularprocesses need to be untangled, as they are for the sum on the left-hand side of equa-tion 10.94 for example. Also, as alluded to in the caption comments, the fold-line infigure 11.8(a) should in principle be constrained to the ‘half-time’ point to accuratelyrepresent the degeneracy count of figure 11.7. Further, we have considered the de-313eneracy count, based on particular sequences of fields leading to a particular finalstate, to represent a measure of the probability P ∝ D + D − for a particular process.However, in order to actually represent a probability this count needs to be determinedrelative to the total degeneracy for all possible final states, which will provide theoverall normalisation and which so far we have not taken into account. In looking toaddress these points we recap how a particular final state is extracted and an eventprobability determined in the context of all possible outcomes in the framework of aQFT, with the aim of establishing a more precise link with similar calculations for thepresent theory.As described in section 10.3 in QFT the initial state | i i evolves through aperiod of field interactions into the state | Ψ( ∞ ) i = S | i i according to the S -matrixof equation 10.35. This evolution is governed at each moment by the equation ofmotion expressed in equation 10.28 in which the interaction Hamiltonian H int ( t ) con-tains all possible field interactions. Hence | Ψ( ∞ ) i in turn contains all possible finalstates. Since H int ( t ) is Hermitian the evolution of the state in equation 10.28 is aunitary transformation, and hence if the initial state normalisation is chosen with h i | i i = 1 this is preserved such that h Ψ( t ) | Ψ( t ) i = 1 at any time t . On inserting asum over a complete orthonormal set of similarly normalised final states | f i we have P f h Ψ( t ) | f ih f | Ψ( t ) i = 1, and in particular in the aftermath of the interaction, we have: X f |h f | Ψ( ∞ ) i| = 1 (11.45)as a mathematical identity. Hence the objects |h f | Ψ( ∞ ) i| , in the sense of consistingof a set of positive real numbers that sum to unity, do have the property of represent-ing probabilities, and in a structure which implicitly contains information about allpossible final states.A relationship between the degeneracy D ( T,
0) of equation 11.41 and the secondorder term of U ( t, t ) of equation 11.42 was described for a particular field sequenceleading to a particular final state, as pictured in figure 11.4. However, in general adegeneracy count associated with all terms of the entire S -matrix is desired in order toexpress everything that can happen, according to the field redescriptions permitted bythe constraints of equations 11.29 in place of an interaction Hamiltonian, and henceincorporate all possible outcomes. This suggests a ‘complexification’ of the probabilitycalculation based on the degeneracy count such that the unitarity constraint, that is SS † = in QFT, might effectively be employed to normalise the total probability forany outcome to unity.The subcomponent degeneracy counts D + and D − , originally considered toprovide a measure of the probability P ∝ D + D − , are each real numbers. The proba-bility for any process is a positive real number P ∈ R from 0 to 1, as for any probability,and as for the square root of this quantity p = √ P . However in principle it may bepossible to consider a complexification of the underlying calculation, represented by p → ˜ p ∈ C , such that P = ˜ p ∗ ˜ p . This is considered to be essentially the case in quantumtheory where unitary symmetry is used to model the properties of probabilities, andin the case of QFT the role of the above complex quantity ˜ p is played by the transitionamplitude M fi .Specifically, the likelihood of an event in QFT is proportional to the squared314odulus of the transition amplitude, as extracted from the terms of equation 11.45 viaequation 10.6, and as introduced in equation 10.3. With the cross-section for a HEPprocess, for example, linked to the imaginary part of the forward scattering amplitudevia the optical theorem expression of equation 10.96 and this latter object, as thereal number Im( M ii ), correlated with a degeneracy count D + D − , as described forfigure 11.8, we have the following chain of associations: P ∝ D + D − ∼ Im( M ii ) ∼ |M fi | (11.46)Here, in order for calculations in the present theory to converge with the formalism ofQFT, the process probability on the left-hand side is linked with the QFT calculationon the right-hand side via the mediation of Im( M ii ). The provisional connection onthe side of the present theory with D + D − has been described above and the connec-tion through the optical theorem with |M fi | on the side of QFT was described insection 10.5.While the structure of QFT on right-hand side of equation 11.46 exhibits thebasic property of probability conservation, via equation 11.45, the input from thepresent theory on the left-hand side provides an explanation of the underlying physicalnature of the probabilities in terms of the relative degeneracy of the field redescriptionsinvolved – that is the ‘number of ways’ in which the event may happen. Essentiallythe progression from left to right in equation 11.46 represents a complexification ofthe calculation in order to employ unitarity to gather a normalised expression of thedegeneracy count from which particular final states might be extracted with a combinedprobability of unity.The fact that the degeneracy count for field redescription sequences may becorrelated with Feynman diagrams, as described for figure 11.8, together with the factthat the optical theorem can be demonstrated order by order in perturbation theoryvia the analysis of Feynman diagrams, as described in section 10.5, suggests that thestructure of equation 11.46 might be explored further for low orders of perturbation.Indeed the assumption of perturbation theory, provided the coupling constant is suf-ficiently small, is that only the first few terms of the expansion of the S -matrix ofequation 10.35 are required for precise calculations.In principle here it might be possible to work backwards from QFT Feynmanrules, such as those in table 10.1 based on the Fourier expansions of quantum fieldssuch as ˆ φ ( x ) in the interaction picture, via the construction of the Feynman propagator∆ F ( x − y ) as implied in the right-hand side of equation 11.42, and use the analogybetween the left-hand side of that expression and equation 11.41 to make a detailedconnection with the present theory. This connection, employing also the optical theo-rem, should also provide a guide for deducing a more rigorous mathematical expressionfor the underlying conceptual picture of the present theory, with the spacetime geom-etry G µν ( x ) constructed in terms of fields such as A µ ( x ) and ψ ( x ) as one of manypossible solutions.On understanding the parallels between QFT and the present theory and mak-ing the connection from the right-hand side of equation 11.46 the aim would be toextract from the constraints of equations 11.29 effective Lagrangian terms within theframework of the QFT formalism, expressed in the flat spacetime of special relativity.On importing aspects of the present theory into QFT in this way, with field redescrip-tions expressed in terms of the algebra of creation and annihilation operators, the315im would be to follow through calculations such as cross-sections using the familiarmachinery of QFT.In this section we have largely considered the alternative route beginning withthe provisional picture described in figures 11.4, 11.6 and 11.7 for the present theoryleading to the simple relation P ∝ D + D − for a process probability, with D + , D − ∈ R .Through comparing the structure of figures 11.7 and 11.8(b), via figure 11.8(a), andmaking the association D + D − ∼ Im M ii this calculation might be ‘complexified’ asguided by the optical theorem of QFT. In particular a unitarity constraint could beemployed to effectively normalise the process probability calculation for all possibleoutcomes, as expressed in terms of an amplitude M fi ∈ C . This complex transitionamplitude may then in turn be determined as described in the previous chapter, andin particular in terms of the Feynman propagator ∆ F ( x − y ) and Feynman rules, suchas those of table 10.1.This approach is anchored in left-hand side of equation 11.46, with the aim offirst motivating all development from the perspective of the present theory in itself.On establishing a link with the framework of QFT various techniques, such as theemployment of unitarity in probability calculations, might be extracted from QFTand adapted for use in the framework of the present theory. It may also be possiblelearn from the relation of QFT to phenomena in condensed matter physics, as weallude to in the following section. Here the aim is to understand the nature of physicalparticle states and determine cross-sections and other observable quantities within theenvironment of the present theory, for which the spacetime geometry accompanyingempirical phenomena is not flat. However in a suitable limit the present theory mayapproximate to the form of a QFT in flat spacetime.The plausibility of either approach, from the left or right side of equation 11.46,rests on the identification of connections between the present theory and QFT whichstraddle the parallel development of the theories. Such a correspondence will be sum-marised in points 1) to 7) below. The ultimate aim here would be to comprehendand follow through a complete calculation in the present theory, without any arbitraryreference to standard QFT, and to establish a direct connection with HEP empiricalphenomena. However, using the canonical approach to QFT as a close guide is a rea-sonable strategy since it has been used widely and successfully in practice to obtainresults for comparison with experiment.In the present theory there have been two distinct considerations:(a) The nature of field redescriptions and an understanding of the permitted elemen-tary exchanges, such as depicted in figure 11.2, according to the various equationsof constraint in the theory. This was the topic of the previous section.The fields such as A µ ( x ) and ψ ( x ) are not introduced onto a pre-existing 4-dimensional manifold M , rather spacetime itself, with the spacetime geometry G µν ( x ) = f ( A, ψ ), is shaped by the possibilities of the field descriptions. Hencefigure 11.2 should not be interpreted too literally but rather a more dynamicalmathematical expression of field redescriptions is desired. This might take theform of equations 11.22 or 11.23 (or 11.30 for the spinor case) in terms of retardedor advanced Green’s functions, provided these expressions are compatible withconstraints deriving from equations 11.29.316b) The calculation of the probability of observable processes, for example in HEPexperiments, based on a count of the possible internal field degeneracies under-lying the process, as depicted for example in figure 11.7. This has been the topicof the present section.Again here the sequence of A µ ( x ) and ψ ( x ) fields in figure 11.7, superposed as ifupon a pre-existing spacetime, presents a somewhat naive and mechanical picturefor the degeneracy count. A more conceptually and mathematically rigorousexpression of this count may be required to describe the multiplicity of ways inwhich the geometry of spacetime G µν = f ( A, ψ ) may be fabricated out of thesefields.One of the initial aims has been to establish a correspondence between thebasic elements of the present theory and those of calculations in QFT. In QFT theconstruction of the transition amplitude M fi generally breaks down into very simpleelements as described by the Feynman rules, as listed in table 10.1 of section 10.5 forthe scalar model. Hence the goal is to explain how the ‘number of ways’ approachof figures 11.4, 11.6 and 11.7 leads to the Feynman rules which determine the quan-tity M fi , and understand why |M fi | should determine the probability for variousprocesses as expressed in cross-section or decay rate calculations.The parallels identified between the present theory and QFT are listed here.The first six items below loosely correlate with the respective Feynman rules of ta-ble 10.1 and the subsequent discussion in section 10.5.1) The number of ways a series of field redescriptions may unfold through a one-dimensional temporal progression with degeneracy D , with terms such as equa-tions 11.41 and 11.44, is analogous to the perturbative expansion of the timeevolution operator U ( t, t ) of equation 10.31 in QFT. The ‘number of ways’ in-tegral sum is naturally normalised by the linear uniform flow of time, with ‘oneway’ for each equal discrete temporal interval ∆ t i in equation 11.40 taken to thecontinuum limit ∆ t i → F ofequation 11.43, taking the form of equation 10.72, arises as an effective momen-tum space prior probability distribution when the calculation is restructured asfor QFT.As simply a set of real parameters in the expansion of a field into Fourier modesthe variables k ∈ R , which may be interpreted as 4-momentum under T µν := − κ G µν , as described in the previous section for G µν = f ( A ) in leading fromequation 11.6 to equation 11.16, may also appear in factors relating to processprobabilities as a result of calculations based on underlying field degeneracy. Thisis the case for cross-section calculations in QFT with factors of the Feynmanpropagator ˜∆ F ( k ) = 1 / ( k − m + iε ) effectively appearing as a weight factor, asfor example in equation 10.47. Hence in the restructuring of process calculationsfor the present theory, via the introduction of T -ordering in equation 10.32 andthe resulting Feynman propagators, such prior probability distributions shouldalso appear through this connection with QFT.317) The redescription expansion is moderated by the need for consistency with theconstraint equations. These include the higher-dimensional form of temporal flow L (ˆ v ) = 1 with D µ L (ˆ v ( x )) = 0 and the original form of the external geometry G µν ( x ) = f ( Y ) with G µν ; µ = 0 throughout; as listed in equations 11.29 andall effectively acting as selection rules for field interactions. Collectively theseconstraints are analogous to a Lagrangian, including in particular the L int termsin QFT as associated with the vertices in Feynman diagrams. For the presenttheory the ‘number of ways’ a process may occur is taken to be proportional tothe couplings implicit in the constraints, such as the factors of ˙ s f in the termsof D µ L ( v ) = 0 in equation 11.33, as also discussed after figure 11.5.In QFT the structures correlating with (a) and (b), listed above for the presenttheory, are seemingly inextricably linked. The interaction Lagrangian, which isclosely associated with the selection rules provided by D µ L (ˆ v ) = 0 for example in(a), appears explicitly in the S -matrix, through equations 10.27 and 10.35, whichis used in the determination of event probabilities for item (b) above. That isin QFT the mathematical structure of possible field interactions is embeddedin the structure of event probability calculations. Effectively this is achievedthrough the mechanism of ‘quantisation’ itself, with the expansion of the fields interms of creation and annihilation operators, which essentially converts a classicalcomposition of fields in an interaction term into a selection rule for contributionsto the S -matrix.In calculations of the transition amplitude M fi the commutation relations, suchas equations 10.16, ensure the correct matching and avoid unwanted cross-termsin compositions of the interaction Lagrangian or Hamiltonian H int ( t ) in the termsof equation 10.31 and its time-ordered form in equations 10.32–10.34. The se-quences of creation and annihilation operators placed between vacuum statesalso ensures causality in QFT calculations in the sense that any intermediatestate must always be created before it is annihilated to yield a non-zero matrixelement S fi . Sequences of creation and annihilation operators from the interac-tion Lagrangian embedded in S fi ultimately determine relative probabilities inthe context of all possible processes.A similar method of ‘quantisation’ might be employed in the present theoryin order to incorporate the constraints of equations 11.29 as selection rules forchains of field redescriptions between initial and final states in a degeneracycount, through the structure of R ( t ) in equations 11.41 and 11.44 for example.3) A free field solution for A µ ( x ) under G µν = f ( A ) in the form of equation 11.1may be expanded in terms of e ± ik · x Fourier modes as described in equation 11.6,as consistent with Maxwell’s equations under G µν ; µ = 0. Exchanges betweenfields such as A µ ↔ ψγ µ ψ are considered in terms of the complex Fourier modesof the fields. Similarly for QFT calculations as presented in chapter 10 usingthe interaction picture, as discussed after equation 10.26, between the initialand final plane waves of the form e ± ik · x the state evolution is mediated by anexpansion of free fields of the form in equations 10.13–10.15, which are solutionsof the Klein-Gordon equation for the scalar model.In the canonical quantisation approach to QFT, as described in chapter 10, anni-318ilation and creation operators, such as a ( p ) and a † ( p ), are associated with theFourier modes e − ip · x and e + ip · x of the field respectively, as seen in equation 10.51for ˆ φ + ( x ), equation 10.52 for ˆ φ − ( x ) and equation 10.13 for the complete freescalar field ˆ φ ( x ). In a QFT calculation the complex plane waves of the form e ± ik · x representing the incoming and outgoing particle states are linked by achain of creation and annihilation operators for a variety of fields to determinethe transition amplitude as described for example in equation 10.36. This struc-ture, employed throughout the calculations in the interaction picture, provides aclose analogy with the present theory.The quantum field ˆ φ ( x ) of equation 10.13 does not represent a solution of theequations of motion given an interaction, nor does it represent a physical entityin any context. Rather this expansion ˆ φ ( x ) carries the potential for all possibletransitions for the corresponding classical field in terms of Fourier components.This is the interpretation in the present theory, for which such quantum fieldexpansions might be employed in the construction of chains of field redescriptions,expressed in terms of complex Fourier modes and employed in a degeneracy countfor any process.4) The geometric constraint G µν ; µ = 0 over the external 4-dimensional spacetime,with energy-momentum T µν := − κ G µν , implies the conservation of 4-momentumfor all possible field redescriptions (in the flat spacetime limit considered here, asdiscussed before equation 11.14). In QFT calculations the time integral R dt overthe interaction Hamiltonian H int is replaced by a manifestly Lorentz invariantspacetime integral R d x over the interaction Lagrangian density L int via equa-tion 10.27 which, as seen for example in the lines of equations 10.46, leads to theconstraint of 4-momentum conservation for each interaction vertex as expressedby the δ -functions.Whether spacetime integrals, as a generalisation of purely temporal integrals,might feature in a generalisation of the field redescription degeneracy countfor solutions underlying a particular geometry G µν ( x ) is open to consideration.However here the field exchanges have been considered to take place purelythrough a temporal progression, consistent with the notion of a fundamentalone-dimensional progression in time that underpins the conceptual basis of thewhole theory. In any case, in the present theory 4-momentum conservation is en-sured through the prevailing relation T µν := − κ G µν and the identity G µν ; µ = 0which hold throughout spacetime and in particular for local exchanges of theunderlying fields. For such exchanges applied to the Fourier modes such as A µ ( x ) ∼ e − ik · x and ψ ( x ) γ µ ψ ( x ) ∼ e − ip · x e − ip · x for example the 4-momentumconservation in a A µ ↔ ψγ µ ψ field redescription takes the form of the mutualcondition k = p + p . This is essentially implied in the requirement that locally the spacetime geometry G µν ( x ) itself is unchanged for such an underlying fieldredescription.5) In the present theory an infinity in the degeneracy count occurs when for exam-ple the intermediate A µ ( x ) field state in figure 11.4 is augmented for a furtherintermediate redescription in terms of a pair of fields, such as A µ → ψγ µ ψ → A µ A µ ( x ) ∼ e − ik · x is replaced by the field ψ ( x ) γ µ ψ ( x ) ∼ e − ip · x e − ip · x up to a mutual freedom in the share of the total 4-momentum be-tween p and p , accounting for an infinite degeneracy of solutions, as describedafter equation 11.44.This is closely analogous to the ambiguity in the 4-momentum carried by aninternal loop in a Feynman diagram, such as that in figure 10.9 leading to thedivergent momentum integral R d r in equation 10.86, and as frequently encoun-tered in QFT. In both cases a means of ‘renormalisation’ is required in order toobtain a finite calculation. By matching such infinities in the present theory withthe analogous quantities in QFT a similar program of renormalisation might beobtained for the present theory, although with a different interpretation as willbe described in the following section. Indeed, the degeneracy count for any givenprocess in any case stands in need of a ‘normalisation’ with respect to the countof the number of ways in which anything can happen.6) Various combinatoric factors due to permutations of interactions for higher-orderfield redescriptions, or symmetries between identical particle states, will need tobe assessed for the present theory and related to the corresponding factors basedon the analysis of Feynman diagrams in QFT. Discrete sums over field degreesof freedom such as spin in QFT also reflect the number of ways a process mayoccur.7) The need to match both the e − ik · x and e + ik · x complex Fourier modes of thefields, through independent chains of degeneracies D + and D − , underlying areal expression of L (ˆ v ) = 1 and G µν = f ( Y, ˆ v ), means that an overall eventprobability is of the form P ∝ D + × D − as described for figure 11.7 (rather than P ∝ D alone from ‘item 1)’ above). For practical calculations it is the relativeratios of the degeneracies for the range of possible processes that is needed toobtain actual probabilities with P F P F = 1, for a sum over all possible finalstates F arising from an initial state interaction, including the case for which thefinal state is identical to the initial state.The calculation of D + D − is correlated with the determination of Im( M ii ) inQFT, as described for figure 11.8, which via a complexification of the calculationand the optical theorem is then closely related to the amplitude squared |M fi | in QFT as described for equation 11.46. Expressed this way the unitary symme-try applying to the complex amplitudes M fi models the conservation of the totalprobability, implicitly normalising the degeneracy count for all possible processes.The fact that renormalisation is required in QFT shows that this application ofunitarity is only partially successful, and does not necessarily automatically nor-malise the degeneracy count completely. Indeed even for a renormalisable QFTfinite calculations might not be achievable at a very high order of perturbation,and in general a more watertight method of normalisation might be sought forthe present theory.For a complete calculation in this theory, putting all of the pieces together, theactual value of the probability P F for a process yielding the final state F is determined320y the relative , rather than absolute , number of ways in which it can occur, essentiallyas is the case for the probabilities of classical physics. For example degeneracy countsover the infinite possibilities in the timing of field redescriptions, such as those inequation 11.41, may be independent of the choice of the external fields, as for examplein figure 11.4 which may describe the leptonic or quark final states for figure 11.5(a)or (b). More generally the infinities in the count of the number of ways will be incommon for a range of competing processes and will cancel in the calculation of physicalquantities such that the total probability for any outcome will necessarily satisfy therequirement P F P F = 1. Some care will then be needed in this theory to deal withinfinities that arise in the stages of such calculations. However, since all probabilitiesare normalised by the total degeneracy for any process the bound 0 ≤ P F ≤ F willresult in a probability P F = 1, which may be problematic in terms of comparisonwith the corresponding empirical value, but it is not possible for the theory to yield anonsensical infinity for the calculated value.The calculation of probabilities via a complexification may prove an effectivetechnique to apply for the present theory, once the relation between the underlying realnumber measure of degeneracy and the QFT calculation through equation 11.46 hasbeen fully understood. In this translation of the calculation a ‘unitarity’ condition willmodel probability conservation, consistent with kinematic factors appearing throughthe propagators, as described for ‘item 1)’ above, provided the ultimate expression forthe probability is a dimensionless quantity.While the seven points listed above express a close parallel between structuresin the present theory and perturbative calculations in QFT, as well as obtaining theFeynman rules for M fi the full cross-section expression is needed for comparison withempirical data. For QFT the structure of the cross-section σ was introduced in equa-tion 10.3 as a product of three factors, namely the amplitude squared |M fi | togetherwith the initial state flux factor and the final state Lorentz invariant phase space d Φ. Various normalisation factors such as the volume V and time interval T of theinteraction cancel in forming this expression.The probability for a process, whether expressed in terms of a degeneracycount or not, should be a dimensionless quantity, as is the transition amplitude M fi for the two-body final states considered in chapter 10. In general M fi need not bea dimensionless quantity provided the cross-section has the dimension of a lengthsquared, as for example in equation 10.12, and as described in the discussion followingequation 10.86.The present theory may involve a different breakdown across the three factorscomposing the expression for the cross-section, compared with that displayed for ex-ample in equation 10.7, with the form of the appropriate normalisation for all threefactors, including those for the initial state flux factor and final state phase space,possibly differing also from the QFT case. For the present theory, as for QFT, it isultimately the calculated cross-section that is required to be of the appropriate formin the context of equation 10.1. The normalisation of factors required for consistencywith the cross-section having the mass dimension D = − classical notion of probability. In this section we have argued forthe replacement or interpretation of the central term |M fi | in this expression in theform of a quantity representing an underlying measure of the degeneracy of ways inwhich the process may occur, in terms of sequences of field redescriptions, and henceconstituting a further purely statistical factor having essentially the same character asa classical probability.Explicitly, in this section we have considered field interactions in terms of pos-sible field redescriptions, involving the e ± ik · x Fourier modes of the fields, as expressedfor example in equations 11.22 and 11.30 of the previous section, causally linked to-gether to mediate observable processes, conceived as a field sequence such as depictedin figure 11.4 or 11.6 and combined as for figure 11.7, and as allowed by the form ofconstraint equations 11.29 such as D µ L (ˆ v ) = 0 and G µν ; µ = 0. This however leadsto a picture of the extended spacetime geometry G µν ( x ) itself constructed as one so-lution out of a myriad of possible ways based on local field description degeneracy,again subject to the constraint equations, not only for HEP processes but everywherethroughout the 4-dimensional spacetime world.This implies a conception of HEP phenomena, such as an e + e − → µ + µ − event,supported by the underlying field exchanges which seamlessly also support the macro-scopic physical world including the detector apparatus itself. In turn the physics ofquantum mechanics is seamlessly connected to the world of classical physics. In thesection 11.4 we further explore this conceptual picture within which quantum and par-ticle phenomena are found alongside macroscopic objects and gravitation in a unifiedframework.While the above seven points provide a useful guide into the workings of suchcalculations ultimately a stand-alone approach within the present theory may be de-sired. In this way the aim is to achieve explicit calculations for comparison with HEPprocesses such as e + e − collisions for the full theory. To make a detailed comparisonbetween the present theory and HEP data ultimately the particle concept, and in par-ticular the nature of the ‘in’ and ‘out’ states at a collider experiment, will need tobe understood within the context of the present theory. This will require an under-standing of the nature of physical particle states propagating in spacetime in general,relating to a fully ‘renormalised’ expression of field exchanges, rather than representingparticle states in the form of simple e − ik · x plane waves as for QFT. This direction willbe explored in the following section. The relation G µν = f ( A ) derives from the internal U(1) Q ⊂ E action within theisochronal symmetry of L ( v ) = 1, and is expressed explicitly in equation 11.1 asdetermined through the analogy with Kaluza-Klein theory as described in section 5.1.In deriving directly from the basic structure of the theory, through equation 2.30applied to the full symmetry group, the internal gauge field component A µ ( x ) itself,which appears in expressions such as D µ L ( v ) = 0 for the broken full symmetry, can322e considered as a ‘bare’ or elementary field at the ‘microscopic’ level from the point ofview of QFT. This same field, implicit in equation 11.1 and hence satisfying the relation (cid:3) A µ = 0 of equation 11.2, in the Lorenz gauge, is also essentially the classical gaugepotential of Maxwell’s electrodynamics of 1864, associated with directly observablelaboratory effects. In this sense, again from the point of view of QFT, the gaugefield A µ ( x ) can be considered as a ‘dressed’ or renormalised field at the ‘macroscopic’level. The question then arises as to how these two views of the same field A µ ( x ) areconsistent.For a non-Abelian gauge field Y µ ( x ) the relation G µν = f ( Y ) of equations 11.29,that is the classical field expression of equation 5.20, contains self-interaction terms asdescribed in equations 11.36–11.38. Hence, even from the perspective of gauge fieldsalone, the macroscopic form for G µν = f ( Y ) will be necessarily shaped and correctedas a consequence of the multiple solutions for the spacetime geometry, as built upona degeneracy of underlying gauge field redescriptions, with the constraint G µν ; µ = 0holding throughout the base manifold. However, in the full theory the gauge field A µ ( x ), associated with the internal Abelian U(1) Q gauge group, is also not free sinceit couples to fermions through the constraints of equations 11.29, as seen in the terms ofequation 11.33 for example. Through field exchanges as considered for equation 11.22Maxwell’s equation is modified to the form of equation 11.26, with a source term de-riving from the fermion components. Hence it is necessary to consider the macroscopicform of the spacetime geometry G µν ( x ) = f ( A, ψ ), and understand how this relates tothe original classical expression for G µν ( x ) = f ( A ) and also to empirical phenomena.Empirically electromagnetic waves are observed to propagate ‘at the speed oflight’ effectively according to Maxwell’s equation (cid:3) A µ = 0, with solutions such as thatin equation 11.6 for the transverse polarisation states r = 1 or 2 and with k = 0.Hence the overall form of the function G µν ( x ), on the left-hand side of equation 11.12with G µν ; µ = 0 implied in equation 11.14, appears to be completely transparent to un-derlying exchanges of indistinguishable fields, with possible intermediate stages similarto those of figure 11.6 or 11.7, which percolate down through higher orders with thespacetime geometry G µν ( x ) always preserved over the possible field redescriptions.That is, unlike the general case, the underlying gauge-fermion field redescriptions ap-pear to make little or no impression on the spacetime geometry associated with anelectromagnetic wave – with G µν ( x ) = f ( A, ψ ) ≃ f ( A ) which takes the shape of T µν := − κ G µν as depicted in figure 11.1 for example.In QED these higher-order solutions are described in terms of photon self-energy contributions, as shown for example in the Feynman diagrams of figure 11.9.Figure 11.9: A series of possible Feynman diagrams which ‘dress’ the original ‘bare’photon propagator, which itself corresponds to the first diagram alone.The particles observed in experiments correspond to renormalised states of thefields. The quanta of the electromagnetic field are massless, even though the higher-order corrections to the photon propagator in figure 11.9 contain virtual particles such323s e + e − and d ¯ d pairs. In QED the preservation of the bare photon mass m γ = 0, andhence the equation of motion (cid:3) A µ ( x ) = 0, for the renormalised field is explained interms of Ward identities (see for example [70]). This observation in QED is analogousto the transparency of the geometry G µν = f ( A ) to higher-order microscopic fieldredescriptions in the present theory, maintaining the macroscopic field condition k = m = 0, and a correlated mathematical explanation might be sought here.In the standard theory of QED the behaviour of the field A µ ( x ) deviates fromthat in classical electrodynamics due to the properties of low energy e + e − pairs. InHEP experiments an effective internal structure of the photon is manifested in ‘two-photon collisions’, such as the process γγ → c ¯ c induced and observed at e + e − colliders.In such experiments the photon expresses itself in revealing the internal structure of itsdressed state. Equivalent empirical effects are expected to arise from the principles ofthe present theory, with the internal structure of matter composed of endless possibleinternal ‘bare’ field redescriptions. Here for example solutions for G µν = f ( A, ψ )may take the effective macroscopic form of an electromagnetic field alone, such as thewave solution in equation 11.6, while implicitly containing a myriad of possible fieldcomponents and hence carrying the potential for the associated interactions as seenfor example in two-photon collisions.The mathematical divergences associated with higher-order loop diagrams inQED are tamed by accepting the non-physical nature of quantities such as ‘mass’ and‘charge’ in the bare Lagrangian and instead aligning the physical parameters of therenormalised theory with empirical values of mass and charge, as described brieflyfollowing equation 10.86. The effect of combining an empirically measured genericcoupling parameter g with quantum corrections determined in theory, through themachinery of renormalisation in QFT, leads to the observable phenomenon of ‘run-ning coupling’ in which the parameter g is found to depend on the energy scale E asdescribed by the ‘renormalisation group equation’: dd ln E g ( E ) = β ( g ( E )) (11.47)The function β depends upon the particular theory. In the Standard Model β is posi-tive for the U(1) Y gauge group and negative for the non-Abelian internal symmetriesresulting in the running coupling shown qualitatively in figure 11.10.The energy dependence of the coupling g , representing the general case inequation 11.47, is independent of the bare Lagrangian parameters, and also indepen-dent of the regularisation method and parameters used to temporarily suppress thedivergences in the process of renormalisation.As described in section 10.5 generally a quantum field theory is renormalisable,and finite results may be obtained for comparison with experiment, if the couplingparameter g is of mass dimension M D with D ≥
0. All of the couplings for the StandardModel, such as g and g ′ in equation 7.40, have D = 0 and the corresponding QFT is just renormalisable. Even here though for the renormalised Standard Model divergencesremain in the sense that the expansion series for equation 10.35 in equation 10.29does not generally converge at higher orders, although the problem does not becomeapparent until terms of approximately order 137 in the case of QED for example ([26]p.681, this is the point alluded to at the end of ‘item 7)’ in the previous section), farbeyond the first few orders needed for calculations in practice.324igure 11.10: The running coupling g ′ , g and g s = √ πα s respectively for the U(1) Y ,SU(2) L and strong SU(3) c gauge interactions in the Standard Model. Extrapolatedfrom their laboratory values over a number of orders of magnitude in energy scale, viaequation 11.47 with conventional β functions, the three parameters mutually intersect,although not simultaneously, at around 10 –10 GeV ([70] p.787).The structure and tools of QFT have a broad scope of applications and do notnecessarily describe the fields or particle states of a ‘fundamental’ theory. An effective quantum field theory is one which is only valid as a physical theory below a certainenergy threshold and describes particle states appropriate within that energy range.Such an effective QFT, for example a theory for nucleon-pion scattering, is necessarilyan approximation to nature, with different physics and new particle states observed athigher energy. The interpretation of particles associated with an effective field theory,such as nucleon and pion states, as ‘fundamental’ particles is hence unsatisfactory.Renormalisable QFTs such as the Standard Model are also considered to be lowenergy effective field theories. The form of the renormalisation group equation, andcontact with empirical observations, is insensitive to high-energy, short-distance phe-nomena, which are also unknown. Hence QFT provides a phenomenological frameworkfor particle physics with fields in the Standard Model Lagrangian transforming underthe SU(3) c × SU(2) L × U(1) Y gauge group describing the types of particles that areobserved in high energy physics experiments. The theory applies over a wide energyrange and provides a unifying framework incorporating weak and strong, in additionto electromagnetic, interactions. The corresponding quanta of the Standard Modelquantum fields describe the particle states of leptons, quarks, gauge bosons and theHiggs, all of which from an empirical point of view appear to be elementary. Howeverthis is not a conclusion that can be drawn from the QFT for the Standard Model itself.A more fundamental theory is needed to ascertain the true elementary struc-tures of nature. The renormalisation for the QFT of the Standard Model has had great325ragmatic success in particle physics but, as well as being insensitive to the method bywhich divergences are ‘cut-off’, in general has very little to say regarding the structureof an ultimate high energy theory. Hence the results of the Standard Model renor-malised QFT are plausibly consistent with an underlying theory for which interactionprobabilities are fundamentally expressed in terms of a degeneracy count of possibleredescriptions of the underlying field function as proposed in this paper. The presenttheory aims to describe the actual nature and behaviour of physical entities down toarbitrarily short distances and up to any energy scale.Indeed the present theory is intended to be a fundamental, rather than aneffective, theory, in contrast with the Lagrangian approach, as has already been em-phasised in section 5.2 and as will be discussed further in section 15.2. The presenttheory is also completely ‘renormalisable’ in an essentially trivial way since probabil-ities are constructed simply in terms of the relative ‘number of ways’ field solutionsmay be obtained. These involve nested sequences going down through higher ordersof field exchanges, as depicted for example in figure 11.11(a), which itself represents ahigher-order extension from the form of figure 11.6 for a single e − ik · x field componentsequence, here depicted alongside the associated Feynman diagram.Figure 11.11: (a) A higher-order sequence of field exchanges for the process e + e − → µ + µ − together with (b) the correlated Feynman diagram with internal loops. Thisfigure is similar to figure 11.8, except here with differing initial and final states andwithout a fold or cut line.Even considering the degrees of freedom of the field redescription timings t i the sum of possible ways is infinite, as described following equation 11.44. Further,the internal 4-momentum freedom for the d ¯ d field state, for example, in figure 11.11(a)translates into the divergent momentum integral for the corresponding d ¯ d virtual par-ticle loop in the Feynman diagram of figure 11.11(b), as described in ‘item 5)’ ofthe previous section. For yet higher orders this structure implies a nested productof infinite sums and integrals which would appear to more and more dominate cal-culations for more and more ‘dressed’ diagrams. However it is conceivable that suchinfinite degeneracy counts largely cancel, resulting in a non-trivial finite calculation ofcross-sections or branching ratios.For example, by relabelling the final state, figure 11.11(a) can be considered326o represent a field sequence underlying either an e + e − → µ + µ − or e + e − → d ¯ d event,amongst other possibilities. Since the intermediate redescriptions in figure 11.11(a) areapplicable for both processes e + e − → µ + µ − and e + e − → d ¯ d the relative ‘branchingratio’ to obtain the final state d ¯ d is simply:BR( e + e − → d ¯ d ) = sum of ways for d ¯ d sum of ways for { d ¯ d or µ + µ − } → ‘ (cid:16) ∞∞ (cid:17) ’ (11.48)For either final state there is an infinite degeneracy of intermediate states owing tothe implied unconstrained 4-momenta for example. These infinities clearly cancel incalculations such as equation 11.48 since there is a similar, in fact here identical, ‘degreeof divergence’ in each case. Indeed generally in forming measurable branching fractionscancellation between common factors will provide the main source of normalisation.Further normalisation factors will be involved in deriving event rates and cross-sectionssuch as σ ( e + e − → µ + µ − ), as described towards the end of the previous section.A similar situation arises in QFT with for example the 4 th order Feynmandiagram in figure 10.9 together with the same diagram with the final state relabelledby X + X − , for the processes X + X − → Y + Y − and X + X − → X + X − respectively, bothof which contain loops with infinite degrees of freedom in terms of the correspondingmomentum integrals. In QFT the methods of renormalisation lead to finite cross-sections and branching fractions for comparison with the empirical data. In fact inQFT tree level diagrams, such as figure 10.4, already give a good approximation forthe rates of such processes, provided the coupling constant is sufficiently small. This isthe case for QED in which the cross-section calculation based on the tree-level diagramin figure 10.3 gives a good approximation for the process e + e − → µ + µ − as describedtowards the end of section 10.2.For the present theory based fundamentally on degeneracy counts the inter-pretation of equation 11.48 may be contrasted with the case of Newtonian calculus inwhich the ratio δy → δx → has a well defined meaning and value since δy and δx tend tozero in a related manner through a continuous function y = f ( x ). Here, in a similarand yet complementary situation, the limit of the ratio P y →∞ P x →∞ in equation 11.48 givesa finite and well defined result due to the close relationship between the divergence inthe numerator and that in the denominator.This a very literal notion of (re)normalisation in calculating probabilities. It isanalogous to everyday cases such as the probability of hitting the ‘20’ on a dartboard.There are an infinite number of ways in which the point of the dart can land on thesurface of the 20 segment. However this infinity is normalised by the infinite numberof ways of landing in any other region such that the total probability is finite andapproximately (for a suitably random dart thrower). Alternatively the sum overpoints may be quantified as a finite integral over surface area, rather like the finiteintegral over possible field redescription times t i in equation 11.41 for figure 11.4 as ameasure of the sum of ways to describe the underlying field function.The above analogy demonstrates the close association of classical and quantumprobabilities in the present theory as will be discussed further in the following section.The cancellation in equation 11.48 not only applies for the infinite degeneracy of fieldredescription times t i but also for the unrestricted internal momentum freedom, im-plicit in the d ¯ d internal state of figure 11.11(a) for example, which is also infinite interms of a real-valued R d k measure. 327n practice calculations of branching fractions and cross-sections may be muchmore readily performed by noting the symmetry of the system (analogous for exampleto the equal sizes of the twenty segments on a circular dartboard in the metaphordescribed above). In the case of QFT unitary symmetry, in calculations based oncomplex amplitudes, is applied to model the conservation of probability; and yieldssuccessful results when supplemented by the techniques of renormalisation. Howeverthese calculations, founded on postulated complex-valued entities, miss the physicalmeaning of the infinities as a real-valued degeneracy in the number of ways a processcan occur. Hence in the present theory renormalisation based on a real degeneracycount is expected to be closely related to QFT renormalisation based on complexobjects, such as amplitudes and propagators, with similar conclusions except withfinite results necessarily to all orders in the present theory.While generating finite results when normalised for specific processes it is plau-sible that the sums and integrals over the myriad of continuous possibilities, such asfor figure 11.11(a), and for an endless range of higher-order field sequences, may haveresidual effects such as the dependency on the energy scale of physically measurableinteraction strengths as described by the running coupling in figure 11.10. Underly-ing differences in branching ratios such as equation 11.48 will then depend directlyupon differences in the ‘bare’ couplings associated with the field redescriptions suchas A µ ↔ ψγ µ ψ for example. These include the ˙ s f real coefficient factors of magnitude1 or for the U(1) Q coupling in equation 11.33, applied for the outer layer of fieldexchanges, that is in the external vertices of the Feynman diagram as for example infigure 11.5(a) and (b) and as described there in the subsequent text, and will applyhere for the final field redescription at time t = t in figure 11.11(a). (In QED thereare Ward identities which both preserve the bare value of the photon mass m γ = 0, asalluded to earlier in this section with reference to figure 11.9, and also which preservethe ratios of charges through renormalisation, and again a correspondence might besought with the structures of the present theory.)The field redescriptions underlying the many solution possibilities are profuselydiffused throughout spacetime, from the temporal origin of the universe in the BigBang, shaping the initial conditions for the evolution of the cosmos as considered inthe following two chapters, to the quantum effects observed in laboratory experimentssuch as that represented in figure 10.1 and described further in the following section.As well as the photon ‘self-energy’ contributions of figure 11.9 the field redescriptions‘dress’ the initial and final state particles for an event observed at a collider experi-ment. These higher-order solutions include the final state processes suggested by theFeynman diagrams in figures 11.5(c) and (d). Since the d ¯ d fields undergo strong SU(3) c interactions, producing an observed final state π + π − pair for example, objectively itmight be expected that many more spacetime world solutions with a d ¯ d comparedwith a µ + µ − final state might be identified, in the context of a grand ensemble of allpossible G µν ( x ) = f ( Y, ˆ v ) solutions on M . This consideration would suggest that thebranching ratio of equation 11.48 should effectively be unity, owing to the apparentrelative infinity of ways to produce a d ¯ d rather than a µ + µ − final state. As pointedout in the discussion after ‘item 7)’ in the previous section such a conclusion for thepresent theory, although being internally consistent, would appear to be drasticallyincompatible with empirical phenomena. 328owever laboratory phenomena, as for all observations, subjectively evolve pro-gressively in time. At the time t of the final field redescription in figure 11.11(a) thelikelihood of a field exchange will depend upon the U(1) Q coupling ˙ s f regardless ofwhat has happened before or what can happen after. Hence the charge value of 1or will dominate the cross-section. Subsequent field redescriptions and interactionsfor the final state produced, as represented in terms of the Feynman diagrams in fig-ures 11.5(c) and (d) for example, will not affect the branching fraction calculationfor equation 11.48 other than through their implications for a final state phase spacefactor, which in the present theory correlates with the range of spacetime geometriesassociated with a particular set of final state particles.More generally out of the grand ensemble of all possible G µν = f ( Y, ˆ v ) solu-tions it might be expected that a typical world would be dominated by strong SU(3) c interactions and corresponding forms of matter, since a relatively much larger range offield redescriptions are possible, via the set of eight self-interacting gluons, comparedwith other kinds of Standard Model interactions. However we do not apprehend a full4-dimensional universe all together in its full temporal extent as a given object, ratherwe subjectively sample a possible world progressively through time. The correspond-ing progressive accumulation of probabilities selects a type of possible G µν = f ( Y, ˆ v )solution which is extremely rare in the context of the full ensemble, with a sparser moreopen form of matter shaped by a more democratic contribution from the componentsof SU(3) c × SU(2) L × U(1) Y gauge interactions.That is the G µν = f ( Y, ˆ v ) solution that we observe is selected with all proba-bilities oriented with respect to an underlying one-dimensional temporal flow from thepast to the future, moulding the matter content and laws of physics for such a universe,with the structure of causality built into the world we perceive. As for perception ofthe world in space and time itself, this subjective causal aspect of observations is afurther necessary a priori structure through which we experience the world, as will bediscussed further in chapter 14.While the accumulation of probabilities along a causal path through a choiceof world solutions shapes the macroscopic properties of matter on the large scale, theprobabilities locally determine the relative likelihood to achieve different outcomessuch as for example the event e + e − → d ¯ d or e + e − → µ + µ − in a HEP experi-ment as described above. Once the final state particles, such as π + π − or µ + µ − , areformed and propagate through spacetime to the extent that the macroscopic shape of G µν = f ( Y, ˆ v ) diverges the relative degeneracy count of field redescriptions for differ-ent processes under the same G µν ( x ) geometry no longer applies. That is a branchingratio such as equation 11.48 is determined by the relative number of world solutions ef-fectively within a local finite spacetime volume (similarly as represented by V T for theQFT calculations described in section 10.2) with a common local geometry describedby G µν ( x ), regardless of what can happen after the final states form.In quantum mechanics the causal sequence of probabilities is reflected in theevolution of the wavefunction Ψ in equation 11.51 below as punctuated by apparent‘collapses’ of the wavefunction, as will be discussed in the following section. As willalso be described further in the next section the underlying statistical origin of quan-tum phenomena in the present theory is very similar in nature to that for a classicalstatistical system, with outcomes essentially determined by the ‘number of ways’ in329hich something can happen. The causal accumulation of probability, that is thetemporal ordering property as described above for quantum phenomena in the presenttheory, naturally also applies for systems of classical physics. In the classical worldthe temporal ordering of probabilities underlies the second law of thermodynamics forexample, which will be considered in relation to the very early universe in section 13.2.The statistical approach underlying quantum phenomena in the present the-ory, fundamentally based on a real-valued degeneracy of field possibilities, has a micro-scopic structure analogous to that studied in the classical physics of critical phenomena.There the forces and behaviour of basic elements of condensed matter systems, suchas magnets or fluids, are sufficiently well known to be modelled and parametrised.There is also a close relationship between such systems and quantum field theory atthe phenomenological level – in fact a correspondence can be identified between renor-malisation in QFT and the theory of critical phenomena which leads to a principle of universality for statistical fluctuations, which is equivalent to the cut-off independencein QFT ([70] p.268). However, although the empirical tests in HEP have been verysuccessful, in the case of the QFT for the Standard Model the short-distance physics,only provisionally represented by field parameters in the bare Lagrangian, is essentiallyunknown, as alluded to earlier in this section.Potentially the present theory extends the analogy between HEP phenomenaand critical phenomena conceptually as well as mathematically, with the microscopicworld being ‘modelled’ on the idea of underlying field redescriptions. This makes acloser relation to the theory of condensed matter systems than for standard QFT,with the latter founded pragmatically on calculations based on complex transitionamplitudes.In principle the present theory reaches down without limit into the microworldrevealing an internal structure in terms of nested multiple field solutions continuing in-definitely in almost fractal-like manner, analogous to the perturbative expansion of theQFT time-evolution operator expressed as an infinite series in equation 10.31. On theother hand the scope of the theory in principle also feeds upwards and seamlessly intothe phenomena of condensed matter physics itself, with magnetic and fluid propertiesemerging at the macroscopic level, and into the realm of classical physics and classicalprobabilities, as alluded to above and described further in the following section.Out of the construction of the spacetime geometry over sequences of field de-generacies, of arbitrary high order, it is suggested that the phenomena of particle statesthemselves arise, apparently propagating through field configurations in spacetime inthe fully ‘renormalised’ theory and mutually interacting, accounting for the phenom-ena observed in HEP experiments. This picture of particle states brings to mind theexcitations of ‘phonons’ in the medium of a solid state device, with here the colourfulvariety of Standard Model particle types arising out of the variety of underlying inter-nal field interactions allowed by the broken form of L ( v ) = 1 and D µ L ( v ) = 0 andthe constraint equations 11.29 on M in general.As for the Standard Model, in the present theory particle masses arise throughthe interactions of the corresponding field with a ‘Higgs’ field. Here a vector-Higgs fieldis associated with the components of h ≡ v ⊂ v of equation 9.46 projected ontothe local tangent space TM , with the effective Higgs phenomenology provisionallyidentified as described in subsection 8.3.3. As well as the selection of the external TM F (h O ) here ‘spontaneous symmetry breaking’ is also realisedin terms of a particular choice of vector field v ( x ) which may ‘point’ in an arbitrarydirection at any given location x ∈ M . This structure may be closely relate to thestatistical methods employed in spontaneous symmetry breaking for critical phenom-ena, as for example associated with the properties of ferromagnetism. Recalling thatthe Higgs mechanism was developed from the early 1960s through analogy with spon-taneous symmetry breaking phenomena as originally conceived in condensed matterphysics this observation sees the Higgs concept returning to familiar territory.Low energy effective phenomena might also arise and be related to the StandardModel, which itself may considered to be an effective field theory as discussed afterfigure 11.10. In this case while some components of v ∈ F (h O ) such as the Diracspinors ψ might correlate directly with elementary fermion states, the vector-Higgs v components may correlate less directly with the empirically observed scalar Higgsparticle. In the Standard Model this latter state is itself treated as a ‘fundamentalparticle’ in the effective theory with symmetry breaking modelled by a scalar Higgs field φ in the contrived potential of equation 7.53 as described in section 7.2. In the presenttheory the degree of freedom | v | is considered as a candidate for a field underlying theobserved Standard Model scalar Higgs particle, which hence does not correspond to afundamental scalar field in the components of L ( v ) = 1 projected over M . Whilethe scalar condensates of technicolor models, described shortly before equation 8.73 insubsection 8.3.3, are analogous to BCS pairs of electrons bound through interactionswith phonons in solid state devices, a different relation to condensed matter systemsmight be sought for the present theory since here technicolor gluons are not requiredto bind the scalar Higgs together.While the microworld is infused with field function redescriptions, such as A µ ↔ ψγ µ ψ , in the multiple solutions under G µν ( x ) physically transmitted real parti-cle states, such as electrons and photons, as detected in HEP experiments propagateover macroscopic distances with measurable and regular properties such as mass m ,charge e , spin s and average lifetime τ . These features, which define the particletypes, are regular and reproducible and hence must to some degree arise as the prop-erties of self-sufficient discrete entities, in the sense of being generally independentof the conditions under which they are produced and the environment within whichthey are observed. Such real propagating particle states are associated with a distinctimpression in the spacetime geometry, that is the form of G µν = f ( Y, ˆ v ), itself. Inpropagating over macroscopic distances particle states, such as photons and electrons,are revealed through their observable apparent interactions between each other andwith the elements of macroscopic apparatus.An electron state in the e − beam of a particle accelerator for example is in con-stant interaction with the electromagnetic fields produced by the accelerating, bendingand focussing components of the machine, via the elementary field exchanges depictedin figure 11.12(a). Even for a freely propagating electron interactions with an electro-magnetic field A µ ( x ) are present in terms of internal ‘self-energy’ possibilities, similarto those for the photon in figure 11.9, as shown here for a free electron state in fig-ure 11.12(b).Both situations depicted in figure 11.12 are submerged within a saturation ofmultiple solution possibilities for G µν = f ( Y, ˆ v ) such that the empirically observed331igure 11.12: Exchanges between the electron field, described in terms of the compo-nents of the fermion field ψ ( x ), and the electromagnetic gauge field A µ ( x ), for (a) aninteraction with experimental equipment via an external photon and (b) a self-energycontribution for a free electron in terms of an internal photon.electron state emerges out of this myriad of interactions as an apparently robust andreproducible discrete entity. Such a particle entity may be guided and to some de-gree localised, propagating in a 4-dimensional spacetime expression of the underlying1-dimensional temporal flow with properties shaped out of the full higher-dimensionalform L (ˆ v ) = 1. The particle states exhibit probabilistic behaviour, of the form mod-elled by quantum theory, as inherited from the probabilistic nature of the underlyingdegenerate set of possible field configurations, as described in the previous section.For the case of the plane wave electromagnetic field A µ ( x ) of equation 11.6 anexplicit form of the spacetime geometry was derived in equation 11.12, with G µν ∼ κ k µ k ν V k via the coefficient C extracted from equation 11.16. This geometry for the fieldin a spatial volume V was provisionally associated with a ‘photon’ of 4-momentum P µ = k µ and k = 0. In this naive picture the photon propagates as a kind of‘microscopic gravity wave’, as suggested by equation 11.12, consisting of purely Riccicurvature as described after figure 11.1 and suggesting a metric g µν ( x ) of a form similarto equation 11.13. The 4-momentum carried by such a ‘particle state’ is naturally‘quantised’ in the sense that the parameter k ∈ R in the e ± ik · x Fourier modes appearsin the expression T µν := − κ G µν = f ( A ), that is equation 11.12, since G µν ( x ) is afunction of the spacetime derivatives of the gauge field A µ ( x ).The actual nature of physical particle structure is expected to be rather moreelaborate than initially suggested by this picture of plane waves in a given volume,which was initially motivated in part by analogy with QFT as recapped at the end ofthe previous section. For the present theory, unlike the external states the intermedi-ate states of field redescriptions, over which the form of the local spacetime geometry G µν ( x ) is unchanged, may however indeed involve independent complex wave com-ponents. As described in section 11.2 a hybrid set of e − ik · x and e + ik · x mode fieldexchanges in such interactions, as depicted in figure 11.7, correlates with the appar-ent ‘amplitude squared’ rule for the associated interaction probabilities. On the otherhand it is in the nature of a ‘particle’ to possess properties quite distinct from planewaves. Whether or not considered in terms of wave packets or within a volume V a‘particle’ here is also not considered to be a ‘point-like’ entity, but rather a state offields as a function on M dynamically prescribed through the conditions of G µν = f ( Y, ˆ v ) and L (ˆ v ) = 1. Higher energy particle transitions may be possible in eversmaller effective volumes V , correlated with higher 4-momentum k , without limit,with an apparent ‘size’ or structure never observable for the initial and final state332entities’ in processes such as e + e − → µ + µ − . Indeed such ‘particle interactions’ aremanifestations of field redescriptions which effectively apply throughout a finite volume V simultaneously, as indicated in and described for figure 11.4 for example, withoutreference to any point-like particle structure at all.The fact that particle phenomena become apparent for interactions on veryshort distance scales, relative to macroscopic laboratory equipment, may be due tothere being a greater likelihood for field functions to be indistinguishable for smallspacetime volumes. On the other hand the idealised case of transition amplitude M fi calculations in QFT effectively considers plane waves defined in the limit V → ∞ , withfactors of V cancelling for observable quantities such as cross-sections. In the presenttheory the role of an apparent volume V with regards to particle interactions, and thediscrete ‘quantised’ nature of particle states and interactions more generally, requiresfurther understanding.The factor of A in the event rate formula of equation 10.7, from the expressionfor the luminosity in equation 10.2, makes intuitive sense when picturing the incomingbeam components as ‘bunches of particles’. However, here the question is how a greaterintensity of field interactions, apparently corresponding a smaller area A , increasesthe production probability for final state particles, with the particle concept itself deriving from the underlying field interactions. The relation of the macroscopic tothe microscopic world through a program resembling renormalisation will be key toaddressing these questions.As described in section 5.2, and reviewed in the opening of the following section,the generalisation from the classical solution G µν = f ( A ), closely relating to Kaluza-Klein theory, will modify the macroscopic form of G µν ( x ) in way that incorporates thecharge density σ ( x ) in the current J µ = σu µ of equation 5.40, the material density ρ ( x ) in − κ G µν = ρu µ u ν from equation 5.39 and the structure of matter T µν := − κ G µν more generally.From this point of view elementary particle states, such as the electrons andmuons observed in HEP experiments, can be considered as quantum transitions withinthe macroscopic world, which is geometrically described by G µν = f ( Y, ˆ v ). With grav-itation encompassing quantum phenomena this describes an environment one layeroutside the traditional approach to QFT for which the particle states are simply given as the initial and final states of particle interactions. We hence return to the con-ception of the physical world as described section 10.1 for the experiment depictedin figure 10.1 for example. In the meantime, in chapter 10, we have dismantled theQFT cross-section calculation in order to identify a correspondence with the presenttheory, as summarised in points ‘1)–7)’ of the previous section; with the ultimate aimof reassembling such calculations in light of the present theory and fully accountingfor the observed particle phenomena.Together with the observations of chapters 8 and 9 for the breaking of higher-dimensional forms of L (ˆ v ) = 1 we may hope to gain some insight into the reasons forthe observed properties of the various particle types without having to merely writethem in by hand based on empirical findings. The abstract Fock space of QFT isnot required, with creation and annihilation of particles through mutual exchangesnow being firmly grounded in the field state of the macroscopic world. Such a statemay consist in the physical components of experimental apparatus themselves, which333xhibit essentially classical behaviour, providing a framework to make firm calculationsand predictions for the properties of the apparent particle transitions recorded.An electron state is then consistent both with the idea that nested multiplefield solutions, generalising from figure 11.12(b), continue indefinitely down on themicroscopic level together with the spacetime geometry satisfying G µν = f ( Y, ˆ v ) and G µν ; µ = 0 at the macroscopic level, with a form of ‘renormalisation’ relating the twolevels. In the case of the electromagnetic field the massless ‘renormalised’ field has aclose resemblance to the bare field A µ ( x ) of equation 11.6, as described in the openingof this section. Further, from the perspective of Kaluza-Klein theory, the externalgeometry is directly related to the internal gauge fields through G µν = f ( Y ) in theform of equation 5.20 for example. On the other hand, in the absence of an expressionof the form G µν = f ( ψ ), the physical fermion particle states of an ‘electron field’ forexample appear to have a somewhat more distant resemblance to the bare ψ ⊂ v subcomponents with which they were originally identified through the action of E onthe components of F (h O ) broken over TM , as summarised in equation 9.46.That is, rather than being described directly by the ψ ( x ) ⊂ v ∈ F (h O )field components projected onto M the form of G µν = f ( Y, ˆ v ) for an e − particle stateobserved in HEP experiments will be shaped through interactions with other fields,such as A µ ( x ), resulting in a ‘renormalised’ or ‘dressed’ state. This was suggestedtowards the end of section 9.3 where it was also hinted however that fermion particlestates might be identified more directly through interactions of a spinor ψ ( x ) fieldand the vector-Higgs v ( x ), initially shaping a geometry more simply of the form G µν = f (ˆ v ), as will be described in section 13.1.The ψ ↔ v field exchanges between the spinor and vector-Higgs fields, consis-tent with the constraint L (ˆ v ) = 1 of equations 11.29, are also proposed to give rise tothe generation of ‘mass’ for the fermion states. The corresponding interaction terms,as described for equation 9.48, are reminiscent of Yukawa couplings of a fermion fieldto the Higgs field in equation 7.69 for the Standard Model. However while such a ‘baremass’ might be identified at the level of L (ˆ v ) = 1, the physical mass, and indeed the concept of mass itself, as an observable quantity is only defined for the macroscopicdressed state as described in terms of the energy-momentum T µν := G µν . For a freeelectron state in the complete theory the aim will be to identify the correspondingmacroscopic form of G µν = f ( Y, ˆ v ), and to understand how k = m e arises as a robustobservable quantity for such a state, as deriving from the underlying interactions ofthe ‘bare’ fields. Further light will be shed on the nature and origin of mass in thepresentation of cosmology in the context of the present theory, in particular towardsthe end of section 13.1 and opening of section 13.2.As well as carrying energy-momentum density in T µν := G µν particle phe-nomena are observed through the transfer of discrete values of 4-momenta k , everenveloped within a spacetime geometry and consistent with G µν ; µ = 0, such that thetotal initial and total final momenta match in processes such as e + e − → µ + µ − . Afull understanding of the nature of such interactions, as provisionally described in sec-tions 11.1 and 11.2, is of course intimately related to an understanding of the nature ofthe particle concept itself. This may require a full exploration of the relation betweenthe present theory, quantum field theory and condensed matter physics as alluded toin this section. 334n general terms to understand what is an electron state or what is a muonstate, as observed in HEP processes such as displayed in figure 10.1 or 10.2, it is neces-sary to think of the full 4-dimensional spacetime picture in relation to the underlyingfield component redescriptions. This will be described further for figure 11.13 in thefollowing section and connects to the broader question concerning the incorporation ofa theory accounting for the quantum properties of fields and particles alongside generalrelativity in a consistent framework, in the form of the theory presented in this paper.The conception of particle phenomena for the present theory will also be discussedfurther alongside figure 15.2 in section 15.2 of the concluding chapter, with particlestates correlated with the emergence of discrete topologies for geometric solutions for G µν = f ( Y, ˆ v ) in the near vacuum limit. In the present theory we begin with a 1-dimensional temporal progression and henceneed to build a 4-dimensional spacetime M with geometry G µν ( x ) out of the structureand symmetries of the underlying multi-dimensional form of temporal flow L (ˆ v ) = 1.The degeneracy of possible solutions for the ways in which this may be achieved re-sults in the indeterminacy of empirical observations in our world and other apparent‘quantum’ phenomena, as studied for example in HEP experiments.Beginning with the electromagnetic field A µ ( x ) in section 11.1 the possibilityof alternative solutions involving the fermion field ψ ( x ) underlying the spacetime ge-ometry were expressed in terms of the field redescriptions of equations 11.22 and 11.23.These equations satisfy equation 11.26 in which the current j µ = ψγ µ ψ may be consid-ered as a source term. Such ‘microscopic’ field redescriptions via the mutual exchanges A µ ↔ j µ are incorporated into the spacetime geometry, generalising from the classicalrelation G µν = f ( A ) of equation 11.1 as originally derived through association withKaluza-Klein theory in section 5.1.The observable world is awash with the interchanges between the A µ ( x ) and ψ ( x ) fields, together with higher-order redescriptions through which the fields may in-teract, saturating the world, as described alongside figure 11.12 in the previous section.This gives rise to a rather fluid mathematical creation of matter as perceived throughthese exchanges and hence the properties and forms of the ‘macroscopic’ materialworld are conditioned by them. This describes the general relativistic limit pertainingto tangible physical objects that take shape on M over the collective contribution ofthe internal fields, such that the apparent composition of the Einstein tensor may bewritten simply as: G µν = − κT µν ( Y, ˆ v ) (11.49)This is equation 5.32 of section 5.2 expressed in a form which emphasises the implicitfield composition of the material world. The effective energy-momentum tensor T µν on M may take different forms in terms of the apparent macroscopic matter distributionon the manifold, but it must always be fundamentally composed out of the interplay ofthe underlying fields, mutually subject to the constraint equations 11.29. For examplethe energy-momentum tensor might describe a perfect fluid and the Einstein tensor willbe macroscopically composed as described in equation 5.37, as we shall consider for the335osmological scales of the universe, alongside equations 12.2 and 12.3, in the followingchapter. The general form of that equation for the structure of G µν ( x ) incorporatesmacroscopic ‘pressure’ p ( x ) terms and defines the scalar field ρ ( x ) which in generalrelativity is identified with the familiar notion of ‘matter density’. For the case of apressureless perfect fluid we have: − κ G µν =: T µνǫ = ρu µ u ν (11.50)that is equation 5.39, with T µνǫ interpreted as the effective energy-momentum ten-sor for a pressureless fluid. In the original formulation of general relativity such anenergy-momentum tensor, through the above field equation, would be interpreted asthe ‘source of curvature’ on the manifold. This correspondence with general relativitywas explored in more detail in section 5.2 where it was described how equation 11.50leads to the geodesic equation of motion for this form of matter, that is equation 5.36,owing to the Bianchi identity G µν ; µ = 0, without the need to introduce the geodesicconstraint as an additional postulate of the theory.The extension of the classical field relation of equation 11.1, which impliesthe homogeneous Maxwell equation (cid:3) A µ = 0 of equation 11.2, as shown for equa-tion 5.30, with the inclusion of the charged matter term ρu µ u ν on the right-hand sideof equation 5.41 is an example of a break away from the pure Kaluza-Klein relationof G µν = f ( A ). This deviation from a free electromagnetic field alone is here con-sidered at the level of macroscopic phenomena, which overlays the microscopic fieldinteractions which led to equations 11.21 and 11.26 in section 11.1 and as describedin the opening of the previous section. Applying G µν ; µ = 0 to the full expression inequation 5.41 led to the incorporation of a charged current J µ , with (cid:3) A µ =: J µ = σu µ as defined in equation 5.40, and to the identification of the Lorentz force law of equa-tion 5.43 as a deviation from the purely gravitational geodesic flow. Here the chargedcurrent J µ corresponds to that observed in macroscopic classical experiments, typicallyfor the non-relativistic limit such as performed by Faraday in the 19 th century. Hencein addition to the apparent matter density ρ in equation 11.50 effective macroscopicphenomena also involve the charge density σ in J µ = σu µ . Both the macroscopic andmicroscopic currents are conserved, with J µ ; µ = 0 as described following equation 5.43in section 5.2 also applying for j µ = ψγ µ ψ of equation 11.26 (as originally expressedfor equation 3.101 in section 3.5 for the Lagrangian approach).Within this limiting case of general relativity described above, that is neglect-ing explicit quantum phenomena, if the approximation of a flat spacetime for which G µν ( x ) ≃ G µν ; µ = 0, since the energy-momentum tensor T µν is identified with the spacetime geometry G µν , regardless of themagnitude of the spacetime curvature. As described in the opening of section 5.2 thisobservation applies in particular in approaching the flat spacetime limit with T µν,µ = 0,and will also apply for the further limit of the non-relativistic case; with the corre-sponding energy-momentum conservation also encompassing all underlying quantumphenomena in all cases. 336icroscopic transitions of internal fields such as A µ ↔ ψγ µ ψ , and quantumprocesses in general, may be recorded in macroscopic devices, generally in the form ofamplified electronic signals. All such macroscopic equipment is also itself composedover field interactions in the form of equation 11.49 and effectively described by anappropriate classical energy-momentum tensor T µν , for the solid state devices typicallyemployed, and at a basic level a tiny ‘detector recoil’ will accompany any productionor detection of particle states as a consequence of 4-momentum conservation. Theequations governing the evolution and interactions of the microscopic world hencemerge into the equations of motion for ‘classical’ objects, such as described by geodesictrajectories or the Lorentz force law. This framework will then shed some light on akey question concerning the relation of quantum mechanics to the world of classicalphysics.In the previous section it has been outlined how empirically observed particlestates, such as electrons and muons, might be identified in parallel with a programof ‘renormalisation’ for the present theory, and merge seamlessly into the state ofthe macroscopic environment. In figure 11.13 a typical high energy physics process,as described in section 10.2 and already depicted in figure 10.2, is contrasted with atypical experiment involving non-relativistic quantum theory.Figure 11.13: (a) The process e + e − → µ + µ − as observed in HEP experiments forwhich the cross-section can be calculated in QFT. (b) The double-slit experiment inwhich a single electron is detected on the screen according to a probability distributiondetermined in non-relativistic quantum mechanics.In figure 11.13(a) a particular event is detected, a final state µ + µ − pair at anangle θ , presumably mediated by one of many possible intermediate sequences of fieldstates such as represented in figure 11.7. In figure 11.13(b) an electron is detectedat a particular location A out of a continuum of possibilities including B, C . . . . Instandard quantum theory both of these processes are assumed to take place against aflat background of space and time, which for figure 11.13(a) is Minkowskian and forfigure 11.13(b) is Newtonian. In the present theory however the base manifold curva-ture, although smooth, is finite and non-flat essentially everywhere in 4-dimensionalspacetime, with both processes depicted in figure 11.13 representing particular featuresof a global G µν = f ( Y, ˆ v ) = 0 solution.Events of the kind sketched in figure 11.13(a) are readily observed by experi-ments of the kind depicted in figure 10.1 for example. In this case both the macroscopic337LD detector and microscopic e + e − → µ + µ − interaction are uniformly envelopedwithin a particular solution for G µν = f ( Y, ˆ v ). The interaction region of such an ex-periment for such an event will locally have a spacetime geometry G µν of a similarform to that for T µν := G µν pictured in figure 11.1 and as represented by the wavylines in figure 11.13(a). As described in section 11.1 the associated metric solution g µν ( x ) will have properties closely relating to the metric of equation 11.13; and the un-derlying field redescriptions, as represented for example by figure 11.7 in section 11.2,will necessarily respect this external physical geometric form.Similarly the system described in figure 11.13(b) will be enveloped within aparticular G µν = f ( Y, ˆ v ) solution, with a non-flat metric g µν ( x ) description. For asufficiently high intensity source with a stable interference pattern observed on thescreen a wave-like solution for G µν ( x ) will permeate the spaces between the elementsof apparatus. The lower intensity case, with a single electron detected on the screenas indicated in the figure, will correspond to a different 4-dimensional world solu-tion for G µν ( x ). While both wave-like and particle-like solutions are shaped by anenveloping geometry with G µν ; µ = 0 uniformly throughout space and time, the under-lying indeterministic character of the field redescriptions become evident as discreteparticle phenomena emerge at low intensity. In all cases the direct identification of − κT µν := G µν = f ( Y, ˆ v ) implies that the field equation of general relativity is faith-fully reproduced, even for the case of a single particle state exhibiting the underlyingquantum behaviour.While a solution G µν = f ( Y, ˆ v ) envelopes the full 4-dimensional system de-picted in figure 11.13(b), including the macroscopic apparatus, the indeterminacy ofthe single particle process lies in the perfect symmetry of possible field solutions un-derlying the smooth function G µν ( x ) locally at the source S , which is the same for anypossible outcome. This situation is then very similar to that in figure 11.13(a), withthe source S corresponding to the interaction region, as represented by the rectangularbox, and with the range of outcomes A, B, C . . . corresponding to the angular range0 < θ < π . The comparison is even more direct if the intermediate double-slit screenis removed from the apparatus in figure 11.13(b).In all cases the full 4-dimensional solution G µν = f ( Y, ˆ v ) encompassing theentire system is intrinsically shaped through a 1-dimensional causal accumulation ofprobabilistic outcomes wherever the geometry G µν ( x ) is locally expressible in termsof a degeneracy of underlying field functions. The inclusion of the double-slit screenin figure 11.13(b) is accompanied by a more complicated spectrum of single particlesolutions, as might be expected since the full system is more complicated. In this casethe relative probabilities, while depending crucially on the underlying field degeneracyat S , turns out to be weighted by the interference pattern as shown.The spacetime curvature itself is far too small to be directly detectable, forexample by geodesic deviation, although the fact that G µν ( x ) is non-zero in theselaboratory experiments is crucial in the present theory. As described above the localspacetime curvature associated with the interaction region in figure 11.13(a) will beanalogous to that for the free electromagnetic wave as derived in equation 11.12 andpictured in figure 11.1. This curvature will naturally be higher in cases of higher energydensity such as at the interaction region of the LHC, where it remains also far too smallto be observable. 338ith T µν := G µν the spacetime curvature is also indirectly made apparentthrough the presence of energy-momentum. For example, since energy-momentumis everywhere conserved in line with the identity G µν ; µ = 0, a small recoil of theelectron source S in figure 11.13(b) will causally precede the detection of an electronat A . Indeed, in principle ‘elementary’ particles might be observed with detectorsin a way analogous to ‘Brownian motion’ with macroscopic matter ‘recoiling’ againstthe elementary transitions of the fields within which it is immersed, bringing out theproperties of both the particles and material objects themselves.Considering a thought experiment with a very lightweight source S situatedat a very long distance from the detector screen in figure 11.13(b) in principle anobservation of the momentum recoil of S could precede the detection of the signal at A (and for apparatus consisting of the source and screen alone the prediction of thehit location on the screen would be very direct). The total momentum of the system,including the source, double-slit screen and detection screen, will be conserved. Thesame quantum interference pattern would still appear on the screen given a largenumber of such events.With the momentum recoil of the macroscopic source S too small to be measur-able in practice for the process depicted in figure 11.13(b) the first and only sign of theevent will be through the amplification of an electronic signal at A . Pragmatically thepossible observable outcomes can be represented in terms of an electron wavefunctionΨ( x ) evolving according to Schr¨odinger’s equation until collapsing to zero at B, C . . . at the moment when the electron is detected at A . In the present theory such a de-scription in terms of an apparently non-local action of wavefunction collapse representsour knowledge of the state of the system rather than its underlying physical evolution.In standard non-relativistic quantum mechanics the basic principle of the con-servation of energy and momentum is considered to hold together with the constraintthat no signals may be transmitted faster than light. For the case of the experimentdepicted in figure 11.13(b) this leads to the question concerning the location of the‘conserved energy’ during the intermediate period between the emission of a particleof a given energy at S and the later detection of a particle of the same energy at A . The corresponding energy-momentum cannot be carried by the wavefunction forexample, due to the discontinuous nature of the wavefunction collapse.In the present theory the ‘energy-momentum’ is distributed throughout interms of the 4-dimensional geometry T µν := G µν . Energy-momentum conservationis everywhere implied in the identity G µν ; µ = 0, with nothing being transmitted fasterthan the speed of light – as defined by the light cone structure which arises through theprojection of the full form L (ˆ v ) = 1 onto the manifold M as described in section 5.3.Since a solution G µν ( x ) = f ( Y, ˆ v ) primarily describes the shape of a particular space-time geometry it may have a highly counter-intuitive distribution when interpreted through T µν := G µν as an apparent flow of ‘ matter ’ through space . Some forms of ge-ometry do possess a form with a natural interpretation in terms of energy-momentum,as expressed for the macroscopic example in equation 11.50 for a pressureless fluid.However, more generally rather more arbitrary geometries are permitted, provided G µν ; µ = 0, and some form of continuous geometry G µν ( x ) will be associated with thesingle particle process depicted in figure 11.13(b).For the present theory the question concerns the manner in which everywhere339ontinuous solutions for a geometry G µν ( x ) can be apparently channelled in certain discrete and localised ways which give the impression of ‘particle’ transitions. Thatis, locally the energy-momentum T µν := G µν can be interpreted as the emission ordetection of a discrete particle, for example at S or A respectively in figure 11.13(b).The answer presumably lies in the nature of the fixed constraint equations 11.29, suchas D µ L (ˆ v ) = 0, which channel the underlying field redescriptions in a limited numberof ways and in turn determine the properties of the particle transitions which emerge,as alluded to in the previous section.All empirical phenomena, whether naturally occurring or constructed in phys-ical experiments such as that in figure 10.1, will be enveloped under a geometry G µν ( x ) = f ( Y, ˆ v ). Since in the laboratory setting this non-trivial geometry, thatis any deviation from Minkowski spacetime, is completely unobservable the conse-quences of this perspective may be pursued by considering more extreme cases, suchas the thought experiment described earlier for figure 11.13(b) with a very lightweightsource S far removed from the detector screen, as well as by analysing phenomenaphysically realised in practice. Any beam of electromagnetic radiation carries energyand is hence associated with spacetime curvature as for the case of standard generalrelativity and as depicted in figure 11.1 for example. In a further thought experimentintense beams of light, for example produced by lasers, could in principle be config-ured such that the tiny geodesic deviation of a suitable test body projected throughthe curved spacetime associated with the laser beam and over a large distance mightbe observed, without any photons of the beam being detected.This situation may be contrasted with the empirical observation of the deflec-tion of light itself in the gravitational field of the sun, as first reported just a few yearsafter the formulation of general relativity. In these cases, for both the above thoughtand practical experiments, there is an ‘interaction’ between light and gravity without the detection of any photons or the need to appeal to any properties associated withquantum theory. For the above thought experiment similar observations would holdfor an intense beam of particles such as electrons in place of the lasers, and leads tothe conclusion that the electron field associated with the event of detecting even singleelectron in figure 11.13(b) will indeed be accompanied by a small, although utterlyundetectable, spacetime curvature.As described in section 11.2 the spacetime curvature G µν ( x ) is always a realvalued tensor field but may be constructed out of a hybrid of complex components, suchas the e ± ik · x Fourier modes, of the underlying fields such as A µ ( x ) and ψ ( x ) as depictedin figure 11.7 for example. The matching of both the e − ik · x and e + ik · x parts comingtogether into a real-valued function for a single field such as ψ ( x ) may correlate withdetection events, such as at A in figure 11.13(b), through which apparently propagatingparticle states, such as electrons, are revealed. More generally the concept and natureof elementary particles needs to be fully addressed, as was discussed in the previoussection and will be further elaborated in section 15.2.While the understanding of the nature of particles as observed in the laboratoryrequires further work, it is clear in the present theory that there are no ‘graviton’ statessince the gravitational field itself is not quantised in any sense. In fact general relativityprovides a classical description of the geometry of the external perceptual frameworkof the world which fully accounts for the phenomena of gravitation. There is no given340 at G µν = − κT µν ( Y, ˆ v ), that is equation 11.49above. Only the right-hand ‘matter’ side of this expression is effectively quantised,as a consequence of the degeneracy of field redescriptions, involving interchanges ofgauge and fermion fields for example, which underlie the solution. To attempt toimpose ‘quantisation rules’ on the left-hand ‘geometry’ side of this expression wouldbe to quantise the same object G µν ≡ T µν twice in two different ways. The externalgeometry G µν ( x ) itself implicitly incorporates a choice of T µν ( Y, ˆ v ) and effectivelythe identification of this 4-dimensional spacetime geometry is itself the mechanismof quantisation for all non-gravitational fields. That is, the possibility of multiplesolutions of the form G µν = f ( Y, ˆ v ) underlying the external geometric framework forperception of objects in the world is the reason why the fields implicit in the energy-momentum tensor T µν := G µν are quantised, with no similar argument applying tothe degrees of freedom of the spacetime geometry itself.Quantum field theory is however formulated against a flat spacetime back-ground and we may also consider the corresponding limit for the present theory. Forthe respective theories of general relativity and quantum fields the geometry of thespacetime manifold and that of the internal gauge fields are independent construc-tions. In the present unifying theory the relation between them is identified througha larger, all encompassing, symmetry group ˆ G for the full general form of the flow oftime L (ˆ v ) = 1, linking the external and internal forces of nature, as for example seen inequations 5.20 and 11.1. Their distinctive, complementary, features arise in the break-ing of the full symmetry group over the base manifold M , as depicted in figure 5.1for the L ( v ) = 1 model. Considering the full forms L ( v ) = 1 and L ( v ) = 1 inturn the surviving local gauge symmetry and resulting field interactions in this theoryhave been compared with corresponding features of the Standard Model in chapters 8and 9. The properties of these interactions will be drawn out and made apparentthrough discrete particle phenomena, which themselves can only be fully explored inthe present theory when the associated minute deviations from a flat geometry arefully embraced, as described above and in the previous section.In a curved spacetime there are generally no preferred choices of Lorentz framesand through the local freedom in l ( x ) ∈ SO + (1 ,
3) general relativity can be interpretedas having some relation to gauge theory, as alluded to towards the end of section 3.4.However, in the limit of a flat linear connection, with Γ( x ) → TM , with a single choice of l ∈ SO + (1 , l ( x ) ∈ SO + (1 ,
3) has effectively been broken to the much morerestricted freedom of a global symmetry on M .341n fact, in the spirit of the present theory as introduced in section 2.2, therequirement of perception implies that the local Lorentz symmetry freedom of thelocal reference frames as a function of x ∈ M acts globally over macroscopic scales toa good approximation and hence is essentially broken down from a local to a merelyglobal symmetry, and hence with far fewer degrees of freedom. This is the reverse ofthe usual case seen in gauge field theories in which a global symmetry is generalised tobecome a local symmetry leading to the interactions described in the Standard Modelof particle physics for example.It is this assumption of what is essentially a hole in the full symmetry ˆ G of L (ˆ v ) = 1 carved out by the global SO + (1 ,
3) symmetry on M that allows thedeployment of a global Minkowski coordinate frame on the base manifold that in turnallows the expansion of each field as a sum over the linearly independent functionsof a Fourier series on the base manifold, as described for the electromagnetic fieldfor example in equations 11.3–11.6. The question concerning the relation between theinteractions of such fields in the present theory in this limit and calculations performedin quantum theory has been considered in the previous sections of this chapter and isfurther elaborated in the following.Generally in physics there are numerous examples in which observable quan-tities parametrised by real numbers are analysed through expressions involving thealgebra of complex numbers. To take a simple example an oscillating quantity suchas the electric current in a wire of the form I = I cos ωt ∈ R can be expressed as I = I Re( e iωt ) ∈ R . The straightforward mathematical properties of objects such as e iωt ∈ C , under multiplication and differentiation for example, may then be exploitedin calculations before the underlying physically real (in the sense of ‘existing’) part isextracted in terms of the mathematically real (in the sense of R ) part at the end ofthe calculation.In quantum theory complex analysis is used directly from the foundations. Viaeither canonical quantisation or the path integral approach as the starting point forQFT, Feynman rules and the complex transition amplitude M fi may be constructedon the way to extracting real-valued cross-sections or decay rates at the end of a calcu-lation. Similarly the postulates of non-relativistic quantum mechanics are couched interms of complex mathematical objects from the beginning – with a complex wavefunc-tion Ψ( x ) or state vector in a Hilbert space completely defining the dynamical state ofa quantum system and empirical predictions obtained in terms of the real eigenvaluesof Hermitian operators.These structures for quantum theory appear quite distinct from other applica-tions of complex analysis in physics which, as for the example of the electric current I = I Re( e iωt ) above, begin with real-valued quantities. In this sense, by comparison,quantum theory appears to hang in the air, apparently lacking a more tangible concep-tual foundation. The present theory aims to supply such an underlying physical basisfor quantum theory in terms of the relative frequency of possible solutions for fabri-cating the 4-dimensional spacetime M itself, with the geometry G µν ( x ) = f ( Y, ˆ v ),as provisionally expressed in terms of the probability P ∝ D + D − of equation 11.46.Linked to the QFT probability |M fi | via the structure of the real-valued quantityIm( M ii ) and the optical theorem, as discussed for figure 11.8 and summarised inpoints ‘1)–7)’ of section 11.2, the aim is to build the theory up from beneath QFT342hrough a complexification of the underlying probability computation, which is basedon the degeneracy of solutions, on the way adopting some the mathematical machineryof QFT itself.Historically QFT was developed in the late 1920s on the coat-tails of the origi-nal quantum mechanics by promoting the wavefunction to an operator field (which wassometimes called ‘second quantisation’, although there is still only one quantisation).By Fourier analysing the vector potential A µ ( x ), as a free-field solution of Maxwell’sequations, into normal modes and applying a quantum mechanical harmonic oscilla-tor treatment to each mode independently photons, as quanta of the electromagneticfield, were the first ‘particle states’ to be studied in a QFT. Since in the present theorywe began by making contact with QFT in the environment of HEP experiments theconnection in the other direction, with quantum mechanics arising as a limit of QFT,should also be considered. For example, the retarded propagator or Green’s functionof QFT, which enters the present theory as described for equation 11.22, can be takento the non-relativistic limit in which it is found to be identical to the transition am-plitude for single particle transitions in quantum mechanics. (The possibility of sucha connection can be inferred from the relation of the function ∆ R ( x − y ) to ∆ + ( x − y )through equations 10.75 and 10.79 and the structure of ∆ + ( x − y ) in equations 10.58and 10.60 in comparison with equation 10.8 as discussed after equation 10.63).While the path integral approach has not been found useful for establishingthe link between the present theory and QFT, as alluded to before equation 10.46 insection 10.3, the relationship between QFT and QM is perhaps most readily seen interms of the path integral approach for which the same basic postulates apply in bothcases. The transition amplitude K is treated as a fundamental object and identifiedwith the sum over ‘all possible paths’ of a phase factor e iA/ ~ , where A is the classicalaction associated with the path (see for example [10] chapter 8). The mathematicalproperties of this phase factor are exploited in the structure of the theory, with thetransition probability postulated to take the form P = | K | such that the basic lawof probability conservation is upheld. Both QFT, with a mathematical formalismfor generating expressions associated with Feynman diagrams through higher-orderfunctional derivatives, and the single particle theory of QM for the non-relativisticcase, which can be generalised with higher-order Green’s functions to describe multi-particle systems, may be derived from the path integral approach. The two cases ofspontaneous symmetry breaking, in condensed matter physics and the Higgs sector ofthe Standard Model, alluded to in the previous section may also be described in verysimilar ways mathematically using the path integral approach, which is otherwise hereseen as a useful formal method of performing calculations rather than relating to theconceptual basis for the present theory.In the present theory objects such as the Schr¨odinger wavefunction Ψ( x ) in QMand transition amplitudes, such as M fi for QFT, are also considered as mathematicalconstructions for pragmatic use in the relevant calculations of real observable quantitiessuch as event probabilities. As complex representations none of these mathematicalobjects directly represent physical entities such as fields or particle states, althoughcomplex Fourier modes of the fields have been employed in the degeneracy count asrepresented for example in figure 11.7.For the present theory the fundamental objects in spacetime are the real-valued343elds directly drawn out from the components of L (ˆ v ) = 1, and its symmetry actions,over the base manifold M , as described in the opening of this section. The energy-momentum possessed by such fields is strictly defined through T µν := G µν = f ( Y, ˆ v ),as described throughout this chapter, in terms of the 4-dimensional spacetime geom-etry description via the Einstein tensor effectively composed of the underlying fields.Field interactions and transitions follow from the degeneracy of possible solutions.This definition of energy-momentum in T µν is independent of the field content, apply-ing to the quantum as well as classical physics case, with 4-momentum conservationcorresponding in all cases to the identity G µν ; µ = 0.While in the present theory we begin with the form L (ˆ v ) = 1 and then identifythe spacetime geometry over an extended manifold M , in QFT the starting point isa flat spacetime manifold itself. From this point of view in standard quantum theorythe presence of energy-momentum T µν = 0 alongside the flat spacetime assumption G µν = 0 not only directly contradicts the central field equation of general relativity butalso evades any possibility of a unifying theory of quantum mechanics with gravitation.That is, since in QFT a flat 4-dimensional spacetime is a given background arena theconceptual origin of indeterminate quantum phenomena as a degeneracy of solutionsfor the underlying spacetime structure itself is entirely missed.Hence in quantum theory wavefunctions and amplitudes are introduced at theoutset and unitary symmetry imposed in order to model the probabilities of suchphenomena. This approach dates back to matrix mechanics, presented by Heisenberg in1925, in resorting to a mathematical framework aimed at coherently linking observablephenomena without an underlying conceptual and physical motivation as the basis.On the one hand the present theory provides such an underlying basis for quantumphenomena in terms of a degeneracy of field solutions for the spacetime geometry, andon the other hand it should also be able to account for the original quantum mechanicsof Heisenberg and Schr¨odinger in the non-relativistic limit.Although the curvature of the spacetime geometry G µν = − κT µν = 0 is unob-servably small by many orders of magnitude on the scale of atomic or HEP phenomenathe standard equations of quantum mechanics and QFT, in assuming a flat spacetimebackground, do depend on the existence of a smooth continuum of spacetime points x ∈ M with the structure of a global Minkowski metric η ab on the manifold, sincethis is required to give meaning to the location of wavefunction or operator field val-ues in these theories. In the present theory this continuum takes the full metric form g µν ( x ) of general relativity, which is determined by the fields themselves, describing aspacetime which is only approximately flat.In the mathematical formalism of QFT the points x of an independent flatspacetime background are mapped into operators x → ˆ φ ( x ) which are defined by theiraction on the states of the system. Quantisation rules are imposed on the dynamicaldegrees of freedom of the field ˆ φ ( x ) itself, as described for equations 10.13 and 10.16 insection 10.3, while the spacetime location x is simply a parametrisation for the field interms of a set of real number coordinates { x } . In non-relativistic quantum mechanicsthe operator ˆ x a , appearing in the discussion below equation 10.78, represents thespatial location of a particle state, while there is no operator corresponding to time.However in QFT, which is invariant under the transformations of special relativity,there are no operators corresponding to either time or space. Since these quantities344re clearly ‘observables’ quantum theory as it stands is not a universal theory, ratherextended spacetime provides an ‘external’ background arena for QFT, as it does forclassical mechanics in the non-relativistic limit.General relativity is the theory of external space and time and is itself not atheory standing in need of quantisation, either on empirical or necessary theoreticalgrounds. While some approaches to ‘quantum gravity’ seek to include gravitation andspacetime geometry within the framework of an extended quantum theory, here inthe present theory the phenomena of quantisation arise beneath the surveillance ofgravitation, with the geometric degrees of freedom associated with general relativityhence outside the domain of quantum theory. As a consequence, for example, thereare no gravitons for this theory, as discussed earlier in this section.The concept of time plays a central role both in relativity theory and in quan-tum mechanics. In general relativity the proper time, with the interval dτ of equa-tion 5.48 for infinitesimal displacements, can be used to parametrise a series of eventson the manifold, such as those that map out the spacetime trajectory of a physicalobject with 4-velocity flow u µ = dx µ /dτ as described in section 5.3. In quantum me-chanics the temporal evolution of a state is determined by the Hamiltonian operator H (as introduced before equation 10.27), which also describes the energy of the state,with the wavefunction Ψ( t ) for example in the time-dependent Schr¨odinger equationsatisfying: i ∂∂t Ψ = H Ψ (11.51)(as exemplified in equation 10.28 for the evolution of the state vector in QFT, andalluded to after equation 10.52 with H ≡ ˆ E in quantum mechanics). One of the maindifficulties with background dependent approaches to quantum gravity that apply thesuperposition principle of quantum theory to spacetime geometries, or make quantumtransitions from one to another, is that, owing to the principle of general covariance ingeneral relativity, there is no well defined way to map points in one spacetime to thoseof another. Labelling the points with coordinates does not help since under generalcovariance coordinates are of no physical significance, as described in section 3.4. This,in particular, means that there is no unique way to specify a map from a temporalderivative on one spacetime manifold to a temporal derivative on another. Hencethe temporal evolution of a quantum state in equation 11.51 cannot be transferredin any meaningful way between different spacetimes. This absence of a well definedindependent temporal parameter is known as the ‘problem of time’ in quantum gravity.It is a problem which does not arise within the present theory since here gravityitself is not quantised and there is no ‘superposition of spacetimes’. As for classicalgeneral relativity, here the emphasis is on complete four-dimensional solutions for thespacetime geometry satisfying G µν = f ( Y, ˆ v ), with indeterminacy and the probabilis-tic nature of quantum phenomena inherent fundamentally in the degeneracy of manypossible field solutions which underlie the world geometry. Hence there is no difficultyin identifying a universal one-dimensional time parameter (such as the proper time inthe local frame of any given observer) and the ‘problem of time’ does not arise here,as it does for theories which place the temporal evolution as conceived in a quantumtheory at the forefront. Rather here quantum effects arise underneath gravity, andcan be consistently parametrised in terms of coordinates on the single spacetime back-ground of perception, with the ready availability of unambiguous temporal derivatives.345he structure of quantum theory is hence fused within the structure of general rela-tivity, with 4-dimensional spacetime infused throughout with a 1-dimensional causalprogression in time as employed for example in equation 11.51.Geometric structures, including the causal structure of spacetime, are describedby degrees of freedom expressed in the metric g µν ( x ) and tetrad e µa ( x ) fields, consis-tent with the Riemann tensor R ρσµν ( x ), on a single spacetime manifold M . Indeed,this manifold is itself constructed out of the more fundamental underlying notion ofprogression in temporal flow s as expressed through L (ˆ v ) = 1. The projection ontothe manifold results in relative time dilation phenomena as described in section 5.3.In the 4-dimensional continuum each observer carries a clock which provides a timeparameter which may be applied in quantum experiments in the laboratory or for ob-servations in general within the spacetime arena; with temporal parameters for mutualobservers simply related by relativistic transformations.The probabilistic nature in terms of what can happen as the outcome of a lab-oratory experiment involving quantum phenomena motivates the construction of thequantum state or wavefunction locally parametrised through a 1-dimensional progres-sion in time according to equation 11.51 for each local observer and the employmentof the associated quantum theoretical tools.In the present theory we begin with real fields such as A µ ( x ) of equations 11.3–11.6, with a complex decomposition into parts such as e − ik · x which seem to resemblea complex wavefunction Ψ( x , t ). Indeed, similarly as applied to the positive frequencymodes of the quantum field component ˆ φ + ( x ) as described after equations 10.51 and10.52, the Hamiltonian operator H of equation 11.51 can be applied to the complexcomponent of the classical electromagnetic field in equation 11.17 resulting in: H A µ ( x ) = k A µ ( x ) (11.52)with eigenvalue k . This is identical to the energy of the normalised electromagneticfield within the volume V as described following equation 11.16, which was extractedthrough the relations − κT µν := G µν = f ( A ). Equation 11.52 exemplifies how the‘operator plus wavefunction’ description can offer a concise way to extract propertiessuch as the 4-momentum from the field as a useful tool for calculations. Althoughthe real-valued field A µ ( x ) of equation 11.6 precisely describes the actual field, if itmay be reconstructed uniquely as the ‘realification’ of a complex component such asequation 11.17 (that is, by adding that equation to its complex conjugate) then thelatter in principle carries all the information concerning the real physical field, and alsosatisfies the same equation of motion as the real field as described after equation 11.20.However, in quantum mechanics the wavefunction Ψ( x ) represents a single ob-servable particle, applying to a physical electron state in figure 11.13(b) for example,and it remains to be seen explicitly how such particle states may be described in termsof underlying fields and their interactions in the present theory. Hence an understand-ing of field renormalisation and the nature of observable particles, as discussed in theprevious section, will need to be further developed in order to establish the full con-nection between the mathematical objects of the present theory and the pragmaticdevices of quantum theory.While representing a single particle a wavefunction Ψ( x , t ) is in general a con-tinuous function of the spatial coordinates with intrinsically non-local properties in346erms of the temporal evolution as a measurement is made and therefore exhibits anon-particle-like structure itself. In a measurement of position the quantum particle isobserved to be in a particular spatial location which is determined by the wavefunction,to the extent that the squared modulus of this complex function | Ψ( x , t ) | determinesthe probability to detect the particle at that location, as alluded to after equation 10.8,with the wavefunction immediately ‘collapsing’ to that measured point. While carryinginformation concerning various physical quantities, when combined with the appropri-ate quantum mechanical operator, the wavefunction Ψ( x ), unlike the field A µ ( x ) doesnot itself carry energy or any other physical attribute and hence there is no physical discontinuity for the experiment of figure 11.13(b) when the electron is observed ata particular location. Rather, in the interpretation of the present theory, the energyis contained in components of T µν := G µν which is continuous everywhere in theseexperiments, as described earlier in this section.From this point of view the quantum mechanical wavefunction reflects our best knowledge of the range of world solutions our current empirical situation is consistentwith; it evolves in a determined way U through passage of laboratory time, as gov-erned by equation 11.51 for a given Hamiltonian operator, in such a way as to yieldprobabilistic predictions for which particular solution state we shall find ourselves ob-serving at the time of the next measurement (see for example [26] p.592). Since thewavefunction is a non-physical entity, the so-called ‘collapse’ or ‘reduction’ R of thewavefunction merely represents the change in our knowledge when such an observationis made, and is not itself a constituent property of the physical world.Hence the apparent conceptual difficulties concerning ‘wavefunction collapse’are a somewhat artificial feature of quantum mechanics since the change in evolutionlaw from the unitary U for the wavefunction Ψ, describing a superposition of states,to reduction R selecting an eigenstate Ψ i in the measurement, is just a pragmaticdevice for calculation (similarly as for the employment of the transition amplitude M fi in QFT ) and does not directly describe the behaviour of a physical entity, suchas represented by the gauge field A µ ( x ) for example.From the perspective of the subjective laboratory view with a sequence of eventsseemingly evolving in time upon a given 4-dimensional background manifold somequantum phenomena appear mysterious, such as the ‘spooky action at a distance’ aspredicted and observed for Einstein-Podolski-Rosen (EPR) experiments. Such exper-iments demonstrate that quantum phenomena cannot be accounted for by an under-lying theory which is both local and deterministic, as constructed in terms of ‘hiddenvariables’ for example. In the present theory however the phenomena of EPR corre-lations and quantum entanglement in general are all sown into the fabric of the full4-dimensional spacetime solutions under the geometry G µν ( x ). These observationsare hence in principle naturally accounted for without any non-local interactions orbehaviour and without hidden variables but with indeterminacy fully embraced as amanifestation of the local degeneracy of possible fields, including the gauge field A µ ( x )and fermion field ψ ( x ), necessarily featuring in solutions for G µν = f ( Y, ˆ v ). Herethe underlying fields such as A µ ( x ) are intimately involved in the construction of thespacetime geometry, rather than introduced separately as classical waves spreading outover a pre-existing M background.The local causality in the present theory incorporates the restriction that sig-347als cannot propagate faster than the speed of light, with special relativity holdinglocally as for general relativity. (In principle a form of the ‘equivalence principle’, asdescribed in section 3.4, might be adopted, but the employment of a ‘torsion-free’ ex-ternal geometry is a simplifying and provisional assumption both for general relativityand for the present theory, as discussed in section 5.3 and also section 13.3). Here‘causality’ means of course that the range of probabilities for possible future states,and not the actual future state itself, is determined locally by the present state.Although the local redescriptions of the fields such as depicted in figures 11.4,11.6 and 11.7 are arbitrary within the constraints of equations 11.29 the overall theoryis ‘deterministic’ in the sense that all possible worlds, all solutions, potentially exist.On the other hand events in the single solution of our world do necessarily appearindeterministic – ‘God does play dice’ from the point of view of observations in ouruniverse.In the case of Schr¨odinger’s famous thought experiment the outcome can onlybe to perceive an alive or a dead cat ([26] p.808), while an entity described by thequantum state ‘ | alive i + | dead i ’ cannot be observed. The present framework incor-porates a theory of perception through which each of the two possible macroscopicstates corresponds to a separate G µν ( x ) world, each necessarily observed subject to G µν ; µ ( x ) = 0 and constructed out of the flow of time with L (ˆ v ) = 1, such that we can-not perceive both large scale states simultaneously since they describe different worlds.The more practical experiments with an electron being detected at A, B, C . . . in fig-ure 11.13(b), or the muon detected at an angle θ figure 11.13(a), are associated witha spectrum of different worlds.Whatever the relative probability of the two alternative outcomes as deter-mined by the apparatus of a ‘Schr¨odinger’s cat’ type experiment, it is possible toconsider two sets of worlds each of which consists of a ‘coarse-grained’ ensemble char-acterised by one of the two possible outcomes. More generally we inhabit one of a muchlarger ensemble of possible worlds, each distinguished by the resolution of a vast num-ber of locally indeterministic processes intrinsic to the 4-dimensional world solutions.With the range of worlds resulting from the many ways to construct G µν = f ( Y, ˆ v )over a 4-dimensional base manifold each solution, each universe, is as real as ours.(This statement carries the caveat that each universe should support observers, in thesense described in chapter 14).The availability of ‘many solutions’ for G µν ( x ) in spacetime responsible for theindeterminacy in such experiments is reminiscent of the ‘many worlds’ interpretationof quantum mechanics. However, here the theory has many solutions by nature, thisfeature is not an interpretation of the theory. In the many worlds interpretation ofquantum mechanics the wavefunction is taken literally as a real entity with the aboveobservations of both ‘an alive and a dead cat’ effectively interpreted as a bifurcationof our world as one of many such divisions in a ‘branching universe’. In the presenttheory the other worlds might be thought of existing ‘out there’ in a realm of pos-sible mathematical solutions, unlike the more intimate picture of the many worldsinterpretation.Here there is also no essential observer participation in ‘wavefunction collapse’in the sense of the ‘many minds’ interpretation of quantum mechanics, rather thewavefunction, as a non-physical entity, is our own pragmatic construction employed to348redict the likelihood of future events. On the other hand in the present theory theobserver does have an innate role in shaping the overall theory through the subjectivenature of perception on the base manifold, which implies the breaking the full L (ˆ v ) = 1symmetry and the ensuing physical structures. This perspective is influenced by theKantian philosophy concerning the a priori nature of perception in the form of space,time and causality, as will be further elaborated in chapter 14 and in particular in theopening paragraphs of section 14.2.During the early history of quantum mechanics the meaning of the formalismin terms of the ‘Copenhagen interpretation’, was a natural, pragmatic and provisionalway of addressing the conceptual difficulties raised. This also marked a relativelyconservative break away from the world of classical mechanics, combining the quantumwith the classical aspects of the world in a way that upheld the classical behaviour ofexperimental apparatus and the classical notion that physics exclusively studies theproperties of a single universe, although now, however, one with an intrinsic elementof uncertainty. While the postulates and mathematical structure of quantum theoryhas remained essentially intact and unchanged since the 1920s, the debate over the interpretation of the theory continues into the 21 st century.The main difficulty with the Copenhagen interpretation is the ‘measurementproblem’ concerning the grey area of interface between classical apparatus and thequantum system under investigation and the nature of the apparent ‘wavefunctioncollapse’. This issue is highlighted by the ‘Schr¨odinger’s cat’ thought experiment andhelped motivate the later many worlds interpretation alluded to above. In the presenttheory the measurement problem is resolved through the seamless employment of aclassical notion of probability, defined in terms of the number of ways an event canhappen, all the way down from the macroscopic apparatus to the underlying micro-scopic field redescriptions. This theory hence unifies the notion of probability for theclassical and quantum domains, as applies for example to the experiments depicted infigure 11.13.As well as having a common underlying origin the meaning of the probability ofan outcome for a quantum process (involving for example an experiment in figure 11.13or the fate of Schr¨odinger’s cat) and for a classical process (such as the roll of a diceor the toss of a coin) is subjectively the same, in terms of for example how we mightmake choices dependent upon such outcomes. In both the quantum and classical casesthe outcome probability is calculated based on our knowledge of the set-up of thesystem before the experiment is performed. However there is also a significant objectivedifference in the nature of quantum and classical chance even in the context of thepresent theory. The difference is that in quantum theory the outcome is fundamentallyunknowable in advance, whereas for the classical case the probability merely representsthe practical limitations of our knowledge and our ignorance of the precise details of theinitial conditions. The actual outcome of such classical experiments would in principlebe calculable and fully determined if we could gather sufficient data and muster thenecessary computational power (the improving accuracy of weather forecasting withimproving technology provides an example). On the other hand, although in themany solutions there are many worlds and essentially everything that can happen does happen in some universe, quantum phenomena from our perspective in our world areobjectively and inherently indeterministic.349or a given observed event, for a process such as e + e − → µ + µ − picturedin figure 11.13(a), the question can be asked whether a particular sequence of fieldexchanges actually mediates the process between the initial and final states. In termsof the field sequence ψγ µ ψ → ϕγ µ ϕ in figure 11.6 for example this corresponds tothe question of whether there are particular values for t , t , t and t , whether theintermediate ψγ µ ψ field state represents a µ + µ − , d ¯ d or other fermion pair between t and t , and the value of the corresponding unconstrained internal 4-momentumdegrees of freedom. In turn there is an endless list of possible field sequences, withfield exchanges separated by intervals of time down to δt → distinct descriptions of the overall process that contributes to the total probabilityto observe the event which is statistically measurable. In a similar way that one par-ticular outcome of many possibilities is observed, such as a µ + µ − or τ + τ − final stateat an angle θ to the incoming e − beam in figure 11.13(a), from a philosophical pointof view it is consistent to think of the internal process as following one particular se-quence, such as via a µ + µ − or d ¯ d internal fermion state in figure 11.6 for example, withparticular values for the continuous degrees of freedom described above. (Althoughsince there is an endless number of infinitely nested possible field redescriptions, asalluded to in the previous section, the idea of singling out ‘one’ such sequence may bepoorly defined). This is again analogous to the classical case in which the outcome ofthe roll of a dice, for example, is the result of one particular dynamical path taken bythe dice out of an infinite range of possibilities – a path which although not predictable is , however, observable to within practical limits of precision for the classical system.This interpretation is of course required to also be consistent with all observa-tions of quantum phenomena. These include interference effects, such as described infigure 11.13(b), apparently well accounted for in terms of a superposition of wavefunc-tions, which in turn feature in the course of the calculations involving complex numberalgebra, but which don’t individually generally represent a particular ‘way’ in which aprocess occurs. It will be necessary to trace a path from the many solutions picture ofdegeneracy in the present theory to the QFT Feynman rules for cross-section calcula-tions based on the amplitude M fi , through equation 11.46 as described in section 11.2,and further to the postulates of quantum mechanics and the construction of the wave-function Ψ( x ) for the non-relativistic limit, in order to see how such phenomena (andtheir quantum mechanical description) are compatible with the present theory.The QFT calculation for the event rate at an e + e − collider, for processes such asdepicted in figure 11.13(a), was presented in equation 10.7 and described in section 10.2.A doubling of the incoming luminosity, for example by doubling the bunch crossingfrequency f in equation 10.2, or a doubling of the available final state phase space,in the final term of equation 10.7, leads to a direct doubling of the observed eventrate. On the other hand on adding new intermediate processes interference betweenthe complex amplitudes M fi may lead to a reduction of the event rate. Indeed, inpractice the phenomenology predicted as a result of adding new hypothetical processesin such a calculation is sometimes studied in order to explain the observation of a lowerthan expected cross-section. However, according to the basic principles of the presenttheory the addition of new processes will only add to the ‘number of ways’ throughwhich to bridge an initial to a final state and always serve to increase cross-sections350nd decay rates.The question then may be asked how apparent interference phenomena arisein the present theory with probabilities based on degeneracy counts which alwaysaccumulate in a positive sense. However, it should be noted that there is no one-to-one correspondence between components of the degeneracy count D and contributionsto the transition amplitude M fi . Rather these two means of calculating the total probability are collectively related by a correspondence of the form of equation 11.46,which in particular implies a mechanism for normalising the degeneracy count througha complexification of the calculation.Interference phenomena in quantum theory are more explicitly presented inthe experiment of figure 11.13(b). As alluded to above this system can be analysedin terms of two wavefunctions, each emanating from one of the two intermediate slits,and added together to form the pattern of constructive and destructive interferencegenerating the probability distribution for events observed on the final screen. Againthere is no direct analogue of the ‘superposition of wavefunctions’ in the present theory,and again there is no one-to-one correspondence between wavefunctions and elementsof a degeneracy count.In the present theory such a degeneracy count is also not based on the ‘numberof ways’ in which an electron, as a particle state, could pass through the slits, but ratheron the number of underlying field solutions for G µν = f ( Y, ˆ v ) given the degeneracy offield redescriptions underlying the common geometry G µν ( x ) for the source S . Particlephenomena themselves arise as an apparent feature of these solutions. In fact, strictlyspeaking it is the phenomena of particle emission or detection, for example from thesource S or at the point A on the screen in figure 11.13(b), that emerge in thesesolutions, with no continuous trajectory of a particle-like entity ever observed. Onlythe particle-like interactions are ever actually directly recorded.Even for the events of sophisticated experiments such as depicted in figure 10.1the apparent ‘tracks’ of particles are reconstructed from a series individual detectorhits, in particular in a tracking chamber. ‘Joining the dots’ in this way creates anillusion of continuous particle trajectories, as was presumed for the incoming and out-going particle states sketched in figure 10.2 for example. The theory is hence requiredto explain how field solutions for G µν = f ( Y, ˆ v ) incorporate apparent particle emissionand detection phenomena, which in many cases create the impression of intermediateparticle trajectories – as an interpretation in part based on a close analogy with theproperties of classical particles. Since the effective local field interaction volume canbe arbitrarily small the associated elementary particle states have no apparent size,consistent with a point-like interpretation.In conclusion, for the present theory particle effects and the probabilistic natureof quantum phenomena generally arise out of the merging of two necessary featuresof the world. On the one hand the world we inhabit must be perceivable , as expressedmathematically in terms of geometric structures on an extended manifold such as M . On the other hand all such mathematical structures derive from a fundamentallyone-dimensional temporal progression which may be expressed in terms of a generalmulti-dimensional form L (ˆ v ) = 1 together with its symmetries. Resolving these tworequirements in a compatible manner leads to the equations of motion and physicalproperties of the tangible material world as perceived in spacetime and incorporating351he phenomena of ‘quantum mechanical’ transitions deriving from the degeneracy ofunderlying field solutions.While the underlying field components of L (ˆ v ) = 1 on M are in principlesubject to the full symmetry degrees of freedom of the multi-dimensional form of timethe geometrical interpretation needed to support the perceptual frame of the worldrequires the identification of a Riemannian geometry on the base manifold of theappropriate mathematical form with a lower symmetry. Here, as for quantum theoryin general, symmetry rather than scale is the key to quantum processes; although(as discussed shortly after figure 11.12 in the previous section) with a higher degreeof field symmetry more likely to be encountered on a ‘microscopic’ scale quantumphenomena are most frequently associated with such dimensions. The spirit of theprinciples of quantum mechanics is hence preserved in this new theory in unificationwith gravitation, with the identification − κT µν := G µν expressing the field equationof general relativity.The similar nature of the interplay between the larger symmetry and the brokensymmetry in the present theory to the situation in quantum mechanics can be exem-plified by the Zeeman effect. The energy levels of the hydrogen atom are split by thepresence of a uniform magnetic field, as a preferred direction in 3-dimensional space reducing or breaking the rotational symmetry of the system from SO(3) to SO(2).Passing a beam of electrons through a magnetic field configured to select a certainspin state provides a further example. Generally, in all cases of a measurement of aquantum mechanical system a structure of lower symmetry, such as the configurationof laboratory equipment, is imposed upon the intrinsically higher symmetry of theunobserved state.In the present theory quantum phenomena arise through the unavoidable a pri-ori imposition of the lower symmetry of 4-dimensional spacetime upon the general flowof time as a prerequisite for perception and observation in the world itself. Throughour innate faculty to organise and interpret our experiences in the world through acoherent global geometrical manifold M (playing the part of the directional magneticfield in the analogy with the Zeeman effect) the full E symmetry of L ( v ) = 1 is bro-ken down to the local external symmetry SO + (1 ,
3) together with the internal gaugegroup SU(3) c × SU(2) L × U(1) Y (which, as the surviving symmetries, collectively playthe part of SO(2) in the Zeeman analogy). However, while in the Zeeman effect themagnetic field direction is a particular choice of experimental setup, in perception theLorentz frame, within an approximately global SO + (1 ,
3) symmetry, is a necessaryform for all physical experience of the world and hence applies to all experiments andobservations.Further, while the SO(2) symmetry of the uni-directional magnetic field im-posed on a hydrogen atom with SO(3) symmetry results in a discrete splitting of theatomic energy levels, the surviving SO + (1 , × SU(3) c × SU(2) L × U(1) Y external andgauge symmetry of the 4-dimensional perceptual field imposed over the full flow oftime L ( v ) = 1 with an E symmetry will be correlated with a discrete set of possibletransitions of the microscopic world which determines the spectrum of elementary par-ticles. (Strictly speaking the ‘surviving symmetry’ is SO + (1 , × SU(3) c × U(1) Q sincethe electroweak symmetry SU(2) L × U(1) Y is itself broken down to U(1) Q through itsaction on the external spacetime components of the ‘vector-Higgs’ h ≡ v ∈ TM . As352escribed in section 9.2 the electroweak symmetry is also yet to be explicitly identifiedin terms of E generators). In general the resulting phenomena will be exhibited in theobserved properties of particles in HEP experiments as well as in the non-relativisticlimit of quantum mechanics itself, as exemplified in figures 11.13(a) and (b) respec-tively. While the physical structures of both gravitational and quantum theory areever present in nature it is possible to consider the limiting cases of the present theoryas applicable to the corresponding empirical observations. The limit in which classicalgeneral relativity emerges on the one hand and a complementary limit through whichan apparent quantum field theory emerges on the other hand can be described interms of two significant symmetries for our world with the external Lorentz group H = SO + (1 ,
3) (in the notation of section 2.3) as a subgroup of ˆ G = E , with thelatter being the symmetry of the full form of temporal flow L ( v ) = 1 as describedin section 9.2. These alternative limits can be characterised by the role of the linearconnection Γ( x ) on the spacetime manifold M , as described in table 11.1.Symmetry GR limit QFT limit H = SO + (1 ,
3) local symmetry on M global symmetry on M generally Γ( x ) = 0 can take Γ( x ) = 0 exactlyˆ G = E effective macroscopic matter local E / SO + (1 ,
3) symmetry T µν ( x ) := G µν ( x ) = f ( Y, ˆ v ) ⇒ Y ( x ) gauge fieldsTable 11.1: Limits in which general relativity and quantum field theory arise. Theemployment of a local or global freedom for Lorentz frames with l ( x ) ∈ SO + (1 ,
3) wasalso discussed earlier in this section in relation to gauge theory.The fact that GR and QFT emerge as almost exclusive complementary limitsis not surprising given the notorious incompatibility of the respective mathematicaltheories and difficulties in uniting them under a single framework. However there isnecessarily a trace of overlap even in the limiting cases. In the GR limit quantumeffects are always locally present underneath the effective energy-momentum tensorwhich describes the apparent matter distribution, with macroscopic material proper-ties shaped by the underlying quantum world. Similarly in the QFT limit particleinteractions are clearly associated with regions of matter density and hence a minutebut finite spacetime curvature is involved, which is a critical observation from theperspective of the present theory.As well as shedding light on the respective limits, the present theory may alsoaddress conceptual problems for physical systems where both gravitational and quan-tum effects are significant. For example the difficulties seen in some approaches toquantum gravity such as the ‘problem of time’, as described earlier in this section, andthe non-renormalisable nature of quantised gravity, as implied in the discussion follow-ing equation 10.86 in section 10.5, are avoided here since gravity itself is not quantised.While one aim of the present theory is to explore particle physics phenomena in theflat spacetime limit with Riemann curvature tensor components R ρσµν ( x ) →
0, as353n approximation to laboratory conditions to test the theory, the case for ‘quantumtransitions’ and ‘particle effects’ for R ρσµν ( x ) = 0, and in general for a highly curvedspacetime, is intended to be fully accounted for in this inclusive theory.The general form of the relation T µν ( x ) := G µν ( x ) = f ( Y, ˆ v ) in table 11.1will be applicable even in locations of the universe with extreme spacetime curvature,such as in the vicinity black holes and during the ‘Big Bang’ epoch. For example,an environment in which both gravitational and quantum effects are expected to besignificant arises for the phenomenon of the emission of Hawking radiation (1974) inthe highly curved spacetime in the proximity of a black hole, and similarly for theUnruh effect (1976) in which an observer undergoing a uniform high acceleration inthe ‘vacuum’ of a flat spacetime can detect thermal radiation. Quantum and particleeffects should be calculable with the present theory in such environments, and also forBig Bang cosmology – which will be discussed in the following two chapters.In QFT the Fock space representation is generally only valid for free fields inflat spacetime. The Fourier expansion of the field ˆ φ ( x ) in equation 10.13 relies on thePoincar´e symmetry of flat spacetime for the preferred basis of normal modes e ± ip · x and a corresponding preferred vacuum state | i . Particle excitations are built uponthis ground state via the operators a † ( p ) and a ( p ). In flat Minkowski spacetime onlyglobal inertial frames of reference are used for which the particle content of a state,implied in the Fourier components, agrees for all observers.This construction is not possible in curved spacetime for which the referenceframes of global coordinate systems are necessarily non-inertial. For QFT in curvedspacetime there is generally no unique set of normal modes, which results in differentinequivalent expressions of a particular QFT without a unique vacuum state, andthe particle interpretation in turn becomes ambiguous. Hence in general there is noobjective possibility of identifying either a vacuum or specific particle state for QFT ingeneral relativity. However, interference between normal modes expressed in differentgeneral coordinate systems has the physical consequence that real particles may becreated by gravitational fields.Indeed physical particle states produced by gravitational fields or, equivalently,by accelerated observers are in principle detectable and hence do represent real objec-tive phenomena which in principle should be consistently accounted for in a completetheory. Similarly the particle states observed in high energy physics experiments areempirical objective entities. In all cases the detection of particle effects hinges on thenature of particle or field interactions , without which the particles could not be ob-served. In the present theory it remains then to fully understand the nature of particlephenomena, and their apparent physical interactions in general, as emerging out of theunderlying interactions of fields, as represented by a degeneracy of redescriptions, aswe began to address in the previous section and will further consider in section 15.2in the discussion of figure 15.2.In the present theory the use of the Fourier transform expansion in equa-tion 11.6 is merely an effective approximation that arises in the limit of a flat Minkowskispacetime, and in which the apparent particle effects might most simply be analysed.Elementary particles are not fundamental entities out of which the world is built, theyare a robust phenomenon that arises in the flat spacetime (and near vacuum) limit,as alluded to in the opening of section 10.1 and as studied in experiments such as de-354icted in figure 10.1. The properties of ‘particles’ may be less robust in highly curvedspacetime, and more difficult to calculate than in the fixed limit of flat backgroundmanifold, but there is no fundamental conceptual difficulty.The field redescription ψγ µ ψ → A µ → ϕγ µ ϕ of figure 11.4 is presumed tobe locally enveloped in a spacetime geometry G µν ( x ) which takes a form resemblingthat of T µν := G µν in figure 11.1. If such an interaction takes place in the prevailingenvironment of a highly curved spacetime, for example in the proximity of a black hole,then to a certain extent all of the fields, ψ ( x ), ϕ ( x ) and A µ ( x ), will be ‘bent the sameway’ and hence processes such as depicted in figure 11.4 might be largely unaffected.Similar underlying field redescriptions in combination with immense gravitational tidalforces might then provide a description of black hole evaporation in the context of thepresent theory.The question can also be asked concerning the nature of phenomena for yetmore extreme spacetime curvature, such as in the region of a black hole ‘singularity’ orgenerally corresponding to a yet higher scale of energy. In the context of figure 11.10the GUT scale, at around 10 GeV, in marking a point of gauge coupling unificationought to be of significance for the present theory in terms of the phenomena of theinternal forces, while the external gravitational field will be treated in the same manneras for the low energy phenomena. Further, in the present theory gravity itself is not‘quantised’, there are no ‘graviton’ particles, and the Planck scale at around 10 GeVmay just be a dimensional quirk with no particular significance. Hence arbitrarily highenergy densities and arbitrarily high spacetime curvature might be considered in thepresent theory in a continuous manner without limit.In summary, from the point of view of the present theory the postulate in quan-tum theory that an event probability is determined by the square of the absolute valueof an ‘amplitude’, with unitary symmetry imposed to ensure the structure is consistentwith the basic laws of probability, should be considered as a provisional constructionstanding in need of an underlying conceptual basis and physical explanation. Such anexplanation would be preferred in place of any theoretical ‘postulate’, and here it lies inthe idea of the natural degeneracy inherent in the number of ways local field solutionsmay be found for G µν = f ( Y, ˆ v ) for processes such as those observed in figure 11.13and more generally.This is the key to combining general relativity and quantum phenomena ina single complete and unified theory. Indeed, given the prohibitive conceptual andmathematical difficulties encountered in attempting to unify these two pillars of 20 th theoretical physics it seems likely that a significant concept or postulate on at leastone side must yield some ground. Here the definition of probability in terms of am-plitudes in quantum theory seems a reasonable place for this, with the amplitudesand wavefunctions of quantum theory then representing calculational tools employedin an intermediate complexification of a computation. This approach has been exem-plified by unravelling the QFT event rate calculation of equation 10.7 and making thecase for replacing the contribution from the amplitude M fi by a quantity based on adegeneracy count D via the associations of equation 11.46.This foundation also unifies the notion of probability with the classical con-cept in the sense of essentially referring to the ‘number of ways’ that a process canoccur given a particular initial state or situation. However, while classical probabili-355ies concern the number of ways that things can happen in spacetime M , quantumprobabilities concern the more fundamental question of the number of ways in whichthe spacetime manifold M itself can be constructed with a world geometry describedby G µν ( x ) = f ( Y, ˆ v ). Further, in principle this approach to quantum phenomenaalso leads to a clarification of the meaning of ‘renormalisation’ as discussed for equa-tion 11.48 in the previous section.For theories which postulate extra spatial dimensions, such as the Kaluza-Klein theories described in chapter 4, our 4-dimensional spacetime world is containedwithin the larger space, for example as a 4-dimensional brane embedded within thehigher-dimensional bulk manifold or with the extra dimensions being ‘compactified’, asdiscussed in section 5.4. For the present theory founded on one-dimensional temporalflow the extended physical world is perceived through the structure and symmetries ofthe multi-dimensional form L (ˆ v ) = 1, with the degeneracy of solutions for constructingsuch a 4-dimensional world underlying the phenomena of quantum theory while theexternal spacetime geometry itself conforms with the structure of general relativity.As well as combining general relativity and quantum theory in a consistentframework within which the two theories are separately preserved in essence, the com-plete conceptual theory is based on sound intuitive principles, founded upon the everpervading multi-dimensional form of temporal flow L (ˆ v ) = 1 rather than upon seem-ingly arbitrary, mysterious or purely pragmatic assumptions. The theory should ofcourse also be able to make predictions and be found to be in full agreement withall observations. Such a correspondence has been initiated in chapters 8 and 9 withregards to comparison with the Standard Model of particle physics. Further, on incor-porating all physical scales, including that of HEP phenomena, in principle the presenttheory is expected to be profusely testable.All the underlying fields in nature, which underlie for example electron andphoton particle states, are in continual interaction through mutual indistinguisha-bility under the external geometry G µν ( x ) – from the interaction region of an HEPexperiment such as that in figure 10.1, to atoms and molecules, through to biologi-cal organisms, planets, stars and galaxies, with the underlying processes moulding asmooth and continuous geometry G µν = f ( Y, ˆ v ) with all the quantum phenomena em-bedded within and in turn, through T µν := G µν , shaping the structure and apparentphenomena of the material world.As well as the extreme environment of a highly curved spacetime alluded toabove, the complementary question concerning the nature of the ‘vacuum state’ canalso be considered. Even in the apparent vacuum, away from tangible physical matter,in general a form of G µν = f ( Y, ˆ v ) must be present throughout M in order to describethe spacetime geometry. This structure might in principle implicitly include a formof effective ‘vacuum energy’, incorporated into the spacetime geometry and describingthe effects of a cosmological constant Λ, at least to a good approximation, and hencein turn accounting for observations of the large scale structure of the universe. Indeed,more generally, as well as terrestrial laboratory phenomena the present theory has alsobeen developed with the cosmological scale in mind, and hence in the following twochapters we review aspects of cosmology in the context of the new theory.356 hapter 12 Cosmology
While the previous chapter focussed on the application of the present theory to thesmallest observable scales, regarding in particular the quantum field and particle phe-nomena studied in high energy physics experiments, here we return to consider generalrelativity and gravitation, continuing the thread from sections 5.2 and 5.3 in the lightof the intermediate chapters, as applied up to the largest empirically accessible scaleof the observable universe and beyond. In the context of the large scale structure of4-dimensional spacetime the right-hand side of equation 5.32 can generally be con-sidered to describe the effective macroscopic form of apparent matter terms, with − κT µν := G µν = f ( Y, ˆ v ) for this equation, that is in the GR limit as summarised intable 11.1, with the practical normalisation factor of − κ inserted. However an under-standing of the impact upon the spacetime geometry of the underlying microscopicfields and their interactions will also be directly relevant both for the universe at thepresent epoch as well as in its much earlier history. Indeed since the energy den-sity in the early universe reaches and surpasses that attainable in high energy physicsexperiments, the environment of the immediate aftermath of the Big Bang may itselfprovide a possible test arena for theoretical particle physics through any imprint whichthe corresponding phenomena may leave in the structure of the cosmos which is stillobservable today.In the following two sections we review some of the main features of standardtextbook cosmology, as deduced from and motivated by empirical observations. Inthe following chapter we then collect and describe a series of observations concerningthe present theory which, at a qualitative level at least, correlate with a number ofaspects of modern cosmology. Without making a quantitative argument in terms ofcosmological parameters these aspects include the dark sector of implied matter andenergy in the universe and the origin and nature of the Big Bang and the very earlyuniverse itself.The rather direct application of the conceptual scheme described in the previouschapters to the cosmological scale will first be outlined briefly in this section. This357pplication is possible since the general picture of the standard cosmological model ofthe evolution of the universe according to Einstein’s field equation of general relativity,given broad underlying assumptions concerning the large scale structure of spacetime,is naturally compatible with the present framework.Based on the translation symmetry represented in figure 2.2 we described insections 2.1 and 2.2 how the perceptual background of a flat SO(3) symmetric spatialmanifold M could be effectively derived through the structure and symmetries of theflow of time expressed in the form L ( v ) = 1 of equation 2.14. On extending thismodel to the case of a full ˆ G = SO(5) symmetry of L ( v ) = 1 projected over M , asdescribed for figure 2.7 in section 2.3, a finite external (and also internal) curvaturewas obtained. Subsequently a ˆ G = SO + (1 ,
9) model for the case of a 4-dimensionalspacetime M as pictured in figure 5.1 was described in section 5.1, again introducingminor distortions from a flat geometry corresponding to the effects of general relativity.These geometric distortions are presumed to be undetectable in everyday experience –that is out of the degrees of freedom of the full symmetry group of L (ˆ v ) = 1 projectedonto the base manifold M we require the local SO + (1 , ⊂ ˆ G subgroup to be brokendown to an approximately global symmetry of the 4-dimensional spacetime manifold,incorporating an approximately Euclidean 3-dimensional space, forming the backdropfor our perception of physical objects in the world.This requirement is borne out by our observations of the world around us onthe scale of the solar system for which the non-Euclidean effects of general, as wellas special, relativity are indeed imperceptible . The non-Euclidean effects such as thedeflection of starlight passing close to the sun are well beyond the reach of casualobservation. On the other hand local observations such as the accelerating fall of anapple from a tree might at first sight be ascribed to a ‘force of gravity’ active within aflat arena of space and time, rather than to an effect of a curved spacetime arena itself.The apparent flatness of the local geometry both from the point of view of our everydayexperience of the world and also for most scientific experiments accounts for the factthat the existence of a non-Euclidean element of 3-dimensional space combined with1-dimensional time was not recognised, through centuries of scientific developments,until the early 1900s.Carrying the same principle of our innate requirement of perception in theworld to the largest scale in which we encompass everything in our observable universeit seems natural to ask how it could be possible for our existence and experiences toinfluence in any way the shape or form of the universe over regions measuring billions oflight-years across. However, a central point of the work presented in this paper is thathere we consider the whole universe to be the physical manifestation that is createdthrough and within the possibility of our experiencing it and is therefore shaped bythe necessary form of that possibility, as we shall describe further in chapter 14. Theinitial naive picture that hence comes to mind is then based upon the assumptionof an approximately Euclidean background extending to the largest observable scale,neglecting the (generally imperceptible) local variations from flatness, with the flow oftime propagating through a 4-dimensional manifold as depicted in figure 12.1. Thispicture represents the largest scale realisation, for our own 4-dimensional universe, ofthe general idea introduced in figure 2.3 of section 2.2 for the model 3-dimensionalworld. 358igure 12.1: Propagation of galaxies, clusters of galaxies and large scale physical struc-tures through the M spacetime manifold, with the temporal dimension directed fromleft to right and one spatial dimension suppressed.We further recall that in the full theory the components of the 4-dimensionalvector field v ( x ) on M are considered to be locally embedded in a higher-dimensionalform of temporal flow L ( v ) = 1 via the space of v ≡ X ∈ h O matrices throughequations 8.4 and 8.5 of section 8.1, and in turn within the form L ( v ) = 1 via theelements x ∈ F (h O ) in the form of equation 9.46 as described in section 9.2. From apurely mathematical point of view the intermediate 4-dimensional case is readily by-passed in generalising from a 1-dimensional temporal progression to higher-dimensionalforms, here represented by a 27-dimensional and on to a 56-dimensional form of timewith a full ˆ G = E and ˆ G = E symmetry respectively. However, in order to physically experience or perceive any structures implicit within the general form of time a lower-dimensional part, with mathematical properties isomorphic to the geometrical formsrequired to define the perception of objects in the world, is projected out, or syphonedoff, from the full general form of temporal flow.In our world this has been taken to be achieved through extracting v ≡ h ∈ h C ⊂ h O ⊂ F (h O ), with the quartic form L ( v ) = 1 for v ∈ F (h O ) having anE symmetry, and projecting the vector component v ⊂ v onto TM . The form L ( v ) = det( h ) = h has the symmetry group SL(2 , C ) ⊂ E which, as describedin section 7.1, is the double cover of the external Lorentz group acting on Lorentz4-vectors. While the representations of the complementary internal symmetry uponthe components of L ( v ) = 1 and L ( v ) = 1 are reminiscent of Standard Modelproperties, as described in chapter 8 and section 9.2 respectively, an extension forexample to an E symmetry of a form L ( v ) = 1, as outlined hypothetically in359ection 9.3, may be needed to fully incorporate the structure of the Standard Model.In principle this projection, on employing the associated 4-dimensional trans-lation symmetry of the form L ( v ) = 1, opens out the local Lorentz subgroup intoan approximately global symmetry on the M manifold, breaking the full local E symmetry while at the same time actually generating the spacetime manifold itself.That is the M manifold is created in the act of the symmetry breaking projection from v ∈ F (h O ) → v ∈ TM ≡ h C , with v a ( x ) for a = 0 , , , local Minkowski coordinate frame on the manifoldas described in section 5.3. In terms of the initial picture, deriving from the trans-lation symmetry as described for figure 2.2, the metric g µν ( x ) = diag(1 , − , − , − globally . This continues to be the case to a good approxima-tion in practice even when planets, stars and galaxies are incorporated as depictedin figure 12.1. However, while maintaining the approximation of neglecting the localdistortions correlated with matter in these forms, it will not be possible to adopt aglobal Minkowski frame when considering the overall cosmological point of view.That is, while compatible with an approximately flat SO + (1 ,
3) frame locally,on the scale of the solar system for example, on larger spacetime scales through tothe vast arena of the universe studied in cosmology there is no longer a necessity forthe geometric form of spacetime to describe a perceptual frame even in approximation.Further, extrapolating beyond our possible experience or observation of the world the 4-dimensional geometrical interpretation may itself at some point break down altogether.This may apply to extreme regions such as black holes at any epoch and the structureof the very early history of the universe and the Big Bang. The effects of particlephysics studied in an Earth-bound laboratory, and in the previous chapters, are alsolikely to play a significant role when extrapolated to the extreme conditions of a highlynon-Euclidean spacetime geometry, as alluded to in the previous section.In the following section the standard cosmological model and a range of possiblelarge scale metric solutions will be reviewed, before turning to the very early universein section 12.3. This will provide a basis for the perspective of the present theory tobe presented in the following chapter.
While the Standard Model for particle physics has been constructed in recent decadesin parallel with the findings of high energy physics experiments, the underlying tools ofquantum field theory were originally developed in the 1920s through to the 1940s. Theframework for cosmological models was originally developed over a similar period fol-lowing soon after the publication of general relativity in 1915 and through to the 1930s,although again here the ‘standard model of cosmology’ has only become established inrecent decades in the light of the empirical data revealed with modern observationaltechnology. In this section we examine the picture of the cosmos and the standardcosmological model that has emerged out of this work (see for example [5, 73]).The standard approach incorporates general relativity, as reviewed in sec-tions 3.3 and 3.4, for which the empirical observation that spacetime curvature isstrongly correlated with the presence of matter is expressed through the field equa-360ion G µν = − κT µν . This equation postulates the equality of the Einstein tensor G µν = R µν − Rg µν with the energy-momentum tensor T µν , to within a constant ofproportionality. This approach is here summarised in terms of three quotes from [74]: • ‘We wish to relate the curvature of spacetime to the presence of matter, sincegravity appears in the neighbourhood of matter’ ([74] p.232). The proportionalityconstant is determined for weak fields by comparison with Newton’s theory ofgravity and found to be κ = 8 πG N , where G N is Newton’s constant, as describedfor equation 3.75. • ‘It will be assumed that the metric in a nearly empty universe is nearly Minkowski’([74] p.229). Essentially this implies that a flat spacetime arena M is presup-posed before the introduction of matter. Within relatively local portions of theuniverse a flat Minkowski spacetime can act as a boundary condition in regionssufficiently far from matter, as for the example of the Schwarzschild solution inequation 5.49. • ‘...the vanishing of the divergence of G µν as a mathematical identity implies thevanishing of the divergence of T µν ’ ([74] p.232). That is in light of the contractedBianchi identity G µν ; µ = 0 this conclusion follows immediately given that theEinstein field equation is assumed to hold.The divergence-free nature of T µν can be interpreted as the conservation ofenergy and momentum in the limit of an approximately flat spacetime, as describedin the opening of section 5.2, since for a suitable choice of coordinates with linearconnection Γ → T µν,µ = 0. On the other hand, this equation for theconservation of energy and momentum is often cited as a starting point and then expressed in a general curved spacetime as T µν ; µ = 0, and it is this observation thatthen justifies the introduction of T µν on the right-hand side of the field equation itself,with the Einstein tensor on the left-hand side, since it happens to be also the casethat G µν ; µ = 0 as the contracted Bianchi identity. Consistent with this requirementan additional divergence-free term may be postulated, associated with a ‘cosmologicalconstant’ Λ, as may be necessary to account for the empirically observed evolution ofthe universe, yielding the full standard field equation as quoted in equation 3.84 andreproduced here (where we turn here to a convention of generally using lower indicesin such expressions): G µν + Λ g µν = − κT µν . (12.1)In 1922 Aleksandr Friedmann made two classes of assumptions in order toobtain solutions for the spacetime structure of the universe as a whole. The first classrequired that the gravitational field should satisfy the equation 12.1, that is the Einsteinfield equation including the cosmological constant term (Friedmann considered the casefor both arbitrary Λ as well as Λ = 0), with matter represented as a pressureless fluidwith energy-momentum tensor T µν = ρu µ u ν where ρ is the proper density of matter.In 1927 Georges Lemaˆıtre, working independently of Friedmann, considered the moregeneral case by including a spatially isotropic pressure term and hence treating matteras a perfect fluid with an energy-momentum tensor in the form of equation 5.37, thatis: T µν = ( ρ + p ) u µ u ν − pg µν (12.2)361here p is the pressure and here u µ represents the 4-velocity of the flow of galaxies, asdepicted for example in figure 12.1. This energy-momentum is then substituted intoEinstein’s equation 12.1 to give: G µν + Λ g µν = − κ ( ρ + p ) u µ u ν + κpg µν . (12.3)We now know that the contribution of radiation pressure to the evolutionary dynamicsof the universe is most significant for around the first 10,000 years of its history, withthe contribution of the matter density becoming comparable around 50,000 years afterthe Big Bang and subsequently increasingly dominating over the radiation term. Hencethe idealisation of Friedmann, treating the flow of galaxies as a dust or pressurelessfluid with p = 0, makes a very good approximation for modelling the cosmic evolution,particularly since the epoch of the ‘decoupling’ of matter from radiation around 372,000years [44] after the Big Bang, still relatively early in the 13.8 billion year history ofthe universe.The second class of assumptions made by Friedmann in order to obtain a so-lution concern the nature of preferred coordinate systems and more direct restrictionson the form of the metric deriving from symmetries imposed on the spacetime. Basedon the picture of galaxies pursuing non-intersecting world lines, for which figure 12.1represents only a particular special case, 3-dimensional spacelike hypersurfaces, or-thogonal to and parametrised by a global timelike coordinate t , are assumed to have auniform t -dependent 3-dimensional scalar curvature R ( t ) independent of the locationon a given 3-dimensional spatial surface. The unambiguous cosmic time t is taken tobe the proper time as measured for any given galaxy. The ‘Copernican view’, thathere on Earth we do not inhabit a central or preferred location of the universe, is sub-sumed into the ‘cosmological principle’ which asserts that at any given cosmic time t the universe on large scales is spatially homogeneous and isotropic about any location.From an observational point of view at the present epoch the assumption ofhomogeneity may be justified by the smallness of fluctuations in the distribution ofgalactic clusters on scales larger than a few 100 Mpc (megaparsecs, where 1 parsecis around 3.26 light-years) in an observable universe with distance scales of up to theorder of the Hubble radius: R H := c/H ≃ h − Mpc (12.4)with h ≃ . H defined below for equation 12.13. Similarly,the assumption of isotropy may be justified by the evenness of the cosmic microwavebackground (CMB) radiation to within of order 1 part in 10 as observed over the fullcoverage of the sky from the Earth.The mathematical basis for the assumptions of the cosmological principle wasstudied thoroughly by H.P. Robertson and independently by A.G. Walker in the 1930s.The 3-dimensional hypersurfaces for constant t are everywhere orthogonal to a con-gruence of geodesics given by the integral curves of the vector field ∂/∂t . For eachsolution the hypersurface curvature R ( t ), while it can vary with time, remains alwayseither positive (3-sphere), negative (hyperboloid) or zero (for a spatially flat universe).The Robertson-Walker line element is the most general spacetime metric compatiblewith homogeneity and isotropy and can be expressed in terms of intervals of proper362ime τ as: dτ = dt − a ( t ) (cid:20) dr (1 − kr ) + r ( dθ + sin θdφ ) (cid:21) (12.5)where the parameters a ( t ) and k will be described below. With the world line of anygiven idealised galaxy expressed in terms of constant 3-dimensional spatial sphericalcoordinates { r, θ, φ } the full 4-dimensional set { t, r, θ, φ } describes a comoving coordi-nate system with the cosmic time parameter t equivalent to the proper time τ elapsedfor the galaxy.As for any metric for 4-dimensional spacetime here the convention is to take thecomponents of g µν ( x ) to have the dimension of length squared, that is [ g µν ] = [ dτ ] = L (with the dimension of length L equivalent to that of time T since c = 1, while inthe notation of the discussion following equation 10.86 the mass M dimension of g µν is D = − g µν ] = L − .(We note that in the present theory the internal Killing metric components, such as g αβ in equation 4.4 for the case of a Kaluza-Klein metric, are not interpreted as representinga physical length in a higher-dimensional space in the present theory, however here weare dealing with purely external metric components in 4-dimensional spacetime). Thisconvention is consistent with the principle of general covariance in general relativity, asdescribed in section 3.4, which implies that in general no physical significance can beattached to a set of coordinates, which consists of numerical parameters of dimension L . As implied in the name, only when the manifold is endowed with a ‘metric’are lengths defined. In fact all parameters on the right-hand side of equation 12.5,including the scale factor a ( t ), can be considered to be dimensionless quantities. Sincean implicit factor of g = 1 carrying the dimension L accompanies the dt term inequation 12.5, the cosmic time coordinate t may be interpreted as having the dimensionof T ≡ L . A similar interpretation might be applied to spatial coordinates in certaincases, in particular for Euclidean coordinates { x, y, z } in the limit of a flat spacetimewith Minkowski metric.The sign of the dimensionless real number k in equation 12.5 indicates thesign of the 3-space curvature. For k = 0 the spatial hypersurfaces are flat, althougheven in this case the 4-dimensional curvature will generally be finite. For non-zerovalues of k the coordinate r may be redefined as r → r/ | k | , and the scale factor as a → a | k | , such that the thus normalised values of k = +1 , − g µν ( x ) down to a single real parameter a ( t )along with a discrete set of three possible values for k in equation 12.5. Together a ( t )and k characterise the Robertson-Walker line element which itself represents a trialsolution for cosmological models. The specific form of the line element will be deter-mined by the dynamics provided by equation 12.3, which depends in turn on the choiceof cosmological constant Λ and the ‘equation of state’ relating ρ and p in the energy-momentum tensor on the right-hand side of Einstein’s field equation. Equation 12.2,with ρ and p functions of t only, is in fact the most general form of energy-momentumtensor consistent with the requirements of homogeneity and isotropy as expressed inthe cosmological principle, which is also respected by the Λ term in equation 12.3, with g µν ( x ) in the form of equation 12.5. The resulting differential equations in the single363ndependent variable t may be solved for a , ρ and p , each of which is a function of t only owing to the homogeneity assumption.The Einstein tensor is constructed from the Riemann curvature tensor in termsof the components of the Ricci tensor as G µν = R µν − Rg µν , as introduced afterequation 3.71 and via equation 3.74; the Riemann tensor is a function of the linearconnection Γ as expressed in equation 3.73, and the torsion-free Levi-Civita connectionof equation 3.53 is employed as also described in section 3.3. The components of themetric tensor g µν implied in equation 12.5 are: g = 1 , g = − a (1 − kr ) , g = − a r , g = − a r sin θ (12.6)These can be substituted into the above chain of relations, via the linear connection,to determine the components of the Ricci tensor R µν and scalar curvature R = g µν R µν as (see for example [75] p.151): R = 3 ¨ aa (12.7) R = − ( a ¨ a + 2 ˙ a + 2 k ) / (1 − kr ) R = − ( a ¨ a + 2 ˙ a + 2 k ) r R = − ( a ¨ a + 2 ˙ a + 2 k ) r sin θR = 6 (cid:18) ¨ aa + ˙ a a + ka (cid:19) (12.8)with both g µν = 0 and R µν = 0 for µ = ν , and the notation ˙ a = da/dt and ¨ a = d a/dt has been employed. Further following the standard procedure and completing the chainof relations from the metric g µν to the Einstein tensor G µν the above expressions for R µν and R are substituted into the field equation 12.3, which includes the cosmologicalterm and energy-momentum in the form of a perfect fluid, with components of thegalactic flow 4-velocity u µ = g µν dx ν dτ = (1 , , ,
0) in the comoving coordinates, to findfor the G and G components respectively:˙ a a + ka −
13 Λ = κ ρ (12.9)2 ¨ aa + ˙ a a + ka − Λ = − κp (12.10)Only the above two independent non-trivial equations result since the equations forthe G and G components are each identical to that for G in equation 12.10,due to the symmetries of the cosmological principle, while the set of six equations for G µν with µ = ν are identically zero on both sides. In equations 12.9 and 12.10 theparameters a , ρ and p are functions of the cosmic time t while Λ, κ and k are constants.Multiplying equation 12.9 by a , differentiating the full resulting expressionwith respect to t and replacing the left-hand side by equation 12.10 multiplied by ˙ aa leads to the relations: ddt ( ρa ) = − p ˙ aa = − p ddt a (12.11)that is: dda ( ρa ) = − pa T µν ; µ = 0 for theperfect fluid energy-momentum tensor of equation 12.2 given the metric componentsof equation 12.6 (see for example [75] pp.152–153, with the same result holding if a Λ κ g µν term is included in T µν since (cid:0) Λ κ g µν (cid:1) ; µ = 0). Alternatively the constraint T µν ; µ = 0can be combined with equation 12.9 in order to derive equation 12.10.This apparent redundancy between the Einstein field equation and the expres-sion T µν ; µ = 0 is expected since, as alluded to above, the form of the field equation G µν = − κT µν can itself be motivated by the divergence-free identity which applies toboth sides and contains equivalent information. In fact in defining − κT µν := G µν ,which is the interpretation implied in the third bullet point near the opening of thissection, the identity T µν ; µ = 0 is simply a copy of the contracted Bianchi identity G µν ; µ = 0 which is an intrinsic property of the Einstein tensor (see also for example[6] p.729).The apparent ‘conservation law’ T µν ; µ = 0 can not be directly interpreted asthe ‘conservation of total energy-momentum’ unless the 4-dimensional geometry in-volved is asymptotically a flat Minkowski spacetime, which is not generally the casefor the Robertson-Walker line element of equation 12.5. In general a suitably flatspacetime can be identified for local regions of the universe, as suggested in the secondbullet point near the opening of this section, and more specifically leads to energy-momentum conservation when applied in the laboratory setting, such as for the QFTlimit summarised in table 11.1 towards the end of section 11.4.As described above in comparison with the contribution from radiation pressurethe universe has been matter dominated since a relatively short time after the Big Bang.Hence considering the pressure-free case of dust with p = 0 equation 12.11 implies that ρa is constant in time. In this case the matter density at any epoch can be written as ρ = ρ a a , where a subscript ‘0’ on a quantity such as the density ρ or scale factor a denotes the present day value at cosmic time t = t . Generally the boundary condition a (0) = 0 will be employed, with the cosmic time t = 0 designating the origin of theuniverse at the ‘Big Bang’. For such a cosmology the present cosmic time t hencedenotes the current ago of the universe.The physical spatial distance d ( t ) between any two galaxies at a given cosmictime t is simply d ( t ) = a ( t )∆Σ where ∆Σ is the comoving ‘coordinate distance’ betweenthe galaxies (when interpreted with care for the length L dimension as introducedthrough the metric as described above, similarly as discussed for the interpretation ofthe Schwarzschild solution around equation 5.50). The physical speed of one of thegalaxies relative to the other is v = ddt d ( t ) which leads directly to the expression: v = ˙ a ( t ) a ( t ) d ( t ) = H ( t ) d ( t ) (12.12)where H ( t ) := ˙ a ( t ) a ( t ) is the Hubble parameter, which generally varies with time. Equa-tion 12.12 expresses Hubble’s law which states that at any given epoch t the relativespeed between any two galaxies on the corresponding spatial hypersurface is directlyproportional to the distance d ( t ) between them, with the constant of proportionalitygiven by the Hubble parameter H ( t ) at that cosmic time. Hubble’s law is a directconsequence of the form of the Robertson-Walker line element of equation 12.5 withvariable a ( t ) and says nothing about the actual dynamics, that is the function a ( t ) it-365elf, for the universe. At the present epoch the Hubble parameter is called the Hubbleconstant (since it is the same everywhere in space) H = H ( t ) = ˙ a a and is empiricallyfound to take the value: H = 100 h km s − Mpc − (12.13)with h = 0 . ± .
012 [44], as employed above in equation 12.4.The functional form of a ( t ) itself may be determined from equation 12.9, whichis also called the Friedmann equation. While considering the case with p = 0 if thecosmological constant is also neglected by setting Λ = 0 the Friedmann equation forany cosmic time t may be written as: H + ka = κ ρ (12.14)The particular value of ρ = ρ c = H κ is called the ‘critical density’ and correspondsto a solution with k = 0, that is a universe which is spatially flat at any epoch. Thissolution is known as the Einstein-de Sitter model and describes an ever-expandinguniverse with scale factor a ( t ) ∝ t , as listed in table 12.1 (in contrast the radiationdominated case with p = 0 and equation of state p = ρ results in a dynamics with a ( t ) ∝ t for k = 0, as also listed in the table).FLRW model: G µν = − Λ g µν Matter ρ = 0 Radiation p = 0 R µν = λ ( t ) v µ v ν p = ǫρ : ǫ = − +1 a ( t ) ∝ e q Λ3 ( t − t ) a t t t ρ ( t ) ∝ constant t − ∼ a − t − ∼ a − t − ∼ a − Table 12.1: Four FLRW (Friedmann-Lemaˆıtre-Robertson-Walker) cosmological modelsfor k = 0 with an energy-momentum T µν = − κ G µν in the form of equation 12.2, with p and ρ related via the equation of state p = ǫρ , corresponding to a universe dominatedby a cosmological term (see below), matter, radiation and through R µν = λ ( t ) v µ v ν (forsection 13.1) respectively. The evolution of the scale factor a ( t ) and effective matterdensity ρ ( t ) are obtained as solutions for equations 12.9 and 12.10.Equation 12.14 can be rearranged in the form: ka = H (Ω M −
1) (12.15)on introducing the matter density parameter Ω M = ρρ c = κρ H . For Ω M > k = +1, and the evolution equation for a ( t ), that is theFriedmann equation 12.9 with Λ = 0, shows that the universe will inevitably collapseback down to the condition a = 0, while for Ω M < k = −
1, and the evolution equation for a ( t ) shows that the universe will expand foreveras for the Ω M = 1 case, with the latter then representing the critical value upon whichthe ultimate destiny of the universe depends. While the Einstein-de Sitter universewith Ω M = 1 describes the unique spatially flat case with ρ = ρ c = H κ for a given H ( t ) at any cosmic time t , for the spatially non-flat cases with k = +1 and k = − ρ > ρ c and ρ < ρ c respectively, for anygiven values of ρ c and t .At the present epoch the density parameter for ordinary baryonic matter alone,which is largely readily visible in the form of galaxies of stars and clouds of dust andgas, is observed to have a value of Ω B = 0 . ± .
002 [44] which, being much lessthan unity, would imply that we inhabit a universe with spatial curvature k = − D = 0 . ± .
011 [44], implying a total matter density parameter at thepresent epoch of Ω M ≃ . k = 0.Since Ω M falls well short of the total value needed to account for the observedspatial flatness, and since this quantity is only sensitive to gravitating matter associ-ated with clustering up to the largest scales probed, a significant contribution fromrelativistic particles or a vacuum energy term is implied. With a negligible contribu-tion from the CMB radiation itself of Ω R ≃ . × − [44] (and with an even smallercontribution predicted for relic neutrinos from the Big Bang) we continue to assume p = 0 but allow the cosmological constant Λ to take a finite value in equation 12.9,which can be divided by H = ˙ a a and rearranged in the form:Ω M + Ω Λ = 1 + k ˙ a (12.16)with Ω M = κρ H and Ω Λ := Λ3 H (12.17)Analysis of the Hubble diagram for distant supernovae of type SN Ia independentlydetermines a value of Ω Λ = 0 . ± .
017 [44]. Hence, as can be seen from equa-tion 12.16, the empirical observations of Ω M + Ω Λ ≃ .
000 and of spatial flatnessconsistent with k = 0 from the CMB anisotropy are in excellent agreement.However while these observations are mutually consistent it appears coinciden-tal that the contributions from Ω M and Ω Λ are of the same order of magnitude atthe present epoch. In particular from equation 12.17 and the empirical values of thedensity parameters the present overall matter density in the universe is approximatelyhalf that of the vacuum energy, with ρ ≃ Λ /κ . Since ρ was much larger in the earlieruniverse and is ever decreasing into the future, and since the matter density ρ describesmatter which is heavily clumped into clusters of galaxies and the stars within, whileΛ /κ (which may be generically termed ‘dark energy’) is of an apparently very differentnature, both constant in time and uniformly distributed in space, the approximatecoincidence of their present average values, within a factor of two, is notable. It is alsoobserved that within the Ω M contribution itself the dark matter content is around fivetimes that of the baryonic matter, which is assumed to be a feature largely independentof cosmic time. An understanding of the origin of the above empirical observationswill require a theoretical understanding of the nature of the dark sector itself.A solution for the large scale cosmic geometry must also be consistent withequation 12.10, which can be employed to further analyse the dynamics. Substituting367 a a + ka from equation 12.9 into this second dynamic equation leads directly to therelation: ¨ aa = − κ (cid:18) ρ + 3 p (cid:19) + Λ3 (12.18)An era of accelerating expansion of the universe, that is with ¨ a ( t ) >
0, provides aformal definition of cosmic ‘inflation’. From the above equation it can be seen thatthis is the case for ( ρ + 3 p ) < > ρ and p are relativelysmall, or some combination of these factors. The dynamics can be described in termsof the ‘deceleration parameter’, defined as q := − a ¨ a ˙ a . Taking the case p = 0, usingequation 12.17 and dividing equation 12.18 by H the deceleration parameter is foundto be related to the density parameters as: q = Ω M − Ω Λ (12.19)At the present epoch, as for the previous several billion years, with the value of Ω M < Ω Λ and q < Ω M > Ω Λ with q > a ( t ) as a function of cosmic time t , up to and beyond the present epoch t .In the future as the matter density ρ ( t ) and the value of Ω M decrease with theexpanding universe the cosmological constant Λ will increasingly dominate the largescale evolution of the cosmos. With p = 0, k = 0, Λ > ρ → a = Λ a . Hence if such a cosmic epoch begins attime t = t Λ the scale factor increases as a ( t ) = exp (cid:16)q Λ ( t − t Λ ) (cid:17) a ( t Λ ), which is alsoconsistent with equation 12.18. For a cosmology entirely determined by a cosmologicalconstant then t Λ = 0 and, setting a (0) = 1 for this scenario, this describes the de Sittermodel with line element: dτ = dt − e At d Σ k =0 (12.20)368here A = 2 q Λ is a constant and d Σ k =0 represents the 3-dimensional spatial partof the line element in equation 12.5 for the spatially flat case with a ( t ) = e At . Sinceif Λ = 0 ordinary matter on the right-hand side of equation 12.1 does not yield asolution in the form of equation 12.20 this special case for FLRW cosmology withan exponential expansion factor was originally considered to represent a matterlessvacuum with Einstein equation G µν + Λ g µν = 0, which is equivalent to the Ricci tensorbeing constrained to the form R µν = Λ g µν with constant Λ.However, since the ‘vacuum’ Einstein equation can be written with the cos-mological term on the right-hand side as G µν = − Λ g µν ≡ − κT µν (Λ) the exponentialexpansion observed for our universe at the present epoch is generally attributed to‘vacuum energy’ or ‘dark energy’, in contrast to ‘dark matter’ and as alluded to in thediscussion following equation 12.17 above. By direct comparison with equation 12.2the object T µν (Λ) may be interpreted as a non-standard form of energy-momentumfor a ‘fluid’ with an energy density ρ Λ = Λ /κ and pressure p Λ = − ρ Λ = − Λ /κ whichare constant in time as well as space even as the universe evolves. This substitution,replacing Λ with effective values of ρ Λ and p Λ , can also be applied directly in the evolu-tion equations 12.9 and 12.10, from which the accelerating expansion may be deducedvia equation 12.18 since ( ρ Λ + 3 p Λ ) < > ρ Λ and p Λ remain constant in timeeven if an energy-momentum tensor with T µν = 0 for ordinary matter is included inthe field equation 12.1, as is the case in equation 12.3.The above de Sitter model of equation 12.20 was introduced in 1917 and origi-nally thought to represent a static solution until it was shown how test particles wouldfly apart from each other in such a universe. In the same year a truly static universemodel was proposed by Einstein, also with p = 0 and Λ > ρ tuned to solve equations 12.9 and 12.10with the constraint ˙ a = ¨ a = 0. The solution for the Einstein model requires a positivecurvature k = +1 and a constant density ρ = κ Λ for ordinary matter fixed for all timeas the universe neither expands nor contracts.From an observational point of view an initial data set of measurements ofsignificant redshifts for a number of nebulae was observed by V.M. Slipher as early1917, that is the same year the above models were proposed. In the early 1920s thebrightest nebulae were resolved into stars, including those of the Cepheid type allowingEdwin Hubble to estimate their distances out to several million light-years. At thistime it was established that the nebulae are in fact further distant galaxies comparablein size to our own and the visible scale of the cosmos was vastly augmented. Thatgalaxies are receding away from our own Milky Way with velocities proportional totheir distance from us, consistent with equation 12.12, was first discovered by Hubblein 1929.Following the empirical conclusion from the 1920s that the universe is expand-ing and Eddington’s theoretical observation in 1930 that the static Einstein model isunstable a policy of dropping the cosmological constant term Λ was generally adopted.This led in particular to the Einstein-de Sitter model of 1932, with Λ = 0, k = 0, p = 0and ρ = ρ c = H κ evolving in time, as described above following equation 12.14. Asdescribed alongside equations 12.16 and 12.17 observations in cosmology dating fromthe 1990s have resulted in the reintroduction of a Λ > ρ ≃ κ Λcontrasts with above finely balanced Einstein universe for which ρ = κ Λ). While theimpact of the cosmological constant on the more recent evolution of the universe isclearly visible in figure 12.2 the much earlier radiation dominated period, while alsoforming a key part of the standard model, in spanning a period of less than 50,000years after the Big Bang is far too brief to feature on the linear scale adopted in thisfigure. In the following section we motivate and review some of the theoretical ideasapplied to the yet far earlier universe.
The redshift z observed for distant galaxies by Hubble, and through to present dayobservations now extending out across several billion light-years, is defined by therelation: 1 + z ≡ λ λ e = a ( t ) a ( t e ) (12.21)where t e is the cosmic time of the emission of radiation from a distant galaxy withwavelength λ e (which can be deduced from well-known patterns of spectral lines) and t is the present cosmic time at which we detect the radiation and measure the wavelengthto be λ in our galaxy. With the value of z = 0 for the present epoch and adopting theconvention a ( t ) = 1 there is a simple relationship between the redshift z at an earlierepoch and the corresponding scale factor a ( t e ) at that time. Since for our universe a ( t )is an ever increasing function of time, as can be seen in figure 12.2, the value of theredshift z can be used to label the earlier epochs of our expanding universe. Hubble’sobservations of a positive redshift are explained via equation 12.21 by the simple factthat a ( t e ) was smaller in the past, while an increasing value of a ( t ) at any given time t implies a positive value for the Hubble parameter defined in equation 12.12.This cosmological redshift does not arise from the Doppler effect, which onlystrictly applies in a local or extended flat region of spacetime, but from the passageof light through a curved t appear to be very flat.While the hypothetical Einstein universe was found by Eddington to be balanced pre-cariously, as alluded to at the end of the previous section, there is also an apparentinstability concerning the state of the universe we actually observe. If the value ofthe total density parameter Ω is not exactly equal to one in a matter or radiationdominated universe, such as we have described for the first few billion years of ourown cosmos, this value will diverge away from unity as the universe evolves. Takingequation 12.15, generically replacing Ω M by a density parameter Ω and with H = κρ ,370s for the form of equations 12.17 for example, leads directly to the relation: (cid:0) Ω − − (cid:1) ρa = − kκ (12.22)from which different conclusions may be reached depending on the equation of statefor the apparent matter density ρ , given that the right-hand side of this expression isa constant. In particular for an expanding FLRW universe that is matter dominatedor radiation dominated the quantity ρa decreases with cosmic time in proportion to a − or a − respectively, as can be seen from the columns of table 12.1. Hence it canbe seen from equation 12.22 that a value of Ω = 1 will diverge further from unity assuch a universe evolves.That is any small deviation of the density parameter Ω from the value ofunity at an earlier epoch with a large redshift z will have been greatly amplified bythe present day, such that in going back to the extreme case of the Planck epoch of t ≃ − seconds after the Big Bang an apparent fine tuning of the density parameterto about 1 part in 10 is required in order to be consistent with the present dayobservation of spatial flatness for the universe ([5] p.323). The need for an explanationof this precise tuning of the initial spatial flatness condition arising out of the Big Bangis known as the ‘flatness problem’.That the 4-dimensional geometry can be highly curved even for a spatiallyflat cosmology with k = 0 is particularly evident in the early universe. For the matterdominated case it can be seen by substituting terms from equations 12.9 and 12.10 intoequation 12.8 that the scalar curvature of the spacetime is simply R = κρ (as wouldbe expected from the paragraph following equation 5.35). Hence as t →
0, in principleto an epoch even earlier than the Planck time, with the scale factor a ( t ) → ρ → ∞ the scalar curvature R diverges to infinity at what is referred to as the ‘initialsingularity’ at t = 0. It is sometimes noted that the standard cosmological modelhence predicts the paradoxical origins of the universe in such an initial singularity,studied by S.W. Hawking, G.F.R. Ellis and R. Penrose around 1965–70, a point atwhich general relativity, which governs the model itself, breaks down. However anyconclusions drawn from the structure of gravitation at the Planck scale are inevitablyuncertain given the as yet unknown role of quantum phenomena in such an extremeenvironment. As described in section 11.4 for the present theory gravitation itself isnot quantised and hence in principle the Planck scale will be of less significance andnot represent a barrier to further extrapolation to arbitrarily early times, as will beconsidered in section 13.2.It is also informative to write the Robertson-Walker line element of equa-tion 12.5 with the cosmic time coordinate t transformed to a conformal time parameter η = R t dt ′ a ( t ′ ) as: dτ = a ( η ) (cid:2) dη − d Σ k (cid:3) (12.23)where d Σ k represents the spatial part of the line element inside the square brackets ofequation 12.5 with k = +1 , −
1. Hence by adopting the conformal time coordinate η in equation 12.23 the scale factor a ( η ) can be seen as a special case of a conformaltransformation, which more generally takes the form g µν ( x ) → f ( x ) g µν ( x ) where f ( x )is an arbitrary real function of spacetime (a very different example of which was con-sidered in equation 11.13). For the case k = 0 the Robertson-Walker line element is371ence related to a flat 4-dimensional spacetime via a conformal transformation. It canalso be shown, using a further suitable coordinate transformation, that the geometryfor each of the k = ± C ρσµν ( x ) = 0, even though components of the Ricci cur-vature R µν ( x ) part of the Riemann tensor may attain arbitrarily large values in thevery early universe.The initial singularity of the Big Bang is a spacelike boundary of spacetime inour distant past, represented by the horizontal wiggly line in the conformal diagramof figure 12.3. In such a diagram all null-rays, that is with proper time line element dτ = 0, are drawn at 45 ◦ and hence the causal properties of the spacetime are madeapparent. The vertical axis of such a diagram is linear in the conformal time η withthe horizontal axis representing comoving coordinate distances ∆Σ, consistent withequation 12.23. In figure 12.3 epochs on the vertical axis are labelled by the cosmictime t , although of course not to scale, and intervals of the horizontal axis at anygiven epoch can be converted to physical proper distances a ( t )∆Σ, as described beforeequation 12.12. A ray of light emitted at time t = t e and reaching us now at t ≡ t will have travelled the comoving coordinate distanceΣ p ( t , t e ) = ∆ η = Z t t e dt ′ a ( t ′ ) (12.24)where δt ′ /a ( t ′ ) is the coordinate distance traversed in a small interval of cosmic time δt ′ . Hence any signal emitted beyond this distance at time t e will not have beenable to reach us yet and hence in turn Σ p ( t , t e ) is termed the ‘particle horizon’. Forany given t e the particle horizon grows with time t = t from the perspective of theobserver b in figure 12.3. The particle horizon can also be defined as the proper distance R p ( t , t e ) = a ( t )Σ p ( t , t e ) on the 3-dimensional spatial hypersurface at the time t ,that is R p ( t , t e ) = a ( t )∆ η , where ∆ η is the conformal time elapsed between t = t e and t = t .At the present epoch t the largest particle horizon R p ( t ) corresponds to sig-nals emitted at the time of the Big Bang. Setting t e = 0 the integral in equation 12.24converges provided the equation of state is such that ρ ( t ) decreases at least as fast as a − ( t ), as it does for a matter or radiation dominated universe as seen in table 12.1.For various cosmological models the particle horizon, obtained from equation 12.24, isgenerally greater than t itself since a ( t ) tends to be smaller for earlier times t < t .For a radiation dominated universe the particle horizon has a value of R p ( t ) = 2 t ,while for the matter dominated case the value is R p ( t ) = 3 t . (For the Einstein-deSitter model with k = 0 the age of the universe can be expressed as t = H and theparticle horizon is twice the Hubble radius, R p = 3 t = H = 2 R H , with the latterdefined in equation 12.4.)For our own universe the present particle horizon is determined to be R p ( t ) ≃
46 billion light-years, which is greater than 3 t , where t ≃ . R p ( t ) represents the edge of the observable universe interms the present distance to objects on the 3-dimensional spatial hypersurface at372igure 12.3: Conformal diagram depicting the past light cone from our present location b at cosmic time t = t extending back to the Big Bang singularity at t = 0. Therecombination era t = t rec ( ≃ t / , η versus comoving coordinate distance Σ the diagram is a 2-dimensional representationof a 4-dimensional spacetime.cosmic time t = t , not of course as we actually might see them via light emitted inthe distant past. While R p ( t , t e ) = a ( t )Σ p ( t , t e ) is the present particle horizon forobserving events from time t = t e , the proper distance to such an event on the horizonat the time of signal emission was a ( t e )Σ p ( t , t e ). For comparing particle horizons atdifferent epochs comoving coordinate distances ∆Σ, that is intervals of the horizontalaxis in conformal diagrams, will be move convenient, as we describe in the following.As well as the Big Bang at t = 0 and the present era t = t the time of‘recombination’ t = t rec is also labelled in figure 12.3. This is the epoch around 372,000years after the Big Bang, with a redshift of z ≃ t = 0). Since there is onlyan extremely small interaction between an external electromagnetic field and neutralatoms this also marks the era of decoupling between radiation and matter alludedto after equation 12.3. Photons from this decoupling epoch have effectively beenpropagating freely since t = t rec , relatively early in the 13.8 billion year history of theuniverse, until detected in the present as the observed CMB radiation now redshiftedto a temperature below 3 K.From our perspective photons composing the CMB radiation were emitted fromanywhere on the 2-sphere of our past light cone in 4-dimensional spacetime at the time t = t rec . Two points u and v on the continuous surface of a 2-sphere can be arbitrarilyclose together, unlike the points u and v in figure 12.3 on the past light cone of this 2-dimensional representation of spacetime. If the comoving coordinate distance between u and v at t = t rec is greater than twice the particle horizon Σ p ( t rec ) then the twospacetime points have never been in causal contact. Hence from our perspective b ,373ith both u and v observed on our particle horizon Σ p ( t , t rec ), there is no reason toexpect a homogeneity of physical quantities such as the CMB temperature as measuredand compared for such regions u and v which have not been in causal contact with eachother. In fact the particle horizon at the recombination era Σ p ( t rec ) only subtends oforder 1 ◦ in the sky from our present perspective b on Earth. The difficulty in contrivingan assumption of homogeneity as an initial condition of the hot Big Bang to accountfor the observed uniformity of the CMB temperature to within 1 part in 10 over allangles of the sky is known as the ‘horizon problem’.In place of postulating homogeneous initial conditions across causally separatedspatial regions of the very early universe the only means by which the temperaturesat u and v might be related through a process of thermalisation is to arrange for thepossibility of causal contact in their past. This requires a mechanism through whichthe Big Bang epoch effectively retreats back further below the recombination era in theconformal diagram of figure 12.3, as demonstrated in figure 12.4. This in turn can beachieved by a sufficient rescaling of proper spatial distances with a ( t inf − ǫ ) ≪ a ( t inf + ǫ ),where ǫ may be a very short time interval, effectively ‘miniaturising’ 3-dimensionalspace during the epoch t < t inf . In this case the horizontal displacements in figure 12.4labelled by comoving coordinate intervals ∆Σ now represent much shorter physicalproper distances a ( t )∆Σ for t < t inf and a given null-ray propagating for a givencosmic time interval ∆ t occupies a somewhat larger portion of the vertical axis whichis linear in conformal time intervals ∆ η ∼ ∆ ta ( t ) . Hence the Big Bang epoch at t = 0is pushed back in the conformal diagram to accommodate this rescaling. Hence inturn the comoving coordinate distance traversed by null-rays in a fixed cosmic timeinterval during this early epoch before t = t rec can in principle comfortably encompassthe present particle horizon Σ p ( t , t rec ) at t = t for signals emitted at t = t rec (see forexample [26] pp.744–747).Figure 12.4: Conformal diagram depicting a similar cosmic history as figure 12.3 withthe same three values of t = 0, t = t rec and t = t but with the addition of a furtherepoch t = t inf during which the scale factor a ( t ) is ‘inflated’ by an enormous degree ina short period of cosmic time. 374volving forwards in time from the Big Bang the rapid expansion of the uni-verse scale factor a ( t ) at the epoch t inf , which in principle solves the horizon problem,is termed ‘inflation’, as a particular case of an accelerating expansion described gener-ically after equation 12.18. The question then still remains regarding the physicalmechanism behind such a radical transformation of the spacetime geometry at thatvery early epoch. Guided by the de Sitter model with the line element of equation 12.20describing an exponential expansion with scale factor a ( t ) ∝ exp (cid:16)q Λ t (cid:17) one way toachieve inflation is with a very large, but only temporarily active, cosmological termof the form Λ g µν in Einstein’s field equation 12.1.On introducing a new scalar field ϕ ( x ) (which is unrelated to the scalar Higgsfield φ ( x ) of the Standard Model of particle physics described in section 7.2) a falsevacuum state obtained for a suitable potential V ( ϕ ) can model the effect of a cosmo-logical term via an energy-momentum tensor T µν with a term of the form V ( ϕ ) g µν (such as in equation 12.25 below). That is, a potential V ( ϕ, T ), as a function of thefield ϕ ( x ) and temperature T ( x ), may be contrived such that the high temperaturevacuum state ϕ = 0 becomes a ‘false vacuum’ as the universe achieves a ‘supercooled’condition below a certain critical temperature T c in the very early universe. The phasetransition to the new true vacuum state with ϕ = 0 for T < T c may involve eitherquantum mechanical tunnelling through an intermediate potential barrier (‘old infla-tion’) or a gradual roll down a potential slope (‘new inflation’). In either case thepotential function V ( ϕ, T ) may be suitably contrived in order that the true vacuumis not immediately attained and the energy of the false vacuum state dominates thecosmological evolution equations for a brief period of time. This cosmic time periodof t inf ∼ − —10 − seconds can be long enough for the scale factor a ( t ) to increaseby a factor of ∼ or more, effectively solving the horizon problem by the rapidinflation of a small homogeneous region of the very early universe (see for example [5]chapter 11).While the de Sitter model of equation 12.20 assumes a spatially flat universewith k = 0, the evolution of the scale factor a ( t ) resulting from a cosmological termΛ g µν in the field equation can also be determined for the cases of k = ± k = +1 , − a ( t ) ∝ cosh (cid:18)q Λ3 t (cid:19) , exp (cid:18)q Λ3 t (cid:19) and sinh (cid:18)q Λ3 t (cid:19) respectively, and hence the k = ± k = 0 andconstant Hubble parameter H ( t ) = q Λ3 ([5] p.326). This convergence towards a den-sity parameter Ω of unity can also be seen from equation 12.22 since the equation ofstate for a cosmological term implies that ρa ∝ a , as can be seen from table 12.1,which hence rapidly increases during inflation, driving Ω →
1. Hence during inflationsolutions for a ( t ) with finite spatial curvature rapidly approach the purely exponentialexpansion solution with k = 0, that is the de Sitter model for a flat universe withthe Λ g µν term simulated by the energy of the false vacuum during the inflationary pe-riod. The brief inflationary era t inf described in the previous paragraph, and depictedin figure 12.4, is sufficient to suppress any non-zero spatial curvature by a factor ofaround 10 or more, hence in principle solving the flatness problem described afterequation 12.22, in addition to solving the horizon problem.Inflationary theory was initially proposed by Alan Guth in 1980, precisely375o address the horizon problem while also accounting for the flatness problem. Infact the strong bias towards spatial flatness is sometimes considered to have been asuccessful prediction of the theory. The hypothetical period of inflation at t inf drives thetotal density parameter Ω extremely close to unity in the early universe such that thesubsequent radiation dominated era of thousands of years and matter dominated eraof billions of years have been insufficient to prise the value of Ω away from the value ofone, as described following equation 12.22, to any measurable degree. The more recentand increasingly dominant effect of the apparently presently active cosmological termΛ g µν is again tending to bind the density parameter yet closer to unity, although thiseffect has thus far been too weak to account for the observation of spatial flatnesswithout the much earlier and much more dramatic inflationary epoch.However unlike the cosmological term Λ g µν which accounts for the present dayrelatively pedestrian accelerating expansion of the universe the much earlier periodof rapid inflation is required to terminate, and such a change in conditions is gener-ally ascribed to a phase transition as introduced above. The original ‘old inflation’model employed a first order phase transition via quantum tunnelling from the false tothe true vacuum once the temperature had dropped sufficiently to allow penetrationthrough the potential barrier. However the quantum nature of the transition resultsin bubble formation and corresponding large inhomogeneities that are not observed.This ‘graceful exit problem’ can be solved by ‘new inflation’ which ends via a transitionfrom the false vacuum at a local maximum in the potential at ϕ = 0, that is through asecond order phase transition, which proceeds more nearly simultaneously throughoutthe universe. An almost flat potential around ϕ = 0 can result in a ‘slow roll’ down tothe true vacuum at the potential minimum, still allowing sufficient time for a dramaticinflationary expansion.Amongst a range of inflationary models proposed ‘chaotic inflation’ in principlealso solves the graceful exit problem. In this model the potential of the scalar field cantake a much simpler form such as V ( ϕ ) = m ϕ or V ( ϕ ) = λϕ with a single minimumat ϕ = 0. Under a large range of possible initial conditions in the primordial chaos insome regions the value of ϕ ( x ) may be far from the minimum. Such a value, required tobe essentially uniform over a region of space of order the present day Hubble radius, canstimulate an inflationary period. A large inflation factor is possible provided that theconstant λ for example is chosen such that the potential function is sufficiently shallowto allow a sufficiently delayed roll down to the true vacuum value at ϕ = 0. As thetrue vacuum is attained and inflation ends our observable universe is contained withina single bubble, one of many resulting from the initial chaotic conditions. Even if thescalar field ϕ begins with a value close to the minimum at zero quantum fluctuationscan drive this value further from the minimum resulting in a self-sustaining ‘stochasticinflation’, or even motivating consideration of an ‘eternal inflation’ model.For any of the above inflationary models an energy-momentum tensor can bederived from a standard Lagrangian for a scalar field, namely L = ∂ µ ϕ∂ µ ϕ − V ( ϕ )including a kinetic as well as the potential term, via Noether’s theorem as described forequation 3.102 (some care is needed with the interpretation of translation invariancesince here we are clearly not dealing with a globally flat Minkowski spacetime, howeverequation 3.102 may be applied for sufficiently small spacetime regions by the strongequivalence principle described in section 3.4 and then generalised for the result below376n replacing η µν by g µν ) leading directly to ([5] p.329): T µν = ∂ µ ϕ ∂ ν ϕ − ∂ ρ ϕ∂ ρ ϕ g µν + V ( ϕ ) g µν (12.25)In addition to the cosmological term for a temporarily finite (and uniform at leastover the spatial extent of the observable universe) value V ( ϕ ) ≡ Λ κ , with an effectiveequation of state p Λ = − ρ Λ (= − V ( ϕ )), driving the exponential expansion, there arealso kinetic terms in the derivatives of the scalar field ϕ ( x ). An equation of motion for ϕ ( x ) can be derived as the Euler-Lagrange equation for the stationarity of the action S = R L p | g | d x which, since the metric g µν ( x ) incorporates the scale factor a ( t ), isfound to include a Hubble drag term of the form H ˙ ϕ ([5] p.331).If after the Planck time the universe is initially radiation dominated then asthe temperature drops below the critical temperature T c inflation begins to dominateand the radiation is rapidly redshifted. During the vacuum driven expansion the uni-verse is essentially devoid of matter and radiation, with the scalar field ϕ completelydominating towards the end of inflation, however any coupling between ϕ and matterfields leads to a further drag term in the equation of motion for ϕ . As the minimum of V ( ϕ ) is approached the dynamic equations drive rapid oscillations, which are damp-ened by the drag terms. This in turn fuels a reheating in the post-inflation era as thevacuum energy is converted into interacting particles, including the familiar states ofthe Standard Model. This period of transition to essentially zero vacuum energy, inwhich the energy is transferred from the scalar field ϕ to ordinary matter and radiationvia their mutual interactions, may also be the time during which any mechanism thatgenerates an asymmetry between matter and antimatter, as still manifestly observedtoday, may act. The origin of dark matter might also turn out to be associated withthe termination of inflation. This epoch then merges into the beginning of the radi-ation and then matter dominated FLRW periods of the standard cosmological modelas described in the previous section, with the initial conditions set by the inflationaryexpansion.In de Sitter spacetime, as for that of inflation, the event horizon (which is dis-tinct from the particle horizon) is of finite size, as for the case of back holes. Thismeans that the conditions for producing Hawking radiation, as alluded to towards theend of section 11.4, are also present during inflation. In turn the possibility arises thatquantum fluctuations can become frozen into residual classical deformations in thelatter stages of inflation. In turn these classical fluctuations will modulate the densityof the radiation and matter produced at the end of inflation, seeding the evolution oflarge scale structure as eventually manifested in galactic formations. Similar fluctu-ations during the inflationary epoch are also predicted to generate a background ofgravitational waves which still propagate today and which, although being much moredifficult detect, are in principle observable through the large scale CMB anisotropieswhich may provide a signature for the metric distortions of the gravity waves.A significant degree of fine tuning is required for any model of inflation basedon the properties of a postulated scalar field ϕ ( x ) in order to achieve a match witha range of empirical observations, which is somewhat unsatisfactory since inflation-ary theory was designed to avoid the necessary fine tuning as initially implied by thehorizon problem and flatness problem. There is also no understanding of the originof the vast difference between the magnitude of the effective cosmological term associ-ated with inflation due to V ( ϕ ), which is of O (10 − ) in natural units, and apparent377osmological constant Λ of the present epoch, which is of O (10 − ) in natural units.Indeed the unaccounted for magnitude of the latter number itself, the ‘cosmologicalconstant problem’ is one of the biggest puzzles in physics, as already alluded to brieflyat the end of section 4.1.A further significant issue regarding the standard model of cosmology, which isnot addressed by inflation, relates to the origin of the very special conditions of the BigBang in that the entropy of the early universe must have apparently been extremelylow, despite the high degree of thermalisation achieved for the degrees of freedom ofthe electromagnetic field. The degrees of freedom of the gravitational field may bedescribed by the Weyl tensor ([26] section 28.8), although both the Ricci curvatureand Weyl curvature parts of the Riemann tensor exhibit the effects of gravity. TheWeyl curvature and its distorting tidal effect tend to increase as matter gravitationallyclumps into dense regions, diverging to infinity in the neighbourhood of a black hole.The entropy associated with a black hole is correspondingly extremely high, attainingvalues much higher than that associated with ordinary thermal entropy. On the otherhand, as described following equation 12.23, in the idealisation of the FLRW cosmolog-ical models the spacetime is conformally flat with zero Weyl curvature. This suggeststhat if the universe originates in a state very close to an FLRW model the initially lowentropy may correlate with the very low Weyl curvature, both of which then tend toincrease as matter progressively clumps together as the universe evolves.More generally the ‘Weyl Curvature Hypothesis’, proposed by Roger Penrosein 1979 ([26] section 28.8), asserts that C ρσµν ( x ) = 0, or is at least very close to zero, asa constraint on the initial singularity. Hence the universe shares at least this property,of conformal flatness, with the FLRW models in the early stages. (In principle thisconstraint might be further augmented by the condition k = 0 as the universe evolvesinto a spatially flat model due to a subsequent period of inflation). This hypothesis ofzero Weyl curvature for the initial singularity of the Big Bang is then in stark contrastto the situation for the terminal singularities of black holes as alluded to above.This very special condition of the Big Bang represents an enormous constraintof low entropy on the initial conditions which in turn provides a suitable point ofdeparture for the second law of thermodynamics. Gravitation, in comparison to allother fields, hence appears to have had a very special status, aloof from thermalisationin the Big Bang, with the second law of thermodynamics only later exercised throughthe gravitational degrees of freedom. While inflation, as described for figure 12.4,provides the breathing space for ordinary matter, including the electromagnetic field,to reach thermal equilibrium in the aftermath of the hot Big Bang, the question remainsto explain why gravitation should apparently be treated in such a radically differentmanner to the other forces of nature. The theory presented in this paper may shedsome light on these questions since, as discussed in the previous chapter, here thegravitational field itself is not quantised and is hence different from all other fields inthis respect.Further, while for a range of given initial conditions inflationary theory cansolve the horizon problem, which was introduced in figure 12.3, by opening up a suit-able spacetime volume to allow points such as u and v to exhibit the same temperaturethrough thermalisation, as described for figure 12.4, the structure of these diagramsindicates that there may be a more fundamental difficulty with this picture. Namely,378ince any two different points such as x and y on the spacelike surface of the initialsingularity at t = 0, as indicated in figure 12.3 for example, have clearly never beenin causal contact with each other it is difficult to conceive how the Big Bang could beeffectively ‘triggered’ simultaneously across this potentially infinite 3-dimensional hy-persurface. This observation applies even if initial properties, such as the temperature,are very different at x and y . It also applies in exactly the same way for figure 12.4and inflation is of no help in addressing this ‘start-up problem’.On the other hand if the Big Bang can be considered as a ‘spacelike event’,encompassing the points of a large region of the initial spatial hypersurface, then thereseems no reason to suppose that the simultaneous ‘cause’ of the Big Bang at pointssuch as x and y in figure 12.3 might not also ‘cause’ them to have the same propertiessuch as temperature. Indeed, the notion of a simultaneous start-up along the t = 0spacelike surface which endows different points with different properties, implyingthe application of a range of possible start-up conditions and resulting in an uneventemperature distribution, seems somewhat more contrived. That is, it seems any twopoints like x and y on the initial singularity must be related in order for the universe tostart-up at both, and any solution to this problem may well itself entail a high degreeof homogeneity in the very early universe and solve the horizon problem without theneed for inflation. A source of later fluctuations and inhomogeneity will then still beneeded to account for the origin and formation of the galactic structures seen today.However, even without the issue of the uncertain role of quantum phenomenaunder the extreme gravitational conditions of the very early universe, care is neededin the extrapolation to the earliest epoch. For most FLRW models as the cosmic timeapproaches the moment of the Big Bang t → a ( t ) →
0, indeed the boundary condition a (0) = 0 is adopted for various dynamicsolutions, as described before equation 12.12. In this limit any finite comoving coor-dinate distance ∆Σ corresponds to a vanishing physical proper distance a (0)∆Σ = 0.With the horizontal axes in figures 12.3 and 12.4 representing coordinate distances thisnaive analysis implies that the observable universe at present came from a physicallyvanishingly small region of the initial singularity. In turn the initial singularity, rep-resented by the wiggly line in these figures, might perhaps be interpreted as a fullycausally connected entity in the limit t = 0, amending the strict causal structure ofthe conformal diagrams in this extreme case.However, since the spacelike coordinate distances are unlimited in magnitudeeven as a ( t ) → t → happens in the Big Bang, what causes it to happen and even why there should be a universe at all. 379 hapter 13 A Novel Perspective onCosmological Structure
Within the context of the present theory the external geometric structure of the worldis intimately associated with a subjective perceptual requirement, forged out of amulti-dimensional form of temporal flow expressed as L (ˆ v ) = 1, rather than beingan apparently arbitrary feature of an objective universe independent of the need forperception. Indeed the specific identification of 3-dimensional spatial expanses withan approximately global SO(3) symmetry would seem to be a somewhat redundantand unnecessary feature of such an inanimate mathematical entity. On the other handthe extent of spatial flatness for the observable universe, as described in the previouschapter, goes far beyond that utilised for perception by sentient beings on the planetEarth. Further, given the observed Hubble constant of equation 12.13 at the presentepoch, in a period of 100 years the fractional change in the scale factor is ∆ a /a ≃ . × − . Hence on the scale of a human lifetime the Robertson-Walker line elementof equation 12.5, for the k = 0 case, describes a flat Minkowskian spacetime to within1 part in 10 , with the expanding universe seemingly hanging suspended as a vastspatial expanse through a given human interval of cosmic time. For the horizontaltime axis representing a duration of 100 years figure 12.1 would then represent anaccurate snapshot of our universe at the present epoch.However the breakdown of global Lorentz symmetry beyond our 100 year thickslice of the universe is readily observed in the cosmological redshift. This redshift,defined in equation 12.21 and as first observed by Hubble and others and now probingdistant galaxies reaching back over billions of years in cosmic time, uncovers the non-Euclidean geometry of the cosmos as summarised by the evolution of the scale factor a ( t ) depicted in figure 12.2. 380he question then is the extent to which the present theory might accountfor the observations of such large scale structure in cosmology, and the phenomenaof the dark sector more generally as summarised in section 12.2, as we shall explorein this section. In section 12.3 it was described how the origin of spatial flatness andthe cosmological principle of homogeneity and isotropy, beyond the pragmatism ofassumptions employed for FLRW models, can in principle be accounted for by thetheory of inflation in the very early universe. In the following section the evolution ofthe very early universe and the nature of the Big Bang itself will be considered herewithin the context of the projection of spacetime out of the general form of temporalflow for the present theory. In section 13.3 the extent to which cosmological andother physical parameters might be explicitly constrained by the theory will also beconsidered.The pure flow of time s , underlying the multi-dimensional form of temporalflow through L (ˆ v ) = 1, is directly related to the proper time τ elapsed from the pointof view of any timelike trajectory through spacetime, as described in section 5.3. Timedilation effects for τ , as implied in the metric g µν ( x ) such as that for the Schwarzschildsolution of equation 5.49, are directly equivalent to those for s . A similar observationapplies for the Robertson-Walker metric of equations 12.5 and 12.6 and hence foran idealised galaxy based observer, with constant comoving coordinates { r, θ, φ } , thefundamental time parameter s in being to proportional to τ is in turn equivalent tothe cosmic time parameter t . Only in this special case under the assumptions of anFLRW model might s be associated with a preferred universal temporal parameter,namely the cosmic time t , for observers attached to idealised galaxies in the contextof such a model.However, the fundamental temporal flow s itself does not represent a uniqueuniversal parameter. In the context of large scale structure a local parameter s , subjectto each observer, depends upon the relative motion of the observer with respect to agalaxy or the relative finite peculiar velocity of the galaxy itself, in precisely the sameway as the proper time τ in special relativity. Similarly the parameter s will dependupon the location of the observer with respect to a local source of gravity, such as anymassive body or even a black hole, again exactly as for the proper time τ , in this caseas for general relativity.The relative time dilations for a community of N observers, each of whom isassociated with a personal flow of pure time s I ≡ τ I (for I = 1 . . . N , generalisingfrom the case of ‘twin A ’ and ‘twin B ’ described at the end of section 5.3), distributedanywhere in the universe dovetail together in a mutually consistent manner. Theparticular temporal parameter s I for a given observer describes the ‘fundamental’ flowof time underlying the mathematical structure of the multi-dimensional form L (ˆ v ) = 1through which the physical processes of the universe unfold from the perspective of that observer. In this sense each s I is a universal temporal parameter, as noted in thediscussion of the ‘problem of time’ in section 11.4 following equation 11.51.Locally the flow of time s ≡ τ parametrises the evolution of fields, such as agauge field Y ( x ) or fermion field ψ ( x ) and microscopic quantum phenomena generally,as well as the dynamics of macroscopic entities, such as a dust cloud described by theenergy-momentum tensor T µν = ρu µ u ν or classical matter generally. Either quantumor classical processes may be utilised in the construction of a physical clock which381ay in turn be employed to measure the proper time τ itself and hence observe timedilation effects. With the microscopic quantum properties of matter underlying, andin harmony with, the macroscopic geometry of gravitational phenomena there is no‘problem of time’ in this picture, as described in section 11.4, with gravity itself not quantised.In general relativity local coordinates can always be found such that for any4-dimensional metric, such as that in equation 12.5, the line element can be expressedthrough a local Minkowski metric with dτ = η ab dx a dx b . In the present theory such alocal structure derives from a 4-dimensional form of temporal flow ds = η ab h dx a dx b ,that is equation 5.47 which is equivalent to equation 5.46, that is the expression: L ( v ) = h (13.1)This latter structure is embedded within a higher-dimensional form such as L ( v ) = 1or L ( v ) = 1 as described in chapter 8 and section 9.2 respectively. It is the higher-dimensional form which both sets the normalisation for the temporal flow s and givesrise to a range of many possible solutions for an extended 4-dimensional spacetime,with geometry G µν = f ( Y, ˆ v ) as described for equation 5.32, incorporating quantumphenomena in the degeneracy of solutions as described in chapter 11.Hence with the geometry G µν ( x ) and the spacetime manifold M itself togetherdrawn out of the structures implicit in L (ˆ v ) = 1, with solutions such that G µν = f ( Y, ˆ v ) = 0 in general, there is no presumption of taking a flat background manifold asa starting point or expectation of obtaining such a Minkowskian spacetime geometry.With the external curvature related to the internal curvature as the symmetries of L (ˆ v ) = 1 are projected over M , as conjectured in section 5.1 in comparison withKaluza-Klein theory, there is a solution with both zero external and zero internalcurvature, as implied in equation 5.20 for example with G µν = f ( Y ) = 0. Even inthis case the assumption, as applied in section 5.3, that the value of L ( v ) = h ( x )of equation 13.1, as projected out of L (ˆ v ) = 1, is constant throughout spacetime isrequired to obtain a flat spacetime manifold. The consequences of a variation in thevalue of h ( x ), as alluded to at the end of section 5.3, will be considered shortly andwill contribute, along with the freedom of the gauge fields and quantum transitions,to a solution for G µν = f ( Y, ˆ v ) which is non-zero in the general case.Our a priori predisposition to mentally project a flat background of space andtime onto the world in order to perceive objects in it will be consistent with the abovemathematical structure provided an effective assumption of G µν ( x ) = 0 is a sufficientlygood approximation at least for the region of the world we locally inhabit. As discussedin section 12.1 this means for example that the local observation of a falling apple canbe accounted for in terms of a ‘force of gravity’ superposed upon an apparently flatarena of space and time, which in practice is both as precise as and much simpler thana full explanation in terms of spacetime curvature. On the global cosmological scale theobserved accelerating expansion of the universe not only contradicts the assumptionof a flat ‘vacuum’ geometry, but is also counter-intuitive given the terrestrial bias ofassociating gravity with a universal force of attraction.In the present theory the question does not concern what needs to be added to aflat background manifold to produce the effects of terrestrial gravity or the introductionof an apparent vacuum energy to account for the accelerating expansion of the universe,382ut rather, in all cases involving gravitation, to ask what is the form of G µν = f ( Y, ˆ v )in general. This observation applies to both everyday material objects such as applesand trees and also in the apparent absence of tangible matter in the case of the darksector for cosmology. This approach in the present theory can be summarised in thefollowing three points (which may be contrasted respectively with the three pointslisted near the opening of section 12.2 for the standard theory): • Rather than beginning with a flat spacetime G µν = 0 and then introducingterms such as T µν or Λ g µν through Einstein’s field equation 12.1 as an apparent source of curvature, with matter in some sense actively perturbing the otherwiseflat geometry, here the energy-momentum tensor is defined through the Einsteinequation, that is − κT µν := G µν , with the external geometry itself determinedthrough the relation G µν = f ( Y, ˆ v ) out of the underlying flow of time in the form L (ˆ v ) = 1 (as for the example of equation 11.12 and figure 11.1). • Hence there is no flat spacetime background, acting as a boundary condition,as an apparent consequence of the absence of matter. Originating from ourapparently innate bias to conceive of such a flat spacetime as a given entity, thisassumption in part underlies the apparent mystery of the cosmological constant,requiring the term Λ g µν to be added to the field equation in a seemingly ad hocmanner to account for the empirical observation. • On the third point quoted from [74] in section 12.2, a similar interpretationapplies here. The identity T µν ; µ = 0 follows trivially from the definition of T µν := G µν given the geometric Bianchi identity G µν ; µ = 0. Indeed, the reverseinterpretation of the Einstein equation with G µν := T µν implying that mattersomehow causes spacetime curvature is more problematic since an independentjustification is then required for the relation T µν ; µ = 0 in a general curved space-time, while the identity G µν ; µ = 0 does not require any such external support.Regarding the accelerating expansion of the universe the question then boilsdown to what in the structure of G µν = f ( Y, ˆ v ) might account for this observation.Ultimately a full understanding will be required for the general macroscopic form for G µν ( x ) constructed over a degeneracy of underlying local field exchanges δY ↔ δ ˆ v ,in principle incorporating some of the machinery of a quantum field theory as de-scribed in chapters 10 and 11. Both the matter density ρ and radiation pressure p , forequation 12.2 substituted into equation 12.1 to obtain equation 12.3, represent possi-ble macroscopic forms of G µν ( x ) which, while also entailing classical thermodynamicphenomena, are dependent upon the statistical range of possible exchanges for the mi-croscopic fields. Arising out of the degeneracy of possible field solutions the conceptualorigin of quantum and particle phenomena in the present theory differs to that in stan-dard QFT as described in chapter 11. Correspondingly the notion of a ‘vacuum state’is also different. Indeed the failure of calculations of the value for the vacuum energyin QFT to match the empirical value for Λ (typically by 120 orders of magnitude, asdiscussed towards the end of the previous section, see also for example [70] pp.790–791) provides a further argument for the need to reassess the underlying structure ofQFT itself, in particular in relation to the theory of gravitation. The possibility of383ddressing the cosmological constant problem within the context of the present theorywas raised at the end of section 11.4.While a number of features of the broken E action on the components of F (h O ) projected over M explicitly match features of the Standard Model of par-ticle physics, as described for equation 9.46 and summarised in the bullet points insection 9.3, in this chapter we shall describe more qualitatively potential connectionsbetween features of the present theory and those of the standard cosmological modeland theories of the very early universe.As alluded to above a correlation between the external curvature and internalgauge fields Y ( x ), expressed generically as G µν = f ( Y ), via the action integral ofequation 5.18, was described in section 5.1 through a comparison with the frameworkof Kaluza-Klein theory. Further, towards the end of section 11.3 it was implied thatan external geometry of a form which might ideally be expressed as G µν = f ( ψ ),corresponding for example to the electron field ψ ( x ) for figure 11.13(a) and (b) insection 11.4, may arise from the fermion components within the space F (h O ) forthe 56-dimensional vectors under L ( v ) = 1 through interactions with the gaugefields or more directly via an expression of the form G µν = f (ˆ v ). Similarly, withoutyet having a fully developed quantised theory, the possible physical manifestation offurther components in the space F (h O ) may be considered.In addition to the Lorentz vector v and Lorentz spinor components of anelement of F (h O ), transforming under the external subgroup SL(2 , C ) ⊂ E ⊂ E ,identified in equation 9.46 there are four Lorentz scalar components α, β, n and N which may also contribute to shaping the external geometry through G µν = f ( Y, ˆ v ).In principle any of these four scalars, or even the scalar magnitude | v | projectedonto M , could contribute to the macroscopic geometry. Each of the Lorentz scalars α, β, n, N and | v | also transform trivially under the internal SU(3) c × U(1) Q gaugegroups identified in section 8.2 and incorporated into the E symmetry in section 9.2.Hence, while the specific nature of SU(2) L × U(1) Y actions on these, or any other,components of F (h O ) is not yet known, in lacking both strong and electromagneticinteractions any of these scalar fields might contribute to the dark sector in cosmology.For example a constant value for a scalar field such as α, β, n, N or | v | pro-jected over spacetime might be associated with the constancy of the scalar Λ for aneffective cosmological constant term Λ g µν in the field equation 12.1 deriving from atleast one of these fields. Interactions between α, β, n, N and | v | implied under theterms of the constraint L ( v ) = 1 may underlie empirically observed gravitational ef-fects, in particular with the first four of these scalar fields coupled to the vector-Higgs v in this way. Similar interactions under L ( v ) = 1 also relate to the fermion massesas described for equation 9.48, and in particular for the low neutrino mass alongsideequation 9.49.The identification of these scalars in the components of the full form L ( v ) = 1projected over M is analogous to the appearance of a multiplet of scalar fields derivingfrom the components of a higher-dimensional metric in some forms of Kaluza-Kleintheory, via a non-Killing metric Φ on the gauge group G as alluded to towards theend of section 4.3, as the geometry is ‘reduced’ over a 4-dimensional spacetime. In thepresent theory there is no higher-dimensional physical metric but, as for the scalarsof Kaluza-Klein theories, here also scalar fields deriving from the breaking of the full384orm of temporal flow L ( v ) = 1 may have implications for cosmology.While ordinary matter, subject to the Standard Model internal gauge symmetrySU(3) c × SU(2) L × U(1) Y , clumps together with an energy density ρ ( x ) an essentialrequirement for a cosmological term is that, while locally having a much lower energydensity than ordinary matter, it should have a largely even effect over cosmologicalscales in the apparent ‘vacuum’ of spacetime. While here not making a quantitativeor specific argument for the Λ g µν term in the field equation the presence of a numbera scalar fields in the theory, any of which may impact upon the external geometry,provides a source for investigation.If a scalar field deriving from a component such as N in F (h O ) does give riseto a geometry of the form G µν = − Λ g µν the effective energy density T µν := − κ G µν inthe form of a perfect fluid with constant energy density ρ Λ = Λ κ , and equation of state p Λ = − ρ Λ , might appear as a form of ‘dark energy’ arising as an apparent vacuumstate, as described after equation 12.20 in section 12.2. The dynamical implications ofsuch a term, as implied in equation 12.20 and summarised for the first FLRW modellisted in table 12.1 in section 12.2, are well known to qualitatively match the empiricalobservation of the accelerating expansion of the universe at the present epoch.Regarding the projection of the components of v ∈ F (h O ) onto the basemanifold, and again without here making a rigorous argument, a symmetric rank-2energy-momentum tensor could be constructed as T µν = κ λv µ v ν , where λ is a real con-stant and v µ ( x ) = g µν v ν are the components of the Lorentz 4-vector v ⊂ v ∈ F (h O )projected onto TM with magnitude | v | = h . This proposal is also motivated by anal-ogy with the energy-momentum for dust T µν = ρu µ u ν , as contained in equation 12.2for p = 0, with the 4-velocity u (with | u | = 1) representing the motion of idealisedgalaxies in the FLRW models. The field equation for T µν = κ λv µ v ν can be written as G µν + λv µ v ν = 0, which has a similar appearance to the field equation G µν + Λ g µν = 0for the de Sitter model. On assuming the timelike flow of v to be aligned with thegalactic flow, as is the case for the 4-velocity u , the components of v are simply v µ = hu µ , which are numerically the same as v µ = ( h, , ,
0) on employing the metricof equations 12.5 and 12.6 and the comoving coordinates { t, r, θ, φ } . The substitutionof T µν = κ λv µ v ν into the field equations 12.9 and 12.10 then leads to the identicalsituation as the matter dominated case except here with an apparent matter density ρ = κ λh . For constant h ( x ) these equations do not lead to a solution unless λ ( t ) isallowed to vary with cosmic time as for the parameter ρ ( t ), in which case this modelis identical to the matter dominated case as listed in the middle column of table 12.1.Alternatively, since the full geometry is described by the Riemann tensor (whichfor example is also directly correlated with the internal curvature through relations ona bundle space such as equations 5.2 and 5.13 in the manner of a Kaluza-Klein the-ory) the Ricci tensor, defined with components R µν = R σµνσ , might be consideredto be geometrically more fundamental than the Einstein tensor in terms of having adirect link with the underlying fields such as N ( x ) or v ( x ) deriving from the com-ponents of F (h O ) in equation 9.46. For the case of a constant scalar N giving riseto a cosmological constant Λ postulating the relation R µν = Λ g µν implies directlythat G µν := R µν − Rg µν = − Λ g µν , which is identical to the case of the first modelin table 12.1 already considered above. On the other hand postulating the relation R µν = λv µ v ν as a possible vacuum limit does lead to a new scenario. Substituting this385xpression, with v = ( h, , ,
0) again aligned with the comoving coordinates, intothe R component obtained from the Robertson-Walker line element in equation 12.7leads immediately to the relation 3 ¨ aa = λh . This expression describes an exponen-tially expanding universe for constant λ > λv µ v ν term in the field equations.However, a solution is of course required to be consistent with all componentsof the field equation. The fundamental role of the Einstein tensor is essentially due tothe contracted Bianchi identity G µν ; µ = 0. The relation R µν = λv µ v ν implies in turn G µν := R µν − Rg µν = λv µ v ν − λh g µν which via the definition T µν := − κ G µν leadsto an effective energy-momentum tensor in the form of equation 12.2 for this modelwith the equation of state p λ = ρ λ = − λh κ . This contrasts with the de Sitter modelwith a Λ term for which p Λ = − ρ Λ = − Λ κ , as reviewed above. However the differingsigns means that for the case of R µν = λ ( t ) v µ v ν a solution for equations 12.9 and 12.10is only possible if λ is negative (that is p λ = ρ λ >
0) and allowed to vary in time, withthe result listed in the final column of table 12.1 in section 12.2. Hence rather thanbeing able to account for an accelerating expansion this hypothesis describes a moreextreme deceleration than either the matter or radiation dominated models. Only thefirst case listed in table 12.1 describes an accelerating expansion, with ( ρ + 3 p ) < T µν ; µ = 0was highlighted by equation 12.11, and similarly here the identity T µν ; µ = 0 itself for T µν = κ λv µ v ν or T µν = − κ ( λv µ v ν − λh g µν ) prohibits a constant value for λ . Howeverin principle a full solution for G µν = f ( Y, ˆ v ) may involve a range of contributionsindividually in the form of those in table 12.1 as well as others besides. In this casethere will be a string of terms effectively composing the energy-momentum tensorwhich collectively are required to satisfy T µν ; µ = 0, a relation which in general mayno longer hold for a particular contribution. This is very similar to the situation asdescribed for equation 5.41 in section 5.2 for which a synthesis of charged matter andthe electromagnetic field led to the Lorentz force law under the constraint T µν ; µ = 0.Here effectively a synthesis of several terms may arise within G µν = f ( Y, ˆ v ) on thecosmological scale.The 4-vector v in a λv µ v ν term could also be considered to have non-zerospatial components which might in principle relate to the formation of large scalestructure in the universe and open up possibilities not available for a purely scalardegree of freedom in a Λ g µν term. However this in turn would imply the complicationof loosening the FLRW assumptions of homogeneity and isotropy in the definition ofthe metric in equation 12.5. Even within those assumptions the possibilities with finitespatial curvature k = ±
1, as well the purely k = 0 solutions of table 12.1, might befurther considered. More generally, if the general structure of − κT µν := G µν = f ( Y, ˆ v )on the largest scales of the universe can be established it will be a case of refitting thecosmological data with the parameters of the new model.However, unlike the need to provisionally postulate explicit terms such as Λ g µν or λv µ v ν in the Einstein equation, as potentially effectively arising from a Lorentz scalarsuch as N or the Lorentz vector v in the components of v ∈ F (h O ) projected over M , there is a much more direct and intrinsic way in which this projection can shapethe 4-dimensional spacetime geometry. We describe this observation, and its possible386mplications for the large scale structure of the universe, for the remainder of thissection.Earlier in this section, as for the discussion in section 5.3, the gravitationaltime dilation effects for s ≡ τ have been considered to result entirely from the metric g µν ( x ) as might be obtained through the Einstein equation 12.1, such as the case ofthe Schwarzschild solution of equation 5.49, that is essentially for cases with a knownform of energy-momentum tensor. So far a constant magnitude has been assumedfor | v | = L ( v ) = h in equation 13.1 in the projection of v ⊂ v onto TM .However in principle all fields on M may vary, within the necessary constraints suchas L ( v ) = 1, similarly as the internal gauge field Y ( x ) can vary under the constraintthat the action integral of equation 5.18 should remain stationary, that is δ ˜ I = 0,for example. Since the components of v ∈ TM represent the injection of the puretemporal flow s into the base manifold M any variation in | v | will itself have someimpact on the spacetime geometry. Here we begin by considering this impact upon anotherwise flat manifold.Hence we first return to the translation symmetry of the form L ( v ) = h under the four degrees of freedom { x , x , x , x } ∈ R , as originally depicted for the3-dimensional case in figure 2.2. Here the constant vector field v = ( h, , , h and with v = dx /ds = h , is aligned withthe global Lorentz frame as represented in figure 13.1(a). This first figure depicts theuniform translation symmetry implicit in the form L ( v ) as described in equation 2.13,which contrasts with the case in figure 13.1(b) in which the magnitude h ( x ) of the 4-vector v ( x ) is free to vary.Figure 13.1: The vector field v subject to L ( v ) = h ( x ) for (a) the original translationsymmetry over R ≡ M with constant h ( x ) and global Lorentz symmetry and (b) with h ( x ) variable and only local Lorentz symmetry. In both cases the flow v is alignedto the timelike coordinate x while x i with i = 1 , , g µν ( x ) is projected from the form L ( v ) = η ab v a v b = h , framing the local injection of temporal flow into the basemanifold. However with the local coordinate x of figure 13.1 representing the funda-mental flow of time according to the relation δs = δx /h , the expression L ( v ) = h ,subject to the constraint L (ˆ v ) = 1, also sets the scale for temporal flow in the localframe. That is, the x coordinate representation of time will vary with the value of387 . With δs = δx /h the pure time s will effectively flow more slowly in regions oflarge h , corresponding to the vectors v with a larger magnitude in figure 13.1(b), andmore quickly in spacetime regions with a lower value of h . More generally the relation L ( v ) = h can be rearranged in the form ds = η ab h dx a dx b , that is the final relationin equation 5.47, with the spacetime metric extracted as: g µν ( x ) = 1 h ( x ) η µν (13.2)when expressed in the global coordinates on the extended manifold M (see also thediscussion of equation 13.3 below). This physical metric g µν , related to flat spacetimethrough the conformal transformation η µν → η µν /h ( x ), describes a non-Euclideanmanifold incorporating time dilation effects. As for general relativity, while generalcoordinate systems are arbitrary and unphysical, local inertial frames with g µν ( x ) = η µν and ∂ σ g µν = 0 do have physical significance. Such an inertial frame may beidentified globally for figure 13.1(a) but only locally for figure 13.1(b). By the strongequivalence principle the laws of physics according to special relativity apply in a localinertial reference frame. As described in section 3.4 the weak equivalence principle issufficient to incorporate the notion that all gravitational effects can be transformedaway in a sufficiently small spacetime volume, and can be interpreted as implying thatthe torsion vanishes.While the unphysical nature of coordinate systems in general relativity is en-capsulated under general covariance, as also described in section 3.4, any coordinatesystem can be represented by the parameter space grid of figure 3.6(a). In the spe-cial case of Minkowski spacetime such a coordinate grid can be mapped onto the4-dimensional manifold such that the metric has constant components g µν ( x ) = η µν ,as is the case for the spacetime underlying the constant vector flow depicted in fig-ure 13.1(a). In this case a unique family of coordinate charts are identified through theparameter space of translation symmetry of the form L ( v ) = h , as described in equa-tion 2.13, and related to each other via global Lorentz transformations. On the otherhand in projecting the coordinate grid of figure 3.6(a) onto the spacetime underlyingfigure 13.1(b) the simplest expression for the metric takes the form of equation 13.2.For the metric of either figure 13.1(a) or (b) obtained in this way through theunderlying injection of temporal flow s into the spacetime manifold, as for the case of ametric determined as a solution to Einstein’s equation as considered in section 5.3, theproper time τ recorded by physical clocks is again tied to the fundamental flow of time s . This is the case since the laws of physics, including those utilised by the structureof clocks, unfold through the underlying temporal flow s and hence the proper time τ ≡ s exhibits the equivalent time dilation effects due to variation in L ( v ) = h , aswas the case for other sources of temporal dilation. The question then concerns themore specific nature of this relation between τ and s , as originally discussed at the endof section 5.3 and earlier in this section.The fundamental temporal flow s is modelled by the real line and hence canbe represented by the values of a pure real number s ∈ R , intervals of which can beexpressed in terms of a set of real parameters of arbitrarily high dimension, as describedfor equation 2.4, which is an essential observation for the present paper. On the otherhand the proper time τ represents intervals of 4-dimensional spacetime on the manifold M and is expressed by a real number associated with the dimension of length L (which388s equivalent to the dimension of time T since natural units are employed, and in asense it would be more appropriate to use T as we are ultimately dealing with multi-dimensional forms of temporal flow). Hence the constant factor γ relating the pure1-dimensional temporal flow s to a corresponding empirically observable progressionin proper time τ = γs is one which carries the dimension of length L . Hence in turnproper time intervals for the spacetime geometry underlying figure 13.1(a) or (b) canbe expressed through: dτ = γ ds = γ h η µν dx µ dx ν (13.3)Here then the metric g µν = γ h η µν explicitly carries the dimension of L , as was de-scribed for the general case in the discussion following equation 12.5 in section 12.2.Since the underlying temporal flow s is not directly observed, and the scale of the realline parametrising s is in any case arbitrary, once empirical units, such as metres, arechosen for τ the coordinate parameters can in turn be chosen such that the metrictakes a convenient form. For example in the case of figure 13.1(a) pseudo-Euclideancoordinates can be chosen such that g µν ( x ) = η µν everywhere.While the value of the factor γ is of no meaning, its significance lies in rep-resenting a constant relation τ = γs . In practice setting γ = 1 can be interpretedas choosing the arbitrary scale of s ∈ R to match the practical parametrisation ofthe proper time τ . In this case the basic metric g µν from equation 13.3 is that ofequation 13.2. With h ( x ) varying the constancy of γ in equation 13.3 implies that ingeneral it is not possible to find any coordinates such that g µν ( x ) = η µν globally forthe scenario in figure 13.1(b), although this relation is always possible locally, as alsosuggested by the equivalence principle.Hence variation in the value of h ( x ) on M directly modifies the effective metric g µν ( x ), warping the spacetime geometry that underlies the vector field in figure 13.1(b)for example. Assuming the geometry to be described in terms of a torsion-free linearconnection the corresponding Levi-Civita connection Γ can be constructed as a functionof the metric of equation 13.2 via equation 3.53 and in turn the components of the fullRiemannian curvature tensor R ρσµν of equation 3.73 computed. In turn the Einsteintensor, for the conformal geometry g µν = θ ( x ) η µν with a real scalar field θ ( x ) = h − ( x ),is found explicitly (and cross-checked with a related calculation in [9] pp.42 and 76)to take the form: G µν = − θ − ∂ µ θ∂ ν θ + 34 θ − ∂ ρ θ∂ ρ θ g µν + θ − ∂ µ ∂ ν θ − θ − (cid:3) θ g µν (13.4)A similar expression, with a different set of coefficients, is obtained as a function of h under the substitution θ → h − , as for any other scalar field related to θ by a simplepower expression. The derivation of this expression for G µν follows the same chain ofrelations that led to the form of G and G , appearing alongside the correspondingΛ g µν terms on the left-hand side of equations 12.9 and 12.10 respectively, given themetric form of equation 12.6 and via the Ricci tensor R µν and scalar curvature R .However here equation 13.4 represents a direct warping of the spacetime geometry dueto the variation in L ( v ) = h which implies equation 13.2, without the need to employfurther assumptions regarding the form of an energy-momentum tensor in Einstein’sequation in order to extract a solution. 389ence this construction can be contrasted with the usual determination of ametric g µν in general relativity. There the metric is extracted as a solution to theset of second order differential equations contained in the Einstein field equation 12.1under assumptions of symmetry regarding both the matter distribution and the formof the metric itself. This was the approach taken for the Schwarzschild solution ofequation 5.49 and also for the cosmological models based on the Robertson-Walkerline element of equation 12.5. Here in contrast the form of the metric g µν = h η µν implies a linear connection Γ and Riemannian curvature R and hence leads to theEinstein tensor G µν = f ( v ) as a consequence of the variation in L ( v ) = h ( x ) underthe constraint L (ˆ v ) = 1, rather than as an equation to solve for the metric.In practice the distribution h ( x ) might be constrained by observations of thecorresponding gravitational effects, in a similar way that the constant k and scale factor a ( t ) of the Robertson-Walker line element of equations 12.5 and 12.6 are determinedthrough empirical observations, found to be consistent with k = 0 and deducing thestructure depicted in figure 12.2 for example. In this sense the procedure to constrainthe actual function h ( x ) is very similar to the standard approach for general relativity,that is by matching equation 13.4 with empirical observations. On the other hand inthis case it may also prove possible to calculate both the typical value of h ( x ), andthe typical range of variation in this magnitude, as constrained for example under therelation L (ˆ v ) = 1, within the theory itself.In addition to the warped spacetime G µν = f (ˆ v ) of equation 13.4 geometriesof the form G µν = f ( Y ), relating the external curvature to the internal gauge fieldsas described in section 5.1, are also possible. The combined general expression G µν = f ( Y, ˆ v ) can be interpreted to incorporate a contribution from the gauge fields Y ( x )which determine the metric via the differential field equations G µν = f ( Y ), while thegeometry G µν = f (ˆ v ) concerns the direct impact of L ( v ) = h ( x ) on the metric in theform of equation 13.2. As described in chapter 11 ordinary matter exhibiting quantumphenomena will arise out of an underlying degeneracy of solutions for G µν = f ( Y, ˆ v )given the field exchanges such as δY ↔ δ ˆ v allowed according to the selection rulessummarised in equations 11.29.Since the fermion components ψ ( x ) in F (h O ) are correlated with the compo-nents of v ≡ h in F (h O ) under the constraint L ( v ) = 1 as described for equa-tion 9.48, fermion terms may explicitly appear through field exchanges of the form δ v ↔ δψ . That is, these interactions may directly give rise to the effective geometry G µν = f ( ψ ) alluded to earlier in this section and towards the end of section 11.3, andas applied for the external geometric structure associated with the fermion fields un-derlying the processes depicted in figures 11.13(a) and (b) for example. In general thefull set of microscopic field redescription possibilities, consistent with the constraintequations, will need to be taken into account to determine the form of macroscopicgeometry G µν = f ( Y, ˆ v ) as shaped through a local degeneracy of field solutions asdescribed in chapter 11.The gravitational time dilation effect, that is the relative slowing of time in thevicinity of a massive object, described by the Schwarzschild solution for the metric inequations 5.49 and 5.50, can be ascribed to the presence of the massive object itself ingeneral relativity. Accordingly the situation for regions in figure 13.1(b) with relativelylarge values of h = | v | , and hence a relative slowing of the flow of time, might be390reverse engineered’ to identify an apparent presence of ‘matter’ in such a region. Thatis, for the geometry G µν = f ( v ) in equation 13.4 it is possible to define an associatedenergy-momentum tensor through − κT Dµν := G µν = f ( v ). Here T Dµν does not thenrepresent ‘ordinary matter’ which is built upon a degeneracy of gauge and fermion fieldsolutions for the geometry G µν ( x ) over M , and in particular made ‘visible’ throughthe U(1) Q electromagnetic interactions, but rather an underlying warping of spacetimegeometry itself. Hence, while not describing baryonic matter, the implicit energy-momentum T Dµν is a candidate for the ‘dark matter’ of the universe.With T Dµν defined in this way for the Einstein tensor G µν of equation 13.4deriving directly from the metric g µν of equation 13.2, this in fact follows the procedurefor obtaining solutions for Einstein’s equation by cataloguing ( g µν ( x ) , T µν ( x )) pairs asoutlined towards the end of section 5.2. Here however it is the form of the metric thatis physically motivated and not arbitrary while the resulting energy-momentum tensorneed not necessarily correspond to any known form of matter.The term ‘dark matter’ implies a kind of ‘phantom source’ of gravitation, whichis only detectable through its manifestation as a structure of spacetime geometry,and indeed the above definition of T Dµν essentially describes a purely gravitationalphenomenon. In general a structure described in the relation − κT µν := G µν may ormay not be detectable as ‘matter’ and may or may not be detectable as ‘gravity’. Forexample ordinary baryonic matter in the form of stars or planets is both visible asmatter T µν and evident as gravity G µν . On the other hand baryonic matter in theform of tables and chairs, while clearly exhibiting a number of properties of matter,does not give rise to any detectable gravitational effects. Contrary to that situation‘dark matter’ in the form of G µν = f ( v ) might produce very significant gravitationalphenomena without being associated with any apparent material effects at all. Forexample, as alluded to above, the variation in h ( x ) might be determined throughobservations of galactic rotation curves and gravitational lensing effects rather thanan explicit empirical detection of a ‘dark matter’ distribution (as would be possible forexample for a cloud of dust on a galactic scale). A fourth case is conceivable in whicha definite mathematical form of − κT µν := G µν = f ( Y, ˆ v ) has evaded detection bothas a material and a gravitational entity.Since the material effects of the local ordinary matter distribution present them-selves more immediately than the corresponding gravitational phenomena, historicallythe sense that universal gravity is a property to be associated with matter was a natu-ral point of view to adopt. In turn the Einstein equation G µν = − κT µν , influenced bythe Newtonian gravity which arises in the appropriate limit, was initially interpretedto imply that in some sense matter ‘causes’ the curvature of spacetime . That interpre-tation is considered to be a ‘reverse engineering’ from the perspective adopted in thispaper in which the energy-momentum tensor is simply defined through − κT µν := G µν ,with the spacetime geometry determined primarily as a solution for G µν = f ( Y, ˆ v )subject to the constraint equations (see also the discussion in the opening paragraphsof section 5.2).In terms of the spacetime solution in the particular region of the early universe,through the mutual gravitation of dark matter and ordinary baryonic matter the effectsof G µν = f ( v ), as a network of creases in the underlying fabric of spacetime, mighthave guided the formation of galaxies and galactic clusters. The properties of these391tructures are then visible today through the motions of galaxies within clusters andthe rotation curves of stars within galaxies, all still in mutual gravitational interactionwith the dark matter. This interplay between baryonic and dark matter is depictedin figure 13.2, where the final stage labelled (e) corresponds to the kind of structuresobserved through to the present epoch as also represented in figure 12.1.Figure 13.2: (c) Fluctuations in the magnitude h ( x ) of the vector field v , representedby the vertical arrows (as for figure 13.1), in the early universe (d) gravitationallymerge along with baryonic matter, represented as points of dust, as the universe evolvesleading to (e) the formation of large scale galactic structures as observed through to thepresent epoch. (Earlier epochs will be represented in figure 13.3(a),(b) in the followingsection).In the standard cosmological model it is known that the dark matter cannotbe baryonic due to the abundances of the light elements resulting from nucleosynthesisin the early universe. Dark matter composed of relic particles from the Big Bangmust also be weakly interacting in order to have evaded direct detection. In the caseof ‘cold dark matter’ (CDM) the relic particles have a low thermal velocity leadingto a hierarchical formation of structure through the merger of smaller initial unitsbeginning in the early universe. This description is consistent with the picture infigure 13.2, except that for ‘dark matter’ in the form of variations in | v | there areseemingly no associated ‘particle’ phenomena at all.Since v ( x ) is a 4-vector field, as well as fluctuations in | v | , ascribed to thetemporal component v in figures 13.1(b) and 13.2, in principle there may be variationsin the spatial components v i also (as suggested earlier in this section for the case inthe final column of table 12.1 for a spacetime geometry incorporating a λv µ v ν term392n the Einstein field equation) which could be pictured as a horizontal component forthe vectors in these figures. Such spatial fluctuations are counter to the assumptionof strict homogeneity and isotropy, as indeed are variations in the magnitude h ( x ),but they could potentially be a factor in the observed peculiar motions of galaxies andclusters of galaxies and might even be associated with a ‘dark flow’ if observations ofsuch phenomena were to be established.Further, while fluctuations towards higher values of | v | , that is a larger valuefor L ( v ) = h , correspond to regions of spacetime with an apparent slowing of theflow of time τ ≡ s , and hence associated with ‘dark matter’, regions with a smallervalue for L ( v ) = h will have a complementary spacetime geometry with a fasterrate of temporal flow and in principle the opposite gravitational effect. Such regionsmay hence tend to open up cosmic ‘voids’ between the galactic clusters and play animportant role in the structural evolution process represented in figure 13.2. On thelargest observable scales such a gravitational repulsion might also be a factor in thecomposition of the apparent ‘dark energy’.It is a very familiar idea that a 2-dimensional surface embedded within a 3-dimensional space will generally have an intrinsic curvature, such as the surface of aball for example. Here we are considering the embedding of 4-dimensional spacetimewithin the structures of a higher-dimensional form of temporal flow L (ˆ v ) = 1, andit is again natural to expect that in general a finite intrinsic curvature for the 4-dimensional manifold might result. It is hence proposed that such intrinsic curvaturefor the spacetime geometry, closely correlated with variation in the component valuesof the projected 4-vector v ⊂ ˆ v onto TM , constitutes at least a significant factor inaccounting for the observed phenomena of the ‘dark sector’ in cosmology.In general relativity spacetime curvature might be considered to account forthe origin of mass in general by interpreting Einstein’s field equation essentially as adefinition of energy-momentum − κT µν := G µν , as we have in this paper and as re-viewed above. This is in contrast with the Standard Model of particle physics in whichthe Higgs field φ ( x ) and the Higgs mechanism of spontaneous symmetry breaking, asdescribed in section 7.2, is responsible for the origin of mass through field interactionsfor a theory framed in a flat spacetime background.In the present theory with a continuous variation in the magnitude of theunderlying field v ( x ) on M , implying time dilation effects and shaping the spacetimegeometry, an apparent ‘mass’ might be associated with this field through − κT Dµν := G µν = f ( v ) as described above, and hence the field v ( x ) can be considered as the source of this apparent mass. In subsection 8.3.3, and in particular in the discussionaround equation 8.76, and similarly around equation 9.48 in section 9.2, the same field v ( x ) has been associated with Higgs phenomena in conveying masses to the fermionand gauge boson fields via possible δ v ↔ δψ and δ v ↔ δY interactions respectively,compatible with the constraints summarised in equations 11.29.Combining these observations suggests that the physical mechanism throughwhich a ψ ( x ) or Y ( x ) field interaction with the ‘vector-Higgs’ field v ( x ) results in a‘mass’ for the fermions or gauge bosons respectively is through the effect on the localgeometry due to the projection of the field v ( x ) ∈ TM out of ˆ v ( x ) under the fullform L (ˆ v ) = 1. Such a δ v interaction may locally correspond to a further geometriceffect on top of the continuous v ( x ) variation. In the present theory such external393ravitational effects will be compatible with underlying quantum effects through theabove field redescriptions of the form δ v ↔ δψ and δ v ↔ δY , which add to thelist of possible interaction vertices of figure 11.3 in the correspondence with Feynmanrules for a quantum field theory. In the context of QFT the relation between the‘bare mass’ associated with these field interactions and the measured mass of physicalparticle states, as considered near the end of section 11.3, will depend upon this impacton the external spacetime geometry.In this theory while both dark matter and Higgs phenomena are directly asso-ciated with the field v ( x ) the dark matter is of course not composed of Higgs particles.Rather the Higgs interactions via discrete δ | v | exchanges are closely associated withvisible baryonic matter in the universe, and as observed in high energy physics ex-periments, while dark matter relates to a continuous variation in the underlying field v ( x ). Even if L ( v ) = h ( x ) was constant on large scales, hence with no dark mat-ter phenomena, ‘strongly coupled’ field interactions with the vector-Higgs field v andHiggs particles would still be observable in the laboratory. This observation is compat-ible with the apparently ‘weakly interacting’ effects of dark matter as a gravitationalphenomenon that arises through variations in h ( x ) on the galactic scale. That is, whilethe spacetime geometry resulting from the continuous v ( x ) variation need not itselfbe associated directly with any quantum or particle phenomena, the scalar δ | v | in-teractions of the same everywhere pervading field may give rise to the detected Higgsparticle states.More generally the question remains to understand whether specific quantumor particle effects might be associated with the dark sector, and how the geometricphenomena arising from the injection of the temporal flow into the spacetime manifoldrelate to the properties of the familiar Standard Model particle states. Together thesephenomena shape the world geometry as described collectively under G µν = f ( Y, ˆ v ).The external geometry will involve the conformal transformation of equation 13.2,which generates Ricci curvature and leads to equation 13.4, together with more gen-eral solutions over a degeneracy of underlying internal field exchanges, resulting in acombination and interplay of both Ricci and Weyl curvature in general.Dark matter is empirically observed to be associated with galactic clusters,and hence the value of h ( x ) is expected to be larger in such regions and lower ininter-galactic space, as sketched in figure 13.2(e). Given that copious photons of lightand other Standard Model particles can be detected on Earth after being transmittedthrough such regions, in travelling from distant galaxies, it appears that the propertiesof such particles must be physically robust for small variations of h ( x ) to some degree.The development of the full physical form and consequences of the expression G µν = f ( Y, ˆ v ) will require a greater understanding of the incorporation of quantumphenomena as introduced in chapter 11. The full implications of the theory, derivedeither via direct calculation or simulation, for particle physics as well as cosmologywill depend both on the degree of variation of L ( v ) = h ( x ) and the typical valueof h ( x ) itself at the present epoch. With material properties and the laws of physicslikely to have some dependence on the value of h ( x ) it may be that the StandardModel of particle physics requires a certain apparent ‘tuning’ of this parameter inorder to allow the formation of ordinary baryonic matter itself. This raises the questionmore generally of the possible uniqueness, or otherwise, of the ‘physical constants’ as394bserved in the world, both for Standard Model of particle physics and in terms of thecosmological parameters, as we shall discuss in section 13.3.In the following section we first consider the possibility that both the averagevalue and the fluctuations in h ( x ) may have been very different in the very earlyuniverse, leading to a ‘phase transition’ to an average value of h ( x ) compatible withthe nature and properties of Standard Model interactions in particle physics and whichhas remained stable to the present day. In this scenario the phase transition may marka point of convergence upon the familiar laws of physics in 4-dimensional spacetimemore generally. These may include the second law of thermodynamics expressed interms of the degrees of freedom of familiar interacting particles which are themselvesproduced in the phase transition.In summary, the ‘novel perspective’ in the title of this chapter refers to themanner in which the intrinsic geometry of the 4-dimensional spacetime backdrop forcosmology is shaped through the projection of the extended M manifold out of thefull multi-dimensional form of progression in time L (ˆ v ) = 1. While the present theorybased on general forms of time is very simple there are a number of features suchas the projection of the vector-Higgs field v onto TM , generating the conformaltransformation of equation 13.2 parametrised for example by the dilation symmetriesdescribed in the opening of the following section, and a set of elementary scalar fields α, β, n and N , as described earlier in this section, which potentially correlate with largescale cosmological phenomena in particular associated with the dark sector. Thesefeatures identified within the components of v ∈ F (h O ) for the 56-dimensionalform L ( v ) = 1 are complementary to the features identified for the Standard Modelof particle physics as summarised in equation 9.46 and in section 9.3. Since the knownphenomena of HEP cannot account for the dark sector in cosmology new features,such as identified in this section, are indeed required to account for the cosmologicalparameters. There then remains the question concerning the degree to which themathematical structures described in this section might compare quantitatively withempirical observations of the large scale physical structure of the universe.A first step will be to seek a guide through a comparison between the approachof the present theory and geometric models aimed at accounting for the dark sector ofcosmology in the existing literature. While papers involving conformal gravity (see forexample [76, 77]) may account for elements of the dark sector in geometric terms, suchmodels appear somewhat different to the approach described in this section. In replac-ing the Einstein-Hilbert action by a conformally invariant action based on the Weyltensor these papers do however implicitly incorporate geometric transformations of thekind in equation 13.2 and hence may relate to the structures of the present theory. Thepresent theory both aims to avoid the employment a Lagrangian formalism and doesnot propose a ‘modified gravity’ of any kind. In fact here the Einstein equation is iden-tified as a fundamental feature embedded within the definition of energy-momentumthrough the expression − κT µν := G µν = f ( Y, ˆ v ), which also provides the interpretationof the Einstein equation in the context of the present theory. While one aim of thispaper has been to avoid ‘postulating’ a Lagrangian of any form, the Einstein-Hilbertaction of equation 3.79 for general relativity, for the vacuum case with L = 0 andΛ = 0, has been adapted in section 5.1 to facilitate a provisional connection betweenthe external and internal geometry arising from the symmetries of L (ˆ v ) = 1 broken395ver the base manifold M , as described for equations 5.18 and 5.19 and guided byKaluza-Klein theory.Further, rather than devising a scheme tailored to match empirical observa-tions, here we begin with an underlying conceptual motivation and foundation for thetheory. Once this theory has been sufficiently developed a full cosmological modelmight be established leading for example to a calculation for the density parametersΩ B , Ω D and Ω Λ , as introduced following equation 12.15 in the previous chapter. Inprinciple the cosmological data itself might then be refit within the context of thetheory in order to test these ideas quantitatively. The highest-dimensional form of temporal flow considered in any detail in this paperis the form L ( v ) = 1 with E symmetry, as introduced in section 9.2. If any ofthe four scalar components { α, β, n, N } of v ∈ F (h O ) in equation 9.46 is foundto be associated with an effective cosmological term Λ g µν in equation 12.1 then themagnitude of this component will be correlated with the magnitude of the accelerationof the cosmic expansion, which may be arbitrarily small. Similarly the magnitudeand variation of the projected v ∈ TM components may directly correlate withthe properties of dark matter or dark energy, while as a ‘vector-Higgs’ the field v also generates mass terms for the fermions and gauge bosons and underlies Higgsphenomena in general, as also reviewed in the previous section. Interactions betweenthe scalars { α, β, n, N } and v might also generate massive weakly-interacting scalarstates as a possible contribution to the dark sector.One means of varying | v | can be described via a simple dilation symmetry as asubgroup of SO + (1 ,
9) for the model described in section 5.1 for a 10-dimensional form L ( v ) = 1 projected over M as pictured in figure 5.1. This dilation symmetry actson the components of v such that the magnitudes of the external v and internal v vectors are traded subject to the constraint L ( v ) = | v | + | v | = 1. This variationin h = | v | can also be described in terms of a one-parameter subgroup denotedD(1) X ⊂ SL(2 , O ) acting as a dilation symmetry on the components of v ≡ X ∈ h O in equation 6.16 preserving L ( v ) = det( X ) = 1.In terms of the largest form of temporal identified another possibility for ‘tun-ing’ the magnitude of the v components under the constraint of the 56-dimensionalform L ( v ) = 1 lies in the dilation symmetry, which will be denoted D(1) λ ⊂ E , asparametrised by λ ∈ R and introduced in equation 9.30. This symmetry acts upon all27 components of Y ∈ h O of the 56-dimensional space F (h O ) in a uniform way, andnot only on the v ⊂ Y subset of components in equation 9.46.As an intermediate case a further dilation of components can be identifiedwithin the E symmetry on the h O subspaces of F (h O ) and will be denoted D(1) B ,as generated by the linear combination of boosts ˙ B tz + 2 ˙ B tz and as introduced inequation 8.35. From table 6.6 this E generator as a vector field in the tangent space396 h O has the form: ˙ B tz + 2 ˙ B tz = + p +¯ a − c + a + m − ¯ b − ¯ c − b − n (13.5)Hence under the action of D(1) B on the components of X ∈ h O in equation 6.26 the10 components of X are inflated while the remaining 17 components of θ and n aredeflated, or vice versa. The consistency of this D(1) B action with the preservation of L ( v ) = det( X ) can be seen directly from the form of det( X ) in equation 6.27 togetherwith the generator coefficients in equation 13.5. For this particular linear combinationof boosts the rank-6 E Lie algebra contains a rank-6 subalgebra decomposition, whichin terms of the corresponding Lie groups can be written as:SL(2 , C ) × SU(3) c × U(1) Q × D(1) B ⊂ E (13.6)As mentioned for the same decomposition in equation 8.35 the mathematical structureof this Lie subalgebra is described in [38] (the first example in Appendix D, p.187).In the present paper the subgroup SL(2 , C ) has been identified with the externalsymmetry of spacetime M in section 8.1, and SU(3) c × U(1) Q as the internal symmetrysubgroup within Stab( TM ) in section 8.2, in each case with a corresponding physicalinterpretation. Clearly D(1) B is not a subgroup of Stab( TM ) due to the action onthe v ⊂ X components in equation 13.5, however since D(1) B is independent ofSL(2 , C ) × SU(3) c × U(1) Q , as described in equation 13.6, it may also be of physicalsignificance.Regardless of the means of varying the 4-vector magnitude | v | = h , whethervia D(1) X , D(1) B , D(1) λ or otherwise, the geometric impact of relatively high valuesof L ( v ) = h ( x ) projected out of the full form L (ˆ v ) = 1 over extended regions of thecosmos is considered to form a candidate for the effects of dark matter, as describedin the previous section. As well as these small variations in h ( x ), the implications of amuch larger time-dependent evolution in the scalar value h ( t ), as averaged over the 3-dimensional spatial hypersurfaces as a function of cosmic time t , can be considered forthe very early universe, as we describe in this section. For example we shall considerthe progression from a value approaching zero h ( t ) → t → h ( t v ) = h in a period of time associated with the epoch of the BigBang, with h = | v | hence also denoting the present day average value. On adoptinga normalisation factor of γ = 1 (a further natural option would be to set γ = h )relating the fundamental temporal flow s with the proper time τ , as described forequation 13.3, the basic metric deformation can be described by equation 13.2. Aninterval of proper time dτ can then be expressed as: dτ = ds = 1 h ( t ) (cid:2) dt − d Σ (cid:3) (13.7)where d Σ represents a Euclidean 3-dimensional spatial element. While this line elementhas the form of a conformal transformation dependent upon h ( t ), similarly as for thecase of equation 12.23 for the FLRW models with conformal time parameter η , wecontinue to think of t as the ‘cosmic time’ parameter. Given that the value of h ( t )397s only considered to differ from the present value h for t < t v in the very earlyuniverse, and that elementary physical structures will be unfamiliar during that epochas the nature of the projection of L (ˆ v ) = 1 over that region of M is correspondinglyalso different, the form of any ‘physical clock’ and the measure of time itself will needfurther consideration for this earliest era. Hence in the above mathematical expressionwe keep track of the coordinate time t in place of defining a new temporal parameter.The identification of D(1) B ⊂ E alongside other subgroups in equation 13.6is analogous to the proposed subgroup SU(2) L × U(1) Y ⊂ E , as a candidate forthe gauge symmetry underlying the left-handed weak interactions, as described inequation 9.47 and the subsequent discussion. In the case of SU(2) L × U(1) Y thefact that this symmetry is not independent of Stab( TM ) (while it is independentof SL(2 , C ) × SU(3) c ) leads to the phenomena of ‘electroweak symmetry breaking’through the interaction of the SU(2) L × U(1) Y gauge fields and the v ∈ TM vector-Higgs field, breaking the symmetry down to U(1) Q . For the future development of thepresent theory it will be important to gain an understanding of the interplay betweenelectroweak symmetry breaking (as well as the role of the unification scale describedfor figure 11.10) and the breaking of the D B (1) symmetry action (or other dilationsymmetry) in the very early universe.A ‘gauge field’ associated with the D(1) B dilation symmetry, through interac-tion with the components of v , might itself drive an inflationary effect in the veryearly universe. The very different physical environment associated with a very dif-ferent magnitude of v ⊂ v projection onto TM in the very early universe mightalso in principle incorporate some of the effects of a cosmological inflation. In eithercase, following the inflationary epoch the D(1) B symmetry action would effectively bebroken in a ‘phase transition’ as the value of | v | is stabilised and the parameters ofthe Standard Model of particle physics are established. That is, with field interactionsleading to a mutual stabilisation of both the value of | v | = h and the Standard Modelparameters.In this theory, with the field v also closely associated with Higgs phenomena,these parameters include the masses of the fermions through couplings implied in theconstraint L ( v ) = 1, as recalled towards the end of the previous section. The D(1) B symmetry, as generated by the E Lie algebra element of equation 13.5, applies toboth X and Y ∈ h O in equation 9.46 and with a uniform action on all components of Y ⊂ h O , not just the v ∈ h C subspace. The X and Y components in equation 9.46carry the u -quark and ν -lepton states while the d -quark and e -lepton states reside in the θ X and θ Y components. Hence the stabilisation of the magnitude of v ∈ TM in theprojection out of L ( v ) = 1, as the D(1) B symmetry is broken in the early universe,will establish the observed masses for the u -quark and ν -lepton states in comparisonwith those for the d -quark and e -lepton states. The ‘tuning’ to these values may beautomatic if there is some mechanism underlying the stability for the correspondingvalue of L ( v ) = h (see also the discussion at the end of section 9.2).As described in section 9.3 a yet higher-dimensional form of temporal flow maybe required to fully identify the u -quark and ν -lepton states as SL(2 , C ) fermions aswell as to identify the second and third generation of Standard Model fermions. Thestructure of such a higher-dimensional symmetry of time, possibly involving an E symmetry on a form of temporal flow L ( v ) = 1 of greater than quartic order, may398ence be needed to address the question of the stability of L ( v ) = h ( t ).As well as the fermions the masses for the W ± and Z gauge bosons from theSU(2) L × U(1) Y sector of the theory and the Higgs mass itself will also be establishedat the epoch of this phase transition. Fermion pairs such as e + e − might be producedvia the decay of a heavy gauge boson, as associated with the Feynman vertex offigure 11.3(a), or via other underlying field exchanges of the form δY ↔ δψ generalisingfrom a geometric solution G µν = f ( Y ) for the spacetime geometry. However the energydensity of the early universe in the form of − κT µν := G µν = f ( v ) might also beconverted into fermion states through underlying δ v ↔ δψ field exchanges, under L (ˆ v ) = 1 as implied in the previous section.As well as the production of such fermion pairs new interactions, for exampleunder the full form L ( v ) = 1, may be significant in the high energy density en-vironment of the very early universe. Such a form, involving quintic or higher orderfield composition terms, might involve the production of leptons and quarks in thesame interactions, with potentially a mechanism for creating an asymmetry betweenmatter and antimatter acting during this very early epoch through to the phase tran-sition. A further possibility might involve gauge bosons, for example from a ‘beyondthe Standard Model’ SU(2) ⊂ E subgroup of the full symmetry of time (in principleidentified through an explicit symmetry breaking decomposition in the form of equa-tion 9.51 acting on the components of L ( v ) = 1), which might mediate interactionsbetween leptons and quarks in an analogous manner to the ‘ X and Y ’ gauge bosonsof an SU(5) GUT model. The origin of this imbalance of matter over antimatter re-mains to be understood, but the underlying asymmetry in the directed flow of time,the parity asymmetry arising from in the choice of the v ∈ TM projection out of thecomponents of F (h O ) in equation 9.46, a mechanism for combined lepton plus quarkproduction or even CP violation in the quark sector, all in the context of the presenttheory, may play a part here.In the standard theory by the time the temperature of the universe has cooledto 10 K at around 10 − seconds after the Big Bang quarks and gluons no longer form acomponent of a relatively weakly interacting plasma, along with leptons and photons,but become confined in hadronic states. A proton to photon ratio of around 10 − to one is established at this epoch, with a negligible contribution from antiprotons.Similarly most of the initial electrons and positrons mutually annihilate leaving aresidual e − contribution, balancing the residual p + states, leading to the much laterrecombination era as electrons combine with nuclei forming neutral atoms around372,000 years after the Big Bang, marking the origin of the CMB radiation as observedtoday and as described in section 12.3. An understanding of the origin of the imbalancebetween matter and antimatter states in the very early universe, accounting for thepredominance of ‘matter’ states as still observed today, within the context of thepresent theory may also aid in the identification of the mathematical structure of thecurrently hypothetical E action on the form L ( v ) = 1, augmenting the input fromthe required Standard Model properties as discussed in section 9.3.For the case of X ∈ h O with constant L ( v ) = det( X ) = 1 and given a verysmall initial value of | v | for a projected v ⊂ X a very large value for the scalar n is permitted, as can be seen from equation 6.27. Similarly, in the context of the form L ( v ) = 1 and equation 9.46, with v ∈ TM projected from the Y components, a399ery large value of the scalar field N ( x ) may be achieved if the dilation symmetryD(1) B , with the generator of equation 13.5, is involved in obtaining a very smallvalue of | v | . Further, if this latter value is obtained via the dilation symmetry D(1) λ of equation 9.30 then a very large value for either the scalar field n ( x ) or β ( x ) fromequation 9.46 can result. Hence if a cosmological term Λ g µν is derived for equation 12.1in the early universe, with the scalar Λ closely related to any of the scalars N , n or β then a temporary but rapid inflationary dynamics might be obtained.The action of a dilation symmetry D(1) B or D(1) λ , or some combination, mayslide the value of | v | = h ≃
0, corresponding to an initial unstable ‘inflationary state’in the very early universe, towards a preferred solution under G µν = f ( Y, ˆ v ) with astable value for | v | = h via interactions with other fields on M . In the very earlyuniverse v ( x ) ≃
0, correlated with a very large value for a scalar field such as N ( x )or β ( x ), might account for inflationary phenomena, with the largest inflation drivenfor example by N → ∞ as | v | → t →
0. After the magnitude of v has grownin time the same field with small variations around | v | = h might account for darkmatter effects, as described in the previous section, while a residual, now small valuefor N ( x ) or β ( x ) might account for the dark energy term Λ g µν at the present epoch, asa greatly suppressed remnant of the early inflationary era. A field such as N ( x ) drivinginflation in the very early universe must be weakly coupled at the present epoch inorder to have evaded detection in the laboratory. On the other hand the stable andcomplementary scalar field h ( x ) = | v | is associated with the vector-Higgs field v ,giving rise to phenomena which are evident in experiments.At the time of the phase transition t = t v the energy of the vector-Higgs field v is transferred to fermion and gauge particle states under the external geometry G µν = f ( Y, ˆ v ) solution. In addition to the stable value of | v | = h the masses ofthe fermions will be established under terms of L (ˆ v ) = 1, as described for the case of L ( v ) = 1 in equations 9.48 and 9.49 at the end of section 9.2. Hence the low value ofthe cosmological constant Λ may be correlated with the low value of the neutrino mass,and the pattern of fermion masses more generally, according to the balance betweenthe stable scalar values for | v | = h , n , N , α and β in equation 9.46, although againa full form such as L ( v ) = 1 may be required for the full picture.Under the assumptions applied for FLRW models, as described in section 12.2,while for t > t v a radiation dominated solution for the line element of equation 12.5initially emerges, for t < t v both the scale factor a ( t ) of that equation and the conformalfactor h ( t ) = | v | of equation 13.7 combine together to form the line element: dτ = ds = 1 h ( t ) (cid:2) dt − a ( t ) d Σ (cid:3) (13.8)In general it will be necessary to solve the 4-dimensional geometry G µν = f ( Y, ˆ v )to determine the dynamical form of g µν ( x ) both for t > t v for the evolution from aradiation to a matter dominated universe and on to the era of dark energy dominanceand potentially also for an ‘inflationary’ epoch for t < t v , both in principle involvingan evolution of the scale factor a ( t ) driven by an effective Λ g µν term in equation 12.1induced by a scalar component such as n, N, α or β . As described above such a scalarfield might take a very large value in the early universe as balanced against h ( t ) → t → L (ˆ v ) = 1. As was described for equation 13.7 the parameter t is considered to represent ‘cosmic time’ rather than ‘conformal time’, even for t < t v .400his convention is further justified here with the scale factor a ( t ) incorporated intothe more complete expression in equation 13.8.However, for t < t v in addition to a possible Λ g µν term there are new featuresthat arise in the present theory. The initial low value for | v | = h ( t ) in equation 13.8implies a relatively rapid flow of the fundamental time s through the spacetime man-ifold as parametrised by the cosmic time t . This property is complementary to therelative slowing of time in the later universe associated with small fluctuations to rel-atively high values of | v | distributed in space, with the resulting gravitational effectsascribed to apparent regions of ‘dark matter’ as described for figures 13.1 and 13.2.The complementary case with much lower values of | v | in the very early universe mayimply an effective expansion of spacetime (which might also apply to a smaller degreein regions of the later universe correlating with ‘voids’ between galactic clusters, asdiscussed after figure 13.2 also in the previous section).In the case of the very early universe the conformal scaling via the factor h − ( t ) in equation 13.8 as h ( t ) becomes smaller for t → { t, r, θ, φ } represent greater physical spacetime volumes as t →
0. For this conformal geometry an infinite spacetime volume may be inscribedwithin a finite coordinate boundary (as might be represented for example by the ‘CircleLimit’ woodcuts of M.C. Escher described in [26] pp.33–34).Allowing an infinite passage of time in the past in this way with τ ≡ s → −∞ as t → t = 0,is unchanged by variation in h ( t ) alone. As for the standard approach to solving thehorizon problem it appears necessary to ‘miniaturise’ physical spatial displacementsrelative to temporal intervals at the earliest epoch via for example an inflationarydynamics of the scale factor a ( t ), as has been applied for figure 12.4. Hence althoughin the present theory much more time may be available in the very early universe both h ( t ) and the scale factor a ( t ), which will be mutually correlated in the dynamics, playsignificant roles in equation 13.8.Nevertheless, in the present theory the magnitude of any inflationary effect,in terms of the increase in the scale factor a ( t ), and its period of duration may besomewhat different than in the original theory of inflation. Here the non-uniformity inthe way that the underlying flow of time s is injected into the spacetime manifold as h ( t ) = | v | evolves may have consequences which partially, or even totally, remove theneed for a rapid ‘inflation’. A much smaller value for h ( t ) in the very early universerelative to the present day value will also mean that the properties of physical structuresare likely to be very different, compared with those of the Standard Model for example.These differing structures may also in principle imply uniform characteristics, such as‘temperature’, across the initial singularity, with little if any time required to attain thehigh degree of ‘thermal equilibrium’ as observed today for the CMB radiation acrossthe full spatial extent of the observable universe.In assuming the vector field v ( x ) to take the particular unstable value in thevery early universe with v ( x ) → t → G µν = f ( Y, ˆ v ), will change with the phase transition at thetime at which the stable value | v | = h emerges at t = t v . This potentially abrupt401hange in the nature of the dynamics may be accompanied by a reduction in symmetry,in particular regarding the effective breaking of the dilation symmetry, composed of acombination of the groups D(1) X , D(1) B and D(1) λ described in the opening of thissection for example. In the original inflation theory there is no relation between thepostulated scalar inflaton field ϕ ( x ) and the Standard Model scalar Higgs field φ ( x ) ofparticle physics. In the present theory ‘inflation’ in the early universe is correlated witha very small value for v ( x ), while Higgs phenomena derive from the present stablevector field with | v | = h , with the two values of the same field v related throughthe action of the dilation symmetries in the very early universe.There are however some models in the literature for which inflation in the veryearly universe is correlated with the Standard Model Higgs field, and the propertiesof conformal transformations, in some way, as for example in [78, 79, 80]. Thesereferences typically incorporate a coupling between the Higgs and gravitational fieldsby postulating a new interaction term in a Lagrangian of the form L ∼ ξφ † φR , where φ is the scalar Higgs field, R is the scalar curvature and ξ is a new coupling parameter.In the theory presented in this paper however the Higgs sector is more intimatelyassociated with gravity since variations in the magnitude h ( x ) = | v | of the vector-Higgs field v ( x ) directly impact upon the external spacetime geometry via a changein the metric of the form described by equation 13.2.The picture of the very early universe in the present theory does also have closeparallels with the original inflationary models described in section 12.3. The scalarmagnitude h = | v | for the initial projection of v ∈ TM out of the componentsof ˆ v in the very early universe is analogous to the initial value of the scalar field ϕ in inflationary theory. For example in ‘old’ or ‘new’ inflation the initial value ϕ = 0becomes a ‘false vacuum’ as the potential V ( ϕ, T ) is modified with the dropping cosmictemperature T until subsequently a stable condition with ϕ = ϕ = 0 is achieved. Inthe present theory the consequences can be considered for a state with h = | v | ≃ L (ˆ v ) = 1 over M until the stable value h = | v | = h = 0 is achieved, particularly in terms of theform of the geometric solution G µν = f ( Y, ˆ v ).In the present theory interactions between the components of the vector field v ( x ) and for example the fermion field ψ ( x ) under the L ( v ) = 1 terms, togetherwith gauge fields Y ( x ) via terms in the expansion of D µ L ( v ) = 0, may compose athermal system incorporating an effective temperature dependent potential V ( h, T ).Such interactions will also generate ‘drag terms’ in the dynamics of the evolution of h ( t ) leading to damping effects accompanying a possible period of oscillations aboutthe potential minimum as energy is transferred to Standard Model particle statescreated and ‘reheated’ as the point of stabilisation with h ( t ) = h at time t = t v isapproached. This describes the ‘phase transition’ at the end of an inflationary period,leaving a residual dark energy contribution, arising for example from a much reducedand stable value for the scalar field N ( x ), in addition to the Standard Model particlespectrum as observed today. A radiation dominated FLRW cosmology emerges atthis time t = t v out of the ‘Big Bang’ with the initial conditions of the standardcosmological model having been set.In beginning with L ( v ) = h ≃ L ( v ) = h at time t = t v via interactions under the terms of the full form of L (ˆ v ) = 1 this picture is closely402nalogous to the second order phase transition of the ‘new inflation’ model described insection 12.3, involving a ‘slow roll’ down an effective potential slope V ( h, T ) on the wayto achieving the stable value. For the present theory the ‘potential energy’ associatedwith the original unstable value of h ≃
0, following the analogy of the ‘false vacuum’state in the new inflation model, might itself effectively provide a direct source of aninflationary expansion.The variation of | v | = h ( x ) alone modifies an otherwise flat Minkowski space-time via the conformal transformation g µν ( x ) = θ ( x ) η µν , with θ ( x ) = h − ( x ) fromequation 13.2. Via the Levi-Civita connection Γ( x ) this results in the Einstein tensor G µν ( x ) explicitly presented in equation 13.4. The shaping of the geometry G µν = f ( θ )from beneath in this way contrasts with the geometry G µν = − κT µν ( ϕ ) deriving fromthe energy-momentum source T µν ( ϕ ) of equation 12.25, which in turn was derivedfrom a Lagrangian for the postulated scalar field ϕ ( x ) in the original inflation theory.Despite this difference in origin there is a close similarity between the kinetic terms inthe field ϕ ( x ) in equation 12.25 and the first two terms in the field θ ( x ) = h − ( x ) inequation 13.4, suggesting the possibility of a similar field dynamics for the two models.For inflationary theory, in addition to the kinetic term drag terms are alsointroduced into the Lagrangian, as described following equation 12.25, and relate tothe physical phenomenon of post-inflation reheating during which the energy of thefalse vacuum is converted into interacting particles. A similar effect may arise in thepresent theory, with an effective potential V ( h, T ) and ‘drag terms’ for the new theoryderiving from interaction terms implicit in the form L (ˆ v ) = 1 as described above.The favoured minimum in V ( h, T ) (which may be largely independent of the effectivetemperature T ) will correspond to the stable value | v | = h , without the need to contrive an appropriate form for the potential V ( ϕ, T ) as is the case for inflationarytheory, since all the couplings of the present theory are effectively implied within theconstraint equations 11.29.An alternative proposal for the present theory features initial conditions with | v | ≫ h with potentially large fluctuations in the components of the field v ( x ),perhaps accompanied by a large value for the scalar field α ( x ) from equation 9.30 pro-viding the source of an inflationary Λ g µν term in the very early universe. Amongst themany possible solutions for G µν = f ( Y, ˆ v ) in principle arbitrarily extreme spacetimegeometries may occur, but without necessarily being supported throughout the full ex-panse of the manifold M . Such an extreme structure may describe the initial geometryin the Big Bang, where the conditions may even be somewhat ‘chaotic’ as an extendedspacetime solution is first shaken out of the mathematical possibilities implied in theform L (ˆ v ) = 1 and its symmetries, assuming the present universe to have evolved fromsuch a state. With potentially a large range of possibilities for | v ( x ) | ≫ h the initialvalue for v ( t ) for t → | ϕ | > | ϕ | in modelsof ‘chaotic inflation’, as also described in section 12.3.However, while there may be a range of possible ‘false vacuum’ initial conditionsfor the projection of v ⊂ ˆ v onto TM in the present theory, the ‘post-inflationary’stable value of | v | = h , as coordinated with the parameters of the Standard Model ofparticle physics, may still be uniquely determined. The possible range of initial valuesfor | v | ≫ h in principle implies a range of inflationary effects and a corresponding403ange of properties for the later evolution of the universe, some of which may becompatible with the present day universe as actually observed, and in particular withboth the horizon and flatness problems resolved as for standard inflationary theory.This raises the question of the uniqueness of the present theory, which will bediscussed more generally in the following section. For the case of h ( t ) = | v | → t → θ = h − diverges. Hence even in this case, if θ is interpretedas the inflationary field, the present theory may be interpreted in a manner analogousto chaotic inflation, with θ effectively taking a broad range of large values in the‘primordial chaos’ of the very early universe. Although these options, in relationto chaotic inflation, might be considered further, here we explore in more detail theimplications of taking h ( t ) = | v | → t → v ( x ) →
0, as depicted in figure 13.3(a), towards the stable state with | v | = h ,with a value which may also be determined uniquely, and as represented in figure 13.3at stage (c), may be essentially unambiguous. The state immediately emerging fromthe phase transition in figure 13.3(c), along with ordinary matter represented by thesprinkling of points of dust, was also represented in figure 13.2(c), although with thespatial fluctuations of the field v ( x ) neglected in the earlier figure.Figure 13.3: (a) Beginning with v ( x ) ≃ | v | = h grows, with potentially large fluctuations in bothmagnitude and direction, until the phase transition with (c) a stable value attained for v ( x ) with small fluctuations about the components ( h , , ,
0) for v in the comovingcosmological frame { t, r, θ, φ } . (Later epochs are depicted in figure 13.2(d),(e) in theprevious section).The picture of the phase transition between (b) and (c) in figure 13.3 is anal-ogous to the that associated with the property of ferromagnetism in a piece of iron.404he atoms in the iron can be considered as forming a lattice of a very large numberof randomly oriented magnets for temperatures T > T c above the critical value. Thisis similar to the situation in figure 13.3(b), except that the atomic magnets would berepresented by 3-dimensional spatial vectors of a uniform constant magnitude. Uponcooling to a temperature T < T c it is energetically favourable for neighbouring mag-netic vectors to align, with an analogous phenomena applying for the vector field v ( x )as the stable value | v | = h is attained as depicted in figure 13.3(c), with small fluc-tuations about the average 4-vector value of v ( x ) greatly exaggerated in the diagram.For the present theory the symmetry breaking in the phase transition, bothin terms of the actions of the dilation symmetry for the magnitude of v ( x ) andfluctuations in the orientation of this 4-vector, further suggests a close relationshipbetween v ( x ) and the Standard Model Higgs field. Indeed the stable vacuum value v = ( h , , ,
0) is precisely the same 4-vector as that in equation 8.72 (where apseudo-Euclidean basis { t, x, y, z } for v was employed), with v = h , as describedin subsection 8.3.3. The ‘vacuum symmetry’ is broken as the vector-Higgs field v ( x )takes a magnitude and particular direction in spacetime which, on average, is pre-sumed to be essentially aligned with the preferred cosmological frame parametrised bycomoving coordinates { t, r, θ, φ } .That is, the comoving cosmological frame is aligned with the average distribu-tion of visible matter, which in turn is presumed to have been formed and evolved inline with the underlying flow v ( x ) through the spacetime manifold M . The degree ofcorrelation between local fluctuations in the flow v and peculiar motions on a galacticscale at the present epoch is an open question. While the laws of physics are locallyLorentz invariant actual physical structures clearly are not, and this also applies tothe large scale structure of the universe. For example a directional relative blueshiftand redshift for the detected CMB radiation depends upon the local choice of Lorentzframe for the observer. In our case these shifts are due to our local motion within ourgalaxy, and can be readily corrected for in the CMB maps.As for the inflationary theories described in section 12.3 quantum fluctuationsand potentially Hawking radiation in the inflationary epoch generate inhomogeneitiesin the very early universe which may become frozen as classical fluctuations in energydensity at the end of inflation, ultimately seeding the formation of galactic structures.In the present theory these quantum effects include interactions between v ( x ) andother fields, such as those for the fermions ψ ( x ) and gauge bosons Y ( x ) as well asscalar fields such as N ( x ), all subject to the constraint equations 11.29 in forming theoverall external geometric solution G µν = f ( Y, ˆ v ) in spacetime. Fluctuations in thevalue of h ( x ) = | v ( x ) | directly impact upon the spacetime geometry, as describedfor equation 13.2, and hence in particular may generate large scale structure whenamplified as the scale factor a ( t ) rapidly grows.Fluctuations in the spatial components of v ( x ) could also in principle have alarge effect during the evolution of the very early universe as represented by the stage offigure 13.3(b), particularly for the case of a pre-inflation spatially ‘miniaturised’ world,with a relatively very small value of a ( t ), as described for figure 12.4. A calculation ofhow such fluctuations might stir up the primordial geometry would involve taking intoaccount all components of the field v ( x ) to determine G µν = f ( Y, ˆ v ), rather than justthe magnitude | v | = h = θ − as was the case for equation 13.4. Such a cosmological405odel, with fluctuations in both the magnitude and direction of v ( x ) impacting uponthe large scale structure, would differ from the forms derived in equations 12.9 and12.10, for G and G respectively, for which homogeneity and isotropy were assumed,unless a statistical average is taken for the large scale structure conforming to thoseassumptions.As the vector field v ( x ) stabilises through the phase transition to the stagedepicted in figure 13.3(c) small residual variations in the components of v ( x ) mightstill remain, and be found to be finely grained on the scale of the observable universe.This residual fingerprint of the earlier fluctuations is the network of creases in thefabric of spacetime as has been described in the previous section for the same epoch asalso depicted in figure 13.2(c). At this point, and throughout the remaining evolutionof the cosmos, this small residual variation in the components of v ( x ) might accountfor the phenomena of dark matter, and even a dark flow, as also suggested in the pre-vious section. That is, with sufficient deviation from the assumptions of uniformity ofthe cosmological principle these residual variations might seed the early formation ofgalaxies and clusters of galaxies through gravitational merging into the courser struc-tures observed at the present epoch. This cosmic imprint in the underlying spacetimegeometry, arising from fluctuations in the very early universe, is interpreted as a man-ifestation of ‘cold dark matter’ in particular, as was described in the previous sectionfor structures observed through to the present epoch as depicted in figure 13.2(e).As noted above, on top of these geometric effects of a continuous variation inthe field v ( x ) in the present theory the same field is responsible for the Higgs sectorin particle physics through interactions or exchanges with other fields. More generally,throughout the history of the universe quantum transitions, in the form of δ v ↔ δψ or δ ˆ v ↔ δY field exchanges underlying the multiple possible solutions for the spacetimegeometry G µν = f ( Y, ˆ v ), with the constraints of equations 11.29 applying everywhereon M , will shape the evolution of the cosmos, including the epoch of the very earlyuniverse. This shaping includes both the impact of observable fluctuations as describedabove as well as the physical implications arising from the statistical average of themicroscopic interactions.As described in chapter 11 the direct association of the likelihood of an ob-servable quantum event with the ‘number of ways’ in which the same empirical effectcan be achieved, quantified in terms of the degeneracy of underlying field solutions forthe same external local geometry G µν = f ( Y, ˆ v ), unifies the quantum process notionof probability itself with the classical concept. While a time-ordered accumulation ofprobabilities in the quantum case is relevant for cross-section calculations, in the caseof classical phenomena it gives rise to the second law of thermodynamics as quantifiedby an ever increasing value of entropy for any evolving thermodynamic state. In thepresent theory all such thermodynamic phenomena are played out in time and do notthemselves drive an ‘arrow of time’, as will be clear in the following chapter.As alluded to above the structure of the very early universe may allow suffi-cient breathing space for the thermalisation of the particle degrees of freedom in theepoch before the phase transition in figure 13.3, as is the case for the pre-inflationenvironment in figure 12.4 as described in section 12.3. However some care is neededin applying the principles of thermodynamics and statistical mechanics, familiar fromtheir application in the flat spacetime environment of the laboratory for example, in406he potentially highly curved and dynamic spacetime of the very early universe. Evenbasic notions such as ‘temperature’ or a ‘black body spectrum’ may be hard to definein such an extreme environment. The approach may be justified to some extent byapplying thermodynamics within small spacetime regions which approximate to localinertial frames, and adopting the strong equivalence principle, given a sufficient num-ber of ‘particles’ and ‘particle interactions’, or underlying field exchanges, within sucha region to apply statistical methods. Further, the properties of the ‘particles’ andfields themselves in the era before the phase transition may be very different to thefamiliar Standard Model particles and fields that emerge out of the Big Bang.It is also noted that the universe, and in particular the structure of the veryearly universe, is a single system for empirical study. Hence thermodynamic argu-ments, which consider an ensemble of systems each of which might form a small com-ponent within the universe, as for example employed for laboratory experiments, maynot apply for the potentially unique system composing the precursor to and immediateaftermath of the Big Bang. That is the observable universe today may have evolvedfrom state in the very early universe which is too small or simple to incorporate astatistical average, and which might in fact be dominated by the effect of a single‘fluctuation’.Further, as was described towards the end of section 12.3, beyond the ‘horizonproblem’ there is apparently a ‘start-up problem’ in the need to choreograph a vastnumber of spacelike separated ‘bangs’ along the initial singularity, either in figure 12.3or 12.4, in order to effectively simultaneously trigger the ‘Big Bang’ itself at cosmictime t = 0. Analogous to a synchronised display of fireworks there might be range of‘temperatures’ across the range of ‘bangs’ creating inhomogeneous initial conditions.However the Big Bang is not such a terrestrial event and there seems no reason whyit should not be in the nature of the start-up to generate essentially homogeneousthermodynamic conditions over the entire spacelike hypersurface at t = 0, upon whichlocal fluctuations may be identified in terms field exchanges underlying the multiplesolutions for G µν = f ( Y, ˆ v ). In particular the different nature of gravity, associatedwith the smooth geometry G µν ( x ), compared with the quantum phenomena associatedwith the internal field interactions, may play an important role in this structure.In the immediate aftermath of the phase transition of figure 13.3(c) with the‘vacuum energy’ being converted into Standard Model particles through transitions ofthe form δ v ↔ δψ under L ( v ) = 1, with familiar microscopic quantum properties, many more degrees of freedom may open up. The entropy content of the observableuniverse emerging from this epoch will depend on the reheating effects of the drag termsimplicit in L (ˆ v ) = 1 combined with the kinetic terms implied in equation 13.4, whichwere discussed earlier in this section and similarly as described following equation 12.25for inflationary theory. However, as also noted in section 12.3 the gravitational fieldappears to have had a very special role in the Big Bang and very early universe inbeing aloof from the thermalisation process.The strong equivalence principle (as reviewed in section 3.4 and adopted above)in part demonstrates how the characteristics of gravity fundamentally differ from theother forces of nature. The properties of local inertial frames are key to the structureof general relativity, with all physical phenomena other than gravity behaving in sucha frame as if gravity were completely absent, while gravity itself is described by the407eometry of the extended spacetime. The differences between gravity and other physi-cal phenomena will be significant for addressing issues for the early universe, includingalso the ‘flatness problem’ as well as the ‘horizon problem’ and an understanding ofthe role of entropy.Within the present theory the special status of the external gravitational fieldfurther derives from the fact that it is of a quite different, ‘unquantised’ nature in com-parison with the internal gauge Y ( x ) and fermion ψ ( x ) fields. The external geometry,described for example in terms of the metric components g µν ( x ) or linear connectionΓ( x ), does not partake in the statistical physics of the internal fields which lies be-neath continuous geometric solutions of the form G µν = f ( Y, ˆ v ). In the expression − κT µν := G µν the right-hand side describes the smooth external geometry, with allquantum mechanical properties of matter implicitly underlying the energy-momentumtensor on the left-hand side. Such quantum phenomena, based on an degeneracy offield solutions, can generally be described to a good approximation within local iner-tial frames, as was the case in sections 11.1 and 11.2. While the electromagnetic field,for example, exhibits thermal properties through the underlying field interactions theexternal gravitational field, being aloof from such interactions, has a very different re-lation with thermodynamic phenomena, and also, being unquantised, does not directlypartake in quantum fluctuations.The phenomenon of Hawking radiation, as discussed towards the end of sec-tion 11.4, arises for quantised fields in the classical curved spacetime of a black holeexterior, with the consequence for example that a black hole with mass O (10 ) kg willevaporate in approximately one second. Such phenomena involve the quantum me-chanical description of the vacuum but the gravitational field itself is not quantised,and lead to a study of the thermodynamic and entropy properties of black holes. Sim-ilar properties of the vacuum may arise for the present theory, since gravity is notquantised here, and also be important in the study of the thermodynamic and entropyproperties of the very early universe, in particular during the inflationary period.For the case in which the initial geometry is dominated by variation in thevalue of | v | = h ( x ), with a metric of the form g µν ( x ) = h − ( x ) η µν in equation 13.2,the spacetime geometry of equation 13.4 is conformally flat, even for arbitrarily largevariations in the scalar field h ( x ). For such a geometry the Weyl curvature tensorvanishes, C ρσµν ( x ) = 0, consistent with the proposal of the Weyl curvature hypothesisas motivated and described towards the end of section 12.3. Hence this observationmay account for the ‘cosmological problem’ concerning the extraordinarily special stateof the Big Bang to 1 part in 10 (according to [26] p.777) as required for thelow entropy initial conditions which underlie the subsequent evolution of the cosmosconsistent with second law of thermodynamics.Through interactions and fluctuations of the form δ v ↔ δψ , in particular withthe transfer of energy from the vacuum to Standard Model particle states towards theend of the inflationary period corresponding to figure 13.3(c), a non-conformally flatgeometry will emerge incorporating Weyl curvature, and hence the propagation of grav-itational waves for example, as well as Ricci curvature. As described in section 12.3(with reference to [26] section 28.8) the entropy of the gravitational field might be ex-pressed in terms of the degrees of freedom of the Weyl curvature and hence contributeto the increase in entropy from this time. As also described in section 12.3, follow-408ng equation 12.23, all FLRW models are consistent with C ρσµν ( x ) = 0 but requiresomething like an initial period of inflation to explain why observations are consistentwith k = 0, that is with spatial flatness. Similarly for the present theory an inflation-ary evolution for a ( t ) in equation 13.8 in the very early universe may relate to thisobservation.As described in section 11.3, and depicted in figure 11.10, the three couplingparameters of the Standard Model gauge group SU(3) c × SU(2) L × U(1) Y approxi-mately converge at an energy scale of O (10 ) GeV. This unification scale will marka significant threshold in the early universe, and it will be important to understandhow it relates to the epoch of the phase transition in figure 13.3(c) for the presenttheory. The interplay between the dilation symmetry, such as D(1) B , and electroweaksymmetry, together with the nature of their breaking, will also be key, as describedshortly after equation 13.7 with reference to equation 13.6. The electroweak symme-try SU(2) L × U(1) Y is broken by its action on v ∈ TM , with the stable value forthe magnitude | v | = h arising out of the Big Bang at t = t v . For t < t v , andin particular for t → | v | ≪ h via the action of the dilation symmetry, theproperties of the electroweak symmetry and the Higgs sector more generally will besomewhat different, for example with regards to the pattern of particle masses. Toaddress the complete symmetry breaking picture it will be required to explicitly iden-tify the electroweak symmetry SU(2) L × U(1) Y within the full E , or E , symmetry ofthe full form L (ˆ v ) = 1, in relation to the dilation symmetries, such as D(1) B , and theSL(2 , C ) × SU(3) c × U(1) Q symmetry already identified, completing the developmentof these structures described in chapters 8 and 9.As also alluded to towards the end of section 11.4 for a theory of ‘quantumgravity’, with the degrees of freedom of the gravitational field quantised, significanteffects are expected at the Planck energy scale E P = (cid:16) c ~ G N (cid:17) ≃ . × GeV. For anydescription of the very early universe in the context of such a theory all classical fieldconcepts in turn fail at epochs earlier than the Planck time t P = (cid:16) G N ~ c (cid:17) ≃ × − seconds. However the energy scale E P is considered to be of no special significancefor the present theory and the time scale t P , representing for example the extremelyearly universe, in principle presents no barrier for this theory. Hence the nature of theuniverse down through epochs at arbitrary cosmic times t < t P might be studied withinthe context of the present theory. This leads essentially to two broad possibilities asdepicted in figures 13.4(a) and (b). In these diagrams t = t v (presumably with t v ≫ t P )denotes the epoch of the phase transition at which there is a convergence to the averagevalue L ( v ) = | v | = h , as represented in figure 13.3(c). For either figure 13.4(a) or(b) the epoch of the ‘Big Bang’ can be identified with the time t = t v or more generallywith the period from t = 0 to t = t v and the state emerging at that latter time.In the first case for figure 13.4(a) the time t = 0 can be considered to bethe moment at which an extended 4-dimensional spacetime world first emerges outof the forms of the pure temporal flow s as identified through the geometric relation G µν = f ( Y, ˆ v ). This is the point in time at which extended and potentially infinite3-dimensional spatial hypersurfaces may be identified as an offshoot out of the multi-dimensional form of temporal flow L (ˆ v ) = 1 and a spacetime geometry with metric g µν ( x ) established, although with a significant deviation from flatness possible both409igure 13.4: Two scenarios for the relation between the temporal origin of the universeand the fundamental flow of time with (a) s → −∞ for t < s → −∞ forpositive values of t →
0. The width of each figure for t > a ( t ) as a function cosmic time t , neither of which are drawn to scale.for the 4-dimensional curvature and for the 3-dimensional hypersurfaces. Consideringa time t > t = 0marks the point at which the geometrical interpretation in terms of a 4-dimensionalextended manifold, supported by the mathematical structure and symmetries of theform L (ˆ v ) = 1, completely breaks down.Before t = 0 in figure 13.4(a) the parameter t no longer represents a coordinateon the manifold M , while the fundamental flow of time s → −∞ continuous withoutany limit as expressible through a general form L (ˆ v ) = 1, as always, but without anyprojection of v ∈ TM components onto an extended manifold. Here, as depictedfor example in figure 13.3(a), we have considered the case with v ( x ) ≃ v ( x ) = 0 at t = 0 with | v ( x ) | = h ( x ) generally growing with t > t = 0 could be considered as the epoch at which a fragmentof temporal flow under L (ˆ v ) = 1 is ‘syphoned off’ into the thereby created spacetimemanifold M . However it is also conceivable that this point of spacetime creation at t = 0 can be accompanied by arbitrary values for v ( x ) >
0, in principle even with | v | ≫ h .The width in both figures 13.4(a) and (b) represents the spatial scale factor a ( t )of equation 13.8, under the presumption of a solution with a ( t ) → t → t = t v , not drawn to scale. The behaviourof the ratio a ( t ) h ( t ) , and in particular whether this fraction tends towards zero, infinityor is finite as t →
0, will be significant for understanding the nature of the geometryof the manifold M in this limit, according to the spacetime structure described byequation 13.8. The geometry in this limit will also be important in relation to thehorizon problem and the ‘start-up problem’ as discussed for figures 12.3 and 12.4 in410ection 12.3. This might be best approached via a redefined cosmic time parametersuch as ¯ t with δ ¯ t = δth ( t ) and with the line element of equation 13.8 correspondinglyreplaced by: dτ = d ¯ t − a ( t ) h ( t ) d Σ (13.9)As has been discussed earlier, care is needed for the meaning of ‘cosmic time’ forthe epoch t < t v , whether parametrised by t or ¯ t , since physical clocks will be of asomewhat different nature for the very early universe, and indeed do not exist in anyform for t <
0. In any case a more complete theory is required to avoid the dangers ofspeculating on the number of angels that might be accommodated upon the head of apin, as noted at the end of section 12.3.The above comments also apply for the scenario depicted in figure 13.4(b), forwhich necessarily v ( x ) → M in the past at t = 0. For this second picture the relation between the flow of the fundamental timeparameter s and the ‘cosmic time’ coordinate t is sketched in figure 13.5.Figure 13.5: For the scenario depicted in figure 13.4(b) the projection v ( x ) ∈ TM converges to zero for t → t = 0 marks a coordinateboundary to the 4-dimensional spacetime M the range of the fundamental temporalflow −∞ < s < + ∞ is tucked away and entirely contained within this manifold.Adopting the approximate components ( h, , ,
0) for v in the comoving frame thephase transition t = t v marks the point at which v = | v ( x ) | = h ( t ) = h stabilises.For this scenario if v ( x ) ∈ TM converges to zero in an appropriate manneras t → s → −∞ the spacelike hypersurface at t = 0, potentially an ‘initialsingularity’ as a ( t ) →
0, is never attained and all of the fundamental flow of time −∞ < s < + ∞ is absorbed into the extended spacetime M of the universe. With s → −∞ without limit at the temporal coordinate origin on M the structure for t < t v in figure 13.4(b) and 13.5 might be pictured poetically as the bottomless waterfallat the end of time. With the familiar structures of the Standard Model of particlephysics emerging in the phase transition, from this epoch and for all times t > t v the fundamental time flow s is equivalent to both the proper time τ and also the411osmic time t for idealised observers in the context of an FLRW cosmological model,as described near the opening of section 13.1.For the case of the scenario depicted in figure 13.4(b) the present day universeis, in a sense, infinitely old in terms of the fundamental time parameter s . Howeverfor the picture in both figure 13.4(a) and (b) the physical and mathematical structurescan be traced back to arbitrarily early times for s → −∞ , with the difference beingthat for (a) physical structures are no longer defined for t < t ≤ s . In both casesphysical structures relating to the Standard Model of particle physics arise out of theBig Bang at t = t v . This is the point in time at which we can effectively ‘start theclock’ with s ≡ τ ≡ t , as might be measured through familiar physical processes, nowdetermined to stretch back through around 13.8 billion years of cosmic evolution. Suchan apparent temporal origin for the laws of physics in our 4-dimensional world may benecessary for consistency with an environment supporting biological life at the presentepoch. Here we refer in particular to the second law of thermodynamics which impliesthe universe is still evolving away from the particularly low entropy state conceivablycorresponding to the nature or uniformity of the gravitational field in the very earlyuniverse, as described above.For either scenario depicted in figure 13.4 the cosmic evolution itself is a fea-ture of the full macroscopic 4-dimensional spacetime G µν = f ( Y, ˆ v ), as shaped bymicroscopic field interactions in the form of the local degeneracies of fields underlyingthe possible solutions, consistent with the constraint equations 11.29, as described inchapter 11. As discussed in section 11.4 in combining gravitation with quantum theorythe notion of a 4-dimensional spacetime solution of general relativity takes precedenceover the 1-dimensional propagation of an apparent quantum state, with the latter de-scribed in terms of a local time coordinate, hence also circumventing the ‘problemof time’ encountered by some approaches to quantum gravity. As also concluded insection 11.4 the nature of probability in quantum processes is essentially the same asthat for classical systems, at heart formulated in terms of the ‘number of ways’ thatan empirical effect may be produced.On the large scale, with many underlying degrees of freedom, the interplay ofboth quantum and classical statistical phenomena will contribute to the shaping of thecosmological solution for G µν = f ( Y, ˆ v ). This solution will also incorporate macro-scopic contributions to the geometry in the form of G µν = f ( Y ) of equation 5.20,by comparison with Kaluza-Klein theory as described in section 5.1, and of the form G µν = f (ˆ v ) of equation 13.4 from variations of | v | = h ( x ) = θ − ( x ) in the projectionof L (ˆ v ) = 1 onto M , as described in this chapter. A correspondence with the tech-niques of ‘renormalisation’ in quantum field theory might in principle be developedin order to study the relation between the macroscopic external geometry and theunderlying ‘bare’ fields, as has been described in section 11.3.The question then concerns how the combination of all of the above geometricaland statistical factors in determining a solution for G µν = f ( Y, ˆ v ), with T µν := G µν providing the interpretation of equation 12.1, might collectively account for the ob-served cosmic evolution, compatible in approximation with the assumptions of theFLRW models and the metric form of equations 12.5 and 12.6, together with thelarge scale galactic structures. While observations of the latter structures require an412pparent ‘dark matter’ component, on the largest scale the solution G µν = f ( Y, ˆ v ) isrequired to account for the apparent effects of ‘dark energy’, for example in the form ofan effective cosmological term Λ g µν in the Einstein field equation. As for the earlier in-flationary epoch, the modern era parameter Λ may not be entirely constant, but withany variation such that (Λ g µν ) ; µ = 0 exactly compensated by an apparent effectiveenergy-momentum tensor with T µνǫ ; µ = 0 consistent with G µν ; µ = 0 and equation 12.1.This possibility was alluded to in the previous section in the discussion regarding ta-ble 12.1, and with reference to a similar observation for equation 5.41. In the presenttheory the total energy-momentum tensor T µν := G µν is defined to incorporate anypossible ‘dark energy’ cosmological term, and indeed the full solution G µν = f ( Y, ˆ v ).In describing the overall cosmological evolution in the spirit of the FLRWmodels the metric of the line element in equation 12.5 or 13.8 underlying the full 4-dimensional solution G µν = f ( Y, ˆ v ) will incorporate the expansion of the universe,including that of the present day, in terms of the scale factor a ( t ). The perspectiveadopted here is not that the universe is expanding now because it was expanding inthe past, analogous to the kinematic propagation of the flight of a cannonball from onemoment to the next along its trajectory, in either case raising the question of how it wasset in motion in the first place. Rather here the very early universe is conceived of as oneparticular region of the full four -dimensional spacetime manifold M , which happensto exhibit properties such as a ( t ) → h ( t ) → t → G µν = f ( Y, ˆ v ) external geometry solution.This is analogous to thinking of the Earth as being in orbit around the sun atthe present day not as a kinematic consequence of the fact that it was in orbit one yearago or a billion years ago but since the 4-dimensional spacetime trajectory, featuringan approximately elliptical orbit, exists as a geodesic solution for a 4-dimensionalSchwarzschild spacetime. In fact since the Bianchi identity G µν ; µ = 0 implies geodesicmotion, as described for equation 5.36 in section 5.2, the full spacetime geometry of anentire planetary system can be conceived of as a particular 4-dimensional solution for G µν ( x ). The idealised Schwarzschild solution itself describes an infinite and eternal 4-dimensional spacetime with G µν ( x ) = 0 everywhere, except for the point at the centreof spherical spatial symmetry, with the metric of equation 5.49. While the componentsof this Schwarzschild metric are constant in time but vary as a function of the radialcoordinate r via the factors of (1 − G N M/r ), the geometry of the Robertson-Walkermetric for an FLRW cosmological solution is independent of the spatial coordinatesbut varies with the time coordinate through the scale factor a ( t ). Both cases representfull 4-dimensional spacetime geometries.In the present theory both a ( t ) and h ( t ) in the line element of equation 13.8shape the geometry for the very early universe with t < t v , with a correlated evolutionof these parameters associated with a period of inflation. The comparison, earlierin this section, with the ‘new inflation’ model represents an analogy for the presenttheory, however the ‘slow roll’ down from h ( t ) ≃ t → h ( t v ) = h may or may not end with a series of ‘oscillations’ as the minimum of theeffective potential V ( h, T ) is achieved. In any case, given the correlation between a ( t )and h ( t ), it is conceivable that spatial regions with residual small positive fluctuations h ( x ) > h may have ‘inflated’ a little longer leaving a value of the scale factor a ( x )also slightly larger than the average value at the end of inflation.413or the large scale evolution of the observable universe for any time t > t v , withquantities averaged over each 3-dimensional spatial hypersurface, the value h ( t ) = h remains constant and stable while a ( t ) continues to increase, parametrising theexpansion of the universe as sketched in figure 12.2. However on the local scale ofgalaxies and galactic clusters it is the correlated distribution in space of a ( x ) and h ( x ),initially established at t ≃ t v , that might be associated with dark matter. That is,evolving forward to the present day, the effects of dark matter might be attributedto regions with small fluctuations of h ( x ) > h together with a correlated spatialprofile in a ( x ), rather than simply the conformal scaling alone of equation 13.2 assuggested following figure 13.1 in the previous section. In this way, generalising fromequation 13.8 for spacetime variation of h and a , the line element takes the form: dτ = 1 h ( x ) dt − a ( x ) h ( x ) d Σ (13.10)This structure opens up a greater degree of independence between the temporal andspatial components of the metric, with for example g ( x ) relatively low and g ii ( x )for i = 1 , , h ( x ) > h and a ( x ) h ( x ) arerelatively high, which in this sense is more reminiscent of the Schwarzschild solutionof equation 5.49, and which also may have geometric properties more characteristic ofa distribution of an apparent form of ‘matter’ than variation of h ( x ) alone.While, given an initially flat spacetime, the purely conformal action of h ( x ) onlygenerates Ricci curvature, the metric of equation 13.10 will generate both Ricci andWeyl curvature contributions extended throughout the spacetime manifold M , bothin regions of galactic clusters and the voids between. Having the variation of both h ( x )and a ( x ) in equation 13.10 increases the potential to match the observations of galacticmotions and rotation curves, together with gravitational lensing effects, as a candidatefor dark matter in interaction with the distribution of ordinary baryonic matter. Onthe yet larger scale of cosmological evolution these contributions to the dynamics ofthe universe might also be compared with the measured density parameters Ω D andΩ B , in addition to Ω Λ , as introduced in section 12.2, as part of a global fit to thecosmological data.In summary, the large scale structure and cosmological evolution of the universeare to be identified generally as aspects of a full 4-dimensional solution for the spacetimegeometry G µν = f ( Y, ˆ v ). There is no presupposition of a flat spacetime manifold. Inprojecting an extended 4-dimensional spacetime M out of the full multi-dimensionalform L (ˆ v ) = 1 of the fundamental temporal flow s large scale geometric distortionsmight be expected, which in turn may correlate with the observations ascribed toinflation, dark energy and dark matter, as reviewed in the previous chapter. Thereremains, of course, the need for a more complete theory and a much more thoroughanalysis, but in the meantime the possible variation of the magnitude of the projected4-vector v ∈ TM and the identification of several scalar fields α, β, n and N fromthe components of L ( v ) = 1 indicates the potential for the application of the presenttheory to these cosmological questions.The above discussion applies for the geometry of the 4-dimensional spacetimemanifold M whether in the context of the scenario depicted in figure 13.4(a) or (b).However, compared with the first scenario of figure 13.4(a) that in figure 13.4(b) is414ore symmetric in time in the sense that both the limit for s → −∞ as well as for s → + ∞ is incorporated within the 4-dimensional spacetime solution G µν = f ( Y, ˆ v ),as depicted in figure 13.6.Figure 13.6: As parametrised by the fundamental temporal flow s the spacetime man-ifold underlying the physical universe can be of infinite extent without boundary intime as well as in space for the scenario of figures 13.4(b) and 13.5.We inhabit a region of this eternal and infinite spacetime located within theperiod of several tens of billions of years following the phase transition at t = t v during which complex physical structures supporting biological life can be found, asrepresented in 13.6(e) and corresponding to the epoch of figure 13.2(e). It may be thatboth the far future through to s → + ∞ as well as the far past with s → −∞ maybecome progressively less structurally varied and eventful compared with the presentepoch. For s → + ∞ the universe may evolve into a relatively uneventful interplaybetween slowly evaporating massive black holes and thermal radiation, as depictedin figure 13.6(f), while for s → −∞ there may be an equally uneventful asymptoticprogression with | v | →
0, as depicted in figures 13.3(a) and 13.6(a). In this picture aphysical understanding of the structure of the universe for both s → + ∞ and s → −∞ may be equally open to study.On the other hand there then remains the question concerning the reason why cause of the Big Bang and the nature of the temporal origin of theuniverse itself. However, with everything, including the Big Bang happening in time ,and with all physical structures in the universe for the present theory built entirelyupon the notion of the one-dimensional flow of time s , there will still remain thequestion of the foundation of this apparently fundamental temporal entity itself, aquestion which applies equally for the scenario in figure 13.4(b). This will form thetopic for the following chapter. In the meantime, in the following section, we considerthe extent to which the properties and laws of physics of the universe, as depicted forexample in figure 13.6, might or might not be unique within the conceptual notionsand mathematical constraints of the present theory. In this section we consider several topics concerning the extent to which the particularproperties as empirically observed for the universe might be either necessarily deter-mined or down to chance, within the context of the present theory, beginning with thevalues of the large scale cosmological parameters. Without a full understanding of theirunderlying origin, the fact that the density parameters are observed to take the valuesΩ B = 0 . ± . D = 0 . ± .
011 and Ω Λ = 0 . ± .
017 [44], as reviewed insection 12.2, mutually within an order of magnitude or so of each other at the presentepoch, given the apparent possibility for each to range over many orders of magnitude,is striking. On the other hand given a universe dominated by either a cosmologicalconstant Λ or matter density ρ term the Friedmann equation 12.9, particularly forthe k = 0 case with H = (Λ + κρ ), shows that the Hubble parameter is essentiallydetermined by Λ or ρ respectively, and is clearly not an independent observable.At the present epoch for our universe, which is consistent with k = 0 and withthe cosmological term beginning to dominate, it is then to be expected that Λ ∼ R − H are of the same order of magnitude, where R H is the Hubble radius introduced inequation 12.4. This observation is a direct consequence of the field equation 12.1 whichleads to the dynamical solution for the metric structure of equation 12.5, including thecase of a Λ dominated universe. If the history of the scale factor a ( t ) is such that H − approximates the current age of the universe, which is the case for our universewith the cosmic evolution sketched in figure 12.2, then R H will be of the same orderas the scale of the observable universe hence in turn relating Λ − to this scale giventhe dominance of the Λ term at the present epoch.While the constant Λ in equation 12.1, considered as a geometrical effect, hasthe length dimension of L − the equivalent ‘vacuum energy density’ ρ Λ = Λ /κ hasthe dimension M L − ≡ L − and may be directly compared with the mass density ρ for both ordinary and dark matter. It should be noted though that on substituting ˙ a a + ka from equation 12.9 into equation 12.10, for the Λ dominated case, it is theextra factor of − Λ in the second equation which leads to a positive value for ¨ a in thecase of positive vacuum energy density ρ Λ >
0. This difference can be interpreted as416 consequence of the effective ‘equation of state’ for dark energy, with p Λ = − ρ Λ , asalso implied in equations 12.18 and 12.19.While the Λ term is beginning to dominate, the present day values of ρ Λ and ρ M (with the latter composed of both baryonic and dark matter together) still havea comparable impact on the large scale cosmological dynamic equations. The valueof ρ Λ ≃ . × − kg m − is apparently uniform in space and time, and hence thesame locally as well as globally, and can be compared with the global value of ρ M ≃ . × − kg m − , which includes a contribution from ρ B ≃ . × − kg m − ,at the present epoch. However the value of ρ B changes significantly with the cosmicepoch while local values for density of ordinary baryonic matter, such as for the planetEarth with ρ B E ≃ ,
500 kg m − , are much more stable in time. The magnitude of thestable terrestrial ratio of ρ B E /ρ Λ ≃ then provides a measure of the apparentlyvery different nature of ordinary matter and dark energy.Another well known apparently natural ‘large number’ in physics concerns theorder of magnitude of the Standard Model couplings of particle physics in comparisonto the strength of the gravitational interaction. For example the ratio of the classicalelectrostatic force between an electron and a proton to the classical gravitational forcebetween them has a value of O (10 ) to one. This empirical observation was also al-luded to near the opening of section 5.2 in motivating the need to introduce practicalnormalisation factors in studying the implications of equation 5.20 in the laboratoryenvironment. In the present theory, with general relativity and the Standard Modelrelating to the external and internal structures of L ( v ) = 1 respectively, the relativestrengths of the corresponding interactions in general will be related to the identifi-cation and interpretation of equation 5.20, which in turn is related to the geometricstructures of Kaluza-Klein theory. The fact that the gravitational field is not ‘quan-tised’, and hence does not exhibit the running coupling of figure 11.10 for example,further distinguishes gravity from the Standard model forces in the present theory.The differing strengths of gravitational and internal gauge forces should also beconnected in some way with the relative magnitudes of the components, such as thoseof the vector v or spinors ψ , within v ∈ F (h O ) of equation 9.46 in the symmetrybreaking projection over M . With the forms of matter and dark energy also relatingto structures within L ( v ) = 1 and its symmetries these relative magnitudes for thecomponents of v may also determine the widely differing local values of ρ B E and ρ Λ ,with the value of Λ possibly relating to the value of a scalar field such as N ( x ), n ( x )or β ( x ) projected out of the components of F (h O ). Hence the symmetry breakingpattern of E on L ( v ) = 1 down to an external SL(2 , C ) acting on v ∈ TM togetherwith the internal structures and the details of the projection of the components of v ( x ) over M may underlie the empirical observation of both of the above largenumbers.It is the relative weakness of gravity that allows structures to form on largescales, from the formation of stable planetary bodies through to clusters of galaxies.On the other hand the relative strength of the internal forces shapes the smaller scalestructures from terrestrial geology down through biological and chemical systems to theelements of particle physics. Immersed in the relatively small scale biological structuresour perspective is one of a spacetime which is flat to a very good approximation uponwhich an apparent ‘force of gravity’ is observed to determine the motion of material417bjects such as apples and cannonballs, as described in section 12.1 and before thebullet points in section 13.1.For all of the reasons of the above paragraph a world in which the elementaryinteractions of the Standard Model of particle physics are of a much greater strengththan that of gravitation is ‘anthropically’ favoured. Such a preference may correlatewith a certain value, or range of values, for the magnitude L ( v ) = h ( x ) in the pro-jection of the v ⊂ v components onto TM , and in turn underlie the empiricalobservation of ρ B E ≫ ρ Λ locally on Earth and for concentrations of baryonic mattergenerally. With the global density ρ M of the combination of ordinary and dark mat-ter (assuming ‘dark matter’ to behave in a similar manner to baryonic matter in thisrespect) declining from a potentially divergent value in the initial singularity and seem-ingly asymptotically approaching zero in the future, the observation that ρ M ∼ ρ Λ are of the same order at the present epoch, an apparently arbitrary point in cosmictime, appears to be essentially coincidental.This determination of ρ M ∼ ρ Λ has some analogy with the observation that r m ∼ r s at the present epoch, where r m and r s are the apparent sizes of the moon andthe sun respectively as viewed from the Earth. The value of r m has been decliningsince the formation of the Earth-moon system as the average distance between thesetwo bodies increases by O (1 cm) every year due to the nature of the gravitationalinteraction between the two bodies. Hence the present situation in which the moonis apparently just large enough to create a total solar eclipse is largely coincidental.However there are anthropic arguments, with the distance of the Earth from the sunbeing in the ‘habitable zone’ (not too near and too hot while also not too far and toocold) and similarly for the distance of moon from the Earth resulting in a magnitude oftides which may have aided the early development of biological life, which make such anapparent coincidence much more likely. Similarly there may be underlying anthropicreasons involving the nature of cosmological evolution which make the observation of ρ M ∼ ρ Λ more probable during a cosmic epoch supporting biological life.In summary, in the present theory the observation of ρ B E ≫ ρ Λ is expectedto be correlated with the observation that Standard Model forces are far greater instrength than the gravitational force. Indeed the cosmological term Λ g µν might beconsidered effectively as a geometric perturbation within general relativity as the largescale external spacetime structure M is identified through the projection of v ∈ TM out of L (ˆ v ) = 1, rather than an internal effect underlying the solution G µν = f ( Y, ˆ v ).The relation between Λ and the Hubble radius, described near the opening of thissection, may also hint at a geometric origin for the cosmological term. As well asthe great difference in strength, the rather different nature of gravitational comparedwith internal gauge forces is further emphasised in the present theory by the fact thatthe degrees of the freedom of the gravitational field, describing the external spacetimegeometry, are not quantised here.With dark matter associated with the external geometric consequences of avariation in the magnitude h ( x ) = | v | , as described in the previous two sections,here the dark sector in general is associated with locally ‘weakly interacting’ generalrelativistic effects. In the context of a solution for the full 4-dimensional cosmologicalgeometry ‘density parameters’ such as Ω D and Ω Λ may not have the same meaningas for the standard theory, since for example the above candidate for ‘dark matter’418ay not evolve in time in the same way as the baryonic matter density and the aboveorigin for ‘dark energy’ may not imply a constant value for Λ. In any case the presentobservation of Ω D ∼ Ω Λ may be a consequence of a correlated geometric origin forthe associated empirical effects, collectively arising from the warping of the manifold M in the projection out of L (ˆ v ) = 1, while the proximity of Ω B to these values mayin part be due to an element of coincidence as described above in the analogy with theapparent size of the moon and the sun.In developing the present theory further gravitational or material effects maybe derived in studying the general structure of G µν = f ( Y, ˆ v ) beyond those of theempirically observed baryonic matter and dark sector. It would seem to require asignificant coincidence if all such effects are of a measurable magnitude and henceobservable at the present epoch. If there are physical consequences of the relation G µν = f ( Y, ˆ v ) which have not yet been detected, and which may be beyond the reach ofany practical observation, this itself would partly account for the apparent coincidenceof ρ M ∼ ρ Λ . That is, these two latter quantities may form a subset of effects whichcollectively comprise a list of mutual contributions to G µν ( x ) at present, with a rangeof other potential terms having much lower density parameters and hence remainingundetected. For example if the empirically deduced cosmological term itself had beenjust one order of magnitude smaller it would have been far harder to detect. On theother hand while a contribution to the cosmic evolution of the form R µν = λ ( t ) v µ v ν (as described in section 13.1 and listed in the final column in table 12.1) has not beenobserved such a term, with a sufficiently low value of λ ( t ), might in principle form partof the large scale spacetime solution. With a larger range of such contributions it ismore likely for any two of them, such as ρ M and ρ Λ , to take similar values and bemutually observable.In chapter 11 the degeneracy of multiple possible local field solutions underlyingthe spacetime geometry G µν = f ( Y, ˆ v ) was described as the origin of indeterministicquantum phenomena in general. However in terms of constructing a solution theremay also be a degeneracy in terms of the average projected values of the componentsof for example v ( x ) and ψ ( x ) out of F (h O ) globally over M . In this case theremay be only a small certain range of values which lead to physical properties of mattercapable of supporting life as we know it. Even with this degree of anthropic selection to‘dial in’ certain ratios of the components of v ∈ F (h O ), via the dilation symmetriesdescribed in the opening of section 13.2 for example, since only a small number of ‘free’parameters are involved in the projection of v ∈ TM under the fixed structures of L ( v ) = 1 the theory would still be highly constrained, and hence in principle stillcapable of making predictions which might be tested. In section 13.2 the point of viewwas adopted that the interactions under the constraints of the theory are such that aunique stable value of | v | = h is achieved, resulting in a phase transition in the veryearly universe, implying an even greater degree of predictability for the theory.For the scenario described in figures 13.4(b), 13.5 and 13.6 at the end of theprevious section a unique asymptotic condition with h ( t ) = | v | → t → h ( t ) → h ( t v ) = h was compared with models of ‘new inflation’. It is also possible to consider a range ofstarting conditions for h ( t ) < h as t → h ( t ) > h for t →
0, as might be associated with the scenario depicted in figure 13.4(a), and419voking a comparison with models of ‘chaotic inflation’. In turn a range of long termcosmological conditions might emerge out of the subsequent phase transition, evengiven the same stable value for h ( t v ) = h , and hence in principle with a degree ofanthropic selection implied for our own habitable universe.This raises the question of the degree of uniqueness regarding other aspects ofthe theory. With the general form of the function L ( v ) determined, as described insection 2.1, it is a well defined mathematical problem to identify particular forms andthen consider the reasons why certain of these may be significant for the physical world.Two such significant forms that we have identified are L ( v ) with an E symmetryacting on elements of F (h O ) and L ( v ) with the Lorentz symmetry acting on the 4-dimensional tangent space TM on the base manifold. With respect to the larger formsymmetry breaking over M identifies the smaller form via the chain L ( v ) → L ( v ) → L ( v ) → L ( v ), rather like a sequence of Russian dolls, with a corresponding chain ofsubgroups SO + (1 , ⊂ SL(2 , O ) ⊂ E ⊂ E as summarised in table 9.1 in the openingof section 9.3.Alternatively a progression of forms L ( v ) → L ( v ) → L ( v ) → L ( v )aligned with the subgroup chain SO + (1 , ⊂ SL(3 , C ) ⊂ E ⊂ E might be con-sidered by expanding the Lorentz symmetry action of SL(2 , C ) on v ≡ X ∈ h C inequation 7.35 to an SL(3 , C ) symmetry of the 9-dimensional form L ( v ) = det( X ) = 1.The sl(3 , C ) Lie algebra basis of equation 8.33 explicitly demonstrates how this struc-ture is naturally embedded within the SL(3 , O ) ≡ E action on the 27-dimensionalspace h O at the next stage of the sequence.At either end of this chain it may be asked why these two particular formsare selected out of a large array of possibilities – why the projection should be ontoa 4-dimensional spacetime manifold and why the highest-dimensional form L (ˆ v ) = 1should be represented by a quartic expression in 56 dimensions with an E symmetry.It could be considered whether further worlds, different to our own and of course notobservable by us, could be created out of other possible mathematical forms of L ( v ).That is, whether the forms L ( v ) = h and L ( v ) = 1 are largely identified as choicesthat agree with our world, or whether either or both of these are determined by physicalstability or mathematical symmetry arguments for example.By extension from the 3-dimensional model world of section 2.2 and figure 2.3with an SO(3) symmetry one way to construct a 4-dimensional world would be toembed 3-dimensional spatial hypersurfaces within a 4-dimensional base manifold withlocal tangent vectors v ′ ( x ) satisfying the form L ( v ′ ) = ( v ) + ( v ) + ( v ) + ( v ) = 1(appending one dimension to the model case of equation 2.14) with an SO(4) symme-try. However while geometric curvature and even particle trajectories might be definedin such a world, given the local SO(4) symmetry on M there is no consistent defini-tion and distinction of a ‘temporal’ direction compared with ‘spatial’ displacements.In principle one of the four dimensions could be arbitrarily declared to represent anapparent temporal component, however due to the nature of the symmetry betweenthe four components the causal structure on M would not be well defined.However given that the 4-dimensional manifold arises as a multi-dimensionalmanifestation of the ordered 1-dimensional flow of time itself and the necessity for thetemporal causal structure to be retained on the manifold, the form L ( v ) = ( v ) − ( v ) − ( v ) − ( v ) = 1 of equation 5.1 is naturally preferred. As described in420ection 5.3 the metric of Lorentz signature implied in this form locally defines a ‘lightcone’ structure on the extended manifold, which hence distinguishes timelike fromspacelike directions on M . The symmetry preserving this form L ( v ) = 1 is the non-compact Lorentz group SO + (1 , L ( v ) = 1 over the above ‘Euclidean’ alternative L ( v ′ ) = 1.A more accurate model for chapter 2 would have involved the 3-dimensionalLorentz group group SO + (1 ,
2) acting on the form L ( v ) = ( v ) − ( v ) − ( v ) pro-jected onto the tangent space of M , as a subgroup of the full symmetry ˆ G = SO + (1 , timelike and spacelike vectors and temporal causality on the base manifold M . However dealing with the simplified Euclidean model in chapter 2 was sufficientto demonstrate the relation between the external and internal symmetry in the presenttheory, with the same conceptual ideas applying for the case of the real world with the4-dimensional Lorentz group on the base space M as described in section 5.1, leadingto the connection with Kaluza-Klein theory as also discussed in that section.As well as the local Lorentz symmetry, which also holds to a good approxi-mation on for example the scale of the solar system in the case of our world, we mayalso consider whether the base manifold M n is required to have n = four spacetimedimensions. If we attempt to construct another possible world using similar reasoningto that presented in this paper then we would expect something similar to generalrelativity, that is gravitation, to arise out of the geometrical properties on the basemanifold of the world, independent of its dimension. One important factor may bethat while for a 4-dimensional spacetime base manifold robust, stable planetary or-bits around a massive object, such as a star, are to be found in the solutions to theequations of gravitation, this does not arise for other dimensions of base space.This was shown to be the case for the motion of a body near a massive grav-itating object, as the source of the Schwarzschild solution for general relativity, in an n -dimensional spacetime by F.R. Tangherlini in 1963. While for n = 4 the metricsolution takes the form of equation 5.49 the functional form of g µν ( x ) depends on thevalue of n . Although theoretically a circular orbit may be permitted in some cases for n >
4, the slightest perturbation, for example from the impact of a ‘meteor’ or thegravitational influence of a third body, would cause the ‘planet’ to wander out of orbitand into a path forever receding to larger distances or spiralling inwardly until collid-ing with the central ‘star’. The same conclusion, that a stable orbit is only possiblefor m = 3 spatial dimensions, was also found by Paul Ehrenfest in 1917 for Newton’stheory of gravity in which the gravitational potential is determined as a solution ofthe m -dimensional Poisson equation (which was introduced for the m = 3 case aboveequation 3.75 in section 3.4).Clearly the stability of the elliptical orbit of the Earth around the sun is neces-sary for life on our planet in our world, although this does not imply that the equivalentstability is absolutely necessary for life in another world with n = 4 spacetime dimen-sions. For example the ‘chemistry’ in such a world would be vastly different from ourown and the relative time scale for the development of life structures to the time scale421f planetary motions may also be vastly different – potentially allowing a civilisationto evolve out of the primordial chemical soup stirring on the planet in the time it takesto glance past a star, even assuming such an encounter with a low entropy source inthe form of stellar ‘nuclear’ energy is necessary.Having then decided upon the 4-dimensional Lorentz group on the M mani-fold to break the full symmetry there are still issues concerning the degree to whichassumptions made about the form of the linear connection Γ( x ) on M are necessary.In the present theory the base manifold M derives from the projected form L ( v ) andhence regarding the local geometry there are a range of local coordinate systems atany point on M for any of which the metric has a Minkowski form g µν ( x ) = δ aµ δ bν η ab .In turn, in deriving from an SO + (1 , x ) will be metric compatible, as also discussed in section 5.3.As for general relativity, in a 4-dimensional spacetime there is enough freedomin general coordinate transformations to set ∂ λ g µν ( x ) = 0 at any given point x ∈ M . However, as reviewed in section 3.4 (and also following equation 13.2) generalrelativity goes further by asserting the ‘equivalence principle’ – according to which thegravitational field can be transformed away at any given point, that is, a local inertialframe can be constructed such that not only g µν ( x ) = δ aµ δ bν η ab and ∂ λ g µν ( x ) = 0but also Γ( x ) = 0 for any given x ∈ M . This means for example that there existseverywhere a local coordinate system in which a geodesic trajectory as described inequation 3.76 for the 4-vector u , with components u µ = dx µ /dτ , takes the simple form d u /dτ = 0. Since the torsion tensor T , with the components of equation 3.60, mustbe zero in all coordinate systems if it is zero in any of them, such a linear connectionΓ( x ) is necessarily torsion-free.In the present theory an extended frame of reference throughout which both ∂ λ g µν ( x ) ≃ x ) ≃ ∂ λ g µν ( x ) = 0 andΓ( x ) = 0 at any given x ∈ M , and taking the torsion to be zero may be a verygood approximation. However given the mathematical basis for what we are taking asthe act of perception it seems perhaps artificial to impose the extra restriction on theconnection that it should necessarily be torsion-free or, further, require that the strongequivalence principle in general should hold. It may be that there is a non-vanishingcontribution to physical phenomena from torsion which has so far been beyond thereach of observation – for example any contribution to the connection coefficientsΓ λµν ( x ) asymmetric in the { µ, ν } indices would have no effect on the simple geodesicmotion of equation 3.76 – and neglecting it has therefore been of no consequence.This is also the case in general relativity where setting the torsion equal to zeroacts as a simplifying assumption. Both in general relativity and the present theorythe linear connection Γ is a metric connection with ∇ g = 0, but this does not implythat the torsion should vanish. In the Einstein-Cartan version of general relativity themore general geometry with finite torsion is considered (with extra dimensions such ageneralisation is also significant for the Kaluza-Klein theories reviewed in section 4.2).In this case while the spacetime curvature is still related to the energy-momentum ofmatter through the Einstein equation the torsion is a function of the spin current ofmatter. Unlike curvature the torsion does not propagate in the matter-free vacuum422nd the two theories are identical in such an environment. Further, given that the spindensity is small for ordinary matter in the universe the two theories have been exper-imentally indistinguishable, and this itself justifies adopting T = 0 as a simplifyingassumption.In the present theory it is an open question whether the linear connection Γis necessarily symmetric and torsion-free, and if so to explain why this is the case.More generally the question regards whether the spacetime geometry and forms ofmatter consistent with G µν = f ( Y, ˆ v ) contain the structures of torsion and a spin cur-rent. In the meantime as for general relativity the assumption T = 0 may be adoptedto simplify some of the mathematical expressions, with in particular the Levi-Civitaconnection of equation 3.53 hence being employed. This is analogous to adoptingthe simplifying conditions which underlie the Robertson-Walker line element of equa-tion 12.5 in order to study models for the evolution of the universe as a whole, eventhough the assumptions of the cosmological principle clearly do not hold exactly. Thedegree to which the large scale structure deviates from the conditions of homogeneityand isotropy may itself not be a uniquely restricted property of the universe.Even for the T = 0 case, in constructing the external geometry in terms of theinternal fields, as well as the 10 components of the Einstein tensor in the form of G µν = f ( Y, ˆ v ) itself the full 20 independent components of the Riemann curvature tensor mayalso explicitly depend on those fields with R ρσµν = f ( Y, ˆ v ). This will include the Weyltensor components C ρσµν = f ( Y, ˆ v ) and hence the identity C σµ = C ρσµρ = 0, asdescribed towards the end of section 3.3, as well as the Bianchi identity G µν ; µ = 0,will also apply implicitly for the internal fields. The 10 degrees of freedom of C ρσµν ( x )are still considered to represent the ‘vacuum’ in the sense that they complement the10 degrees of freedom of − κT µν := G µν = R µν − Rg µν in the decomposition ofequation 3.69 for the full Riemann tensor R ρσµν = f ( Y, ˆ v ).In classical general relativity while matter is identified with the content of theenergy-momentum tensor T µν the vacuum geometry with G µν = 0 and C ρσµν = 0still carries energy, in the form of gravity waves for example, as also discussed afterequation 5.44 in section 5.2. Hence energy can propagate through the ‘vacuum’ ofspacetime even when not expressed in terms of any underlying internal fields. Incontrast it is also possible in the present theory that there may be fields on M , forexample from some of the components of F (h O ) underlying the form L ( v ) = 1, atleast in some regions of spacetime, that may not directly contribute to the spacetimestructure of G µν = f ( Y, ˆ v ) at all, and hence which do not carry energy-momentum.While for general relativity the Einstein equation 3.84 can be derived from theEinstein-Hilbert action of equation 3.79, it can be shown, as demonstrated by Cartan,Weyl and others, that the most general divergence-free symmetric 2-index tensor con-structed from the metric and its derivatives up to second order is a linear combinationof G µν and g µν (see for example [82] appendix II). This consideration itself leads toEinstein’s equation 3.84 and 12.1, with Λ a free parameter, as essentially the only ad-missible field equation for a geometric theory of gravity consistent with a divergence-free energy-momentum tensor on the right-hand side. Regardless of the method ofderivation the significance of the Einstein equation derives largely from the contractedBianchi identity G µν ; µ = 0, which then necessarily applies to the energy-momentumtensor. On the other hand symmetries in the apparent distribution of matter can be423mployed to assist the search for solutions, with for example equation 12.2, with ρ and p being functions of cosmic time only, being the most general energy-momentumtensor consistent with the assumptions of the cosmological principle, as described insection 12.2.In the present theory energy-momentum is defined through T µν := G µν . While G µν = f ( Y, ˆ v ) incorporates ordinary matter, a possible Λ g µν term, dark matter phe-nomena and the structure of the very early universe collectively into an apparent T µν ( x )there may be further geometric or material phenomena, arising out of the internal fieldsor their interactions, which have not yet been detected. This possibility was alludedto earlier in this section in the discussion of the observation of ρ M ∼ ρ Λ , as exempli-fied by a potential contribution originating from a term of the form R µν = λ ( t ) v µ v ν ,and now incorporates also the possibility of finite torsion. The potential for new phe-nomena will be of particular interest if yet higher-dimensional forms of L (ˆ v ) = 1 areconsidered, with a corresponding larger symmetry, which may also be needed to fullyaccount for known Standard Model particle phenomena.The Lie group E was originally selected as a candidate symmetry for the fullform of L (ˆ v ) = 1, in the context of the present theory, in part since it is already ofwell known interest in relation to the observed gauge groups of elementary particletheory, as reviewed in section 7.3. However, it was primarily chosen for detailed studyas it acts on a relatively high dimensional vector space, with 27 dimensions comparedwith the four on the base manifold M , and stands out as exhibiting particularly richmathematical structures, involving for example the triality symmetry of the octonionsand three interlocking actions of SL(2 , O ) as described in chapter 6, through whichto channel the temporal flow via the components of v under the constraint of the27-dimensional cubic form L ( v ) = 1. Expressed in terms of the octonions, whichthemselves form the largest of the normed division algebras, this form of L (ˆ v ) = 1provides a unique structure. The existence of elaborate mathematical properties withinthe substructures of the E symmetry acting on h O matrices is perhaps the reason whyE stands out as a kind of significant mathematical resonance amongst other possiblesymmetries of temporal forms in yet higher dimensions. By comparison for examplehigher-dimensional spacetime symmetries SO + (1 , m ), acting on quadratic Lorentzianforms with an arbitrarily large number m of spatial dimensions, arguably exhibit asomewhat less elaborate structure.In section 9.2 the analysis was extended to the smallest non-trivial represen-tation of E realised as an action on the 56-dimensional space F (h O ) preserving acertain quartic form L ( v ) = 1 and incorporating the octonions in two independenth O subspaces. Building upon the properties identified for the symmetry breaking ofthe E action on h O described in chapter 8, the structure of the broken E action on F (h O ) when projected over M has a number of properties reminiscent of the Stan-dard Model, as summarised in equation 9.46. However it is still very much an openquestion as to which other symmetry groups should perhaps be considered and whatobservable effects they may have on our own world. These effects might be manifestedin particle physics phenomena through the prediction of additional states, or the de-termination of the properties of known states, which might be observed in high energyphysics experiments.The hypothetical extension to an E symmetry on a 248-dimensional form424 ( v ) = 1, as described in section 9.3, would in principle be large enough to incorpo-rate the full set of known Standard Model states, including all three generations of thefermions. Given that E is the largest exceptional Lie algebra, terminating the chainof Dynkin diagrams depicted in figure 9.1 of section 9.2, such a form of temporal flowmight uniquely complete the sequence of extensions listed in table 9.1 of section 9.3.As a continuation of that sequence, and also in particular to contain the non-compactLorentz group SO + (1 ,
3) as the local symmetry of the causal structure on M as dis-cussed earlier in this section, this may involve the non-compact real form E − asdescribed for equation 9.50.The present theory is based on the observation that the one-dimensional pro-gression in time, via the elementary arithmetic properties of the real line R , can beexpressed in terms of variables in an arbitrary number of dimensions. In principle thesame observation might be applied to each of the n real components underlying an n -dimensional form of temporal flow L ( v n ) = 1. For the case of the orthogonal groupO( n ) in the limit n → ∞ certain properties related to the octonions make various cal-culations more tractable. In his study of the homotopy groups of the topological groupO( ∞ ) in 1957 Raoul Bott discovered the isomorphism π i +8 (O( ∞ )) ∼ = π i (O( ∞ )). Suchperiod 8 structures, which are also seen for Clifford algebras and known generally as‘Bott periodicity’, are all closely related to the 8-dimensional octonions. Similar peri-odicity structures may become relevant in the exploration of higher-dimensional formsof L (ˆ v ) = 1, for which octonion elements explicitly feature, and might even be impor-tant for calculations relating to the degeneracy of solutions underlying G µν = f ( Y, ˆ v ),involving a higher-order nesting of field redescriptions, which underlie quantum andparticle phenomena.The progression towards higher-dimensional forms of L (ˆ v ) = 1 described abovemay uncover a uniquely determined mathematical structure. Given also that the 4-dimensional Lorentzian form L ( v ) = h projected into M may necessarily providethe means of breaking the higher symmetry the laws of physics observed in our universemight in turn be uniquely determined. Even in this case our universe does not representthe unique manifestation of such a world, but rather one of a vast number of possiblesolutions for the external geometry G µν = f ( Y, ˆ v ), built upon an underlying degeneracyof local internal field descriptions as expounded in chapter 11. While events at a HEPexperiment, such as depicted in figure 10.1, exhibit the intrinsic structure of quantumuncertainty, the spectrum and properties of the particles identified in the laboratorymay be unique. On the other hand in principle there might still be solutions formultiple universes with a range of large scale cosmological structures depending on thenature of the overall G µν = f ( Y, ˆ v ) solution, in particular with regard to the apparentconditions in the very early universe.It nevertheless will be required to carve out of the full form L (ˆ v ) = 1 a universelike ours, such as depicted in figure 13.6 and described at the end of the previous section,incorporating all of the observed large scale structure and the phenomena of the BigBang. Regardless of the degree of uniqueness of such a world, in being constructed outof the multi-dimensional forms of temporal flow, it derives in turn from the priorityof one-dimensional temporal flow as the underlying basis of the universe. Hence theconceptual question remains regarding the origin of this one-dimensional structureitself, as we consider in the following chapter.425 hapter 14 The Origin of Time
The aim of theoretical physics at a fundamental level could be described as a programto uncover the basic scientific principles of the world, the consequences of which en-compass all empirical phenomena. From the objective point of view the existence ofthe universe, and its matter content, began with the Big Bang and evolved accordingto equations of motion, as governed by the fundamental principles, for billions of yearsas the matter condensed into galaxies, stars and planets, some of which are conducivefor biological life, until eventually conscious observers such as ourselves in turn evolved,with the ability to contemplate the world and the cosmos around us. Two of the mostpressing kinds of questions raised by this picture concern the nature of (1) the BigBang and (2) conscious life:(1) What can we say about the universe before the Big Bang? How and why does theBig Bang occur? How is spacetime itself created? Can the ‘initial singularity’be avoided? What determines the particular initial conditions? How is mattercreated and what determines its properties? Why are the laws of physics theway they are?(2) Given that a material universe is created and set in motion subject to the physicallaws, how is it possible to mould the conscious experiences of observers, aware ofthemselves and the world around them, out of inert, lifeless, material substanceof a seemingly qualitatively entirely different nature?It seems inevitable that any physical theory must be founded on a ‘loose end’concerning the basic elements of the theory. This is the case whether these basic entitiesconsist of particles, fields, strings, spacetime, extra dimensions, or some combinationof these or further concepts, and is generally justified on the grounds that ‘one has tostart somewhere’. A similar argument could be made for the present theory foundedon the concept of time. This paper has presented the mathematical development ofthis theory, beginning with the general form of temporal flow L ( v ) = 1 as deduced426or equation 2.9, through the construction of a physical world in space and time forcomparison with observations, leading to a discussion of the possible uniqueness ofthis structure in the previous section – which addresses some of the points of item (1)above. However, no matter how far progress might be made with the elucidation ofempirical phenomena the theory is incomplete so long as there remains the questionregarding the origin of temporal flow itself, as represented by the loose end on theleft-hand side of figure 14.1.Figure 14.1: Beginning with the notion of one-dimensional progression in time, via thegeneral mathematical form L ( v ) = 1, both an extended spacetime manifold and thephysical bodies perceived within it are in turn derived. The two loose ends concernrespectively the origin of time itself and the subjective experience of the observer.With the basic entity having such a simple structure, namely a one-dimensionalordered flow of time modelled by the real line R , this first loose end is particularlystriking for the present theory. By comparison a theory founded for example uponthe basic entities of a set of fields in spacetime begins with a great deal of structure,and can to a large extent be considered as a study of the phenomenology of fields inspacetime. However here since the fundamental temporal flow, represented by the realline, cannot be readily decomposed into simpler elements it is very natural to raise thequestion of its origin, and in turn there is a greater sense of obligation to address theissue of a foundation for the present theory.Given a description of the physical world, whether founded on the notion oftime or other basic concepts, containing bodies which can be observed, the secondloose end, as depicted on the right-hand side of figure 14.1, regards the question ofhow it is possible for an entity to be aware of an observation. This question concernsthe issue of how ‘we’, as beings conscious of observations and thoughts, are embeddedwithin the structures of the world. The physical structure of the organic brain is closelyassociated with this latter loose end as an apparent vehicle for self-reference capable ofencoding subjective experiences within the physical world. In this section we considerhow such a structure might be modelled or explained in terms of mathematical orphysical elements, before returning to the first loose end of figure 14.1.The idea that conscious experience can arise out of physical structures on thespacetime manifold M should not be too controversial since it is essentially impliedin most approaches to fundamental physics. If based on a quantum field theory, asapplied in the Standard Model of particle physics for example, all properties of matterultimately arise from the properties of the basic fields and their mutual interactions.Hence the microscopic properties of matter underlie the structure of macroscopic ob-jects in the world including both inanimate objects such as rocks and pencils as wellas biological structures such as flowers and brains. The self-reflective, self-consciousactivity of the human brain must therefore be supported by the underlying elementsof the theory and the structures which they generate in spacetime. This is essentially427he case for any physical theory, since it is evident that beings conscious of experiencearise in the same world as described by the theory. Both aspects of this world, thatis the subjective mental phenomena as well as the objective material phenomena, arethen in principle amenable to theoretical analysis.On the practical side, since the early history of computing, with devices de-signed or constructed first of mechanical and later electronic components, comparisonshave been drawn between ‘artificial intelligence’ and the workings of the naturally oc-curring physical structure of the brain. Indeed, the design of a computer as envisagedby Alan Turing in the 1930s and 1940s was based on modelling the action of the humanmind with the ambition to ‘build a brain’ out of electronic components. This camewith the significant advance in the design whereby programs as well as data couldbe stored in symbolic form, allowing both to be modified and manipulated by the‘universal machine’. On the more philosophical side Turing demonstrated that thereare questions involving the performance of a universal machine which are intrinsically‘non-computable’ for the device. Turing also came to the conclusion that the actionsof a human brain are ‘computable’; with such thought processes then being amplifiedthrough the actions of the human body.The notion of computability for physical devices has a close parallel in the fieldof pure mathematics, regarding in particular the demonstration by Kurt G¨odel a fewyears earlier that propositions can be constructed in an arithmetical calculus whichare intrinsically unprovable within the calculus. It is this latter analysis we considerhere in order to then describe a model for a self-referencing subjective state.Proposition VI of G¨odel’s 1931 paper, On Formally Undecidable Propositions ofPrincipia Mathematica and Related Systems I [83] can be paraphrased: ‘All consistentaxiomatic formulations of number theory include undecidable propositions’; that is,there are true statements of number theory which its methods of proof are too weakto demonstrate. The argument can be applied to any calculus (that is a formal systemconsisting of a set of axioms and rules of inference) powerful enough to express thebasic arithmetic (with addition and multiplication) of the natural numbers (0 , , , . . . ).Hence any such formal system is ‘incomplete’. The essential idea employed by G¨odelwas to find a way to use mathematical reasoning to explore mathematical reasoningitself (see for example [84, 85]).Following a chain of deductions which begins with a construction known as‘G¨odel numbering’ a formula called G (after G¨odel) is derived which is the mirror im-age within the arithmetical calculus of the meta-mathematical statement that: ‘Theformula G is not demonstrable’. G¨odel was able to show that if in fact G is demon-strable then its formal contradictory ∼ G (i.e. ‘not G ’) would also be demonstrable,leading to an obvious inconsistency. He proved that if the formal system is consistentthen G is formally undecidable ; that is, neither G nor ∼ G can be deduced from theaxioms and rules of the calculus.It can however be seen by meta-mathematical reasoning that G is in fact atrue proposition of the calculus. Hence G is a true arithmetical formula and in factexpresses a certain property of all natural numbers. Hence an arithmetical truth hasbeen discovered which can not be deduced formally from the axioms and rules ofinference of the calculus. Any calculus incorporating arithmetic is incomplete in thisway (in the original historical context this signalled the demise of David Hilbert’s428hallenge to prove the contrary). Although we are free to simply add G as an extraaxiom for the formal system, in this case a different true undecidable arithmeticalformula G ′ could be constructed from the augmented calculus. Again, adding G ′ asan axiom we would still be able to construct a true undecidable G ′′ , and so on; thatis, for any augmented set of axioms and rules it will always be possible to constructfurther undecidable propositions – the calculus is ‘essentially incomplete’.The essential points of G¨odel’s theorem for our purposes are summarised here: • The symbols, axioms, rules, theorems and general expressions of a calculus orformal system capable of expressing arithmetic can be mapped onto a subset ofthe integers by G¨odel numbering. • Meta-mathematical statements about expressions of the calculus are associatedwith a mirror image within the arithmetic itself. • Assuming that the calculus is consistent, formulas such as G can be constructedwhich can be shown to be true while being formally undecidable – it is notpossible to prove either G or ∼ G within the calculus. • Augmenting the calculus with new axioms such as G leads to a new calculusfor which new undecidable formulas such as G ′ can be found; completeness ofarithmetic can not be achieved, it is ‘essentially incomplete’. • The consistency of the calculus can not be proved from within the system, butit can be demonstrated by meta-mathematical reasoning outside the system.We next ask how the above considerations may be of relevance in the theoreticalsciences and in particular in relation to the theory investigated in this paper. Thegeneral mathematical form L ( v ) = 1 was derived in equations 2.1–2.9 on consideringthe notion of progression in time to have a structure isomorphic to the algebra ofthe real numbers R , including the basic arithmetic operations of + and × . Since thenatural numbers N are embedded as a subset of the real numbers the mathematicalcalculus concerned with L ( v ) = 1 is certainly sufficient to express the usual rules ofarithmetic for the non-negative integers. Further, in developing this physical theorycertain mathematical structures arising from the forms and symmetries of L ( v ) = 1have been taken to be isomorphic to the structures that we perceive in the physicalworld. It is a world in which we find both natural and manufactured machines anddevices which are in some cases capable of expressing statements about mathematics,and in particular about the kind of mathematical calculus that underlies the world.Since the physical world can be expressed in mathematical terms capable of describingthe behaviour of objects and devices in the world exhibiting for example structures(such as the human brain) powerful enough to perform arithmetic operations andsupport states of self-reference, then it seems that ‘formally undecidable propositions’must inevitably arise in the application of these mathematical structures. We maythen consider the possibility that the manifestation of such mathematical phenomenain the world is in the form of our own conscious experience of being in an ‘undecidedstate’, with the above list of five points correlated with the corresponding list below:429 There is a necessary isomorphism between the physical structure of everythingin the material world, including brains, and mathematical structures expressiblein the calculus underlying the expression L ( v ) = 1. • The human brain is capable of making meta-mathematical statements aboutstructures deriving from the mathematics of L ( v ) = 1, which therefore necessarilyhave a mirror image in structures deriving from the L ( v ) = 1 calculus itself. • We experience questions we can ask of ourselves in making a choice, such as“Shall I pick up the pen or the pencil in front of me?” as being undecidable (thatis, we cannot predict our own future actions). • In making a choice, for example in picking up the pencil, we find ourselves in anew state for which a further horizon of similarly undecidable questions perpet-ually arise. • Our experiences are organised and synthesised into a self-consistent and coherentawareness of the world.This is indeed, of course, very far from being a definitive analysis of the phe-nomenon of our conscious experience in the world. The intention here is rather merelyto consider the close analogy with the elements that go into the construction of G¨odel’stheorems. That there may be a more significant relation between these two cases issuggested by their close structural similarity, the fact that they are both groundedin mathematical considerations involving self-reference and the fact that potentiallyhighly complicated mathematical expressions arise in both cases. We observe furtherthat in considering a choice it is precisely our ‘undecided’ state that we are aware of.For this preliminary discussion of this phenomenon in the context of the presenttheory we proceed with the following simple experiment. For clarity of exposition thediscussion is presented in terms of my experiences in the world, where my and I referto any individual, such as the person currently reading this text. I can place, forexample, a pen and a pencil on the table in front of me and allow myself to deliberatefor several seconds over the question “shall I pick up the pen or the pencil?”, whilefiltering out other thoughts as far as possible. In performing such an experiment theexperience is one of initially having an awareness of being in an ‘undecided’ state, inwhich I may ‘change my mind’ several times almost as if compelled along on a waveof reasoning guided by practical or aesthetic judgements concerning, for example, theutility of the pencil or the colour of the pen, and then, quite suddenly, as if I haveto let go, I find myself in the ‘decided’ state of having chosen the pencil and hold itin my hand (in fact, the more casually or lazily I make the choice the more it feels determined by the rational course of the world, including subconscious processes, withmy conscious deliberation being a kind of internally reflecting resistance to that flow).That we can readily do this kind of ‘thought experiment’ and attempt to observe what happens when the choice is made serves to emphasise just how central the phenomenonof conscious decision making is in the world. A general physical theory of the worldshould then ideally have something to say about this phenomenon or be able to offera good reason why it does not.Here we comment on the fundamental difference between questions we can askof the kind “will the apple fall off the tree?” and of the kind “shall I pick up the430pple?”. The former question about the external world, not involving self-reference, is‘undecided’ to the extent that we lack the relevant knowledge about the physical stateof the objects concerned – we simply await the resolution of the question as carriedexternally in the inertia of the world (and with a similar interpretation applying forthe outcome of indeterministic quantum processes, as depicted in figure 11.13(b) forexample). For the latter question regarding whether or not to pick up the apple,in attempting to predict our own future action based on our internal thoughts weare conscious of falling over ourselves in search of the answer until we experience theresolution.To proceed further we consider a self-referential mathematical system R whichis assumed to be correlated with a physical brain state. For such a given formal system R in principle a large number n of undecidable propositions G i , with i = 1 . . . n , mightbe formulated; as represented in figure 14.2(a). If any one of these is taken to beabsorbed into the mathematical structure as a new axiom then a new formal system R ′ with a new horizon of internally undecidable propositions G ′ i arises, as depicted infigures 14.2(b) and (c).Figure 14.2: Expansion of a formal system R as the ‘undecidable’ proposition G i isincorporated as a new axiom.From the subjective point of view the system R represents a self-reflective con-scious state of mind, which is in constant interaction with the subconscious mind andthe world beyond, which are also represented by mathematical structures and providea reservoir of information and data in the environment E which might enter consciousthought as represented by figure 14.2(b), effectively corresponding to a realisation ofthe truth of G i . The subjective correlate of a single step R → R ′ is considered tobe the experience of making a choice . The overall mathematical structure with theprogression R → R ′ → R ′′ drawn out through the interaction between the consciousstate R and the broader environment E is outlined in figure 14.3.The essential feature of figures 14.2 and 14.3 is that any change in the system R , due to the interaction between R and the mathematical forms of E , results in a progression . The all-encompassing mathematical environment E in figure 14.3 can bethought of as a static sculpture of mathematical objects. Within this structure theself-referential systems with . . . R ⊂ R ′ ⊂ R ′′ . . . carve out a one-dimensional orderedprogression. A given self-referential state R ′ within this sequence absorbs the state R accompanied by one of its undecidable propositions G i , now included as an axiomwithin R ′ , with respect to which R represents the ‘past’. Similarly R ′ is itself in turncontained within R ′′ with the latter state incorporating a resolution of an undecided431igure 14.3: In terms of mathematical objects the formal system R corresponds to asubset of a larger environment E which provides the source of information and datafor the progression depicted in figure 14.2.proposition of the state R ′ , that is one of the G ′ i , and hence representing the ‘future’.Along with this terminology, with for example the self-reflective state R ′ correspondingto the ‘present’ experience, incorporating R in the past and drawn towards R ′′ in thefuture, the structure represented in figures 14.2 and 14.3 is postulated as a model forsubjective temporalisation .To follow the above analogy with G¨odel’s theorem closely then would be tosay that our experienced state of being undecided finds resolution by absorption into anew state in which a particular choice, or corresponding new ‘axiom’, is included. Thepossibilities to incorporate further new axioms in the attempt to resolve a perpetualstate of undecidability leads to an ordered progression (incorporating . . . G , G ′ , G ′′ . . . into . . . R , R ′ , R ′′ . . . respectively) which is therefore structurally identical to, andproposed as the origin of, our experience of temporality.It is important to emphasise here that it is the mutual association of the R states in this ordered series that has itself a temporal structure. It is not a question ofbeing situated at R ′ for example and asking how it is possible to move on to R ′′ , sincefor something to move presupposes an already existing flow of time with respect towhich the motion takes place. Rather it is the unambiguously ascending logical orderof this series itself which, having a structure that can be mapped onto and modelledby the one-dimensional ordered real line R , reveals the form of time itself. We have‘time’ already in its pure and simplest essence as an ordered progression in this abstractseries. From the objective point of view a state R corresponds to a limited physicalsystem in the world, correlated in particular with features of a physical brain. Viaphysical processes new data can be introduced through the interaction between, effec-tively, the conscious brain and the subconscious brain, as well as with the rest of thephysical world, as will be described further below and in the following section. Futureactions are not fully determined by or contained within the self-reflective state R itself.The physical process of the subconscious intervening in the conscious deliberation, asmodelled by the progression from R to R + G i and attainment of the new state R ′ depicted in figure 14.2, correlates with the subjective experience of ‘letting go’ after aperiod of ‘falling over oneself’ in debating whether to pick up the pen or the pencil.From an internal subjective point of view the conscious mind is ignorant of the choice432 ntil it is made and the individual finds himself holding the pencil rather than the penfor example, contributing to his sense of temporalisation. The passage of time andconscious experience more generally may feel somewhat mysterious since we do notgenerally perceive the objective structures and interactions represented in figure 14.3,only their internal subjective correlate.Evidently our thoughts are not really as clear cut or ‘binary’ as suggested inthe example above when confronted with a simple choice such as “Shall I pick up thepen or the pencil in front of me?”. It is not that we are really considering an isolatedpossible future state corresponding to each alternative G i in figure 14.2(a). Rather,there is an enormous ensemble of possible future states which may be divided into twosets, each with a vast range of members:A) { I shall pick up the pen + X } B) { I shall pick up the pencil + Y } (14.1)where X and Y each refer to possible features of a state of mind in addition to whetheror not I hold the pen or pencil respectively. The idea here is not so much that individualundecidable propositions G uniquely correspond to simple thoughts or actions such as‘I pick up the pen’. Rather it is to be considered that there is a vat of an enormously large number of correlated G -like statements G i (with i = 1 . . . n and n an extremelylarge number) relevant to a particular brain state. A subset of the G i will incorporatethe statement ‘I pick up the pen’ amongst other actions, others will incorporate thestatement ‘I pick up the pencil’ amongst other actions, while still further subsets ofthe G i will represent the cases ‘I pick up both’ or ‘I do not pick up anything’. Each ofthese ghostly undecidable propositions G i points towards a possible extension of myself-reflective state. Such extended systems draw us in and as we progress from onestate to another, augmented, state our sense of temporality is created.In the course of this dynamical stream of temporalisation I shall find myselfcoming into a state of picking up the pen or pencil, depending on the choice of thepossible G i . This set of potential G i is itself of course very dynamic, as representedfor example by the set G ′ i in figure 14.2(c), and evolves in turn with the incorporationof new axioms, or choices, and new information into my system, corresponding to theever evolving set of my possible future actions.That the nature of subjective awareness may be correlated with the mathe-matical notion of the undecidable in self-referencing systems opens the door to a morethorough investigation. However, technically, in the context of the physical world, itmay be that ‘computability’, rather than the closely related notion of ‘decidability’, is amore directly relevant concept to employ, since we know that the laws of physics in ourworld are such that ‘computing machines’ (both artificial and organic) are supported.That is, we are directly dealing with the states of such ‘devices’ in the physical worldrather than with abstract mathematical symbols in a formal system, although there isa close structural parallel between the two cases. The discussion has been framed interms of ‘decidability’ partly due to the similarity of the language used to express the experience of making a choice; that is, in making a choice we are primarily consciousof being in an undecided state. On the other hand given this coincidence of languageterms some caution is needed in order to avoid being misled into taking the connectiontoo literally. 433t is indeed very much open to question how far to take the analogy between thestructures pertaining to G¨odel’s theorem and the subjective process of decision making,although there is some degree of correspondence as indicated by the two sets of bulletpoints listed earlier in this section. With contradictory ‘undecided’ propositions fromsets A and B being simultaneously entertained in equation 14.1, corresponding forexample to G i and G j with i = j , this structure does seem to have some importantdifferences also with the above mathematical correlate, since for G¨odel’s theorem the‘undecidable’ describes the relation between G i and ∼ G i , with the proposition G i representing an unprovable but true statement. Although this implies that to somedegree G¨odel’s construction should be taken metaphorically here, the employment ofa mathematical framework with self-referential structures is still very relevant.As well as the subjective interpretation the structure in figure 14.3 must alsocorrelate with a physical manifestation. From this objective perspective the laws ofphysics must support a kind of inertia in the substructure of the physical brain, corre-sponding to the subconscious mind, that carries the subject into just one of the arrayof ‘true’ states either in set A or in set B of equation 14.1; that is into a new structureof self-reference such that the other options (in particular those in set B or set A re-spectively) become manifestly false propositions. The wiring of the subconscious mindin this sense will govern to a large degree the patterns of behaviour of an individual.Naturally, we are taking this to be a phenomenon that our thoughts are thor-oughly and continuously saturated with, rather than a discrete set of deliberationssuch as “hmmm, shall I pick up the pen or pencil?”. That is, many of our ‘choices’in this sense are simply the train of thoughts at the forefront of our mind that con-tinually bubble up even when we are not trying to think. Most of these thoughts arenot directly accompanied by an external bodily action such as picking up an objector not. For example each process of ‘changing my mind’, as described for the thoughtexperiment shortly after the second set of bullet points above, is also a choice, evenwhen not accompanied by a decisive external action.An analogy between our thought processes and the mathematical structuresunderlying G¨odel’s theorem has been elucidated by other authors. In the prefaceto reference [85] (p.7) Hofstadter refers to elementary expositions involving a self-referencing loop leading to undecidable propositions, such as G considered above, ascontaining:. . . only the most bare-bones strange loop, and it resides in a systemwhose complexity is pathetic, relative to that of an organic brain. Moreover,a formal system is static; it doesn’t change or grow over time. A formalsystem does not live in a society of other formal systems, mirroring theminside itself, and being mirrored in turn inside its “friends”. . . . there is nocounterpart to time, no counterpart to development, let alone to birth anddeath.For Hofstadter, it is the self-referential and mirroring properties of the brain,giving rise to abstract structures similar to the ‘strange loops’ encountered in demon-strating G¨odel’s theorem, that is central to the emergence of an animated conscious ‘I’from the inanimate particles of matter of the brain. As suggested by Hofstadter, forthe present theory also, a mathematical structure somewhat more complicated than434hat required to demonstrate G¨odel’s theorem might be needed to account for thesephenomena.In this paper, we consider that the possibility for such systems to change andgrow is not only something that objectively takes place in time; but moreover it isthe ordered nature implicit in such a series of potentially related states that describestemporalisation itself. It is the possible existence of this ordered progression of sys-tems which, through its simple structural isomorphism to an ordered one-dimensionalmathematical series (that can be mapped onto the real line R ), itself corresponds toour immediate experience of time. The resolution of undecidable propositions from onesystem to another corresponds to the progression of choices we find ourselves making,with varying degrees of awareness. From the subjective point of view these choices are not deterministic, in the usual sense of the word, since they are not something that happens in time ; rather they are the generators of temporality itself.The progression depicted in figure 14.3 only has one direction. This underliesour experience of an apparent ‘arrow of time’ which corresponds simply to the one-waynature of this process (always with the possibility of losing knowledge of the world asour memory becomes frayed at the edges, it being supported by an imperfect physicaldevice and following behind in the wake of our new experiences). The phrase ‘arrowof time’ is somewhat misleading since it implies the possibility of time having theopposite sense, that is flowing in the ‘other’ direction, effectively as if an empirical timeparameter t could be seen to be reversed with t → − t . However this is not the case asthe progression in figure 14.3 possesses only a single direction, which may be associatedwith + t ∈ R . The sequence of self-reflective states R , subjectively experienced as theflow of time, creates the inertia of the external world carrying physical objects. Theseobjects include, for example, the components of a clock which can be used to measurethe ‘time’ t . The fact that we can imagine, or even construct, a physical clock to‘run backwards’, or for example watch scenes of a movie played backwards, creates theillusion of an alternative possible sense for time. However, we can only detect thata physical clock is running backwards since it operates relative to the fundamentalunderlying ordered progression of time.The purely mathematical structure of figure 14.3, encapsulating the experienceof a 1-dimensional temporal progression, can itself be encoded within the physicalstructures of a 4-dimensional spacetime world as depicted in figure 14.4. Here thestructures in M can be considered to represent a static L ( v ) = 1,and in particular the properties of the components v i , with i = 1 , ,
3, of the form L ( v ) = ( v ) − ( v ) − ( v ) − ( v ) with an SO(3) ⊂ SO + (1 ,
3) symmetry, in termsof extended geometrical forms, as was described more generally in chapter 2. Thisincorporates the perception of physical objects in an extended 3-dimensional space, as435igure 14.4: A representation of the progression of the self-reflecting state of figure 14.3as translated into an extended physical environment M .represented for example on the hypersurface planes in figure 2.3. The structures of thatfigure, interpreted as a 4-dimensional spacetime, can be superposed on the manifold M of figure 14.4, within which more complicated mathematical structures also ariseout of the full mathematical form L (ˆ v ) = 1 when projected onto the 4-dimensionalbase manifold. These further mathematical objects, described for example in termsof fields on M , incorporate series of self-referring elements which have a structuralisomorphism with an experience of a directed 1-dimensional flow in time.For the world sketched in figure 14.4 the mathematical structures in spacetimehence have both the necessary mathematical properties to give rise to perception ofobjects in space , that is in a 3-dimensional geometrical volume (represented by 2-dimensional planes in figure 2.3), as well as the experience of events in time , in adirection geometrically ‘orthogonal’ to the 3-dimensional spatial hypersurfaces on themanifold M , which possesses the local SO + (1 ,
3) symmetry of the form L ( v ) = h of equation 5.46. Hence both spatial and temporal forms of perception are encodedin the mathematical structures of the world. That is, we consider not only that thestructures obtained from ˆ G/ SO + (1 ,
3) symmetry breaking for L (ˆ v ) = 1 projected over M can be equivalent to the geometrical shapes we experience in space, but that theyalso incorporate self-referential mathematical structures which may be isomorphic tothe self-reflecting progression that we experience subjectively as the flow of time.In turn this one-dimensional temporalisation itself provides the source of dy-namical laws through the breaking of the multi-dimensional form of temporal flow L (ˆ v ) = 1 over the 4-dimensional spacetime, generating the physical laws on M withwhich physical structures in general, and those depicted in figure 14.4 in particular,must be compatible. Hence the progression R → R ′ → R ′′ identified within the mathe-matical environment E in figure 14.3 must be consistent with the seemingly inevitable progression R → R ′ → R ′′ of states in the physical spacetime environment M infigure 14.4 described as an apparent consequence of the laws of physics.436he inertia of the physical world conforming to these laws of nature carrieswith it both the subconscious and conscious components of the brain and with them a‘decision’ already shaped in the former is swept into a new self-reflecting state of thelatter, for which an ‘undecided state of mind’ is now experienced as being resolved.More generally, information and data in the broader environment of M in figure 14.4,as labelled by E in figure 14.3, which implicitly includes both the subconscious elementand anything else distinguished from the self-reflecting R state considered, can inprinciple contribute to the progression.An analogy can be made between the self-reflective system R in the extendedenvironment M and a thermodynamic system B embedded within the same largerenvironment on the manifold M . While interactions between R and the further struc-tures in M lead inevitably to the progression of the self-reflective state R → R ′ → R ′′ ,for example in terms an increase in subjective ‘information’, the interaction between B and the broader environment leads inexorably to an increase in the total entropy S . Infact both the incorporation of a G¨odel statement G i into R , as depicted in figure 14.2,as well as the case of increasing entropy might be considered as analogies for the phe-nomenon of temporalisation. The first example may also carry some elements above amere metaphor, while for the second example an increase in entropy will accompanythe physical process underlying the subjective experience of time.For a sufficiently complex system such as a human brain the complete immer-sion of the self-reflective state within the wider environment might effectively generatea continuous temporalisation. Indeed, while for figures 14.2–14.4 a series of discretesteps has been described, subjectively we generally experience a continuous flow oftime without any gaps or jumps. For example, while watching a ball roll along a table,essentially obeying Newton’s first law of motion, we observe a smooth progression rel-ative to our internal sense of temporality. It is this continuous subjective experience ofthe flow of time, as modelled by the one-dimensional real line R , that forms the basicentity of the present theory.Indeed, although subjective experience in general exhibits a correlation withobjective phenomena it is not explicitly described by the latter phenomena. For ex-ample, the sensation of ‘green’ is associated with radiation from an interval of theelectromagnetic spectrum with a wavelength of around 500 nm in physical interactionwith the cells of the human eye and the resulting neural activity in the brain. Howeverthe subjective experience of ‘green’ is not explicitly contained in the description at anylevel of detail of these objective physical processes. Similarly here, the subjective ex-perience of a continuous flow in time is associated with the physical structures impliedin figure 14.4. However it is not necessarily the case that a continuous sequence needsto be identified in a physical system based on the progression R → R ′ → R ′′ in orderfor it to underlie a subjective experience of time which can be accurately modelled bythe continuous real line R .The irreversibility of conscious choices, the origin of the ‘arrow of time’, isechoed in the irreversibility of many physical systems which are all governed by equa-tions derived from the general mathematical form of progression in time. For examplethe second law of thermodynamics itself arising as a statistical consequence of a pro-gression of states, as alluded to in section 13.2. An essential difference is that whileentropy increase is solely something which happens in time , the physical progression437 → R ′ → R ′′ of figure 14.4 is directly correlated with a subjective experience oftime which drives the temporal flow itself. In addition to systems of classical physics,quantum phenomena are also subject to the underlying ordered flow of time whichis infused into the base manifold M . Calculations of probabilities and cross-sectionsfor quantum processes depend on the accumulation of the possible field degeneraciesconforming to a causal sequence in time, building upon time-ordered expressions asdescribed for equations 11.41 and 11.44 and more generally in sections 11.2 and 11.3.In addition to the fundamental temporal progression itself there are a largenumber of apparently one-dimensional quantities which may be constructed out ofthe physical structures on M . As well as the example of entropy S these includethe temperature T of a body or even the ‘time’ t recorded by a mechanical clock.However each of these quantities correlates solely with the objective collective actionsof molecular motions in spacetime and each defines a measurable property of the four-dimensional world. Even the time t recorded by the clock is not a 1-dimensionalgeometric entity, but rather signifies a certain coincidence between the hands of theclock and the numerals on its dial in 3-dimensional space.In fact no purely 1-dimensional phenomenon can be objectively inscribed withina 4-dimensional spacetime without reference to the extended M manifold or physi-cal processes within it. While the self-reflective physical structure in figure 14.4 issimilarly diffused in spacetime, objectively in terms of the firing of brain neurons forexample, the subjective experience of time is of a different character. Unlike a physicalquantity such as S , the mental process of experiencing time is a purely not located within spacetime. Theprogression R → R ′ → R ′′ is subjectively fused in mind into a purely 1-dimensionalexperience of a qualitatively different nature to, and hence distinguished from, the4-dimensional spacetime arena.This one-dimensional structure is the origin of time in the world, in the formof subjective temporalisation, and provides the foundation which underlies the generalmathematical form of temporal flow L ( v ) = 1 and the physical laws in spacetimeitself. It is through attempting to address the second loose end on the right-hand sideof figure 14.1 that a source has been identified for the first loose end on the left-handside. For a description of the universe in terms of a purely objective theory a 4-dimensionalbackground arena for events in spacetime, as for the case of general relativity, mightbe postulated as a fundamental entity or perhaps derived from a higher-dimensionalspacetime. This is consistent with the observation that all physical events in the worldhave both a spatial and a temporal location in the universe. For the present theoryit is noted, however, that while we observe such events in spacetime our subjective experience in the world is more fundamentally temporal than spatial. While manyexperiences are accompanied by a sense of both time and space, all appear to exhibita temporal aspect while some, such as the experience of listening to a piece of musicor of simply thinking itself, lack any accompanying sense of an extended spatial arena.438his observation, along with the simplification of founding a theory on one dimensionrather than four, provided a source of motivation for the present theory.The sensation of time that accompanies all subjective experiences may be mod-elled mathematically by a continuous interval of the real line R , which is precisely thesame one-dimensional structure of temporal flow considered objectively as a presencewhich underlies the structure of the entire universe. That is, innate within the expres-sion for this temporal flow in the multi-dimensional form L ( v ) = 1 of equation 2.9,along with its symmetries, the form of the physical universe throughout an expanse ofboth time and space is supported. As discussed towards the end of the previous sec-tion, both temporal causality and a spatial geometry, deriving from the form L ( v ) = 1,are infused throughout the manifold M .At the mathematical level the unfolding of this structure is analogous to somedegree to the properties of the Mandelbrot set, in that a highly complex pattern isidentified through a very simple mathematical expression. A further analogy we con-sider here is the simple differential equation ∂ y∂x + y = 0 with the possible solution y = sin x . This sine wave is typically represented as a graph in the 2D plane incorpo-rating for example a horizontal axis for values of − π ≤ x ≤ + π . However innate in theexpression ∂ y∂x + y = 0 the actual mathematical solution is of course present throughout the infinite real line for −∞ < x < + ∞ , even though we typically only picture a smallportion of this solution. Similarly while the extended ‘spatial’ arena correspondingto the translation symmetry of L ( v ) = 1 is pictured for a finite region in figure 2.2this purely mathematical structure is of infinite extent in all n dimensions. This ob-servation still applies when the construction of the spacetime arena is generalised forthe geometry G µν = f ( Y, ˆ v ), as one of many possible solutions involving differentialequations in, and a degeneracy of, the underlying fields. As for the sine wave above,this purely mathematical solution has no limit for the coordinate parameters on thebase manifold (including in fact the particular solution for G µν ( x ) of equation 11.12which is itself described by a simple sine wave function as represented over an intervalof x in figure 11.1), while as for the Mandelbrot set the structure which emerges ingeneral may be highly complex.These mathematical patterns and structures on M arise through the projectionof the full form L (ˆ v ) = 1 onto the base manifold and the associated symmetry breaking.Through an innate subjective interpretation of certain entities on M there arisesfor us a vivid impression of material phenomena which appear to be detached andhovering outside us in an apparently spatial expanse. This entire perceived worldis however mathematically enfolded within the one-dimensional subjective temporalprogression through which everything in the world is observed. (This description is verymuch influenced by the notion of the a priori necessity for both temporal and spatialforms of experience, and their mutual relation, as elucidated by Immanuel Kant).The observation that spatial structures through the form L ( v ) = 1 can be implicitlyenfolded within the experienced one-dimensional flow of time that accompanies all ofour perceptions in the world completes the initial motivation for the present theorydescribed in the opening paragraph of this section.The form L ( v ) = 1 itself is derived from within the notion of a ‘momentof time’, divided into infinitesimal intervals, as described in section 2.1. From themathematical point of view the solution G µν = f ( Y, ˆ v ) over the manifold M is a439tructure implicit within the full form L (ˆ v ) = 1 and its symmetries, describing thegeometry of an infinite expanse of 4-dimensional spacetime (as for example implied forfigure 13.6) which does not ‘take time’ to unfold across the cosmos, rather it underliesall cosmic structure, similarly as the solution y = sin x is implicit within the expression ∂ y∂x + y = 0 across the full extent of the x -axis. This mathematical structure logicallyprecedes the laws of physics and the properties of physical objects perceived in theworld. As a manifestation of the underlying mathematical structures these physicalproperties include causal relations in general, incorporating for example the dynamicevolution of the fields, on M . The causal and spatial relations between physicalevents unfolding in the world, which do ‘take time’, create the sense of a world outside accommodating all of the apparent material phenomena. All physical structures aresubject to the laws of physics, which derive from the underlying mathematical forms,which apply for example to the phenomena depicted in figures 14.3 and 14.4 which inturn have both an objective and a subjective interpretation.The logical precedence of the elements of the theory described above is un-packed in the following sequence:(1) The objective starting point of the theory is one-dimensional progression in timewith a mathematical structure isomorphic to an interval of the real line R .(2) From the basic arithmetic properties of R a general multi-dimensional flow intime subject to the constraint L ( v ) = 1 can be derived.(3) The identification of extended geometrical structures from the form and symme-tries of L ( v ) = 1 provides a basis for the necessary arena for perception, that isa subjective experience of a spatial expanse.(4) In breaking the symmetry of the full form L (ˆ v ) = 1 over the M base manifold theproperties of material phenomena are sculptured and made visible in conformitywith the resulting laws of physics.(5) The material objects in the world include the complex structures of physicaldevices, such as brains, capable of performing mathematical operations and en-coding a progression of states of self-reference governed by the physical laws.(6) The sequence of self-referential states, linked through a contiguous resolution oftheir associated ‘undecidable propositions’, correlate with subjective thoughtsand experiences, ever accompanied by the sense of an ordered flow in time.(7) The subjective temporalisation may be modelled by an interval of the real line R having a one-dimensional mathematical structure identical to that in item (1).The first four points listed above form the main thrust of this paper from theopening chapters through to and including chapter 13, while the remainder of theabove chain has been the topic of the present chapter. In this paper the self-reflectivestructures depicted in figure 14.4 and discussed in the previous section are proposedas the means through which subjective experiences arise, although this may be a vastsimplification, or even largely a metaphor, for the actual mechanism. In any case, theexistence of a sequence such as that described in the latter four points above, beginning440ith an empirically observed physical world and leading to self-reflective consciousexperience in the world, is incontrovertible to the extent that it is evident that thepresence of conscious beings is amongst the known phenomena of the physical world.This observation applies for any physical theory, as discussed shortly after figure 14.1,although the details of the theoretical mechanism that underlies the subjective thoughtprocess remains open to investigation.For any physical theory built upon essentially any postulated entities, such asfields or particles and a background of spacetime, the universe can be described inmathematical terms as a ‘static’ 4-dimensional object, for example in the form of aspacetime diagram for the entire cosmos, which includes within it the full history ofeach human brain and all other material entities. However this is clearly not the waywe see the universe, rather the 4-dimensional spacetime structure of the brain mustprescribe our subjective perception of the universe as dynamically evolving through aprogression in time. Hence for any such physical theory the above segment of argumentin points (5) and (6) can still be applied, however there then remains dangling theprominent loose end that there is no apparent justification for the origin and propertiesof the initially postulated physical entities themselves, other than that they may becontrived pragmatically, for example in terms of a Lagrangian function in spacetime,to match the empirical data from observations and experiments.On the other hand the key observation for the present theory is that the finallink, item (7) in the above chain, representing the fact that temporalisation is containedas an ever present feature of subjective experience in the world, reconnects the chainto the initial link of item (1) at the top. Hence not only is a mechanism for the originof time conceivable, providing a foundation for the left-hand loose end of figure 14.1,but this temporalisation itself arises through self-reflecting structures, identified in thephysical world itself, which account for our subjective experiences in general and theright-hand loose end of figure 14.1. The chain then naturally closes into the cycledepicted in figure 14.5.It is a feature of the present theory that the two loose ends of figure 14.1 canbe mutually tied up in this way. From a mathematical point of view each of the sixstages in figure 14.5 is contained within the previous stage, supplying a foundation forall of the structures of the theory. This system can then be considered to establish a‘universal foundation’ for the present theory. The entire system is self-supporting in thesense that whenever we ask “where does X come from?”, where X can be time, space,matter, conscious beings, or anything at all, the question can be answered in terms ofsomething else within the system. Without the need for any external foundation orassumptions and entire structure in figure 14.5 hence detaches itself and floats free.This figure does not, of course, express an impossible cyclic chain of cause andeffect relating the six stages in a temporal sense. Indeed time itself is contained asone link within this cycle hence incorporating also the notion of temporal causality within this structure, and in particular for the physical laws in node (4). Rathereach connection between neighbouring stages has the logical nature of a structuralisomorphism, more precisely in the sense that the properties of node ( i + 1 , mod 6)are contained within the structure of node (i), with the net effect of expressing aself-contained and consistent mathematical and physical system. While providing achain of concepts for the benefit of deliberation the six nodes of figure 14.5 can be441 L ( v ) = 1mathematicalform M extendedspacetime matter andlaws ofphysics thoughts andexperienceof time self-referencingstructures R Figure 14.5: A self-contained ‘time cycle’ leading from the notion of progression in time,through the general mathematical form L ( v ) = 1 and perception of physical structuresin spacetime, to self-reflective entities incorporating experience of a progression in timeand hence completing the cycle.considered to collapse down to a single entity. This entity contains an entire universecreated through the temporalisation represented by the structure in figure 14.3 whichitself is entirely enveloped within the same physical world.In the opening of this section it was noted that while empirically everythinghappens in spacetime from the subjective point of view time is a more fundamen-tal mode of experience than space. This observation provides part of the originalmotivation for basing the present investigations on a general form of temporal flow L ( v ) = 1, together with its symmetries, rather than upon a 4-dimensional, or evenhigher-dimensional, spacetime structure. The further observation here that it is in thenature of time itself to provide the link connecting the two loose ends in the theoreticalsciences described in the previous section and in figure 14.1 adds further circumstantialsupport for this approach.While the means of supporting spatial perception arises from a very direct inter-pretation of the geometric forms implicit in the mathematical properties of L ( v ) = 1,as described in chapter 2 and corresponding to nodes (2) → (3) of figure 14.5, themeans of generating temporal experience arises from the far more complex mathe-matical structures represented in figure 14.3, as described in the previous section andcorresponding to nodes (5) → (6) of figure 14.5. The figure as a whole can be seen as aninterplay between 1-dimensional and multi-dimensional forms of time. The underlyingmechanism for obtaining an extended 4-dimensional world out of 1-dimensional tempo-442al flow, summarised in the upper half of figure 14.5, differs from the far more complexstructures required to identify a purely 1-dimensional entity out of the 4-dimensionalphysical world, as summarised in the lower half of the figure. (As discussed at theend of the previous section a simple physical clock, for example, does not possessany intrinsically 1-dimensional geometric structure). It is the very different nature ofthe mechanism for obtaining multi-dimensional forms and extended spacetime from1-dimensional time on the one hand and for identifying temporal flow itself out of thehigher-dimensional and spacetime structures on the other hand, as required for sub-jective perception and experience, that opens up the non-trivial system of figure 14.5.The contrast between the objective features of temporal progression, identifiedas the simplest element of figure 14.5, and the subjective experience of progression intime arising out of the most complex structures in this system, while both aspects oftime, in nodes (1) and node (6) respectively, share the identical structure of an intervalof the real line, underlies the enigmatic quality of the concept of time itself. Referencesto the seemingly more philosophical nature of time in the physics literature are rare butnot entirely absent. Near the beginning of the introduction to his Space–Time–Matter
Hermann Weyl writes ([82] p.1):Since the human mind first wakened from slumber, and was allowed togive itself free rein, it has never ceased to feel the profoundly mysteriousnature of time-consciousness, of the progression of the world in time, –of Becoming. It is one of those ultimate metaphysical problems whichphilosophy has striven to elucidate and unravel at every stage of its history.While the upper half of figure 14.5, that is the chain of nodes (1)–(4), representsthe development of the theory within the traditional scope of physics, the entire scheme,including the lower half of the figure, is fully incorporated within the sphere of scientificstudy more generally. Indeed experiments are performed, dating for example fromthose conducted by the neurologist Benjamin Libet in the early 1980s, concerned withthe relation between the physical brain and conscious actions, that is essentially nodes(5) and (6) respectively in figure 14.5. In such experiments physical cerebral activity isfound to precede a conscious awareness of intention typically by around 300 millisecondsor more.For the present theory a conscious intention, or choice, is associated with theorigin of temporalisation, as described for figures 14.2 and 14.3. This leads to themulti-dimensional form of temporal flow L (ˆ v ) = 1 through which derives the mathe-matical structure underlying the entire physical universe on M , incorporating its fulleternal temporal extent both into the past and into the future. This physical universeincludes in particular the brain state 300 milliseconds before the conscious choice wasexperienced, and indeed at any other time, as embedded within M as representedin figure 14.4. Hence the overall scheme presented here is fully consistent with theexperimental findings of Libet and others. More generally the full cycle of figure 14.5,including all of the nodes and links, is fully amenable to theoretical and scientificinvestigation.As a preliminary discussion the remarks made here on the origin of our tem-poral experience and the phenomenon of consciousness, together with their mutualassociation, are necessarily somewhat speculative. However, it is meaningful to for-443ulate such questions, the worldview presented in this paper provides a new arenathrough which the construction of figure 14.5 seems inevitable, and this provides afirm mathematical basis for a possible scientific enquiry into the nature of subjectivephenomena compatible with the basic structure of the present theory.Most fields of scientific study are rooted in node (4) of figure 14.5, in thatthe natural starting point for any scientific investigation is observation of the physicalworld around us. For the physical sciences the general aim is to deduce the basis of theunderlying structure of the world, extrapolating inwards as for example in the directionof nodes (3) → (2) for the present theory, while on the other hand the biologicalsciences, for example, also study the world at face value and extrapolate outwards,which might include the properties of nodes (5) → (6) in figure 14.5. However, unlikethe case for other scientific theories in general, in the present theory it is natural toextrapolate one step further, both inwards and outwards, to establish the final link innode (1) and hence complete the circuit. Here the overall structure of figure 14.5 thenhas the shape of providing an answer to the general question “why is there somethingrather than nothing?”, rather than merely displacing it.From the mathematical perspective while an exposition of the structures infigure 14.5 could begin with any given node the simple mathematical structure of time,as a 1-dimensional progression modelled by the real line R , provides a convenient entrypoint into the study of the whole system. In particular the unique properties of a realinterval provide an unambiguous structure upon which to develop the theory, as willbe emphasised later in this section. If the properties of the real line R are considered todefine the axioms, which in general underlie all the expressions which may be derivedin a formal system, then the self-contained structure of figure 14.5 might be thoughtof as a mathematical system which ‘contains its own axioms’.From this point of view as a single entity of self-creation the time cycle infigure 14.5 can be considered firstly as a purely mathematical structure which can bedescribed in terms of the six nodes displayed with each one mathematically identicalto, or contained within, the previous node of the chain. This picture can then be‘coloured in’ with both the objective material features of a physical world and thesubjective experienced aspects of self-reflective thoughts and perceptions.The subjective experiences, as much as the objective material phenomena ob-served, are an irreducible component of this system. Indeed it is the experience oftime, as well as of space, that generates necessary links in the time cycle of figure 14.5.Such a world cannot exist unless it is experienced . The two loose ends, left exposedin many conceptual worldviews, relating to the origin of conscious experience and theorigin of the material world are interwoven into one coherent system. Here the em-phasis does not weigh heavily upon a pre-existing material content of any kind, butrather takes an overall more balanced view within which ‘matter’ is identified with aform of experience shaped in ‘mind’.The apparent distinction between mind and body arises in part since the spa-tially distributed matter we experience appears to exist out there , however here theconcepts of ‘mind’ and ‘matter’ are intimately intertwined within one system. Wehave no need to postulate two wholly different kinds of substance and ponder howthey interact, such as through the pineal gland in the brain in the worldview of Ren´eDescartes. Rather mind and matter are different aspects of the same self-contained444ystem: the conscious mind being bound with the structures of mental activity andtemporality embedded in the physical world, while spatially extended matter itself iscarved out of the multi-dimensional algebraic properties of time. Hence both sidesof the philosophical dichotomy between mind and matter are accounted for and thepoints of view of both the idealist and the materialist democratically amalgamatedinto this structure. We are not spirits haunting Earthly bodies, and neither are wemachines in search of a soul.While forming components of one overarching framework both the objectivestructure of the physical world and the subjective forms of experience in the world canbe described in terms of theoretical elements, and each is sufficiently distinctive andwell defined to seemingly take on a ‘life of its own’. From the point of view of thepresent theory the materialist is grounded in node (4) of figure 14.5 and can constructa relatively short, physically motivated, argument to account, via node (5), for therealm of the idealist in node (6). On the other hand the idealist, based upon thesubjective experiences of node (6), is required to make a more lengthy detour, via theconceptual and mathematical structures of nodes (1), (2) and (3), in order to arriveat the materialist’s realm in node (4). This asymmetry in the apparent directnessof mutual explanatory power perhaps in part accounts for the predominance of thematerialist, ‘a spade is a spade’, philosophy that has underpinned most progress in thehistory of science, in addition to its practical utility.Outside the present chapter of this paper, as for the vast majority of work intheoretical physics in general, the focus has been with the study of a mathematicaldescription or model of the physical material world, here through equations such as L (ˆ v ) = 1 and G µν = f ( Y, ˆ v ). However mental phenomena, such as our awareness ofthe physical world and decision making actions, are also very much a feature of theuniverse and in principle equally amenable to theoretical analysis, as discussed above.To recap, in the present theory the mathematical structure described in fig-ure 14.3 models our conscious self-reflective state and ever present feeling of not know-ing for sure quite what we shall do in the next moment. This perpetual uncertaintyas to our own thoughts or actions resolves momentarily in a choice ‘ G i ’ opening upa new horizon of uncertainty, as represented in the progression of figure 14.2. Theself-reflective state is inexorably drawn through the series . . . R → R ′ → R ′′ . . . of fig-ure 14.3 correlating with an internal experience of a sequence of thoughts, aspects ofwhich have a complex mathematical representation, but in all cases associated with asubjective experience of a simple one-dimensional temporal flow.Within this structure the term freewill , as used without hesitation in everydaylanguage, is identified as this ‘experience of choice’ as one feature of the overall systemof figure 14.5. Everything that happens objectively in the physical world follows inthe wake of this subjective temporalisation phenomenon. The historical philosophicaldebate concerning ‘freewill versus determinism’ becomes more strictly a question of‘freewill versus the laws of physics’ in the context of modern day science. The lawsof physics include ‘indeterministic’ quantum phenomena as a feature of the objectiveworld which in the present theory are not correlated with the subjective act of makinga conscious choice. Indeed the intrinsically random transitions of quantum effects areof a wholly different nature to rational decision making or freewill. On the other handquantum properties are a major component of the laws of physics, and it is this full445ackage of physical laws which determine all physical structures. These include thephysical state of the brain which evolves in time according to the laws of physics,exhibiting properties which do correlate with the interaction between the consciousand subconscious mind as implied in figure 14.4 and hence providing the vehicle tocarry self-reflective experiences.The traditional philosophical difficulty in reconciling freewill and the laws ofphysics derives from the observation that an apparently independent objective worldevolving according to a set of deterministic laws (together with random quantum phe-nomena) seems to leave no room for the notion of freewill. However, here in the presenttheory, since the physical world is brought into being through a subjective temporalisa-tion sufficient breathing space opens up for the concept of freewill – not as a secondaryphenomenon on top of a given physical world, but as an irreducible feature in dynamicinterplay with it, as summarised in the time cycle of figure 14.5. An element of thephilosophical confusion concerning these issues arises as there is considerable ambigu-ity in the meaning of the term ‘freewill’ in itself. The present theory provides a contextwithin which the notion of freewill might be more precisely defined. Within the systemof figure 14.5 the means by which the world is experienced in mind is as important asthe empirical forms of matter, with freewill being a property of the former while thelaws of physics are a property of the latter.It seems of course counter-intuitive to suggest that the great expanse and‘weight’ of the entire physical universe might be created through and carried in asingle waking moment of thought. However, as described near the opening of thissection, the mathematical structures underlying a solution for G µν = f ( Y, ˆ v ) innatein the form of temporal flow L (ˆ v ) = 1 are perfectly ‘weightless’ and infinitely deli-cate, effortlessly supporting an entire cosmic history throughout the full expanse ofthe physical manifestation of the universe. If the laws of physics in this spacetime arecompatible with the local evolution of a physical brain as depicted in figure 14.4 whichencodes the self-reflective sequence of figure 14.3 which in turn represents a subjectiveexperience of a one-dimensional temporal progression isomorphic to the ordered realline in node (1) of figure 14.5 then the circuit closes and the experiencing being locateshimself at a particular place in a particular world (in this chapter in this context pro-nouns such as ‘himself’ or ‘his’ refer to a non-gender-specific being in any world). Thisspacetime location will be within the habitable epoch of the cosmological evolution asdepicted in figure 13.6(e), and most likely upon a planet orbiting within the habitablezone of a suitable solar system as considered in section 13.3, for the case of our ownuniverse. The poets have more readily conceived of such a world, as for example inthe often quoted opening lines from William Blake’s Auguries of Innocence of 1803:To see a world in a grain of sandAnd a heaven in a wild flower,Hold infinity in the palm of your handAnd eternity in an hour.Here, not limited by poetic licence, we require only a moment rather than anhour through which the entire universe may be perceived. The contention of the presenttheory sees the world and the heavens, together with an infinite expanse of space andan eternal temporal duration all held within a moment of time. The completion of this446icture is depicted in figure 14.5 with the experience of time itself contained withinthe structures of the physical universe.While the entire physical universe is created through the experience of a singletemporal moment, the moment itself is not unique. The circuit of figure 14.5 can beclosed by any one of a large number of possible local structures representing the pro-gression of figure 14.3, each embedded within the physical world and each associatedwith a moment of experienced time. Indeed if the physical world is capable of sup-porting such a structure at all then in principle there may be many examples. Thisgeneralisation is depicted in figure 14.6. v )=1 3space1time 4matter A A A A A i B Figure 14.6: Rather than the single experience of time represented by node (6) in fig-ure 14.5 in general a large number of physical structures on M may generate momentsof temporalisation, each represented by one of the small boxes labelled by A or B hereand each of which completes the circuit of a time cycle for the same physical world ofnode (4).In particular the set of temporal moments labelled by the series A . . . A i infigure 14.6 might form a contiguous structure in the sense of the embedding of theprogression of figure 14.4 within the physical world on M . The corresponding tem-poral experience correlated with this structure is the sensation of a ‘sliding now’ for aparticular individual. Similarly while the ‘spark’ that creates the universe from withincan be any moment A i associated with such an individual, it could also belong toany other being, such as the temporal experience represented by B in figure 14.6. Ingeneral the experiences of a community of beings A, B, C . . . may be inscribed withinthe same manifestation of a physical world, as depicted in figure 14.7.447igure 14.7: A depiction of the trajectories of several self-reflecting ‘beings’
A, B and C experiencing life in the M spacetime manifold.Any structure of temporalisation, such as those represented by each ‘ X ’ infigure 14.7 can take the place of node (6) in figure 14.5 and complete the circuit whichalso incorporates the physical world itself which a community of beings such as A, B and C cohabit. Each of these individuals observes a time-ordered progression of statesof material entities, from stars and planets to tables and chairs and other individuals,distributed in a 4-dimensional spacetime as originally envisaged in figure 2.3. While agiven observer A experiences the subjective freewill of his own self-reflective state, andan internal temporalisation as represented in figure 14.3, from his perspective both thesubconscious as well as the conscious elements of the brains of the other beings B and C are unambiguously seen to partake seamlessly in the physical flow of events in theworld. That is, the behaviour of the other, progressing in parallel and as representedfor example in figure 14.4, conforms to the basic laws of physics exactly as any otherphysical entity such as the tables and chairs carried along in the inertia of the world.With a perfectly reciprocal account given from the internal subjective point of view of B or C the mutually consistent perspectives of all individuals dovetail together withinthe common physical world.In conformity with this symmetry between A, B and C in terms of a perspec-tive on freewill and the laws of physics each observer carries a personal experiencedfundamental time parameter s . This temporal flow s is equivalent to the proper time τ recorded by a physical clock in the frame of the individual, as related by the constantfactor γ described for equation 13.3. The progression in time s ≡ τ for any givenindividual is related to that associated with each of the other observers through the448ilation effects of both special and general relativity in spacetime, as described for the‘twins’ A and B towards the end of section 5.3 and generalised near the opening ofsection 13.1, again in a completely reciprocal manner.For any individual the seemingly vast potential arena for the flow of time in theuniverse at large contrasts sharply with the observation that we experience time at anapparently brief moment. The concept of ‘now’ can be identified subjectively with the‘present moment’, which consists of a small duration rather than a point in time. Thisleads us to pose the question – given such a vast expanse of time – “why is it now !?”;as opposed to, say, some time last week. This question is particularly challenging fortheories of the world which posit an initial extensive and objective spacetime arenaupon which the laws of physics are mathematically constructed from an independentperspective outside spacetime, such that the physical laws governing all phenomenahave perfect symmetry with respect to translation of location in either space or time.Within such a framework ‘now’ is generally conceived objectively as a point in time,as a mathematical point of the real line. While we have a wide choice over where tomake an observation the fact that we necessarily observe the world as it is now , at this particular point in time, appears to explicitly break the time translation symmetry.The problem disappears when we consider the meaning of ‘now’ within thetheory presented in this paper. Our self-referencing awareness involves the physicalstructure of a small region of the world which is sufficiently complex to support ‘unde-cided states’, but further complex structure carried in the physical world, in particularthat of the subconscious brain, holds the resolution to such states and draws consciousawareness into the wider world in the process of temporalisation. Beyond the brainwe find also the human body, the habitable environment and the entire physical worldunfolding through the physical realisation of temporality creating a situation in whichthe conscious being exists. Since every situation is an experience and every experienceis an experience now the logical meaning of the word ‘now’ in this system is entirelyredundant (although, of course, it has a practical purpose in everyday language). Thefact that it is now , rather than some time last week, is simply that I am this experience,whereas the situation for a particular individual at a particular time last week is that experience. The apparent problem is then largely an issue of the assumptions made inthe use of language regarding the identity of an individual (the ‘I am’) as somethingmore attached to a bodily form than to an experience.The fact that it feels like ‘now’ comes from the fact that the world exists ‘allat once’ – that we can conceive of a past and future progression within which we placeourselves in the present, now . However, past and future are not periods of a pre-existingexternal and independent world-time; rather the past and future refer to locationswithin the universe with respect to the perceiving being whom experiences the situation– it is a description of the experience which partitions a self-reflecting conscious stateinto a concrete past and an uncertain future as a necessary structural form of a thinkingbeing. (This aspect of the worldview being described here is philosophically close to thestandpoint of existentialism, and is influenced in part by the philosophy of Jean-PaulSartre).I have to experience the world now in a similar way that I also find myself here at a particular spatial location in the world. While the ‘body’ of the wholeworld is created through the structure of our being, here is where my eyes, and other449ense organs of the human body, locate me spatially relative to other physical objectsin the world. To necessarily exist here and now is simply the statement of havingto be the centre of reference for an experience in a world. This central vantage-point is essentially the location of the physical manifestation of the associated thoughtprocesses, as represented in figure 14.4, within the extended spacetime manifold M .From the perspective of any individual such as A the universe created throughany given experienced moment, such as A in figure 14.6, not only mutually supportsthe contiguous moments of the ‘sliding now’ and into the span of the current day, butalso the moments of yesterday, and the past in general, and those of tomorrow, and thefuture in general. As well as the spacetime separation between moments experiencedby A and B each individual is also separated from himself in time, corresponding to themoments marked ‘ X ’ on the trajectory of A in figure 14.7 for example. The identicaluniverses generated from A ’s experienced moments on Monday, Tuesday, Wednesdayand so on resonate together into a single life history. Any moment of temporalisationnot only brings the corresponding present self into being but also the physical structurefor all the past and future ‘selves’ in the life history of the same individual. This systemis hence comfortably compatible with the experimental findings of Libet and othersas noted in the discussion following figure 14.5. The mutual relations between any of . . . A , A . . . A i . . . from a single life history dovetail together, as with the moments ofany other beings, such as B and C , in the same physical world.The exhaustive spacetime coverage of the universe created through each tem-poralised moment A i together with the inertia of the derived material processes in theworld maintains the physical manifestation of any individual during non-waking hoursor through different shades of consciousness. In this way historically separated wakingmoments are seamlessly stitched together over periods of years alongside those of otherbeings immersed in the same world.The existence of different shades of consciousness, such as the experience ofdreams, suggests that a rigid geometric framework in space may not be essential forsome forms of perception, although dream sequences are closely associated with wakingexperiences. The question regards whether spatial perception is required in some formin order to complete the circuit of the time cycle in figure 14.5. As discussed insection 12.1 and section 13.1 (before the bullet points) our a piori imposition of anextended 3-dimensional frame for our perceptions in the world does not perfectly matchthe non-Euclidean geometry of the world – which we however effectively interpret asbeing flat while certain phenomena are ascribed to an apparent force of gravity. Wevery rarely perceive solely events within a local inertial reference frame, such as withinan orbiting spacecraft, however such an idealised limiting geometry is not required inorder for us to be able to interpret and organise our perceptions of the world in amanner compatible with the presumption of a flat Euclidean frame of reference.In addition to providing a spatial orientation for vivid conscious experiencesof the world, with material objects obeying physical laws of motion within the per-ceptual framework, the general laws of physics themselves, which shape all materialproperties, arise from the projection of the full form of temporal flow L (ˆ v ) = 1 ontothe base manifold M . Complex mathematical and physical structures which arise inthis breaking of L (ˆ v ) = 1 and its full symmetry over M accommodate the mechanismfor self-reflective conscious experience itself, as described for figures 14.3 and 14.4. The450hysical laws deriving from the symmetry breaking hence not only maintain objectivematerial objects in the world but also images of the same objects which can be main-tained in our subjective thoughts even while the object is not being directly perceived(as for example in a dream or in a waking moment in which we simply look away fromthe physical object while still thinking about it).Hence the laws of physics derived from the symmetry breaking of L (ˆ v ) = 1 overthe base manifold M give rise to both the structure of conscious self-reflective statesand the material phenomena, perceived against the M background, which constituteobjects of consciousness. This then describes the primary requirement of the symmetrybreaking of L (ˆ v ) = 1 in order to complete the time cycle in figure 14.5, that is to openup structures that may be presented as objects of conscious experience together withthe self-reflective elements capable of contemplating such objects. In our world thesestructures are obtained through the projection of L (ˆ v ) = 1 over a locally approximatelyflat 4-dimensional spacetime M which incorporates a 3-dimensional spatial arena forthe perception of objects.In principle we can enquire what it might be like to be immersed in a highlycurved spacetime environment of a different world. Without the support of an effec-tively Euclidean spatial orientation it would be harder to organise our incoming sensorydata and difficult to predict the physical consequences of our own actions and to en-gage in such a world generally. The likelihood of errors of judgement in this respectis much lower in the local environment of an apparently flat spacetime combined withthe very regular patterns of motion deriving from Newtonian gravity, as we encounterin our own world.It seems very natural to us that space ought to have Euclidean properties, aswitnessed by the historical perseverance of the geometrical laws of Euclid formulatedin ancient Greece, which until the early 20 th century were assumed to describe thereal world. While applying to an excellent approximation in the local environment ofthe Earth and solar system, the assumption of a flat spacetime geometry breaks downfor large scale cosmological structures. In general any manifold with two or moredimensions can exhibit arbitrarily large curvature at any point, as is the case for our4-dimensional universe for which the curvature diverges in the proximity of the initialsingularity or a black hole. However the curvature of any 1-dimensional manifoldis trivially zero and the geometry necessarily ‘Euclidean’. Hence an interval of the1-dimensional real line R , as a unique and robust structure, and as a parametrisationof the subjective experience of temporal flow, provides an unambiguous starting pointfrom which the present theory has been developed in this paper.This discussion raises the questions considered in section 13.3 regarding whetheror not the symmetry of L (ˆ v ) = 1 is uniquely required to be broken over a 4-dimensionalspacetime M and whether structures identified in the symmetry breaking are requiredto be compatible with the notion of perception as conceived in our world. Whether acomplete time cycle of the kind in figure 14.5 incorporating self-reflecting beings with-out an a priori spatial perception of any form could exist, or even whether there areconscious organisms within our own world that completely lack any spatial awareness,may be difficult questions to address. Such self-reflecting creatures may still necessar-ily require an M base space to break the full L (ˆ v ) = 1 form in order to physicallyexist (as do all non-sentient biological life forms in our world), yet without employing451 subjective spatial interpretation of the 3-dimensional structures on M . In a similarway we require the extra dimensions of the form L (ˆ v ) = 1 in order to physically existourselves, yet without our being directly aware of them.Here we recall that the term ‘perception’ is being employed not just in thenarrow sense of that which we are visually aware of in the moment. It refers moregenerally to an organising faculty for all the data about the world that enters andour thoughts through all of our senses. This data is accumulated both directly, forexample through the experience of vision or touch, as well as indirectly, for examplevia intermediate objects, tools of experimentation or the accounts of other people.This data concerns aspects of the world in spacetime ranging from our immediatelocality, down to the minute microscopic scales explored in HEP experiments, outto regions very remote from us and through to the limit of observations relating tothe structure and evolution of the cosmos. Perception is a form of knowledge thatencompasses everything we can understand about the world in space and time, inprinciple anything associated with nodes (3), (4) and (5) in figure 14.5.Moulded by this form of perception physical structures as we experience themexhibit the effortless complexity inherent in the breaking of L (ˆ v ) = 1 over the infiniteexpanse of M as depicted for example in figure 13.6. As alluded to in the openingof this section such a structure is analogous to the endless delicacy of the Mandelbrotset, which arises from the iteration of a simple mathematical expression in the complexplane. In both cases an inexhaustible variety of fine detail can be observed whereverwe choose to ‘zoom in’ and examine for example biological forms in the physical worldor the spiralling patterns of the Mandelbrot set. At the shortest physical distancesprobed the properties of elementary particles emerge over an underlying fractal-likestructure of field solutions, as described in section 11.3, while at the other end of thescale, throughout the expanse of the observable universe, we perceive the manifestationof the laws of physics in the swirling patterns of galaxies and galactic clusters. Theobservations of cosmology, on this largest scale, are contained within the physical worldof node (4) in figure 14.5, which in turn provides a context for understanding the ‘cause’of the Big Bang and the origin of the universe more generally within the ‘system ofthe world’ presented in this paper. The big picture for the present theory, as represented by the time cycle of figure 14.5which sees the conscious observer in a dynamic interplay with the entire physicaluniverse as an irreducible, integral component of the world, offers a very differentperspective to the ‘Copernican view’, which sees humanity playing a far less significantrole in the cosmos. The fact that the physical manifestation of humanity representsa tiny contribution to the total matter content of the Earth, which itself is in orbitaround a far more massive sun, which in turn is one of countless stars distributedthrough the galactic structures of the universe all serves to cement the Copernicanworldview concerning our apparent insignificance in the grand scheme of things. Thisis a misconception of the nature of the cosmos from the point of view of the presenttheory. On the other hand here there are potentially a vast number of subjective452xperiences which may complete the circuit for any physical universe, as described forfigure 14.6, and each objective physical universe is one of a potentially vast numbersolutions of the form G µν = f ( Y, ˆ v ) capable of supporting self-reflective temporalisingbeings. In any case the ‘cosmological principle’, as described in section 12.2, is validfor our universe in being sufficiently consistent with empirical observations to providea valuable aid in finding solutions for the spacetime geometry on the largest scalesobservable. Such an entire solution for a physical universe, represented in a space-time of unlimited 4-dimensional extent as depicted in figure 13.6 and with a geometryexpressed as G µν = f ( Y, ˆ v ) in the present theory, is created as a mathematical possi-bility within the system of figure 14.5. The nature of this geometric solution is verymuch in the spirit of general relativity for which a spacetime geometry such as theSchwarzschild solution of equation 5.49, although typically employed to determine aplanetary orbit about a star, represents an infinite 4-dimensional spacetime.Here the possibility of the overall mathematical solution represented in fig-ure 14.5 is the reason why the universe exists. Our local perspective of observing theflow of cause and effect in the everyday physical world leads by analogy to the pre-sumption that the universe itself must have been created either by an event in timeor by an event coinciding with the beginning of time. For any creation event in timethe question then ever remains regarding the cause of that event while for a creationevent at t = 0 the nature of an event without an apparent cause is certainly no lessproblematic from a conceptual point of view. In either case there are an array of fur-ther conceptual difficulties regarding the origin our own universe, such as the ‘start-upproblem’, as described towards the end of section 12.3.In the present theory the creation of the universe is not something that ‘hap-pens’ in the Big Bang, or temporally before it, rather the very early universe and theBig Bang correspond to a certain region of the spacetime geometry at a particularepoch of the full 4-dimensional solution. This early epoch is beyond the horizon ofour direct experience but its existence depends upon the self-reflective temporalisingexperience that arises in the history of the universe, as does everything in the cosmos.All the physical structure and conditions of the universe, including that for all futureas well as past epochs and throughout the vast spatial expanse both within and beyondour observational reach at any epoch, are brought into being through the nature of atemporalising entity, which in turn is supported within the physical world, as depictedhere in figure 14.8.All experience in general is played out through a moment in time, includingour perception of the physical world, with the structure of the universe being math-ematically described by a solution for G µν = f ( Y, ˆ v ) as conforming to the full formof temporal flow L (ˆ v ) = 1. Hence here the structure of the entire universe is derivedmathematically through a moment of time, typically conceived as a duration of or-der one second as represented by a small one-dimensional interval ∆ s , as the windowthrough which it is perceived for example by the observer in the centre of figure 14.8.This contrasts with the standard cosmological models for which the entire observableuniverse evolves physically from a vanishingly small 3-dimensional spacelike hypersur-face of size a ( t )∆Σ → t →
0, as described for figures 12.3 and 12.4 in section 12.3,corresponding to the point at the base of figure 14.8.453igure 14.8: The physical universe contains its own means of creation as perceivedthrough the window of an interval of pure time ∆ s subjectively experienced by theobserver within the world. This picture is in contrast to standard cosmology for whichthe observable universe evolves from a vanishingly small spatial extent a ( t )∆Σ at t = 0.Here the scenario described for figure 13.4(b) has been depicted.454or the standard approach all of the field content, particle properties andphysical laws in general need to be added onto the spacetime in order to determine theevolution of the universe from the initial spacelike state, which is presumed to exhibitsuitable initial conditions. However for the present theory all of the fields and physicallaws derive from the structure and symmetries of L (ˆ v ) = 1 through the necessaryprojection onto M in framing our perception of the world, including the StandardModel particle properties as identified in chapters 8 and 9. Here the apparent ‘initialconditions’ for t → s brings with it the possibility of con-structing a universal foundation as represented in figure 14.5, while beginning witha spacelike hypersurface ∆Σ at t = 0 leaves questions open concerning not only thesource of physical laws and the nature of the initial conditions but also the origin ofspacetime itself. Further the existence of the temporal moment is evident, in fact weare perhaps more intimately familiar with our experience of it than of anything else inthe world, while the hypothetical initial spacelike state of the universe is an extremelyremote entity. Hence overall, the notion of the present theory that everything is ‘per-ceived through a moment of time’ is perhaps not less reasonable than the standardpicture for which everything ‘evolves from a point of space’.The system constructed in figure 14.5, for which figure 14.8 represents a par-ticular manifestation such as our own world, can be considered as being centred fun-damentally upon addressing the question of how it is possible to have subjective expe-riences of a world. As described in the previous section such experiences always takeplace here and now in the world, with everything else we can say about the universe,whether at some distance in space or extrapolated through time into the future or the past, necessarily consistent with the fact that we experience the world here in thepresent moment. The environment we experience in the present incorporates, amongstother things, observations based on a geometrical spacetime manifold; in particular weare able to perceive a world since it is cast against an approximately flat spatial back-ground. However, there is no reason to expect the mathematical preservation of such anapproximately flat pseudo-Euclidean spacetime indefinitely into the past as we extrap-olate beyond the horizon of our direct physical experience of the world. The geometryof the very early universe for example, being beyond our immediate perception, witha potentially extreme spacetime curvature, is not required to be compatible with our a priori imposition of a flat framework of space and time within which to organise ourimpressions of the world and plot our actions within it.Hence neither an approximation to spatial flatness nor any other constraint onthe 3 or 4-dimensional geometry is required for the early universe regions of figures 13.4or 14.8, in particular in approaching t →
0. In fact at earlier times there remains norequirement for the identification of a 3-dimensional spatial or 4-dimensional spacetimemanifold structure of any kind, as is the case for the scenario depicted in figure 13.4(a).However while the identification of the manifold M itself, together with the projection v ∈ TM , may break down at an epoch before the Big Bang the general form of455emporal flow L (ˆ v ) = 1 remains ever valid for any value of the fundamental temporalparameter, even for s → −∞ as described for figure 13.4(a) in section 13.2.From this point of view while the universe can be considered to be infinitelyold, in terms of the value s → −∞ , the cosmic time t = 0 can be considered to be thepoint in time at which a 4-dimensional spacetime manifold M unfolds from the form L (ˆ v ) = 1, as depicted in figure 13.4(a). The familiar laws of physics in 4-dimensionalspacetime, including the second law of thermodynamics expressed in terms of thedegrees of freedom of Standard Model particle states and the gravitational field, mayfirst be collectively applied as they emerge from the Big Bang at t = t v ; for either thescenario of figure 13.4(a) or (b) as also described in section 13.2. While we do notdirectly interact with the very early universe we are intimately connected with it notonly through observations in cosmology but also through the need for the conditionsof both stellar and biological evolution to arise and be consistent with the support ofself-reflective beings at the present epoch.In order to achieve this in addition to the microscopic field and particle interac-tions underlying the macroscopic gravitational structure G µν = f ( Y, ˆ v ), as empiricallyobserved in the high energy physics laboratory and the cosmos respectively, at anintermediate scale the laws of physics implicit in this solution must necessarily becompatible with the development of the structures of molecular biology, such as DNA,which underpin the evolution of life. The complex biological structures implicit in fig-ure 14.4, correlated with the subjective experience of temporalisation, must themselvesarise in the material dynamics of the universe in a manner consistent with the lawsof physics in 4-dimensional spacetime. That is, the physical universe we observe mustsupport not only the formation and history of the solar system but also the evolution ofbiological life on Earth and the birth and development of specific self-reflective beingsas manifested in human form and as represented in the centre of figure 14.8 for ourworld. It could be asked: if the whole universe is brought into being through anexperience of it here and now , why does it appear that biological evolution, leading upto the human race was necessary? Why not have readily formed humans along with theEarth and our local environment suddenly appearing, along with the identification ofthe M manifold itself dating from the ‘cosmic time’ t = 0 just a few centuries or even afew minutes ago? However, the full extent of our spacetime world, including everythingcausally related to us from the past, must conform to the form of our perception inspacetime through the breaking of L (ˆ v ) = 1 over M and the consequential laws ofphysics as implicit in the solution G µν = f ( Y, ˆ v ). The flow of the world in our pastand into the future must obey these laws and also be consistent with our biologicalform as observers in the present.Such an overall solution might be much more likely achieved through a verysimple initial state followed by a prolonged cosmic and biological evolution as shapedby the laws of physics, rather than the apparently more direct route via a highlyimprobable ‘initial state’, in the form for example of a ‘snapshot’ of the universe takena few minutes ago, which may in any case be prohibited through contradiction withthe necessary laws of physics. This would still be the case even if the ‘snapshot’ onlymet the minimal requirement of preserving the complex form of the local environment,in which case the large scale cosmos would also most probably look very different to456ur universe. At the other extreme the universe, as an extended spacetime manifold,may not have a temporal origin at all in the sense that arbitrarily early times withfundamental time parameter s → −∞ might be contained within M . This is theseemingly more natural scenario depicted in figure 13.4(b) for which the ‘initial state’corresponds to the asymptotic conditions as s → −∞ and t →
0, as also described forfigure 13.5.On the other hand the conditions in the universe observable today, even ne-glecting the consequences of the cosmic expansion, could not have prevailed indefinitelyinto the past. The laws of physics, in particular the second law of thermodynamics,demonstrate that it is not possible to sustain an everlasting immortal species on theEarth, and itself implies a necessarily finite lapse of time into the past to an apparentorigin for our physical universe, which is also consistent with the observed expansion ofthe universe. Hence human life forms must have been moulded out of the state of thephysical world at an apparent temporal origin of the 4-dimensional universe, that is thetime at which the familiar laws of physics were established, culminating in a physicalevolutionary process which in our case involves the processes of genetic mutations andnatural selection. This apparent temporal origin must itself have an explanation interms of the overall theory, and is here associated with the phase transition at the endof the Big Bang, that is at t = t v in the scenario of figure 13.4(a) or (b) and as alsoindicated in figure 14.8.That ‘there was evolution’ is a statement from our perspective within the uni-verse, which itself can be considered from an outside perspective as an ‘atemporal’static 4-dimensional entity, about the world as a whole and the structure it must havefor us to exist here and now in 4-dimensional spacetime. To ‘visualise’ the wholeuniverse it is convenient to return to the three-dimensional spacetime analogy andcombine the content of figures 14.4, 14.7 and 14.8. Through the circuit of figures 14.5and 14.6 life draws itself into being out of the ‘mathematical vacuum’. While the lawsof nature on our spacetime manifold are carved out of the general form of progressionin time, the actual physical forms we encounter in the universe, whether in our presentor uncovered from our past, are moulded to conform with the possibility of our ownbodily existence and conscious experience within it.Many features of the world that we observe, such as our existence within acommunity of beings (the experiences of whom mutually dovetail together as describedfor figure 14.7) rather than finding ourselves in isolation, are the way they are sincethe world in which we find ourselves situated must accommodate a physical sequenceof events, including for example an evolutionary and social history, leading up to theform of each individual experience.All matter of the universe is brought into existence through our experienceand perception of it as being mathematically, and hence physically, connected to thenecessity that the experience itself exists. Hence all of our body organs, blood vesselsand so on, as well as the human brain, necessarily come into being through the mech-anisms and processes that give rise to life, in terms of its physical parts, along withthe entire biological world, through the logical and rational requirement that we mustbe physically sustained within the world which we experience. The seemingly greatimprobability of life in terms of the complexity of biological structures such as sensoryorgans and the nervous system is essentially irrelevant. If such a biological system is457hysically possible at all and represents a self-reflective temporalising structure withinthe mathematical system of figure 14.5 then it will draw itself into being and exist asthe realisation of an underlying mathematical necessity.The constraints on the form of such a mathematical solution will be all the morestringent if there are essentially no free parameters in the breaking of the full form oftemporal flow L (ˆ v ) = 1 over a base manifold M n (for an n -dimensional world solution).However, as for any mathematical problem, whether or not a solution actually existsdoes not depend upon the apparent difficulty of the problem. Whether or not a degreeof tuning is possible for the symmetry breaking parameters (and whether or not n = 4),as considered in section 13.3, and regardless of the extent to which the apparent ‘initialconditions’ of the universe might be constrained, life will find a way if any solution forthe structure in figure 14.5 exists, no matter how difficult or how remote the possibilityof such a solution might seem to us.Conscious life draws itself into being, through a self-supporting system, withinthe constraints of the mathematical form of the physical world it engages with. This isnot necessarily a straightforward feat to achieve, in the sense of the non-trivial math-ematical and physical structures required. Indeed, the fact that our ability to physi-cally experience the world relies on the support of a human body which is enormouslycomplex on the scale of the fundamental laws of physics (gravitational and quantumparticle) is itself evidence of the difficulties of embedding the physical manifestation ofa conscious life within a physical world constructed within the constraints imposed bythe underlying mathematical progression in time experienced by the conscious beingsthemselves.As a solution to the cycle of figure 14.5 the physical world of node (4) canbe described in the mathematical terms of the full 4-dimensional spacetime structure G µν = f ( Y, ˆ v ) as moulded in conformity with the simple state of a moment of timein node (1). Alternatively the structure of the universe can be described effectively inthe physical terms of a dynamic evolution from an apparently initial state at t = 0,as discussed for figure 14.8. Expressed in this latter way the physical development ofthe organic form of conscious beings out of a comparatively far simpler physical stateat the apparent temporal origin of the world, according to precisely determined lawsof physics, not surprisingly requires a relatively long period of biological evolution ona planet such as the Earth in a stable orbit around a star such as the sun. Hencethe fact that we find ourselves in a universe at a spacetime location such that the sunis 149 million kilometres away in space and the Big Bang is 13.8 billion years awayfrom us in our temporal past, as depicted in figure 14.8, have similar explanations:both are required of our physical environment in order that we, conscious beings, canconsistently exist here and now . The observed vastness of the cosmos that surrounds usbeyond the solar system is in some sense a byproduct arising from the non-triviality ofrealising a solution for the overall structure in figure 14.5, albeit a byproduct which isentirely ‘weightless’ from a mathematical point view as described towards the openingof section 14.2.Analogous observations would apply to worlds other than our own, drawn intoexistence as a solution for the general form of figure 14.5, insomuch as it would seemsurprising for a ‘simple’ solution to exist. The question concerning the uniquenessof our world, as considered in section 13.3, requires consideration of other worlds458hat could be created by and through other self-reflective beings. Since the 1950sphilosophers in this world have sometimes enquired “what is it like to be a bat?”,which is very difficult to answer since, amongst other things, bats and humans havedifferent forms of sense perception. This kind of question becomes yet much harder ifwe attempt to enquire “what is it like to experience a different possible world to ourown?” Here we refer to a different world with different laws of physics and perhapseven a base manifold with an intrinsically highly non-Euclidean geometry or a differentdimensionality to ours.All the varieties of other possible worlds with different laws of physics stillhave significant features in common, assuming they fall within the general frameworkdescribed in this paper, involving a multi-dimensional form of temporal flow. Thefull form L (ˆ v ) = 1 may in a strict sense represent the greatest possible dissolving ofthe temporal flow via an infinite dimensional channelling through ˆ v ∈ R ∞ , as alludedto towards the end of section 13.3, and hence be unique for all worlds. Symmetriessuch as E acting on the form L ( v ) = 1 with v ≡ X ∈ h O or E acting on theform L ( v ) = 1 with v ≡ x ∈ F (h O ) may also represent significant mathematicalresonances which dominate the actual physics observed in any universe. The physicallaws themselves are then effectively determined in breaking the full symmetry, forexample through the identification of a subgroup acting on the subspace of vectors v n ⊂ ˆ v projected onto the tangent space of an n -dimensional base manifold M n . This smallerspace, together with the symmetry group for L ( v n ), is broken out of the larger spaceand symmetry group of L (ˆ v ) = 1 in the formation of a global background manifoldwhich acts as a geometrical reference frame for events perceived in the world. Thisbackground provides the relief against which apparent material objects are brought tothe attention of the self-reflective beings through the laws of physics resulting from thebreaking of the full symmetry group ˆ G .Whether there is only one such kind of world, of which our own would thenbe a particular manifestation, or several, which might even be catalogued, is likely tobe difficult to determine (perhaps even much more so than categorising all possiblebiological life forms given the laws of physics within our own universe, whether onthe Earth or elsewhere). Certainly for any world to be possible in this framework isequivalent to the statement that it must actually exist , and in this case our variety ofuniverse would not be entirely unique. However, we would not be able to communicatewith other worlds, or the creatures living within them, and there is no question ofinterference with the internal consistency of our own world.While the existence of other worlds with different laws of physics is an openquestion, there will be, according to this theory, many possible solutions for a geom-etry G µν = f ( Y, ˆ v ) on a 4-dimensional spacetime manifold M , apart from our ownworld, which share the same laws of physics and will also internally support consciouslife under circumstances similar to those in which we find ourselves on Earth. Thenotion of ‘many worlds’ as an interpretation of quantum mechanics is distinct from,although implicit within, the overall framework presented in this paper, where herewe are referring to the ‘many solutions ’ embedded within the theory, as discussed insection 11.4. Different solutions for G µν = f ( Y, ˆ v ) involve δY ↔ δ ˆ v field exchanges inprinciple anywhere on the spacetime manifold M , even back to the Big Bang epochfor cosmic time 0 < t < t v , and when considered from a dynamic point of view our459niverse in some sense might be considered to have ‘branched’ from another possiblesolution at each quantum event.Such quantum transitions, which are indeterministic from the perspective of asingle universe, taking place in the very early universe might serve to seed the eventualformation of stars, galaxies and large scale structure generally, as alluded to in sec-tions 12.3 and 13.2. That is, in part due to the causal temporal accumulation of suchprobabilistic events, the impact of a quantum fluctuation for t < t v on the overall struc-ture of the universe might generally be far more dramatic than a similar ‘branching’resulting from a ‘Schr¨odinger’s cat’ type experiment performed at the present epoch.While many solutions for G µν = f ( Y, ˆ v ) might be considered to be mutually relatedby such ‘branching’ events, in the present theory each solution is primarily interpretedas an independent full 4-dimensional spacetime solution in its own right.Much of modern science adopts an essentially materialist worldview in linewith our Newtonian heritage. From this objective point of view with the universe seenas a fundamentally material phenomenon, created in the Big Bang as an inanimatephysical entity with the various seemingly arbitrary parameters of cosmology and par-ticle physics, it appears extremely fortunate for us that such a world can both supportbiological life and lead to the development of our own society, culminating in our ownpersonal human form, through a series of chance events. In particular life itself, as weknow it, would be impossible given a small change in any of a range of the empiricallymeasured physical parameters.As usually presented this means that, for example, the laws of physics arerequired to be such that the chemical elements necessary for life on Earth could bemanufactured in the hot Big Bang – which successfully accounts for the relative abun-dances of the light nuclei, D, He, He and Li, cooked up from a hot soup of protonsand neutrons in the first few minutes – together with the much later generation of theheavier elements through stellar nucleosynthesis. The latter stage is possible thanksto a seemingly fortuitous energy level of the carbon nucleus that allows the three-bodyreaction 3 He → C to proceed at a reasonable rate. In 1953, in a famous caseof anthropic principle reasoning, the necessity of this carbon resonance was predicted by Fred Hoyle, to account for our own presence in the world as a carbon-based lifeform dependent on the heavy elements. Given this motivation the resonance was thenexperimentally observed shortly afterwards.However, for the present theory the universe, through the structure of fig-ure 14.5, is born out the intimate interplay between conscious beings and the physicalworld. The complexity of the resulting physical structures within such a solution cre-ates the illusion of the fortuity of our own existence. Due to the non-trivial nature ofsolutions achieving a completion of the time cycle in figure 14.5 any possible physicalworld is likely to appear highly complex, as discussed above. Hence beings in any suchworld are likely to require a number of parameters to describe empirical findings intheir world, as for the Standard Model of particle physics in our world for example.Hence in turn, with the physical support for known biological life forms apparentlycollapsing under a hypothetical change in the empirical parameters, beings in such aworld might consider themselves lucky. Given the familiarity of our own world as astarting point we can readily conceive of many ways in which a physical world couldnot support life, through small perturbations to the properties of our own universe,460ut it is much harder to conceive of very different worlds with very different solutionsfor supporting the structures of conscious life.Hence here there is a major contrast between the present theory and variousforms of the anthropic principle, which are generally subject to criticism due to their lack of predictive power. For example based on the anthropic principle a theory maypostulate the existence of a very large ensemble of different universes with differentinitial conditions, physical constants or laws of nature – then the fact that our universeis necessarily a member of the ensemble in which the structures for life can formnecessarily greatly restricts the possible structure of the physical laws and conditionsthat we can observe. However the potentially vast range of physical properties forthe worlds of the whole ensemble, most of which are presumably not observed by anybeing, is in no way limited by this principle.Here the present theory is ‘anthropic’ to a more extreme extent in the sensethat the only worlds that exist at all are those that can be brought into being througha conscious, temporalising observer, and in this case we may hope to discover the opposite conclusion that the laws of physics are necessarily determined, or at leasthighly constrained, by this requirement (although naturally there will always be themore trivial anthropic matter of the local selection of a habitable environment, suchas the Earth, within such a world). That is, rather than postulating a large ensembleof typically inanimate physical universes with a range of parameters, one of whichhappens to provide a suitable environment for ourselves, we draw our own world intoexistence, sculpting the physical contents of the world out of the possibilities inherentin perceiving a world through the forms of L ( v ) = 1 as a solution for figure 14.5.The number of parameters needed to describe the projection of v ∈ TM outof the components of X ∈ h O or x ∈ F (h O ), involving for example the dilationsymmetries described in the opening of section 13.2, is much less than the number ofparameters needed to describe the empirical data as observed in particle physics andcosmology. Hence the present theory in principle will be highly predictive even if thereare some possibilities for the variation of certain parameters within the mathematicalconstraints of the theory, that is with an anthropic degree of tuning for the parametersinvolved in the projection of v ∈ TM out of L (ˆ v ) = 1. However, in chapter 13 we havegenerally presumed that interactions between the fields as determined by the constraintequations 11.29 result in a fixed and stable value around | v | = h emerging from thephase transition at t = t v in the very early universe, as described for figure 13.3(c).The potential uniqueness of the laws of physics and particle properties arising at thistime suggests the present theory should be profusely testable.Here the perspective is to consider the ‘early’ universe to be an object of studyas a limiting extrapolation from our present experience in the world rather than asan objective self-sufficient physical state that happens to be the causal origin leadingup to the present conditions in our world, as has been described for figure 14.8. Witha solution for the time cycle structure of figure 14.5 taking priority and founding thetheory, not only is nothing needed as a temporal antecedent of the Big Bang to cause the universe to exist, but the particular conditions of the Big Bang and early universeare shaped by the overall consistency of the solution within the structural constraintsimplied in figure 14.5. These constraints on the apparent ‘initial conditions’ of the earlyuniverse are ultimately manifested in the physical and biological processes required to461upport self-reflective life forms at the present epoch, as described above.Here we take, possibly rather indirect, measurements of cosmological structureincluding that for the earliest epochs of the universe, as for laboratory experiments inparticle physics, as being extensions of our world experience – quantitatively differingfrom the nature of everyday experience in the world more generally, but in all casessubject to the same laws of physics and all within the same system. From the basicexperiences of thinking, listening to music, walking down the street and watchingan apple fall from a tree to performing experiments and studying the structures ofbiology, chemistry and physics on all scales, there is a continuity from the notion ofexperience through to, and incorporating, the practice of experimental and empiricalobservations. In the present theory both the notions of scientific observations andsubjective experiences more generally are drawn together and unified as particularmanifestations of experience in time.If the present theory were to be founded on a purely objective notion of one-dimensional temporal flow, as modelled by the real line R , the sequence for figure 14.5could still be constructed linking the nodes (1) , (2) . . . (6) but without the final linkbetween nodes (6) and (1). For such a theory the subjectively experienced time ofnode (6) would hence be derived from a long chain of non-trivial steps (1) , (2) . . . (6)beneath which the fundamental objective temporal entity of node (1) would be verymuch hidden from our immediate view of the physical world, and would not be an entitywe might directly perceive. However, this unnatural duplication of the concept of timein nodes (1) and (6) is avoided through the actual perspective of the present theorywhich has been developed beginning with the notion of subjective experienced time.Indeed temporal flow is not something that we ‘see’ in the world, as is the case even forthe elementary forms of space, rather it is an innate characteristic of our engagementin the world. That is temporal flow is not a property of the world which we need toset out to discover, as for the ‘hidden’ structures of material phenomena or particleinteractions of node (4) for example. Rather we do directly perceive the underlyingtemporal flow of node (1) since it is identified with our immediate experience of timein node (6), hence in turn completing cycle of figure 14.5.The overall system of figure 14.5 is perhaps best understood by thinking throughthe cycle of six nodes and links in turn, beginning from any point, but the structure canbe contracted down in a number of ways including a more minimal scheme describingan interplay between experience and the empirical, or essentially between subjectivetemporal flow and the objective laws of physics as associated with nodes (1) and (4)respectively. Ultimately the full set of six nodes coincide as six facets of the internalstructure of the possibility of conscious experience, conceived as a unified whole, essen-tially adopting the philosophical outlook of existentialism as alluded to in the previoussection. As discussed in the previous section, from this point of view the possibilityof an experience is a more fundamental concept than the individuals who believe theyhave them, and with the laws of physics, which shape both the physical individualand his environment, also determined through the constraints on the possible formsan experience can take within the system of figure 14.5.From the philosophical perspective of materialism, which is grounded largelyin node (4) of figure 14.5, the ‘problem of consciousness’ arises since the concept ofsubjective experiences seems to be of a qualitatively different nature to anything stud-462ed in the realm of the physical world. While from this point of view consciousnessappears mysterious and beyond the reach of the physical sciences, it nevertheless re-mains the case that conscious experience is a very real phenomenon of the world, andindeed it is the feature of the world with which we are most intimately familiar. Hencean inclusive scientific theory should either have something to say regarding the natureof consciousness or provide a good explanation as to why it should not, as suggestedshortly before figure 14.2. On this basis the speculative structure of figures 14.2–14.4has been studied here in section 14.1. One possible justification for not addressing thisquestion regards the complexity of the human brain, being beyond the current scopeof an exhaustive scientific understanding.On the other hand the nature of subjective experience can be very simple, asexemplified by the ‘thought experiment’ involving picking up a pen or pencil as alsodescribed in section 14.1. This suggests that the broad objective physical correlate ofsuch experiences might also be described in terms far simpler than those required togive an account of the detailed structure of the brain. Together with the practical ex-periments of Libet and others discussed in section 14.2 it is clear that the phenomenaof conscious thought are in any case open to study. Indeed research into consciousnessis a scientific field of study in its own right, although one which is not traditionallyclosely linked with physics. It’s relevance for the present theory lies in the close rela-tionship between the nature of consciousness and the structures proposed to completethe cycle of figure 14.5. In return the perspective of the present theory, in which con-sciousness is closely associated with temporalisation and related to the physical worldthrough figure 14.5, might in principle be of value for the corresponding area of studyin neurology, for which a firmly materialist standpoint is commonly adopted.It is suggested here that consciousness is not something that can be fully ex-plained as a phenomenon arising solely within a pre-existing physical world, as wouldbe required from a purely materialist perspective. Subjective experiences cannot bedirectly described in terms of objective matter, but rather correlate with certain math-ematical structures which underlie the physical world within the context of the systemdepicted in figure 14.5. On the other hand the content of the physical world is notfully contained within the horizon of our conscious observations, as might be the casefor the pure idealist. We can conceive of an infinite expanse of the physical world inspace and time beyond the horizon of our direct experience as supported by the fullmathematical solution for G µν = f ( Y, ˆ v ) implied in the structure of figure 14.5, whichitself provides the context for all the structures of cosmology. Conscious experience isan irreducible feature of the world and the means by which a mathematically possibleuniverse is realised through the intimate interplay between the subjective and objectiveaspects of figure 14.5.Much of the apparent mystery of ‘consciousness’ owes to the fact that noth-ing exists without its support and hence it is impossible to step back and isolate thephenomenon ‘in itself’. Everything that exists or happens does so within the contextof consciousness, even our awareness of a discussion of consciousness itself, with thephenomena of the physical world ultimately inseparable from the experiences of tem-poralising beings. A theory which, on the contrary, attempts to construct a notion ofconsciousness entirely within the limits of a given independent physical world, implyingthat such a world can ‘exist’ even in the absence of such sentient beings, is necessarily463ealing with an incomplete system. Rather, while also supporting the physical corre-late of conscious mental phenomena, the physical world is itself engulfed within thesphere of conscious experience, as implied in the relations depicted in figure 14.5.For the above materialist worldview in addition to the difficulty in constructingan explanation of consciousness upon a given physical world, as alluded to also for theright-hand end of figure 14.1 and discussed more generally in section 14.1, on the otherhand there remains the second major loose end regarding a foundation for the physicaltheory itself.An appeal to ‘beautiful mathematics’ is often made either explicitly or implic-itly as a significant motivating force in theoretical physics, promoting a sense thatnature ‘ought’ to make use of aesthetically pleasing mathematical structures. Whilesome successes may be cited, notably for example regarding the Dirac equation fora fermion field (quoted here in equations 3.99 and 11.31 with a gauge field interac-tion included), the achievements of this approach, in terms of discovering empiricalphenomena that match a beautiful mathematical theory (applied in particle physicsor cosmology), have been particularly limited in recent decades. This approach alsohas serious philosophical difficulties, regarding not least the highly subjective notionof ‘beautiful mathematics’ itself and the means through which physical entities in theworld should relate to the mathematical components of the theory.Alternatively an objective physical theory might be founded upon a concep-tual idea regarding the nature of an inanimate physical world, which will subsequentlybe formulated and developed in mathematical terms in order to derive testable conse-quences for the theory. Examples of this approach include the description of gravitationin terms of a curvature of 4-dimensional spacetime in general relativity, or the prop-erties of discrete particle-like entities interacting in a flat spacetime. However it isdifficult to conceive of any physical concept which does not itself stand in apparentneed of a further underlying explanation. Progress may be proposed, for examplewith gravitation and the geometry of our world in 4-dimensions arising out of a morefundamental higher-dimensional spacetime or with particle phenomena deriving froma field theory, but at some point the basic physical entities, together with perhapsa Lagrangian formalism or a quantisation procedure, is essentially ‘postulated’ as anapparently necessary starting point.The foundations of such a theory can be justified provisionally on the groundsthat ‘one has to start somewhere’, as alluded to in the opening of section 14.1, providedthe theory satisfies a criterion of empirical success. Based upon that success we learnwhat a more fundamental theory should effectively look like in a certain environmentor under certain limiting conditions, such as those for general relativity or quantumfield theory as described for table 11.1 in section 11.4. Whether an objective physicaltheory is founded chiefly upon mathematical, conceptual or empirical grounds (and inpractice in some combination) the foundational loose end is generally accompanied byquestions concerning the nature of the origin of the universe in the Big Bang, which isneeded in order ‘to get the ball rolling’ in the first place, as summarised in point (1)in the opening of section 14.1.The approach of the present theory, with respect to the two loose ends offigure 14.1, is to fully embrace the subjective element of our engagement in the world.With all experience in the world having a temporal aspect the theory is founded purely464n the notion of a one-dimensional flow of ‘time’ as a necessary component of boththe subjective and objective world. Since time is a feature of the world, which weexperience directly without any intermediate interpretation, this offers an extremelyconservative starting point for a theory. To be aware of anything at all is to experiencean irreducible moment in time, as a basic aspect of thought and experience generally.With all thinking having a necessarily temporal dimension we have essentially retreatedto the minimal observation that, with a twist on the famous words of Descartes, ‘I thinktherefore I temporalise’. This provides the mathematical basis for a full physical theorywhich supports the entire structure of the universe as perceived in an experience itself.In its simplicity this starting point is largely devoid of any arbitrary aspects, unlikethe case for most theories which are motivated on mathematical or conceptual groundswhich are purely objective.Through the dual subjective and objective nature of time, both modelled on thesame mathematical real line, this theory can ultimately also supply its own foundation,tying up the two loose ends of figure 14.1 in the shape of figure 14.5. Although here thetheory is motivated from the direction of a conceptual argument, based upon temporalflow, rather than from the direction of ‘beautiful mathematics’, the mathematicalstructure represented in figure 14.5 has itself a degree of elegance in its simplicityand self-contained nature. However instead of beginning with mathematical beautytogether with the presumption of its necessary application to the physical world, herethe realisation of the mathematically elegant structure described in figure 14.5 containsits own inevitability , in that it incorporates both self-reflective intelligent entities andits own foundation.Further, this structure provides a context within which an entire universe, asdepicted for example in figures 13.6 and 14.8 and supported by a spacetime manifold M of infinite extent, forms part of the overall solution, hence incorporating all featuresof the physical world, from the microscopic to cosmological scales, including the BigBang and events in the arbitrarily distant past. Although the subjective aspects arenecessary to conceive of the whole system and help motivate the initial foundation ofthe theory in terms of the flow of time, the structures contained within nodes (1)–(4)of figure 14.5 can be essentially treated as an objective physical theory, as has been thecase for the large majority of the work presented in this paper, which may be measuredagainst observation in the empirical world as for any other theory.While the simplicity and elegance of the mathematical structure of nodes(1)–(4) might itself be considered, in order to fully justify the present theory not onlyon conceptual grounds but also from the perspective of the mathematical elegance offigure 14.5 as a whole a more rigorous mathematical account of the lower half of thechain through nodes (4) → (5) → (6) → (1) might be desirable. However, all elementsof the cycle are open to such an exploration, and in section 14.1 we described a possibleapproach to uncovering a mathematical correlate of self-reflective subjective thoughtsand decision making.There we also noted a close analogy between the mathematical structures re-lating to G¨odel’s notion of decidability and the properties of physical devices relatingto Turing’s notion of computability. Following Turing and the ambition to developartificial intelligence it is conceivable to attempt to build a machine exhibiting theproperties self-reflective conscious experiences and creative thought. The design of465uch a machine might include a complicated arrangement of malleable and adaptableelectronic, and even biological, components capable of internal development, as wellas an array of sensory input devices and means of interacting with the environment.Given the design on paper, for the machine to actually ‘exist’ it would then need to bebuilt, requiring the physical assembly of the necessary technological components. Onlywhen manufactured in this way could we declare, in the words of Dr. Frankenstein,that “it’s alive!”.If the machine could think and have experiences in a similar way that we do,it might also ask itself how the physical universe and its place in the world came intobeing, and might also be drawn to a conclusion in the form of the system describedin figure 14.5. For the case of this artificial intelligence the full physical environmentmust include not only our biological evolution but also the particular human inventorsand technicians with the ability to design and construct the machine.On the other hand if we consider directly the purely mathematical constructionof self-reflecting elements relating to G¨odel’s theorem or a similar theoretical struc-ture, rather than taking the computing route of Turing, the conclusion is somewhatdifferent. In this case we might design a particular mathematical system capable ofdescribing self-reflective states and which also contains its own foundation as sketchedin figure 14.5. This mathematical structure, as for any logically possible mathematicalconstruction, is in principle a free creation for our mind to think about abstractlyand objectively from an independent point of view. While we can discover such alogically coherent structure in this case any ambition to build such an entity wouldbe meaningless (unless it could be mapped onto the design of a practical machine asdescribed above). However, since the kind of structure depicted in figure 14.5 has thecharacteristic that it contains thoughts and experiences of internal elements all withinthe same structure together with its own foundation it is in the nature of this mathe-matical system to spontaneously realise its own existence, detached from any externalsupport. The contention here then is that our own experiences in our own universeare a particular manifestation of precisely such a self-illuminating world.466 hapter 15 Towards a Complete Theory
The underlying unifying principle for the theory is simply the observation that ev-erything takes place through progression in time. Based upon this principle in thispaper we have explored the extent to which the empirical phenomena of the physicalworld might be accounted for. In the previous chapter we have described how physicalstructures in the world might themselves inscribe subjective experience of progressionin time and hence act as the source of temporalisation itself. Regarding the generalstructure of the theory, we first summarise here the main novel ideas presented as thefoundation for the physical world as described in detail in the preceding chapters.The mathematical possibility of a multi-dimensional flow in time is expressedthrough the general mathematical form of progression in time L ( v ) = 1 as derivedfor equation 2.9. The creation of an extended spacetime manifold out of the flow oftime is possible through an innate subjective interpretation of a subset of the algebraicstructures incorporated within L ( v ) = 1 in terms of a geometrical representation.This spatialisation of the world is considered a subjective phenomenon insofar as itis through it that experience of a physical world by sentient beings is possible. Thedescription of the geometry of the resulting extended external spacetime is identifiedwith that for general relativity, as applying for all physical scales.Since the extended frame for perception is constructed out of a substructureof the full form of temporal flow described by L (ˆ v ) = 1 a natural mechanism forbreaking the higher, unifying, symmetry of time arises. Non-gravitational fields andinteractions are induced on the spacetime manifold through the residual componentsof the full form and symmetry of L (ˆ v ) = 1. The possibility of a degeneracy of solutionsfor the external spacetime geometry underlies the phenomena of quantum theory andparticle physics. The breaking of explicit full symmetry groups for candidate formsfor L (ˆ v ) = 1 over the 4-dimensional spacetime base space is found to yield structuresclosely correlating with features of the Standard Model of particle physics.A significant novel feature of this theory is that the spacetime manifold is notpostulated as a starting point, rather it is grounded as a possible structure within the467ulti-dimensional flow of time, arising out of the translation symmetry inherent inthe form L (ˆ v ) = 1. In other background-free theories one main difficulty is to explainthe origin of such an extended spacetime structure. Hence most theories employ apre-existing 4-dimensional manifold, or a higher-dimensional spacetime arena in whichto embed the former, and then introduce fields or other mathematical entities uponthe manifold. Since here we extended the symmetry group of L ( v ) = 1 to act on ahigher-dimensional form of time, absorbing the 4-dimensional one, these ideas couldalso be considered as a theory with extra dimensions . However, here they are not extradimensions of a spacetime, although algebraic forms or symmetries which also havesuch a geometrical interpretation (including the temporal form L ( v ) = 1 with thesymmetry SO + (1 ,
9) considered in figure 5.1 for the model of section 5.1) may happento arise in the mathematics. On the other hand this theory can also be conceived asa rather more economical approach with fewer dimensions , in that the world emergesfrom a one-dimensional progression of time.The physical theory presented in this paper, based on the notion of a fundamen-tal underlying progression in time taking the general form L ( v ) = 1, has progressedalong four main fronts, as depicted in figure 15.1. In this concluding chapter these the-oretical developments are summarised along with a discussion of how they are mutuallyrelated and might be combined together in progressing towards a complete theory. L ( v ) = 1 ✲✛ ✻❄ (2) E Action on F (h O ) L ( v ) = 1, D µ L ( v ) = 0(Standard Model) (1) Isochronal Symmetry G µν = f ( Y ), G µν ; µ = 0, on M (Kaluza-Klein Theory) (3) Many Solutions δY ↔ δ v Redescriptions(Quantum Field Theory) (4)
Large Scale Structure G µν = f ( v ) and Generalised(Standard Cosmology) Figure 15.1: Developed from the original underlying notion of the primary role oftemporal flow these four areas of progress (1), (2), (3) and (4) have been described indetail in this paper in chapters 2–5, 6–9, 10–11 and 12–13 respectively. (In each casethe main guide from established physical theory is appended parenthetically).468he four fronts of the theory described in figure 15.1 contain aspects of the in-terplay between the various forms of the flow of time considered, from one-dimensionaltemporal causality itself up to the largest form L ( v ) = 1, the full symmetry ofwhich is broken over the base manifold M . Individually these four fronts exhibit thefollowing principal features:(1) Motivated by the notion of perception over a 4-dimensional base manifold M fourextended external dimensions are initially identified through translation symme-tries of the full form L (ˆ v ) = 1. Subgroups of ‘rotational’ symmetries of L (ˆ v ) = 1imply the identification of gauge fields on M relating to both the external andinternal geometry and the unifying framework of a principle fibre bundle for gen-eral relativity and classical gauge theory can be constructed. With the externaland internal geometry correlated as the full symmetry of L (ˆ v ) = 1 is broken inthe projection over M this structure, with the four external dimensions iden-tified as above rather than with the ‘extra’ dimensions being ‘compactified’, isreminiscent of non-Abelian Kaluza-Klein theories.(2) Motivated by its mathematically rich structure out of the infinite possible formsof L ( v ) = 1, a 56-dimensional form of temporal progression L ( v ) = 1 witha high degree of symmetry is identified through the action of the group E inpreserving a quartic form defined on the space F (h O ), containing the determi-nant preserving action of E on the space h O . When broken over the external M base manifold the residual internal gauge group contains features of thesymmetry SU(3) c × SU(2) L × U(1) Y acting upon components of F (h O ), includ-ing subspaces identified as spinors under the local external Lorentz symmetrySL(2 , C ) with charges under an internal U(1) Q symmetry, which are reminiscentof the Standard Model of particle physics.(3) Conforming with the underlying one-dimensional causal flow of time the degen-eracy of field solutions for the world geometry G µν ( x ), consistent with the brokenform of temporal flow expressed dynamically on the base manifold via expres-sions such as D µ L ( v ) = 0, selection rules for exchanges between gauge Y ( x ) andspinor ψ ( x ) fields may be obtained. This leads to interaction phenomena with amathematical structure reminiscent of calculations employing the time evolutionoperator U ( t, t ) in a quantum field theory based upon a given Lagrangian.(4) In constructing the base manifold M out of the full form L ( v ) = 1 and itssymmetries variation in the magnitude of the projected subspace vectors v ( x ) ∈ TM , with | v | = L ( v ) = h ( x ), itself generates a non-flat external geometry.The general solution for the 4-dimensional geometry G µν = f ( Y, ˆ v ) might alsoincorporate a cosmological term in principle deriving from the scalar componentsof F (h O ). Collectively the resulting large scale structure of the cosmos maycorrelate with the observed phenomena of the dark sector and properties of thevery early universe, that is in a manner reminiscent of the standard cosmologicalmodel and inflationary theory.Hence the theory represents new directions of research in fundamental physicsbranching into several areas. At the same time the main part of this work sits comfort-ably within the existing infrastructure of theoretical and experimental physics. The469athematical framework has been adopted entirely from that used in much of con-temporary theoretical physics, with the novel input more in the nature of the overallconceptual picture.The essential theoretical ingredients to account for the Standard Model of par-ticle physics and large scale cosmological structure, while sidestepping the Lagrangianformalism and also providing a conceptual basis for the ‘quantisation’ of the fields, arein principle all found in the structures of the present theory. All four of the above frontsare directly related to consideration of the basic idea expressed in the general form oftemporal flow L ( v ) = 1, and are mutually related to each other. The immediate futuredirection and main aim for further study on each front is first summarised here:(1) Use the mutual relationship between the external and internal curvature in orig-inating from symmetries of the same full form L (ˆ v ) = 1 projected over M ,described in terms of the differential geometry of the structure of a fibre bundle,to derive the relation G µν = f ( Y ) in the form of equation 5.20 without anyexplicit application of an action integral such as equation 5.18 as adapted fromKaluza-Klein theory.(2) Determine a higher-dimensional form of temporal flow and corresponding sym-metry to build upon the features of the Standard Model identified in the actionof E on L ( v ) = 1 when broken over M as summarised in equation 9.46. Forexample a presently hypothetical E action on a full form L ( v ) = 1 might besought, the structure of which will be guided by fields and interactions of theStandard Model Lagrangian yet to be accounted for.(3) Use a statistical approach to HEP phenomena with probabilities based upon fielddegeneracy, building upon the relationship with quantum field theory describedfor equation 11.46 and possibly employing the analogy between the propertiesof condensed matter systems and QFT, to develop the theory through to thecalculation of cross-sections and the identification and conceptual understandingof particle states without imposing quantisation rules.(4) Build upon the geometry G µν ( x ) of equation 13.4, deriving from a variation of themagnitude L ( v ) = h ( x ), to a full general form G µν = f ( Y, ˆ v ) incorporating alsoscalar fields and applied for the large scale structure of the universe, in order tomake a more quantitative comparison between the present theory and empiricalobservations in cosmology; with one aim being to deduce which scenario, such asthat in figure 13.4(a) or (b), applies for the very early universe.The main prediction of the theory at present is a mathematical one concerningthe existence of an E symmetry acting upon a quintic or higher order form L ( v ) = 1as alluded to in front (2) above. This structure, as an extension from the E actionon L ( v ) = 1, when broken over M should incorporate further Standard Modelproperties such as three generations of fermions, as motivated in detail in section 9.3.More generally the overall aim is to fuse the above four areas together in a full unifiedtheory, and assess the consequences and possible predictions of the theory that canbe further compared with and tested against empirical data from HEP experiments,cosmology and other observations. We begin here by observing the following relationsbetween the four theoretical branches summarised in figure 15.1.4701+3) The key motivation for front (1) is the identification of a smooth external ge-ometry G µν ( x ) on M as an arena for perception in the world. Since there isno similar requirement regarding the need for a ‘smooth’ internal geometry ofgauge fields it would be more natural to begin with the structure of fronts (1+3)combined, as implied in the relation G µν = f ( Y, ˆ v ) as a possible solution forthe world geometry on M . A finely fragmented and fractal-like structure offield exchanges δY ↔ δ ˆ v underlies the smooth external spacetime arena, with G µν ; µ = 0 maintained as a geometric identity. In this way the degeneracy ofmany possible solutions brings the phenomena of general relativity and quantumtheory together at the same time in the process of identifying the base manifolditself, rather than beginning with a ‘classical theory’ of the form G µν = f ( Y )which is then ‘quantised’.The relation between the initial theoretical ‘bare’ fields and empirically observed‘dressed’ fields was also described in the opening of section 11.3. Indeed, a ge-ometrical relation of the form G µν = f ( Y ) might still be identifiable for macro-scopic fields, such as the empirically observed electromagnetic field. Out of thecomplete framework the standard theories alluded to parenthetically for fronts(1) and (3) in figure 15.1 may be shown to emerge in the appropriate limits:namely Kaluza-Klein theory in a curved spacetime as an example of the macro-scopic field limit of general relativity and QFT in the limit of a flat spacetimefor microscopic fields, as described for table 11.1 in section 11.4.(2+4) In the present theory the phenomena of electroweak symmetry breaking and inparticular the masses of particle states observed in the laboratory arise out ofinteractions between the components of the vector-Higgs field v ( x ) and otherfields such as the fermions ψ ( x ) identified in the components of F (h O ) throughthe terms of the quartic form L ( v ) = 1. On the other hand cosmological struc-ture depends on variation in the magnitude | v | = h ( x ) as v ∈ TM is projectedout of the full form L ( v ) = 1 over M , which itself provides a geometric ex-planation of the origin of mass in terms of an effective energy-momentum tensordefined in − κT µν := G µν = f ( v ) = 0. Hence these two notions of mass areintimately related via the field v ( x ).The dilation symmetries, acting on the components of F (h O ) as discussed in theopening of section 13.2, change the value of | v | and may be significant in relationto the mechanism of electroweak symmetry breaking in the very early universe.The physics of the very early universe may also guide the identification of ahigher-dimensional form of time, such as the hypothetical L ( v ) = 1 with E symmetry. In particular the mechanism for generating a matter-antimatter asym-metry might be determined by interaction terms implicit in the form L ( v ) = 1or involve a further internal gauge field deriving from the E action, as also dis-cussed in section 13.2. Hence the structure of the full form L (ˆ v ) = 1 is closelylinked with an understanding of significant questions in cosmology.(1+2) In equation 6.3 of chapter 6 the generators of the symmetry of a 27-dimensionalform of L ( v ) = 1 were introduced as operators that annihilate the cubic normdet( X ) with v ≡ X ∈ h O . A complete basis for this 78-dimensional Liealgebra of E , as represented by vectors of the tangent space ˙ R ∈ T h O , is listed471n tables 6.6 and 6.7 at the end of section 6.5. Such a ‘static’ generator can bepulled back to a Lie algebra valued 1-form Y µ ( x ) on M , as initially describedin subsection 2.2.3, and appears in ‘dynamic’ expressions on the base manifold.Kaluza-Klein models based on fibres identified with homogeneous spaces werereviewed in section 4.3, and might provide additional insight in comparison withthe closely related theories constructed on principle fibre bundles described insections 4.1 and 4.2.With regards to the model described for figure 5.1 in section 5.1, with the fullsymmetry group SO + (1 ,
9) acting on the form L ( v ) = 1 over M , the structureof the Lie algebra for SO + (1 ,
9) can itself be expressed in terms of vector fieldson the space of 10-dimensional vectors v ≡ X ∈ h O with det( X ) = 1, basedon the opening of section 6.3. With h O ⊂ h O embedded as a subspace a closeconnection is made with the above case for E acting upon the homogeneousspace composed of vectors v ≡ X ∈ h O of unit determinant. The E actionon F (h O ), broken over the 4-dimensional base space M , represents a higher-dimensional extension of this structure, while the full form of L (ˆ v ) = 1 thatprovides the actual setting for a description of the real world is open to furtherinvestigation. Hence branch (1) relates to branch (2) of figure 15.1 essentially inthe choice of L (ˆ v ) = 1 and the corresponding full symmetry group over the basemanifold M .(2+3) Taking the example of the E case, the generators of the internal symmetry action˙ R ∈ T F (h O ) give rise to the gauge fields Y µ ( x ) on the base space while the com-ponents of v ∈ F (h O ) are also intimately related to the base manifold throughthe translation symmetry over x ∈ M as originally described for figure 2.2.Hence since F (h O ) forms the representation space of E the gauge fields Y µ ( x )naturally couple with components of v ( x ), including the spinor fields ψ ( x ).The dynamics of the interaction between the components of v ∈ F (h O ) andthe gauge fields, under the constant form L ( v ) = 1, is subject to the constraint D µ L ( v ) = 0, expressed through the covariant derivative D µ ∼ ∂ µ + Y µ (as forthe E example in equation 11.33). In this way interaction terms similar in formto those introduced for L int in the Lagrangian approach for the Standard Modelare identified. Arising from symmetry breaking over the base manifold M thepossible δY ↔ δψ exchanges of field components are also constrained by the setof degenerate solutions under the same local external geometry G µν = f ( Y, ˆ v ).All observed fermion states interact with at least one gauge boson via terms of D µ L ( v ) = 0, as applied for the electron self-energy interaction in figure 11.12(b)for example. Hence the external geometric structures relating to the ψ ( x ) com-ponents will be shaped by the bare gauge fields such as A µ ( x ) with which theyinteract. With the bare gauge fields subject to G µν = f ( Y ) from the isochronalKaluza-Klein relation the physical fermion particle states will emerge throughmodifications to the geometry G µν ( x ) due to δY ↔ δψ interactions. In turnthe question of the form of G µν = f ( ψ ) for electron, muon and further particlestates might be considered. This form of solution should also extrapolate to thenon-relativistic limit, such as for the implied electron state linking S and A infigure 11.13(b) for example. 4723+4) Given also the non-trivial geometry G µν = f ( v ) from L ( v ) = h ( x ) variationthe implications of further field interactions of the form δ v ↔ δψ under theconstraint L ( v ) = 1 will also contribute to the form of G µν = f ( ψ ). Theseinteractions with the vector-Higgs field v ( x ) are expected to relate to the originof fermion masses, with the details giving rise to the mass difference between theelectron and d -quark states for example. In order to investigate the mass dif-ferences between the three generations of fermions, such as between the electronand muon, a higher-dimensional form such as L ( v ) = 1 may be required. Theequality of the empirically observed electric charge across the generations mayrelate to the role of ‘Ward identities’ in the QFT limit.With the relation G µν = f ( v ) generalised for multiple solution field exchangesunder the form G µν = f ( Y, v ) essentially all matter T µν := G µν is expected tobe associated with quantum phenomena, with the variety material forms observedin the universe shaped according to the probabilistic nature of the underlying fieldcomposition. The relative probabilities of local solutions for G µν = f ( Y, v ) aredetermined through a ‘number of ways’ statistical count of the underlying fieldredescriptions, essentially as for the determination of probabilities for classicalsystems. This leads to a unified approach to quantum and classical thermody-namic properties, which in particular will be significant for studying the evolutionfrom t = 0 to the phase transition at t = t v , as the stable value L ( v ) = h isattained in the very early universe, as described for figure 13.3. This may alsomark an epoch of fermion production via δ v ↔ δψ exchanges as the propertiesof the Standard Model of particle physics emerge in the phase transition.(4+1) While we have considered beginning with the classical geometric relations G µν = f ( v ) or G µν = f ( Y ) more generally these two means of obtaining finite externalcurvature will be combined in a general solution for G µν = f ( Y, ˆ v ). In the fulltheory field interactions of the form δY ↔ δ v , resulting from the action ofthe corresponding gauge symmetry on the external components v ∈ TM , willrelate closely to the identification of gauge boson masses and the phenomena ofelectroweak symmetry breaking generally.In principle the theory might rather begin with the full general form of G µν = f ( Y, ˆ v ), fully incorporating quantum phenomena and completing the programdescribed for fronts (1+3) combined above, as will be required to fully accountfor both the large scale structure in cosmology and the phenomena observed inthe HEP laboratory. While the pure ‘bare’ forms of the relations G µν = f ( v ) or G µν = f ( Y ) may not be found in nature, due to the possibility of underlying fieldinteractions, each of these relations may play a role in an appropriate classicalfield limit.Hence the aim from the developments in figure 15.1 is to generalise from (1)the geometric structure of gravitational and gauge fields deriving from the isochronalsymmetry of L ( v ) = 1 to incorporate interactions with the field components of (2) v itself subject to the dynamic relation D µ L ( v ) = 0 derived from the action of thefull symmetry of E on F (h O ) broken over M , taking into account the impact of (4)variation in the projected value of | v | = h ( x ), to arrive at a general form of solutionfor G µν = f ( Y, ˆ v ) over (3) a degeneracy of ‘quantum’ field redescriptions underlying473n external geometry with G µν ; µ = 0 everywhere, which itself provides one of theconstraint equations 11.29.Collectively progress on fronts (1), (3) and (4) of figure 15.1 can be consideredtogether under the ambition of accounting for the empirical properties of a quantumfield theory without applying standard quantisation rules for the present theory. Thesethree fronts all relate to the identification of a smooth geometry G µν ( x ) constructedin terms of fields extended on the spacetime manifold M , the identification of which,as the background for perception in the world, itself motivates this construction. Thisarea of research, guided by the analogy between QFT and condensed matter systems,might proceed based on a provisional assumption for the full symmetry of the full form L (ˆ v ) = 1 such as the E case.In fact for this purpose a yet simpler, but non-trivial, model could be consideredbased on ˆ G = SL(3 , C ) as the full symmetry of time acting upon elements ˆ v = v ∈ h C such that L ( v ) = det( v ) = 1 is invariant. This structure incorporates a subgroup ac-tion SL(2 , C ) ⊂ SL(3 , C ) on the subcomponents of v ≡ h ∈ h C , identified with theexternal tangent space TM , as described for equation 7.35 at the end of section 7.1.The structure of the resulting symmetry breaking to SL(2 , C ) × U(1) ⊂ SL(3 , C ) overthe base manifold M may be sufficient to study a model accommodating both gen-eral relativity together with a form of quantum electrodynamics deriving from theinternal U(1) symmetry. On generalising from the complex space C to the octonions O the symmetry action SL(3 , C ) is itself contained as a subgroup of SL(3 , O ) ≡ E as explicitly demonstrated by the generator composition of equations 8.32 and 8.33in subsection 8.3.1. In this way the form L ( v ) = 1 naturally takes its place in theprogression L ( v ) → L ( v ) → L ( v ) → L ( v ) discussed in section 13.3.Independently of combining the above three fronts, that is (1), (3) and (4),further progress may be made on the structure of front (2) itself which, although thesubspace of vectors v ∈ h C is associated with the external spacetime, considersthe symmetry structure of L (ˆ v ) = 1 without explicitly projecting the componentsinto fields over M . This further study concerns, for example, the explicit identifica-tion of an internal SU(2) L × U(1) Y ⊂ E subgroup together with a determination ofsin θ W and the study of electroweak properties within the theory based on the form L ( v ) = 1. However the larger ambition for front (2) will be the identification ofthe full general form of temporal flow, involving for example an E symmetry of thecurrently hypothetical form L ( v ) = 1. The progression of table table 9.1 and theknown structure of equation 9.50 together with the general discussion of section 9.3strongly hints towards the real form E − as a candidate to be sought for such a fullsymmetry.A more thorough understanding of quantum phenomena in spacetime and adetermination of the full form of L (ˆ v ) = 1 are hence the two main branches to bepursued en route to the formation of a complete theory incorporating all four frontsof figure 15.1, with the aim to account both for cosmological observations and theproperties of the Standard Model of particle physics through the structure of L (ˆ v ) = 1,and without introducing a Lagrangian or any other arbitrary postulates for any pointof the theory. 474 For contrast with the present theory the general recipe for constructing a standardfield theory is summarised in the following three stages. This involves in particularemploying a Lagrangian, such as equation 3.96 or as described in section 7.2 for theStandard Model, to introduce interactions into the theory in order to describe thephenomena observed in HEP experiments.(a) Together with the Lorentz group for the external spacetime symmetry, a gaugegroup is selected, generally motivated on empirical grounds, to describe the in-ternal symmetry of the model. The field content of the theory, in terms of thefield transformation properties as a choice of the representations of the symmetrygroups, is also determined in order to comply with the findings of experiments.(b) A scalar Lagrangian as a function of the fields is written down, invariant underthe symmetries of the theory, with various caveats on the general form of theterms – for example to ensure the renormalisability of the quantum version ofthe theory. The Lagrangian function is used in conjunction with the principle ofextremal action to determine the equations of motion for the fields.(c) The classical theory can be quantised for example by introducing field operatorsˆ φ ( x ), commutation relations and a Fock space of particle states such as | p i asreviewed in the opening of section 10.3. The framework of QFT is built upon aflat spacetime background as a given entity.From the point of view taken here the introduction of a scalar Lagrangianfunction in item (b) above is conceptually a particularly poorly motivated aspect ofthe standard theory. The roots of the Lagrangian approach originate historically inthe study of classical mechanics for non-relativistic material bodies, reproducing New-ton’s Laws of Motion in a more general framework. Later, further pragmatic progressand empirical success was achieved in generalising this framework to incorporate fieldtheories and also to derive relativistic field equations in the Minkowski spacetime ofspecial relativity. The Lagrangian approach is also employed for the quantised fields ofQFT in a flat spacetime on the one hand, and in general relativity, with the geometric R p | g | Lagrangian term based on the Ricci scalar R for example in equation 3.79, ina curved spacetime on the other hand.However there is no underlying conceptual justification for the invention ofsuch a scalar field, the integral of which over a set of spacetime coordinates shouldbe stationary under field variations, either for a classical or quantum theory. In theQFT for the Standard Model it is the empirical observation of the effects of localgauge groups through their representations on apparent particle multiplets that guidesthe construction the Lagrangian, taylored to generate the desired equations of motion.That the Lagrangian framework should remain valid for a unified theory of quantumphenomena and gravitation is a further assumption built upon an uncertain foundation.By contrast with the Lagrangian approach, in the present theory a fundamentalscalar function which is not only stationary but constrained to a particular scalar valueis readily identified, that is L ( v ) = 1. Although general empirical features, such asthe required rank of a unification group as described in section 7.3, serve as a useful475uide for the study of E and E as a symmetry of time, here empirical details of theStandard Model are uncovered in the structure of the external and internal brokensymmetry action on the components of the spaces h O and F (h O ), as described inchapters 8 and 9. Further, the equations of motion for the fields on M can be derivedpurely as a consequence of the constraints of the theory, which are summarised inequations 11.29. For example Maxwell’s equation 11.26 and the Dirac equation 11.31result from the degeneracy of field solutions subject to the constraints, as described insection 11.1. Hence in contrast to the recipe for a standard field theory listed abovein (a)–(c), the necessary ingredients arise naturally in the present framework as listedbelow:(A) All the main symmetries considered must form a group or subgroup of a symme-try of time, that is of the equation L ( v ) = 1. The Lorentz group is motivated byits pseudo-Euclidean structure as required for external perception, while the in-ternal gauge groups are identified in the breaking of the higher, richer, symmetrysuch as E over the base manifold M . The representations are already essen-tially determined since the Lorentz and E groups are selected by their actionsupon the vector spaces h C and F (h O ) respectively, with the broken internalgauge groups acting upon multiplets of SL(2 , C ) ⊂ E Weyl spinors.(B) Equations of motion are constrained by the fundamental requirement L ( v ) = 1which further implies D µ L ( v ) = 0, as listed in equations 11.29. Further con-straints on the equations of motion for the fields are governed by the relation G µν = f ( Y, ˆ v ), consistent with the Bianchi identities for the external and internalsymmetries. This structure over M naturally arises as required to frame a worldof physical perception, in a geometrical space and time, out of the general formof temporal flow. Field ‘interactions’ are implied at the outset in the form ofthe above expressions over the base manifold, in terms of gauge Y ( x ) and spinor ψ ( x ) fields for example, leading to expressions such as equation 11.33.(C) In the present theory the phenomena of quantisation correspond to the degener-acy of the multiple solutions implied in the expression G µν = f ( Y, ˆ v ), consistentwith L ( v ) = 1, as has been summarised in the previous section. That is, thefields are intrinsically involved in creating the non-trivial geometry G µν ( x ) of thebase manifold itself. It then remains to be described how the particle phenomenaseen in HEP experiments, in particular the nature of the initial and final particlestates, arise out of these field exchanges in spacetime.The non-gravitational fields on M derive from the symmetries and componentsof the ‘extra dimensions’ of temporal flow, in a manner analogous to the employmentof the additional degrees of freedom in theories based on extra spacetime dimensionssuch as Kaluza-Klein theories. Here the equations of motion are simply equationsfor the variation of the mathematical structures which arise as projected onto the 4-dimensional base manifold and parametrised by the underlying 1-dimensional temporalflow. They are not equations of motion for some other body or entity introducedindependently of time itself.The field and particle content of the theory will be determined by the choiceof the full and external forms of temporal flow, here taken to be L ( v ) = 1 and476 ( v ) = h on M with their respective symmetries of E and SL(2 , C ) (with thelatter originally identified as a subgroup of E as described for equation 8.6). Themathematical and conceptual limitations on the choice of these significant forms andthe component normalisation such as h , and hence the observed field and particleproperties induced through the symmetry breaking, were considered in the section 13.3.There questions were raised concerning the uniqueness of the present theory and theextent to which it is constrained given, for example, the possibility of further higher-dimensional forms of temporal flow.Here, with E taken to describe the symmetry of the full 56-dimensional formof temporal flow, for the complete theory the full set of broken L ( v ) = 1 and D µ L ( v ) = 0 terms may be written out. All empirical effects must then be con-sistent with these equations together with the local geometrical forms G µν = f ( Y )and G µν = f ( v ), the latter of which augments the set in equation 11.29, as com-bined globally under the solution G µν = f ( Y, ˆ v ) together with the identity G µν ; µ = 0framing the spacetime manifold. Hence the set of possible field couplings, as expressedthrough causal sequences of degenerate field redescriptions, must conform to this setof equations. These equations, essentially acting as selection rules, are listed in theleft-hand column of table 15.1 alongside examples of possible terms and the associatedfield interactions or empirical effects in the remaining columns.Equations Terms Field Interactions and Phenomena L ( v ) = 1 ∼ vvψψ Yukawa-type couplings for fermion massesinvolving vector-Higgs v components D µ L ( v ) = 0 ∼ vvψY ψ gauge-fermion interactions for internal forcesalso gauge- v coupling for Z , W ± masses G µν = f ( v ) equation 13.4 significant for geometry of dark sectorand evolution of the very early universe G µν = f ( Y ) ∼ F F with F = d Y + [ Y, Y ], equation 3.37, havegauge field cubic and quartic self-coupling G µν ; µ = 0 T µν ; µ ( Y, ˆ v ) = 0 conservation of energy-momentum andconstraint on field equations of motionTable 15.1: The set of constraints in the first column determine the field interactionsand associated field equations of motion, in place of an imposed Lagrangian.The interactions described in the right-hand column bare a close resemblanceto those placed by hand in the Standard Model Lagrangian, however the correspondingfield terms in table 15.1 arise naturally in the present theory. Collectively the con-straints in table 15.1 expressed over the spacetime manifold M replace the need tointroduce a scalar Lagrangian function. With respect to local internal symmetry trans-formations all of the equations in table 15.1 are gauge invariant while they transform477ovariantly under external Lorentz transformations as scalar, vector or tensor repre-sentations. This latter feature, as well as the fact that there are several equations,distinguishes this theory from the scalar Lagrangian approach, and indeed the presenttheory will need to be fully worked out independently of the standard framework.Given a sufficient understanding of how field degeneracy in the present theoryrelates to quantum phenomena it may be possible to deduce effective Lagrangian termsfrom the constraints of the equations listed in table 15.1 and import these structuresinto the framework of a QFT employing a Lagrangian approach. This substitutionof fields and interactions derived from the present theory into the standard proceduresummarised in items (a), (b) and (c) above might be provisionally followed all the waythrough to standard QFT calculations such as cross-sections. However, the alternativeapproach, with the emphasis on a complete understanding of the present theory, wouldbe much preferred in the long term, with the formalism of a QFT Lagrangian lateridentified in a suitable limit of the complete theory.For the present theory the meaning of quantisation itself is to be found inthe degeneracy of field solutions, without following a standard QFT approach suchas attaching creation a † ( p ) and annihilation a ( q ) operators to the field componentsand applying canonical commutation rules. However in the process of calculationthe field couplings arising from the equations in table 15.1 may be associated withvertex diagrams, as was described for a few cases in figure 11.3, as one part of thecorrespondence with Feynman rules described more completely in section 11.2. That is,while the present theory is constructed on a firm conceptual foundation, the empiricalsuccesses of QFT suggests that a complexification of a calculation and the employmentof the mathematical tools of QFT, such as amplitudes and unitary evolution, mightalso be applied pragmatically here. Hence the optimal approach may be to straddleboth perspectives – pursuing the development of the present theory while incorporatingcalculational tools from QFT.Between the macroscopic structure of the external geometry G µν = f ( Y, ˆ v )and the internal microscopic field interaction exchanges, consistent with the equation D µ L ( v ) = 0 for example, nested layers of multiple solutions will shape the physicalmanifestation of the theory in a way reminiscent of ‘renormalisation’ techniques inQFT. While the particle concept and HEP calculations may be motivated from withinthe present theory mathematical tools extracted and adapted from QFT will play animportant role in the development of the complete theory and the establishment of adetailed comparison with empirical measurements.Since the physical couplings and masses measured for HEP phenomena corre-spond to renormalised states it isn’t expected that the full features of the StandardModel should be seen directly in the bare broken terms of E on F (h O ) for example.In QFT the bare Standard Model Lagrangian, with the Higgs field added in a relativelyunnatural way, does mimic the processes of HEP to some extent. For the present the-ory, intended as an underlying fundamental theory, the fact that a number of featuresplaced by hand into the Standard Model Lagrangian have already been reproduced, assummarised in section 9.3, suggests that further specific details of empirical phenom-ena might be uncovered for the complete theory. These empirical details include inparticular the 18 free parameters of the Standard Model as summarised in table 15.2.478M Parameters ψ ↔ v coupling in L (ˆ v ) = 1 termsequations 8.76 and 9.48Gauge Couplings 3 ψ ↔ Y coupling in D µ L (ˆ v ) = 0 termsequations 11.33 and 11.34Higgs Potential 2 v ∈ TM projected from full ˆ v equation 8.72, with | v | = h stableQuark Mixing CKM 4 mass and gauge couplings for 3 generationsmay require ‘E on L ( v ) = 1’Table 15.2: The 18 parameters of the Standard Model and their correspondence in thepresent theory. All essentially originate as couplings implied in L (ˆ v ) = 1 as exemplifiedin the above equation references, including further parameters for the neutrino sector.The QCD θ -parameter, introduced in equation 11.39 and which is consistentwith zero empirically, is not included in the table since the corresponding field interac-tion terms do not arise in the present theory, as described in section 11.1. On the otherhand the new structures presented in this paper may imply new kinds of interactionterms which do have empirical consequences. As well as identifying new processes thepresent theory may be tested through its ability to reproduce the details of knownphenomena through the interactions listed in table 15.1.As noted in that table, these include observations of the large scale structure incosmology, which may relate to variation in the magnitude | v | under G µν = f ( v ).In addition to accounting for the Standard Model particle properties the completetheory would aim to provide a match for the cosmological data, including the densityparameters Ω B , Ω D and Ω Λ introduced in section 12.2, and the structure of the cosmicevolution generally. In particular the Lorentz scalar components α, β, n and N of F (h O ) in equation 9.46, which also transform trivially under the SU(3) c × U(1) Q ⊂ E gauge group while effectively acquiring mass through interactions with the vector-Higgs v under the terms of L ( v ) = 1, may contribute to the dark sector in cosmology, asdiscussed in section 13.1.Other known phenomena are not explicitly expressed in table 15.1. An exampleis provided by the CKM quark mixing parameters alluded to in table 15.2, which can beexpressed explicitly in the Standard Model Lagrangian as described for equation 7.78.In the Standard Model the phenomena of CKM mixing arise for the three generationsof quarks due to the mismatch between the Yukawa and gauge couplings, as describedtowards the end of section 7.2. While fermion masses and gauge couplings arise inthe present theory as indicated in the upper half of table 15.2, the further necessaryingredient of three generations required for CKM mixing may require a further exten-sion to for example an E symmetry acting upon the hypothetical form L ( v ) = 1as discussed in section 9.3. 479urther parameters for three generations of neutrino masses and correspondingmixing phenomena are also needed as a known extension to the Standard Model, andare presumed to have a similar origin as described above for the quark sector in thepresent theory. As also suggested in section 9.3 the SU(2) L internal symmetry mayplay an essential role in distinguishing three generations of fermion states. It will alsobe required to identify neutrino and u -type quark states that transform as SL(2 , C ) Weyl spinors and hence form SU(2) L doublet partners with charged lepton and d -typequark Weyl spinors respectively, which may also involve the identification of a full E symmetry action on L ( v ) = 1.The phenomena of electroweak symmetry breaking arise since the SU(2) L × U(1) Y symmetry action itself also impinges on the components of the external vector-Higgs field v ∈ TM . These interactions of the SU(2) L × U(1) Y gauge fields accountfor the massive nature of the Z and W ± gauge bosons as described in subsection 8.3.3.That is the masses of all particles, fermions and gauge bosons, are here postulated tooriginate through field interactions with the components of v ( x ) ∈ TM rather thanwith a fundamental scalar Higgs field. The large mass of the Z and W ± bosons,of the same order as that of the empirically observed Higgs boson, will need to beunderstood in the context of the present theory. Indeed, the Higgs particle state itselfwill also need to be identified within this theory, echoing the empirical search for theHiggs which concluded successfully in 2012 at the Large Hadron Collider.In the Standard Model the masses for the Z , W ± and Higgs boson can beexpressed in terms of gauge coupling and Higgs parameters of the left-hand side ta-ble 15.2 as described in section 7.2. The scalar Higgs field φ exhibits self-coupling,with terms such as φ † φ and ( φ † φ ) in the Lagrangian potential of equation 7.53, ascontrived to break the symmetry of the vacuum. Within the new approach the scalarHiggs is provisionally identified with the magnitude h ( x ) of the vector-Higgs v ( x ) asprojected onto TM such that the relation L ( v ) = | v | = h is directly identifiedwithin the full form L (ˆ v ) = 1. Cubic and quartic field couplings, within the terms of L ( v ) = 1 and L ( v ) = 1 respectively, involving the components of v (coupled withcombinations of the four scalar fields from the α , β , n and N components of F (h O )for example, as can be seen in equation 9.28), generate an effective potential V ( h, T ),which may be dependent upon an apparent temperature T , as described in section 13.2.For the new approach yet further possible interactions will arise for higher-order fieldexchanges or a higher-dimensional full form of time. An initial unstable value of h ( t )has been considered for the extreme spacetime environment of the very early universeas discussed in section 13.2 in relation to inflationary theory, with the stable value h ( t ) = h achieved at cosmic time t = t v marking a phase transition.In chapters 6–9 of this paper the emphasis has been on the identification ofknown Standard Model properties from within the structure of the present theory, assummarised in the four bullet points and further discussion in section 9.3. The furtherambition is to develop the theory to the point of making new empirical predictionsthat might be tested in existing and future laboratory experiments in particle physicsas well as through observations in cosmology. Such theoretical predictions could beworked out concurrently with the running of the LHC in time to anticipate new effectsthat may appear in the data analysis. The predictions might also influence the designspecifications for the future International Linear Collider.480or the present theory in addition to breaking the full symmetry of L (ˆ v ) = 1through the choice of the projected vector v ∈ TM , with the stable value of | v | = h ,symmetry breaking is also exhibited through the choice of particular components forthe vector-Higgs v in the local tangent space on the 4-dimensional manifold. Thischoice, represented in figure 13.3(c) with exaggerated fluctuations about the meanvalue, is analogous to the choice of component contributions for the Standard Modelscalar Higgs vacuum value in equation 7.54. However, due to the difference in un-derlying structure, differences between the Standard Model Higgs phenomena andpredictions of the present theory might be observable in the laboratory environment.In considering the hypothetical structure of an E action on a form L ( v ) = 1the possibility of identifying the external spacetime vector h ≡ v ∈ TM by fus-ing together a set of two or three right-handed spinors { θ Y L , φ Y L , ψ Y L } ∈ C underSL(2 , C ) ⊂ E was described alongside equation 9.52 in section 9.3. In turn there area number of ways of identifying scalars from the components of the above three spinors,including the scalar magnitude | v | = h . This in principle opens up the possibility ofidentifying additional Higgs-like states, beyond the earlier possible scalar states thatmight be associated with the α , β , n and N components of F (h O ) for the E case. Inaddition to a direct search for such scalar states at the LHC an e + e − collider tuned tooperate as a ‘Higgs factory’ might be sensitive to some of the observable consequences.Since the employment of the three spinors in this way corresponds to the empiricalabsence of a set of three generations of right-handed neutrinos, these structures mayalso impact upon the neutrino sector in a manner beyond the Standard Model.Considered in general terms the extension to an E symmetry itself also suggeststhe possibility of new gauge bosons beyond the Standard Model deriving from theextra SU(2) × U(1) that is appended to the familiar Standard Model symmetry inthe rank-8 decomposition of equation 9.51 in section 9.3. However, the first objectiveis a mathematical one in identifying the predicted E action on a quintic or higherorder form L ( v ) = 1 itself, as highlighted in the previous section, and to assess thefurther extent to which known Standard Model properties might be recovered beforeconsidering additional empirical consequences in great detail. In the meantime thegeneral manner in which particle states might be described from a conceptual point ofview can be further elaborated as we now consider.Under the assumption of a global flat spacetime in the laboratory the Lorentzsymmetry may be augmented to the 10-parameter Poincar´e group and particle statesclassified by their mass m and spin s (or helicity h for m = 0) according to the values of( m ) and ( m ) s ( s + 1) (for m = 0) they take respectively for the two Casimir operators P µ P µ and W µ W µ , where W µ is the Pauli-Lubanski vector. This applies to all particlestates, including hadrons composed of quarks and the Higgs scalar which is presumedto be composed out of the collection of non-scalar field components of the vector-Higgsfield v in the present theory, as recalled above (with an analogous construction fortechnicolor models reviewed in subsection 8.3.3).The four Weyl spinors of equation 8.13 identified in the components of θ insection 8.1 relate to projected components of the larger Dirac spinors, which in turncan be identified within the components of F (h O ) in equation 9.46 under the action ofSL(2 , C ) ⊂ E . The fermions of the Standard Model are Dirac spinors, with differingproperties for the projected left and right-handed Weyl spinor parts as reviewed in481hapter 7. These different properties arise here through the necessarily asymmetricembedding of the vector-Higgs v ∈ TM with respect to the X , Y ∈ h O subspaces of F (h O ) and the resulting asymmetric action of an internal SU(2) L ⊂ E symmetry onthese components in equation 9.46.As described in section 8.2 alongside the U(1) Q symmetry of electromagnetismthe broken E symmetry on the space h O also includes SU(3) c as a pure internalsymmetry, to be associated with massless gauge bosons, the gluons of QCD, in theStandard Model. In subsection 8.3.2 it was described how this U(1) Q symmetry sur-vives the breaking of an SU(2) × U(1) ⊂ E symmetry in a ‘mock electroweak theory’,as a provisional guide towards the identification of an SU(2) L × U(1) Y symmetry withinE or E acting on the full temporal form L (ˆ v ) = 1 in the complete theory.Combining the above external properties under the Poincar´e symmetry withfull set of internal quantum numbers according to the transformation properties un-der SU(3) c × SU(2) L × U(1) Y will lead to a classification of particle states for a morethorough comparison with the Standard Model framework. That the enormous wealthof experimental data in high energy physics all points to a concise and simple table ofa relatively small number of elementary particles, the fermions and bosons, as sum-marised in the Standard Model of particle physics with the 18 parameters of table 15.2above, further motivates the aim to determine such particle properties in the presenttheory by taking a mathematical limit or approximation that mirrors the physicalconditions to be found in such laboratory experiments.In order to make contact with terrestrial laboratory experiments in HEP itwill be necessary to proceed from the ideas presented in this paper through practicalcalculations for processes such as those in figures 10.1 and 11.13 and beyond to moregeneral, and even novel, applications. In the particular case of figure 10.1 out of thegeneral solutions G µν = f ( Y, ˆ v ) over M the emergence of the initial e + and e − states,an intermediate Z boson and the final state particles will need to be described. Outof the annihilation of the particle and antiparticle in the centre-of-mass system infigure 11.13(a) a large number of field transmutations are possible, whether througha photon or a Z boson state, allowing a large number of possible δψ ↔ δY fieldexchanges and further states to be produced. These include the leptonic final statedepicted in figure 11.13(a) as well as the hadronic jets seen in figure 10.1, resultingfrom quark pair production, together with all the particle states within the jets. Theseand further particle phenomena need to be accounted for within the structure andconstraints of the present theory, as has been described in chapter 11.One way of approaching the nature of particle states might be to considerthe simple decay process Z → e + e − via δY ↔ δψ field exchanges resulting in thepropagation of two independent fermions. This would also require an understandingof the Z gauge boson mass in terms of δY ↔ δ v interactions, incorporated intoa solution G µν = f ( Y ) for a massive gauge field with k = m = 0, possessing athird polarisation state ε µ , and which satisfies equation 11.21. Similarly a Higgs decayprocess such as H → e + e − could be studied directly in terms of δ v ↔ δψ fieldexchanges, closely relating to the mechanism for fermion production during the phasetransition at t = t v in the very early universe described in section 13.2.On the other hand a purely QED process might be considered with the electro-magnetic field A µ ( x ) interacting with fermions. Since the photon is massless a possible482pproach would be to take a superposition, or sum, of electromagnetic fields, each inthe form of equation 11.6, mimicking the situation of a two-photon collision and henceable to produce fermion pairs, as alluded to near the opening of section 11.3. Thenature of a single intermediate photon state, effectively with k = 0, in the centre-of-mass frame of an e + e − collider might also be considered. The production of fermionswould be required to proceed through field redescriptions of the form A µ ↔ ψγ µ ψ , asinitially discussed for figure 11.2, consistent with the constraint equations 11.29 undera geometric solution for G µν = f ( Y, ˆ v ).A consistent normalisation of the fields will be required in field exchanges ofthe form A µ ↔ ψγ µ ψ , under the local geometry G µν = f ( Y, ˆ v ) with G µν ; µ = 0, linkingexternal and intermediate field states. This will relate the C coefficient and polarisa-tion vectors ε µr ( k ) for the electromagnetic field, as introduced in equation 11.6, to thespinor coefficients for a Dirac field ψ ( x ). In the standard theory there are four inde-pendent solutions to the free Dirac equation labelled by the 4-component coefficients u , ( p ) and v , ( p ), with for example ψ ( x ) = u ( p ) e − ip · x which may be normalised bykinematic factors of energy and mass (see for example [70] sections 3.3 and 5.2). Sim-ilarly in the present theory the coefficients of the electron field ψ ( x ) for example willcontain energy p and mass m factors which will need to match those for the normali-sation coefficients of the electromagnetic field A µ in Fourier mode expansion exchangesbetween the fields under G µν = f ( A, ψ ). In all cases such ‘kinematic factors’ arise from‘numerical parameters’ such as p ∈ R in the Fourier modes e ± ip · x themselves. As wellas being mutually compatible these normalisation factors will ultimately translate intothe appropriate dimensions for cross-section calculations, as described towards the endof section 11.2In the environment of HEP experiments it is generally assumed that the space-time is flat and a Minkowski coordinate system employed such that the external Lorentzconnection has components A abµ ( x ) = 0, corresponding to a linear connection Γ( x ) = 0by equation 3.51. Transforming under the global Lorentz symmetry the components ofthe 4-component Dirac spinors ψ ( x ) are normalised as alluded to above. The Lorentzconnection A abµ ( x ) acts on a Dirac spinor ψ ( x ) through the associated spinor connec-tion as a representation of the Lorentz symmetry. This structure can also be appliedto the more general case of a curved spacetime, employing a spinor bundle over M toexpress the dynamics of the Lorentz connection, with A abµ ( x ) = 0 in general, on thebase manifold in relation to spinor fields. As described towards the end of the previoussection, a starting point might be to develop a minimal model based on the full sym-metry ˆ G = SL(3 , C ) acting on v ∈ h C leaving the form L ( v ) = 1 invariant. For thismodel fermion states derive from the Weyl spinor ψ L in equation 7.35 in interactionwith an internal U(1)-valued gauge field, in principle describing a model for QED.As well as classifying particle states such as gauge bosons and fermions in arepresentation space according to their transformation properties under the externaland internal symmetry groups and their possible interactions, the structure of tangiblephysical particles in spacetime as detected in experiments can also be investigated.Physical particles evidently transfer energy and momentum, which can be describedby the tensor T µν ( x ) and is presumed to be conserved in 4-dimensional spacetime. Inthe present theory energy-momentum is defined by the relation T µν := G µν (withina practical normalisation factor of − κ ), and hence the transfer of a finite amount of483nergy must necessarily be associated with G µν = 0 and hence a non-flat spacetime,while the identity G µν ; µ = 0 also ensures energy-momentum conservation throughout.In turn this tangible spacetime form of a particle is expressed in terms of the underly-ing fields as a solution for G µν = f ( Y, ˆ v ). This smooth external geometry represents amacroscopic ‘dressed’ or ‘renormalised’ object constructed out of the underlying micro-scopic ‘bare’ field exchanges. Representing the electron beam in a HEP accelerator forexample, observable properties associated with the energy-momentum for the electronfield are carried by the tensor: T µν := G µν = f ( Y, ˆ v ) (15.1)This is equation 5.32 of section 5.2, where a particular vector space representingthe full temporal flow ˆ v may be substituted in. The expression G µν = f ( Y, v ) impliesan underlying innumerably nested sequence of indistinguishable field descriptions under G µν ( x ). This geometry is entirely constructed out of field components derived from L ( v ) = 1 and the corresponding E symmetry actions. However, in this theory, itseems quite possible that some components of the fundamental form L ( v ) = 1 andthe gauge fields may exist on M without contributing to the geometry field G µν . With T µν := G µν this would imply that not all fields in spacetime have energy-momentumin the sense of T µν = 0. This possibility was discussed in section 13.3 and comparedto the case of gravity waves which, while associated with a geometry with G µν = 0,carry energy via a finite Weyl curvature as described after equation 5.44 in section 5.2.Here we consider the measurable phenomena of HEP particle types and properties tobe determined by the mutual constraints of equations 11.29 applied to the underlyingfields and conveyed via energy-momentum in the form of the generalised expression ofequation 15.1, as originally employed for the special case of the free electromagneticfield leading to figure 11.1 in section 11.1.While a significant correlation between the structures of the present theoryand calculations in QFT has been identified as described in sections 11.1 and 11.2,a key question remains regarding the precise conceptual form and mathematical ex-pression of the nature of field quantisation. One major aspect concerns whether theprojected field components themselves are effectively fragmented into discrete elementsdistributed over spacetime and related via δY ( x ) and δ ˆ v ( x ) differences, as has typi-cally been conceived as the theory has developed, with the components of the externalgravitational field composing the only smooth and continuous functions on M . Analternative view might see all fields smooth and continuous on M , with discrete ex-changes only in the local contributions to G µν ( x ) in equation 15.1 consistent withequations 11.29, considered as ‘excitations’ of the fields and giving rise to observa-tions of apparent quantum phenomena. A full understanding of this description ofsuch quantum phenomena in the context of the present theory is one of the two mainbranches to be pursued as summarised at the end of the previous section.With all physical entities described by equation 15.1, subject to constraintssuch as L (ˆ v ) = 1, this includes solutions that incorporate the phenomena of apparentparticle effects, as discussed in section 11.3. These solutions must describe the discreteemission and detection of the same conserved 4-momentum p with p = m and con-served charges, arising from the internal field constraints, giving the rather mechanical impression of an intermediate ‘classical particle’ or projectile of some form. As dis-cussed in section 11.4 the ‘particle tracks’ that we construct by joining up detector484its, as depicted in figure 10.1 for example, reinforces this illusion of an independentparticle-like entity pursuing a continuous trajectory.One way to approach the nature of the actual physical structure underlyingsuch particle-like phenomena is to begin by considering a general state of macroscopicmatter described by G µν = f ( Y, ˆ v ), as represented by the ‘bulky’ geometry of fig-ure 15.2(a), which might represent for example the matter content T µν := G µν ofordinary ‘table and chairs’. Subsequently a progression down to a more minimal fieldcontent underlying a solution of G µν = f ( Y, ˆ v ) can be considered, down to a stage thatdoes not simply gradually fade away towards G µν ( x ) = 0, but rather solutions for geo-metric structure emerge that take on the shape of a discrete set of topologies due to thediscrete constraints on the underlying fields. In this case a somewhat ‘tubular’ struc-ture might arise as the vacuum limit is approached, as represented in figure 15.2(b).These near vacuum conditions correspond for example to the environment created inHEP experiments as described near the opening of section 10.1.Figure 15.2: Representations of 4-dimensional solutions for G µν = f ( Y, ˆ v ) for (a)the general case of ordinary extended matter (b) the discrete structure emerging aspermitted by the underlying field constraints as the vacuum state is approached.The pattern of inner lines in figure 15.2(b) are analogous to the contours ona map representing the altitude of a continuous physical terrain, with the geometry G µν ( x ) being perfectly smooth and continuous, as also for figure 15.2(a) and all othercases. Hence this geometry might more accurately be represented by a continuousshading. Considered as a full 4-dimensional spacetime solution the contour tubes inthe near vacuum region in figure 15.2(b) connect and are continuous with macroscopicentities such as HEP accelerators and detectors, as represented by the outer structurein the same figure. The inner structure in figure 15.2(b), with time directed from leftto right, might represent for example the overall particle interaction process e + e − → µ + µ − , via an intermediate γ or Z state, which is typically pictured in terms of particletrajectories in 3-dimensional space as depicted in figure 11.13(a) and described insection 11.4.While shaped by the discrete enveloping topology the spacetime geometry forsuch a process will also be modulated by a wave-like structure of a form similar toequation 11.12 and figure 11.1, corresponding to a particular 4-momentum transfer.As also described in section 11.1 the spacetime metric g µν ( x ) itself associated withthis modulation is presumed to take a form similar to equation 11.13. In the overallsolution of equation 15.1 for such a process the left-hand side ‘ T µν := G µν ’ of the485quation describes both the kinematic properties of the interaction via the energy-momentum tensor T µν ( x ) and the smooth external geometry G µν ( x ) as for generalrelativity. Through the right-hand side ‘ f ( Y, ˆ v )’ of the same equation all quantumproperties are sown into this structure in the form of an underlying set of discrete fieldredescriptions of the form δY ↔ δ ˆ v , subject to the constraints such as L (ˆ v ) = 1, whichdetermine in turn the possible set of discrete particle types and interactions that canbe observed in HEP experiments.That is, while G µν ( x ) is perfectly smooth and continuous there is both a discreteset of apparent particle types and a discrete set of possible topologies, corresponding forexample to n -particle final states, that may be obtained for the near-vacuum solutions.This structure hence provides a coherent conception of the nature and properties ofparticle states observed in the laboratory. For example a continuous range of conservedmomenta is available for the apparent emission and detection of a fermion state withinthe discrete constraint p = m , corresponding to an apparent particle mass m whicharises from the underlying interactions between the particular fermion field ψ ( x ) andthe vector-Higgs field v ( x ).The metric g µν ( x ) for the external geometry depicted in figure 15.2(b) repre-sents a particular solution for G µν = f ( Y, ˆ v ) on the macroscopic scale of HEP exper-iments, similarly as the Schwarzschild metric of equation 5.49 represents a particularmacroscopic solution on a much larger scale. Unlike the large scale case, for which theprecise trajectory of planetary orbits and the deflection of light passing near the sunis observable, it is clearly not possible in practice to send ‘test particles’ through thelaboratory environment of figure 15.2(b) in order to map out the spacetime curvature(although such a project can be readily conceived in terms of a thought experiment,as for that involving geodesic deviation due to the geometry of intense beams of lightas described in section 11.4).However, crucially for the present theory, this non-trivial external geometrywith metric g µν ( x ) is a physical characteristic of a possible solution for G µν = f ( Y, ˆ v )and the test of this proposal, which will require all elements of the full theory, will reston the ability to identify HEP processes which are actually observed and to predictnew phenomena. This will involve both the determination of the internal quantumnumbers of the apparent particle types, as implied in the underlying field structure f ( Y, ˆ v ) for such a process, and in particular the apparent kinematic constraints on the4-momentum p transferred, where with p = m and T µν := G µν the invariant mass m provides a direct characterisation of the external geometry itself.Within the field constraints more generally a range of topologies which arerather more complicated than that depicted in figure 15.2(b) will arise. For examplethe process recorded in figure 10.1 is identified as an e + e − → Z → b ¯ b event in theanalysis of [68]. Such a process typically involves ‘particle tracks’, as shown in theevent picture, each of which apparently emanates from one a sequence of vertices,each of which in turn is associated with the Z boson itself or a B or D hadron in asubsequent decay chain. With generally five such decay vertices for each such eventmutually separated by typically a few millimetres, within the volume of the detector forwhich the closest devices are a few centimetres from the interaction point, the topologyof the apparent particle-like structure described by the solution G µν = f ( Y, ˆ v ) will berelatively intricate for these processes. 486et other forms of solutions for G µν = f ( Y, ˆ v ) may appear less ‘particle-like’as for the case of an e − state apparently simultaneously ‘passing through both slits’in the experiment depicted in figure 11.13(b). The overall geometry G µν ( x ) for theset-up of figure 11.13(b) for the case of a high intensity electron beam, with the fullinterference pattern clearly observed on the final screen, will be of a macroscopic formas described for figure 15.2(a) above. As the intensity is turned down, corresponding toa transition towards a near vacuum solution as exemplified in figure 15.2(b), an overallgeometry will emerge incorporating the transfer of an apparent single e − particle fromthe source S to the detector hit A in figure 11.13(b) in continuity with the structure ofthe apparatus of the double-slit experiment. The geometry of such a solution serves toemphasise the fact that a ‘particle’ should not be considered as a kind of localised entityin the form of an ‘energy-knot’ propagating in 3-dimensional space (see for example[82] pp.202–204), but rather as an apparent phenomenon associated with a particularkind of smooth extended 4-dimensional solution for G µν = f ( Y, ˆ v ) constructed overthe underlying field possibilities. Similar 4-dimensional spacetime solutions will alsoincorporate the phenomena of quantum entanglement and EPR experiments as dis-cussed in section 11.4. In many cases however a solution for G µν = f ( Y, ˆ v ) will take aform consistent with the notion of a localised propagating particle-like entity.Although in the present theory there are also no fundamental ‘string-like’ ob-jects, there may be some relation to string theory (for which there are also no funda-mental particle entities) in that diagrams with a similar topology to that of the innerstructure in figure 15.2(b) also appear in relation to string theory calculations. Herehowever rather than describing the trajectory and interactions of a set of closed stringsthe tubular contours in figure 15.2(b) purely represent the structure of an extended4-dimensional geometry. In string theory such a diagram correlates with the ‘tree level’process as represented by the Feynman diagram of figure 10.3 for example, while forthe present theory figure 15.2(b) represents the full physical process with arbitrarilynested field exchanges implied under the solution G µν = f ( Y, ˆ v ). However, althoughthe inner structure of figure 15.2(b) in relating to a process such as e + e − → µ + µ − has a very different physical and conceptual meaning to analogous diagrams featur-ing in string theory, some of the mathematical properties of topological structures in4-dimensional spacetime might be jointly applicable.Rather than the phenomena of a discrete spectrum of particles being deter-mined by the vibrations and tension of hypothetical strings, here such phenomena aregenerated by the possibility of actual underlying field redescriptions subject to theconstraints of equations 11.29. As considered above a practical starting point may beto identify QED processes involving electron-photon interactions, such as with Bhabhaor Compton scattering events, in this unified framework alongside general relativity.This study might begin with a model based on the full symmetry SL(3 , C ) for the form L ( v ) = 1 before generalising to the octonion case with a full SL(3 , O ) ≡ E symmetryacting on the form L ( v ) = 1. The action of the internal U(1) Q ⊂ E symmetrygenerated by ˙ S –– l on the spinor components of T h O , as seen for example in equa-tions 11.33 and 11.34 in the terms of the field constraint equation D µ L ( v ) = 0, givesrise to the phenomena of electrodynamics. The precise manner in which the factors of | ˙ s f | = 1 or | ˙ s f | = in this expression translate into the corresponding factor of threein charge ratio for physical renormalised particle states, as discussed for figure 11.5487n section 11.2 in the context of cross-section calculations, will need to be determinedalongside the full understanding of the structure of quantum phenomena and particlestates themselves.The above QED phenomena will generalise for the complete internal symmetryidentified in the breaking of the E symmetry over the extended external M manifold,and then further with the full symmetry of time identified as E or even E on the fullform of temporal flow L (ˆ v ) = 1. The insight gained from the U(1) Q case might thenbe extended for the remaining internal generators to identify further features of theStandard Model and beyond as they arise naturally out of the complete theory. It islikely that the full package will be required with all the features of figure 15.1 combinedtogether, and the full set of possible fields and field interactions incorporated, in orderto determine specific quantities such as the electron mass and the full set of StandardModel parameters as summarised in table 15.2, including the neutrino sector, generally.In conclusion, the field and particle content of the present theory, in terms offigure 15.1, includes the external gravitational and internal gauge fields which arisefrom the symmetries of L (ˆ v ) = 1 and are mutually related as described for ‘front (1)’,together with the fermion and ‘vector-Higgs’ fields identified from the F (h O ) com-ponents studied for ‘front (2)’. Consistent with the gauge invariance of the constraintequations the non-gravitational fields mutually interact to form combinations underpossible solutions G µν = f ( Y, ˆ v ) for the world geometry on M as described for‘front (3)’, taking into account the intrinsic warping of the spacetime geometry due tovariation in | v ( x ) | and the role of the scalar field components as studied for ‘front (4)’.In order to develop this theory further and establish full contact with the results ofHEP experiments, as well as with empirical observations in cosmology and physicalphenomena more generally, the four fronts of figure 15.1 will need to be further devel-oped and combined as provisionally outlined in the previous section. While emphasising the possibilities for progressing outwards from the structure offigure 15.1 the present theory is based upon the multi-dimensional form of temporalflow L (ˆ v ) = 1, at the centre of the figure, which in turn derives from the simplestructure of one-dimensional progression in time as described in section 2.1. With boththe familiar four dimensions of the extended spacetime manifold M and the ‘extradimensions’, which are associated with the properties of physical objects in spacetime,deriving from a single temporal dimension the question concerning the origin of timeitself is inevitable. A naive further reduction down to ‘zero dimensions’ together with acontrived argument to generate one dimension is not considered here to be of any greatvalue. On the other hand the observation that the arithmetic properties of multipledimensions are implicit within the arithmetic structure of the real line R , as describedin section 2.1, provides a natural and major motivation for the present theory. Asecond founding motivation for the entire theory is the apparent necessity for any andevery subjective experience , including our observations of the physical world, to takeplace in time. This conception of the theory itself implies a subjective nature for theorigin of time and leads to the conclusions described in chapter 14, and in particular488o the ‘universal foundation’ for the theory depicted in figure 14.5.This overall structure can be considered as a system rather than just a theory (in the usual sense of the word) – it is intended not merely to represent the world bya model , but rather it aims to describe the way the world actually is , and how it ispossible for it to be . This is in a similar spirit that a biologist, for example, mightdescribe the system of a living organism – although finding such a metaphor for thewhole system is particularly problematic due to its unique and all-embracing nature.It is a system founded upon general experience of living in the world as wellas upon knowledge gained from the high energy physics laboratory together with cos-mology and from scientific observations in general. Indeed all such experiments andobservations are just a refined and specialised form of our experience in the world.While the primary aim has been to demonstrate a unified theory that can accountfor a wealth of scientific data, and thereby also provide a means of verification of theideas, it has also been considered desirable to incorporate the nature of experienceitself in the world. This leads to a unification not only of experimental findings butalso of science as a whole with our experiences of the world in general. Hence althoughmuch of the presentation has involved scientific knowledge, from particle physics tocosmology, the overall conceptual scheme arrived at is that of a world which one canfeel oneself to be immersed or engaged within while walking down the street.While the physical laws and structures of the 4-dimensional world are carvedout of the general flow of time, as filtered by the spacetime form of perception, theactual physical objects we encounter, such as complex organic life forms, are mouldedto conform with the possibility of our actual existence in the world. The apparentstability of the perceived physical forms – from inter-galactic structures to the insectworld on Earth – gives the illusion of a robust universe, independent of consciouslife, constructed upon an independently existing material substratum, a notion uponwhich the early development of science also built its foundations. It is an illusionwhich continues to yield enormous practical advances in navigating our way aroundthe physical world.Both time and space are direct forms of subjective experience of mathematicalstructures in the world, through which the physical world itself is created and sustainedas incorporated in node (4) of figure 14.5. Although a more rigorous mathematicaldescription of all aspects of this structure is to be sought this does not imply thatthe system of the world is itself fundamentally a ‘mathematical object’. Rather, as isthe case in general, mathematics provides a precise and concise means of describingand elaborating both physical and abstract structures. It is conceivable though thatthere may be essential properties of complex entities in the physical world such asthe structure of the human brain which cannot be transcribed into a mathematicallanguage which is both precise and concise enough for an exhaustive and practicaldescription. Such a physical entity is of course ‘still there’ even if it cannot be succinctlyexpressed in mathematical terms, in which case a mathematical approximation tonature might still be employed for practical purposes.For the present theory mathematics offers a precise, quantitative language forthe scientific study of the conceptual, organic interplay between the physical world andconscious observer as represented in figure 14.5. However, while there is considerablescope for further mathematical development of the theory the time cycle structure489an be conceptually and logically coherent even if it may be humanly difficult to com-prehend or develop a precise mathematical description of certain elements, such asfor nodes (5) and (6) of figure 14.5, or if such an element does not directly correlatewith a mathematical expression in a sense that we might recognise from familiar text-book maths. These elements of the theory may be correspondingly harder to bothinvestigate in full detail as well as model in mathematical terms. Regardless of thesepractical difficulties the fact remains ultimately that we do ‘see’ the world through aone-dimensional progression in time (in a similar sense that we see some objects as‘green’ as described towards the end of section 14.1). This continuous temporal pro-gression is inseparably fused together with all subjective experience as a fundamentalcharacteristic of all experiences.Taking the 1-dimensional flow of time in node (1) of figure 14.5 to be modelledaccurately by an interval of the mathematical real line R can itself be considered asa provisional assumption. This can be justified since experience of a moment of timehas the very simple structure of a continuous one-dimensional progression which maybe uniquely and unambiguously represented by the properties of the real line. Thisassumption may be further justified by empirical tests of the consequences of the theoryderived, via nodes (2) and (3), for node (4) of figure 14.5.Again, further stepping around the cycle in this figure, the employment oftractable mathematical language may fall short of providing an accurate and unam-biguous account of the full nature of the self-reflective structures R which are centralto nodes (5) and (6) of figure 14.5. The use of the mathematical structures relating toG¨odel’s theorem and undecidable propositions G in section 14.1 marked a provisionalattempt to model such a structure, although in a manner that seems far too simplistic.However, further mathematical development of this aspect of the theory is bothdesirable and possible, with the aim of identifying a more precise description of theprogression of self-reflective physical states, as crudely represented in figure 14.4, inmathematical terms. This may involve a degree of approximation based on a statis-tical approach to the phenomena of systems composed of many parts, by analogy forexample with the thermodynamic properties of entropy. Even if such a mathemati-cal structure remains somewhat elusive the conceptual ideas regarding the notion ofsubjective temporalisation might in principle be tested to some extent against empir-ical findings in the field of neuroscience. Some of the ideas presented might also beof relevance in the field of artificial intelligence (as initially discussed at the end ofsection 14.3) featuring for example the design of a device as a 4-dimensional entityin spacetime incorporating a structure of internal temporalisation – that is a machinenot just programmed to do things in time but also capable of internally representinga potentially subjective temporal structure itself.In contrast to these more speculative elements of the theory the full mathe-matical expression of the upper half of figure 14.5, beginning with the objective flowof time modelled by an interval of the one-dimensional real line R and leading viathe multi-dimensional form of temporal flow L (ˆ v ) = 1 and its symmetries to the ex-tended spacetime arena of the physical world as constructed through one of a myriadof solutions for the expression G µν = f ( Y, ˆ v ), is in principle highly testable and hasalso been by far the main focus of the present theory. Amidst the resulting quantumphenomena the full theory can be applied to the observations of HEP experiments as490odelled by the techniques of QFT and expressed in the form of the Standard Modelof particle physics, and here arising from the E symmetry of a 56-dimensional form oftime. The external theory of general relativity, describing gravitational phenomena, ishere unified with the internal theories of gauge fields and particle physics through theprojection of the form L ( v ) = 1 and breaking of its symmetry in the identificationthe spacetime manifold M as an arena for perception in the world.Through these ideas the present theory also incorporates the subjective wayin which we experience an apparently classical world of Newtonian material objects.Although having its origins in the fundamental notion of progression in time and per-ception in space the theory has developed with large scale cosmology and the StandardModel of laboratory particle phenomena in mind, resting heavily upon knowledge ac-cumulated by the experimental and theoretical communities over recent decades todraw out the system of the world presented in this paper. The theory is expected tobe profusely testable in terms of determining the extent to which the known form ofthe physical world can be ascertained from the basic conceptual ideas of the theoryin addition to making new predictions for as yet unobserved phenomena which mightbe discovered. Indeed the properties already deduced from the theory, in matching anumber of features of the Standard Model mark a first success for the theory. Thissuccess is summarised in section 9.3 where further progress is proposed in seeking anE symmetry of an appropriate form L ( v ) = 1 as a mathematical prediction of thetheory. The other principle area for study in the next stage of developing the theoryis towards a more detailed understanding of the application of statistical methods andrenormalisation techniques for the present theory in relation to QFT. The phenomenaof ‘running coupling’ will be of relevance here and the extrapolation of the three gaugecouplings from the laboratory energy scale may encounter ‘new physics’ in terms ofnew interactions or states identified in the theory on the way up to the GUT scale.Consistency with the unification of the gauge couplings hence will also provide a testof this theory. The Planck scale seems to be of no great significance for the presenttheory since gravity is not quantised.Returning again to figure 15.1, with the theory developed from the notion ofa multi-dimensional form of time L ( v ) = 1, front (1) has shown how a Kaluza-Kleinrelated unification between gravitational and internal gauge fields can arise naturallyout of an underlying isochronal symmetry, rather than an isometry, for a world per-ceived over a 4-dimensional spacetime manifold. The results presented for front (2)already establish a substantial connection with empirical data in the form of severalbasic features of the Standard Model. Within the same framework, generalised formultiple solutions, front (3) has described how the calculational tools of quantum fieldtheory might be incorporated, again originating out of the basic principles of this newtheory. In addition to accounting for small scale laboratory phenomena, culminating inthe particle concept described for figure 15.2(b), the large scale structure of cosmologyis also addressed in front (4), including the remote reaches of the very early universe,leading to the conception of the cosmos summarised in figure 14.8. While well definedareas of further development have been identified the progress made and propertiesuncovered in all directions, together with the simplicity inherent in the founding notionof the flow of time, add to the overall plausibility of the theory.491 ibliography [1] John C. 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