Generalised Proper Time as a Unifying Basis for Models with Two Right-Handed Neutrinos
aa r X i v : . [ phy s i c s . g e n - p h ] M a y Generalised Proper Time as a Unifying Basisfor Models with Two Right-Handed Neutrinos
David J. Jackson [email protected]
May 27, 2019
Abstract
Models with two right-handed neutrinos are able to accommodate solar andatmospheric neutrino oscillation observations as well as a mechanism for the baryonasymmetry of the universe. While economical in terms of the required new statesbeyond the Standard Model, given that there are three generations of the other leptonsand quarks this raises the question concerning why only two right-handed neutrinostates should exist. Here we develop from first principles a fundamental unificationscheme based upon a direct generalisation and analysis of a simple proper time intervalwith a structure beyond that of local 4-dimensional spacetime and further augmentingthat of models with extra spatial dimensions. This theory leads to properties of matterfields that resemble the Standard Model, with an intrinsic left-right asymmetry whichis particularly marked for the neutrino sector. It will be shown how the theory canprovide a foundation for the natural incorporation of two right-handed neutrinos andmay in principle underlie firm predictions both in the neutrino sector and for othernew physics beyond the Standard Model. While connecting with contemporary andfuture experiments the origins of the theory are motivated in a similar spirit as for theearliest unified field theories. 1 ontents and E . . . . . . . . . . . . . . . . . . . . . . . 273.2 Predicted Role for E . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 A salient feature of the history of physics is the progression in experimental and theo-retical insights delving deeper into the structure of matter on ever smaller scales. Whilethe regularity of chemical elements in the Periodic Table, as originally organised byMendeleev in 1869, provided the first indirect hint of an inner structure for atoms, thefamiliar pattern of elementary particles established in the 1970s in the Standard Modelencapsulates an order in the structure of matter at a further submicroscopic level. Acentral characteristic of this structure is the empirically uncovered distinctive symme-try properties of the elementary particle multiplets. Given the very fundamental levelof these observations the modern-day quest to theoretically elucidate the underlyingsource of these symmetries of the Standard Model, in preference to positing them as‘brute facts’ about the world, is particularly pressing.This quest is analogous then, at a more elementary level, to the pursuit ofan explanation for the Periodic Table that culminated in the discovery of the cen-tral atomic nucleus and the quantum mechanical properties of electron orbital statesaround one hundred years ago. While having something in common with the ‘unifiedfield theories’ of that era, in this paper we describe a new theory based upon a gener-alisation of proper time which has the potential to naturally incorporate not only theStandard Model but also the phenomenology of physics beyond, including that of theneutrino sector. The core arguments for this analysis are summarised in [1] and willbe elaborated along with further observations in this paper as outlined below.2n the first subsection to follow we review the empirically established propertiesof neutrinos and several of the models stimulated by these results. We then presentan historical survey of early unified field theories in subsection 1.2 which will lead to aprobing of the motivation for extra spatial dimensions and the application of a furthergeneralisation for a proper time interval in section 2. (This paper can be read beginningwith section 2, with section 1 for reference). In section 3 the greater suitability ofthis generalisation in directly accounting for features of the Standard Model will beexplained, further justifying the new approach. The manner in which this theorymight naturally provide a foundation for models with two right-handed neutrinos willbe described in subsection 4.1, with broader connections with new physics beyond theStandard Model and further possible tests discussed in subsection 4.2. We return to theinterpretation of theory and its relation to special and general relativity and connectionwith early unified field theories in subsection 5.1, with the underlying simplicity of thetheory evaluated further in subsection 5.2. In section 6 we conclude with emphasis onthe opportunities for the further development of this fundamental theory. The maingoals of this paper are summarised here: • A main theme of the paper is to emphasise the simplicity of this theory basedon a generalisation of proper time, arguing that a unifying basis for fundamentalphysics can be encapsulated correspondingly in ‘one simple equation’. • We describe how the theory can be seen as a natural progression from special andgeneral relativity, while also being complementary to the latter, and is related tothe early conceptions of a unified field theory. • While reviewing the links with the Standard Model here we focus on possibleconnections with the further esoteric properties of physics beyond, in particularestablishing a link with models incorporating two right-handed neutrinos.The theory, originating from considerations of the first bullet point above, willbe shown to provide a connection between the old and the new, from the second andthe third bullet points respectively, by demonstrating the relevance of the theory forcontemporary particle physics and cosmology. With this aim in mind we begin byreviewing the current status of the neutrino sector.
The striking progress in the empirical understanding of neutrino physics in recentdecades has centred upon the compelling observations of oscillations between the left-handed neutrino states. These can be described assuming three-flavour neutrino mix-ing parametrised by the 3 × ν e , ν µ and ν τ and three mass eigenstates ν , ν and ν with respective masses m , m and m .This structure accommodates both the ‘solar neutrino’ oscillations, which arepredominantly described by the mixing probability P ( ν e → ν µ ), as confirmed by nu-clear reactor experiments, via a mixing angle θ sol = θ ≃ . ◦ and mass differ-3nce ∆ m = ∆ m := m − m ≃ . × − eV , as well as the ‘atmospheric neu-trino’ oscillations which can be assumed to be almost completely due to the mixingprobability P ( ν µ → ν τ ), as confirmed by accelerator experiments, via a mixing angle θ atm = θ ≃ ◦ and mass difference ∆ m = ∆ m := m − m ≃ . × − eV (with all data in this subsection obtained from [2] unless stated otherwise). The thirdmixing angle has also been determined as θ ≃ . ◦ , principally from the disappear-ance of ¯ ν e produced in nuclear fission reactors ([2] section 14.12), while an estimateof the phase of the PMNS matrix δ CP ≃ π has been obtained from long baselineaccelerator ( – ) ν µ → ( – ) ν e appearance experiments ([2] section 14.13).The third mass difference is determined from the definitions by the other twowith ∆ m = ∆ m − ∆ m , and with ∆ m ≃ ∆ m since ∆ m is somewhat smallerthan ∆ m . There is however no existing constraint on the sign of ∆ m , that iswhether m is greater or less than m (and m ) – termed the ‘normal’ or ‘inverted’hierarchy respectively. The neutrino oscillation data also does not determine the ab-solute mass scale.The electron neutrino mass m ν e can be defined by m ν e = P i =1 | U ei | m i , where U ei are elements of the top row of the PMNS matrix. The measured parametersabove, given that the lightest neutrino mass state has mass m min ≥ m ν e > .
01 eV for the normal hierarchy and m ν e > .
05 eV for the invertedhierarchy. A direct upper limit on the electron neutrino mass is set by tritium β -decayexperiments with m ν e < . m ν e < . m min = 0 eV the simple sum of the three neutrino masses m tot = P i =1 m i would be m tot = 0 .
06 eV or m tot = 0 .
10 eV for the normal or invertedmass hierarchy respectively (see also [3] section 2.1). The constraints from cosmolog-ical observations, although dependent upon the ΛCDM cosmological model (with Λthe cosmological constant and CDM cold dark matter), imply m tot < .
12 eV at the95% confidence level [5], already putting pressure on the inverted hierarchy. Futureprospects for the three left-handed neutrinos within the ΛCDM model are for the low-est possible value of m tot = 0 .
06 eV to be detectable at the 3–4 σ level in the comingyears ([2] sections 25.4 and 64).While the existing data cannot distinguish between whether neutrinos are Diracor Majorana fermions, any observation of neutrinoless double- β decay would indicatethe Majorana type (that is, with such a neutrino identical to its own antiparticlestate). If such experiments were to determine a non-zero value for the appropriatelydefined effective Majorana mass m ββ for this process with m ββ . .
01 eV, an order ofmagnitude beyond the current sensitivity, then the inverted hierarchy could be ruledout ([6], [2] section 62, [3] section 3.3). Such a measurement with m ββ & .
005 eVwould also indicate that m min > β decay with m ββ & .
06 eV also implying a value for m min > ν R which are ‘sterile’, that is they transform triv-ially under the Standard Model gauge group SU(3) c × SU(2) L × U(1) Y , accounting forthe difficulty of their direct detection (unlike the familiar ‘active’ left-handed states ν L that undergo weak interactions). These right-handed states can be utilised to intro-4uce light masses for the active neutrinos by extending the Standard Model Lagrangianwith further Dirac mass terms. Such terms are similar to those for the charged lep-tons and quarks but with unnaturally small Yukawa couplings to the Higgs field, byaround a factor of 10 − relative to that of the top quark and even by a factor of 10 − or less relative to that of the electron in the same weak doublet as the electron neu-trino. However, owing to their trivial gauge transformation properties, Majorana massterms can also be added for the right-handed neutrinos which, if sufficiently heavy, cangenerate the light active neutrino masses via a ‘seesaw’ mechanism ([3] section 2 andreferences therein). For either of the above cases, since each ν R state can only generateone ν L mass, at least two right-handed neutrinos are required to account for at least two finite active neutrino masses as implied by the established measurements of ∆ m and ∆ m , unless there is a different source for the ν L masses.In the absence of a theoretical argument to the contrary a natural expectationmight be for all three active neutrinos to be massive, via the introduction of three right-handed neutrino states, matching the three generation structure of the otherleptons and quarks. This is the case for the ‘Neutrino Minimal Standard Model’,or ν MSM ([7, 8], [3] section 7), proposed as a simple economical extension from theStandard Model (for which all three ν L states are massless and there are no ν R states).In the ν MSM there are two ν R states with nearly degenerate masses in the rangefrom ∼ CP -violating oscillations of these two sterile neutrinosduring their thermal production can also in principle generate the baryon asymmetryof the universe.Compelling empirical evidence for the mass scale of right-handed neutrinosmay be even harder to establish than for the active left-handed states. However thelower part of the preferred mass range of 1 ∼
100 GeV for two of the three right-handedneutrinos in the ν MSM could be accessible to direct searches for ν R states throughmixing with active neutrino states in laboratory experiments ([3] section 3.4, [9]). Suchmixing is required to generate the active ν L masses via the seesaw mechanism. Further,much lighter right-handed neutrinos can also play a significant role in cosmology [10].In particular, sterile right-handed neutrinos, which apart from the above mixingeffects could only be observable through gravitational interactions, can be considereda natural candidate for dark matter [11]. However the required Yukawa couplings forthis application are very different to those associated with neutrino oscillations. Inthe case of the ν MSM the third ν R state, with a mass of a few keV, acts as a warmdark matter candidate but with a Yukawa coupling too small to make a significantcontribution to the active neutrino masses, hence leaving the lightest active neutrinopractically massless. More generally, for scenarios with three ν R states there is nointrinsic limit on the mass of the lightest ν L mass state.Neutrino models are further stretched if required to accommodate the em-pirical hints for anomalous observations in ( – ) ν µ → ( – ) ν e oscillations, which imply a massdifference of ∆ m ∼ , as well as the anomalies observed in reactor and galliumexperiments. Further neutrino states or other new features are needed to provide aphenomenologically complete description of all neutrino particle physics data (see forexample [12]).On the other hand the compelling neutrino oscillation observations can be5ccounted for by models with only two right-handed neutrinos, which can also ac-commodate a source of the baryon asymmetry (see for example [13, 14, 15, 16]). Forthese models, which imply that m min = 0 eV for the left-handed neutrinos, the tworight-handed neutrinos are typically very massive in the range from ∼ GeV up tothe GUT scale. In this case a possible source of the matter-antimatter asymmetryderives through a leptogenesis scenario via heavy right-handed neutrino decays in thevery early universe (see also [17, 18]). The indications that the PMNS phase δ CP maybe relatively large suggests that CP -violating effects in the oscillations of light activeneutrinos may be significant, and can be linked via the neutrino mass matrices in spe-cific seesaw models with the CP -violating phase in the heavy sterile neutrino sectorthat drives the baryon asymmetry via leptogenesis (see for example [13], [18] section 8,[19]). While models with two ν R states might hence account for active neutrino oscil-lations and the baryon asymmetry, similarly as for the ν MSM, in lacking a third ν R state an alternative candidate for dark matter will be required.In summary, based only upon well-established observations in the neutrinosector the main questions to be addressed concern the absolute value and spectrumof the active neutrino masses, resolving the sign of ∆ m , the number and massesof sterile neutrinos, and whether neutrinos are Dirac or Majorana particles. Furtherquestions concern the PMNS matrix: including the proximity of the value of θ tomaximal 45 ◦ mixing, the need for a more precise evaluation of δ CP , and regarding therelation of the neutrino mixing matrix to the very different CKM mixing matrix in thequark sector.While it is always possible in principle to model a wide range of neutrinophenomena, by extending the Standard Model Lagrangian, a more fundamental expla-nation of the origin of these empirical observations, providing a deeper understandingof the neutrino flavour structure and mass generation mechanism, would of course bedesired. Ideally such an explanation would take the form of a theory that also accountsfor the properties of the Standard Model itself, and which might be empirically testedin neutrino experiments, in the particle physics laboratory more generally or throughcosmological observations. In this paper we propose such a fundamental theory. Thetheory may provide in particular a possible basis for models with two right-handedneutrinos in a unified structure consistently alongside three generations of the otherleptons and quarks, as will be explained in subsection 4.1. While connecting withthese contemporary phenomenological issues, as described further in subsection 4.2,the underlying motivation for the theory is related to that for the earliest unified fieldtheories; we hence next review this background in the following subsection. While the field concept as employed with great success for Maxwell’s equations [20] hadvery much influenced the conception of Einstein’s theory of general relativity [21] halfa century later, after 1915 it was natural seek a unified field theory that would gener-alise the theory of gravity to incorporate electromagnetism, rather than the other wayaround. The reason for this is perhaps that while the gravitational field equations aremore complicated than those for electromagnetism the underlying motivation envisagedfor Einstein’s theory can be considered somewhat simpler. As a particularly elegant6spect of general relativity the assumption of an extended globally flat 4-dimensionalMinkowski spacetime is dropped , with special relativity holding for all non-gravitationalphysics strictly only in the limit of infinitesimally small inertial reference frames by theequivalence principle ([22] chapter 9). Within such a local reference frame with localcoordinates { x a } an infinitesimal proper time interval δs can be expressed, in a forminvariant under local SO + (1 ,
3) Lorentz transformations between such frames, as:( δs ) = ( δx ) − ( δx ) − ( δx ) − ( δx ) = η ab δx a δx b (1)with the Lorentz metric η = diag(+1 , − , − , −
1) and a, b = 0 , , , x , x , x , x ) in whichequation 1 holds for arbitrary finite spacetime intervals anywhere in the Minkowskispacetime, leaving the corresponding finite proper time interval ∆ s invariant underglobal Lorentz transformations between such reference frames. This framework of spe-cial relativity was largely motivated through compatibility with Maxwell’s equations,with the laws of electrodynamics taking the same form in any of these global inertialreference frames [23].In general relativity there are no such global coordinates and inertial frames, ex-cept in the flat spacetime limit. Instead general coordinates { x µ } , with µ, ν = 0 , , , δs ) = g µν ( x ) δx µ δx ν (2)which only locally reduces to equation 1 in suitable local coordinates. The force ofgravity is not present in such a local inertial reference frame but rather can be ascribedglobally to the metric field g µν ( x ) which describes the curved geometry of the extendedspacetime. Through a mutual dynamical interplay the spacetime geometry describedby g µν ( x ) is related to the distribution of matter through Einstein’s field equation (aswill be discussed in subsection 5.1 for equation 43), with test particles postulated topropagate through spacetime along geodesic trajectories. Einstein’s theory of gravity,expressed directly in terms of the more general and flexible structure of a curved space-time, surpassed that of Newton in accurately accounting for gravitational observationssuch as the orbit of the planet Mercury. While Newton’s law of universal gravitationstrictly concerned mathematical relations, and was not based on any hypothesis re-garding the cause of the force of gravity acting at a distance across space, for Einsteinthe curvature of spacetime also succeeded in furnishing an explanation of gravitationalphenomena on relaxing the assumption of spacetime flatness.On the other hand for Maxwell’s electromagnetism, initially formulated in aNewtonian background of absolute space and time, 3-component electric E and mag-netic B fields were first added , conceived of as mechanical states of an underlying‘ethereal medium’ filling all of space ([20] part I, see also [22] section 6(a) and [24]section 12(a) part 1); with Maxwell’s equations constructed on the basis of empiricalobservations, as Newton’s law of gravity had been in the same background arena. After1905, with the theory of electromagnetism readily compatible with special relativity,Maxwell’s equations could be expressed more succinctly in a Lorentz covariant form interms of the antisymmetric electromagnetic field strength tensor F µν ( E , B ), and after7915 could be accommodated within the curved spacetime of general relativity via theequivalence principle. However, it was natural to enquire whether a further generali-sation from general relativity might itself provide an explanation of electromagnetism(which alongside gravity was then the only other fundamental force known) on relaxingfurther assumptions regarding the 4-dimensional spacetime metric geometry.The first proposal for such a unified field theory was made in 1918 by Weyl ([25],[26] chapter IV section 35, [27] chapters 1–3) on dropping the assumption that thelength of a 4-vector, determined by the metric g µν ( x ) of equation 2, should be path-independent when ‘parallel transported’ in spacetime, an invariance which could beinterpreted as a residual of rigid Euclidean geometry still remaining in general relativ-ity. Hence, similarly as a 4-vector direction is propagated in a path-dependent mannerthrough a curved spacetime via a linear connection Γ ρµν ( x ), a unique function of thefirst derivatives of g µν ( x ) in general relativity, Weyl introduced a vector field A µ ( x )to induce path-dependent changes to 4-vector magnitudes – with the metric ¯ g µν ( x )employed in forming inner products defined by the scaling:¯ g µν ( x ) = λ g µν ( x ) with λ = e R x x A µ ( x ) dx µ (3)The length of a 4-vector is path-independent under parallel transport betweenany two spacetime points x and x only when the scale factor λ is integrable. Thisis the case if the new connection field A µ ( x ) can be expressed as the gradient of acontinuous function χ ( x ), that is A µ ( x ) = ∂ µ χ ( x ), which in turn implies that thequantity defined by F µν ( x ) = ∂ µ A ν ( x ) − ∂ ν A µ ( x ) vanishes. In the general case thefield F µν ( x ) was identified with the electromagnetic field strength tensor and A µ ( x )with the corresponding vector potential, within normalisation factors, similarly as theRiemann curvature tensor together with Γ ρµν ( x ) and g µν ( x ) are associated with gravityin general relativity. In this manner Weyl inferred that on the 4-dimensional spacetimemanifold ‘all physical field-phenomena are expressions of the metrics of the world’ ([26]chapter IV section 35). The theory hence demonstrated that electromagnetism couldin principle be accorded such a geometrical significance. However, since scaling lengthsvia λ in equation 3 implies scaling time intervals also in equation 2 Einstein immediatelynoted a fatal flaw of the theory – the sharp lines of atomic spectra observed in thelaboratory are not dependent upon the history of individual atoms.Weyl’s theory did however provide the first step towards non-Riemannian con-nections and gauge theories, with the term ‘gauge’ retained from the length scaling factor in equation 3. In fact both F µν ( x ) and the ‘Action’ defined for the theoryare ‘gauge-invariant’ under arbitrary re-calibrations, that is under local changes ofthe adopted metric scale. Progress was achieved by 1929 on introducing a factor of i = √− λ as a phase factor to beapplied instead to a complex wavefunction Ψ( x ) in the then recently invented quan-tum mechanics ([28], [29] chapter II section 12, [27] chapters 4–5). Correspondinglyin [28] Weyl concludes with the assessment that ‘electromagnetism is an accompanyingphenomenon of the material wave-field and not of gravitation’. Hence with the phasefactor taking values in the symmetry group U(1) electromagnetism could be success-fully described as a stand-alone U(1) gauge theory with a gauge field A µ ( x ), rather thanas a geometric augmentation to general relativity. During the 1950s–1970s such gaugetheories, with field interactions considered a consequence of the gauge-invariance of the8quations, were developed and generalised beyond the U(1) gauge symmetry of elec-tromagnetism to non-Abelian gauge symmetries, ultimately incorporating electroweakand strong interactions also within the framework of the ‘gauge principle’, essentially detached from consideration of the geometry of external 4-dimensional spacetime –which could be taken as the flat background of special relativity to a very good ap-proximation in a laboratory setting.The properties and representations of gauge symmetry groups are central to themodern-day structure of the Standard Model and unification schemes. It is well knownfor example that the branching patterns for the smaller non-trivial representationsof Lie groups such as SU(5), SO(10), E and E on extracting the Standard Modelsubgroup SU(3) c × SU(2) L × U(1) Y bear some resemblance to the gauge multipletstructure of leptons and quarks ([30], [31] section 13, [32] and [33] respectively) as thebasis for a Grand Unified Theory (GUT). While the earlier unified field theories werebased on generalisations of general relativity, for GUT models the focus is on particlephysics with gravity being neglected and deferred for later consideration. However,in the case of the Lie group E a symmetry breaking structure can be correlatedwith a full three generations of leptons and quarks incorporating also transformationsunder the external local spacetime Lorentz symmetry alongside the Standard Modelgauge group [34]. Nevertheless for each of the above Lie group structures the matchwith the symmetry properties of the Standard Model is incomplete and significantproblems remain. Further, while the three largest exceptional Lie groups E , E andE are of particular interest, owing to the high degree of symmetry they describeand the uniqueness of these mathematical structures, the nature of a clear underlyingconceptual origin , whether geometric or otherwise, to motivate the application of thesegroups in particle physics remains an open question.Despite his rejection of Weyl’s theory Einstein himself sought a unified fieldtheory for gravity and electromagnetism based on generalisations of general relativity.From 1925–1955, throughout the last 30 years of his life, Einstein worked on gener-alisations of 4-dimensional Riemannian geometry based in particular on dropping theassumption that the metric tensor g µν ( x ) and/or the linear connection Γ ρµν ( x ) shouldbe symmetric in the µ, ν indices ([22] section 17(e)). The most direct attempt intro-duced a nonsymmetric fundamental tensor ˜ g µν ( x ) with a full 16 real components whichwas proposed to decompose into symmetric g µν ( x ) and antisymmetric ´ g µν ( x ) parts as:˜ g µν ( x ) = g µν ( x ) [gravitational field] + ´ g µν ( x ) [electromagnetic field] (4)For this scheme g µν ( x ) was retained as the original gravitational metric fieldwhile ´ g µν ( x ) was identified with the electromagnetic field strength tensor F µν ( x ), withina normalisation constant. Other attempts involved associating the electromagneticvector potential A µ ( x ) with components of a nonsymmetric linear connection. (In anindependent application the study of linear connections with an antisymmetric parthad been initiated by Cartan in 1922 in the geometric context of general relativity withfinite torsion, later known as Einstein-Cartan theories). While originally motivatedby simplicity Einstein’s unified field theory attempts became increasingly elaborate,lacking the conceptual elegance of general relativity, and none of them led to the freeMaxwell equations even in the weak-field approximation, nor was there any prospectof incorporating nuclear forces into these schemes. During the same period, fromaround 1925, the mainstream physics community was also more focussed upon the9evelopments of quantum theory, with the unification proposals of Einstein seeminglyattracting more attention from The New York Times ([22] section 17(e)). However,while our understanding of fundamental physics has continued to be dominated byquantum theory, the general spirit and motivation for Einstein’s attempts at a unifiedfield theory remains enlightening when transplanted into the context of the present-dayquest for unification, as will be discussed in subsection 5.1.Einstein had also been initially enthusiastic about the potential of Kaluza-Klein theory as also introduced in the 1920s ([35, 36], [22] section 17(c)), upon whichhe worked intermittently himself over a number of years ([22] sections 17(c,e)). In thisapproach to a unified field theory the assumption that spacetime should be limited tothe familiar 4-dimensional arena of general relativity was dropped. With 4-dimensionalspacetime augmented by an extra spatial dimension a 5 × g ( x ) could be definedon the extended spacetime subsuming the original 4 × g ( x ) of equation 2. Inprinciple the four components of the electromagnetic vector potential A ( x ) could thenbe accommodated inside the extended 5-dimensional metric: ˆ g × = (cid:20) g (cid:21)(cid:20) A (cid:21) [ A T ] φ (5)where the further new component φ ( x ), alongside the original metric g ( x ) of general rel-ativity, lacked any clear physical significance. Certain components of the 5-dimensionalLevi-Civita linear connection ˆΓ( x ) could then be identified with the electromagneticfield strength F µν ( x ) = ∂ µ A ν ( x ) − ∂ ν A µ ( x ) as a function of the components A µ ( x ) inequation 5 in the appropriate way, on taking the field values to be independent of thefifth dimension. Maxwell’s source-free equations for the electromagnetic field and theequation of motion for a charged body in an electromagnetic field could be obtainedunder suitable assumptions for the extraction of 4-dimensional physics from the em-bedding in the 5-dimensional spacetime framework. However, while hence providing anelement of formal geometric unification with general relativity, no predictive power ornew phenomena could be determined and the question of the very different propertiesrequired for the fifth dimension remained.In the case of equation 3 with a geometric scale factor λ and that of equation 4with a nonsymmetric metric ˜ g µν no further natural generalisation is possible, howeverfor the case of equation 5 an arbitrary number of further extra spatial dimensions couldin principle be considered. Indeed, despite the lack of empirical support, this thirdmeans of augmenting the 4-dimensional spacetime structure has led in recent decadesto a large number of unification models based upon various approaches to extra spatialdimensions, motivated in part by the elegance and unity of the Kaluza-Klein idea.The realisation that the geometry of such augmented spacetimes could beadapted to incorporate the internal symmetries of non-Abelian gauge theory over4-dimensional spacetime, with a close relation between gauge and coordinate trans-formations described explicitly on a ‘fibre bundle’ manifold, had revived interest inthis approach to unification by the 1970s. (See for example [37], the mathematicsof fibre bundles had been developed since 1935 for the field of topology in differentialgeometry [38] independently of any application in physics). This framework ‘combinedgravitation with gauge theory in the context of a unified geometric theory in the bun-dle space’ ([37] section 9) by employing an extended higher-dimensional metric defined10n the full space. In this manner gauge theory, which had parted company from ageometric context in the 1920s as described above following equation 3, was placed inthe setting of a higher-dimensional spacetime arena, and in particular reattached to thegeometry of an external 4-dimensional spacetime base manifold, in a unifying physi-cal framework that might in principle reach beyond gravitation and electromagnetismalone. While the earliest attempts at a unified field theory may have been premature,given the hindsight of the subsequent century of accumulated knowledge in particlephysics, the quest since the 1970s to accommodate the rich properties of the StandardModel, or even a Grand Unified Theory, within the unifying framework of geometricstructures deriving from extra spatial dimensions over 4-dimensional spacetime con-tinues, as will be discussed in the next subsection. The search now includes the neednot only to account for the Standard Model but also new physics, such as that of theneutrino sector reviewed in the previous subsection. In this paper we shall motivateand build a new unified theory from first principles with the potential to accommodateboth the physics of the Standard Model and that beyond, including the possible featureof incorporating two, and only two, right-handed neutrinos alongside three generationsof the other leptons and quarks. We begin by reassessing the motivation for employingextra spatial dimensions in the following section. Rather than considering generalisations of the global metric g µν ( x ) of equation 2 on anextended higher-dimensional spacetime manifold here we focus upon the local metric η ab of equation 1 associated with a local inertial reference frame. At this most ele-mentary level of purely local structure theories with extra spatial dimensions extendthe metric geometry of 4-dimensional spacetime, with local coordinates ( x , x , x , x ),augmenting the quadratic expression for the proper time interval δs of equation 1 tothe n -dimensional form:( δs ) = ( δx ) − ( δx ) − ( δx ) − ( δx ) − ( δx ) . . . . . . − ( δx n − ) = ˆ η ab δx a δx b (6)where ( x , . . . , x n − ) are ( n −
4) extra dimensions, ˆ η = diag(+1 , − , . . . , −
1) is theextended local Lorentz metric and a, b = 0 , . . . , ( n − x , . . . , x n − ) are considered extra ‘spatial’ dimensions owing to the quadratic struc-ture and the minus signs in equation 6, sharing these properties with the three originalspatial dimensions given the Lorentz metric signature convention of equation 1. Ondividing both sides by ( δs ) and defining the components v a = δx a δs on taking the limit δs → | v n | := ( v ) − ( v ) − ( v ) − ( v ) − ( v ) . . . . . . − ( v n − ) = ˆ η ab v a v b = 1 (7)in terms of the components of the ‘ n -velocity’ vector v n = ( v , . . . , v n − ) ∈ R n . Thequantities δs and | v n | in equations 6 and 7 respectively are invariant under SO + (1 , n − n -component expressions.11he simplest and most direct means of constructing a physical theory based onthis structure is to assume the identification of the four components ( x , x , x , x ) inequation 6 with a set of local coordinates on an external spacetime M , without neces-sarily specifying a mechanism for distinguishing this extended manifold itself in four,and only four, preferred dimensions. The first four components v = ( v , v , v , v )of equation 7 are correspondingly projected onto the local tangent space of the ex-tended 4-dimensional spacetime, with v ∈ TM , upon which a preferred local exter-nal SO + (1 , ⊂ SO + (1 , n −
1) symmetry acts. This breaks the full n -dimensionalSO + (1 , n −
1) Lorentz symmetry of equation 7 and on taking the residual componentsof that equation to form the basis for ‘matter fields’ in the extended spacetime wedirectly deduce the following symmetry breaking structure:SO + (1 , n − → SO + (1 , × SO( n −
4) : external × internal (8) v ∈ R : 4-vector invariant : tangent vector in TM v n ∈ R n → ( v n − ∈ R n − : scalar ( n − M (9)This direct generalisation from the structure of a strictly 4-dimensional propertime interval of equation 1 is depicted in figure 1.Figure 1: (a) A 4-vector field v ( x ) can be constructed from locally Lorentz invariantproper time intervals δs at each point x ∈ M and (b) augmented for time s propagatingthrough a higher-dimensional spacetime, with the corresponding local values of v n ( x )projected onto the external 4-dimensional spacetime M here represented by a plane.We note that although the projection takes place locally on the external man-ifold M the action on v n − ( x ) of the full internal symmetry G = SO( n −
4) impliesthe incorporation of this complete gauge group manifold in a ‘trivial principal fibrebundle’ structure P ≡ M × G , with ‘vertical’ fibres G attached over each point of the‘base space’ M of figure 1(b). (The case for n = 10 in equation 8–9 is depicted in[39] figure 1(b) while the relation of this construction to the geometry of non-AbelianKaluza-Klein theories is described in [40]). However in this simple picture the matterfield v n − ( x ) of equation 8–9 and figure 1(b) in spacetime M , as a Lorentz scalar thattransforms under the ( n − n − n inequations 7–9 and clearly does not help this situation even for models with n → ∞ ,which might be considered in the absence of a natural limit, constituting an extremecase of a ‘waste of space’ given the lack of any apparent empirical connection.12 more sophisticated approach is clearly needed for any attempt to accom-modate the rich properties of the Standard Model within a geometric framework de-riving from extra spatial dimensions while maintaining a reasonably economical levelof structure and assumptions. The most direct possibility might be to consider the14-dimensional spacetime case for equation 8–9 resulting in a G = SO(10) internalsymmetry that might in principle be connected with a corresponding GUT model in-corporating the Standard Model gauge group SU(3) c × SU(2) L × U(1) Y ⊂ SO(10)(see for example [41]). Adopting a different scheme in 1981 Witten [42] utilised su-pergravity in an 11-dimensional spacetime, with the seven extra spatial dimensions‘spontaneously compactified’ over the external 4-dimensional spacetime, as a potentialframework for the unification of the gauge fields of the Standard Model with gravity.However a major obstacle is encountered in incorporating the appropriate quantumnumbers for the leptons and quarks. Particularly challenging more generally is the am-bition to incorporate the Standard Model in a seemingly natural and unique manner,with the search ongoing given the absence of any compelling success.The question of uniqueness becomes more acute for the most technically so-phisticated approach via the extra spatial dimensions incorporated into string theory.String theory was primarily motivated in the early 1970s as a candidate for a quantumtheory of gravity [43]. This provided an independent motivation for the introductionof extra spatial dimensions which were required in order to obtain a consistent stringtheory. The original bosonic string is only consistent in a 26-dimensional spacetimewhile n = 10 is the critical dimension for the fermionic string (see also [44] section 1.2).A major breakthrough came in 1984 with the demonstration that ‘type I string theory’is finite and free of anomalies for the gauge group SO(32), with support then grow-ing for string theory as a promising framework for a unification incorporating particlephysics as well as quantum gravity. The observation that the anomaly cancellationproperty is shared by the Lie group E × E motivated the construction of ‘heteroticstring theory’ [45], combining features of both bosonic and fermionic string theory andincorporating an E × E gauge group over 4-dimensional spacetime for the low energyeffective theory. This framework has been favoured in attempts to connect string the-ory with the Standard Model via a GUT scheme associated with one of the E factors.By 1985 superstrings had become a mainstream activity and a total of five separateconsistent theories (type I, type IIA and IIB, heterotic SO(32) and E × E ) had beendescribed; hence also with an element of concern over the uniqueness of the theorygiven these five branches.During the above ‘first superstring revolution’ of the mid-1980s Witten alsobecame a proponent and played a key role in the ‘second superstring revolution’ of themid-1990s. Marking the latter revolution the five different known types of string theorywere shown to be interrelated by dualities, or equivalences, and subsumed under asingle framework with each obtained as a different limit of an 11-dimensional ‘M-theory’(see for example [46]). The fundamental objects of M-theory include extended higher-dimensional entities called ‘branes’ as well as the original one-dimensional ‘strings’.Combining the five branches of string theory in this way tentatively offered some hopefor demonstrating the uniqueness of the theory as a unification scheme. Howeverthis sophisticated framework provides far from a minimal approach to the question ofaccounting for the Standard Model within a structure of extra dimensions. Indeed thetheory is confronted by the ‘landscape problem’ on attempting to deduce a realistic13acuum solution resembling properties of the Standard Model of particle physics andobservational cosmology out of a vast array of possibilities [47, 48]. Resorting toan ‘anthropic principle’ argument to identify our world out of a ‘multiverse’ of anestimated 10 or more possible string vacua, a range of which may be consistentwith our world, seems barely preferable to positing the properties of the StandardModel as ‘brute facts’ as alluded to in the opening of section 1.In this paper we describe a more explicit and potentially unique means ofuncovering familiar features of the Standard Model through a new fundamental the-ory. As we have described in subsection 1.2 both gauge theories and extra spatialdimensions have their roots in attempts to generalise general relativity dating froma hundred years ago. At that time although a relatively modest generalisation wasrequired to incorporate solely electromagnetism alongside gravity a range of possibleapproaches were conceived for example by Weyl, Einstein and Kaluza/Klein as re-viewed for equations 3, 4 and 5 respectively. Given the fruitfulness and influence ofthat period in shaping modern-day theories of unification, and now with the benefit ofhindsight regarding both developments in the mathematical description of pertinentsymmetry structures and the wealth of empirical data as embodied in the StandardModel, we might reconsider whether there is another possibility concerning a broadergeneralisation from equation 1 or 2 at an elementary level; one that might providemore direct access to the structures of particle physics. In some models extra spatial dimensions are taken to be infinitely extended, withour own 4-dimensional spacetime ‘brane’ world confined to a hypersurface in a larger n -dimensional spacetime ‘bulk’ (see for example [49, 50]). More typically the extraspatial dimensions of the bulk space are curled up or compactified on a very smallscale ranging from of order 0.1 mm, if only gravity propagates in the extra dimensions,down to the Planck length, accounting explicitly for our inability to observe them (seefor example [51] section 2, [52]).However, since we neither perceive nor navigate around the extra dimensionsthere is no compelling argument for the additional components to be either extendedon a global scale, as a higher-dimensional generalisation of equation 2, or even topossess the local structure of the extra components ( δx , . . . , δx n − ) in equation 6 asa quadratic extension to the local 4-dimensional spacetime form of equation 1. Thatis, with the minus signs in equation 6 adopted from the Lorentz metric signatureconvention, the extra components have the ‘spatial’ property of adding quadraticallyto form local ‘lengths’ δ Σ, with for example:( δ Σ) = ( δx ) + ( δx n − ) (10)which via the Pythagorean theorem describes a right-angled triangle structure as abasis for a local Euclidean spatial geometry. While this property is required for thecomponents ( δx , δx , δx ) of the external space dimensions of the world we inhabitthe ‘extra dimensions’ are not observed and there seems no essential reason to restrict the extra components ( δx , . . . , δx n − ) to also possess this locally Euclidean geometricproperty. 14his unnecessary restriction seems all the more artificial on considering large n , or even on taking the limit n → ∞ , since then almost all of the components onthe right-hand side of equation 6 are not required to be of a quadratic ‘spatial’ formas the { δx a } for all a > n → ∞ the left-hand side of equation 6 still describes a simple robust intervalof proper time δs ∈ R , now invariant under SO + (1 , ‘ ∞ − δs is hence pivotal in threadingtogether all of the components on the right-hand side and in defining this structure,and on shifting our focus to the left-hand side we can in fact interpret equation 6 asrepresenting a possible arithmetic expression for a real proper time interval δs ∈ R .We can then ask what further arithmetic possibilities there may be.As an invariant entity proper ‘time’ is in itself something that might be objec-tively measured, as recorded by the readings of a physical clock. Arbitrary intervals oftime are normally conceived of as an additive linear progression, with seconds containedwithin minutes contained within hours and so on. This will be the case for the propertime recorded by the ‘tick-tock’ of a pocket watch carried by a pedestrian standing in astreet with local rest frame spacetime coordinates ( x , x , x , x ) aligned with the localneighbourhood street plan. For the stationary watch an interval of proper time δs , hereconsidered infinitesimal or finite, can be expressed directly as δs = δx , preserving thesimple linear structure. As the pedestrian walks down the street along the x directionthe same proper time interval for the watch will be expressed in the quadratic form( δs ) = ( δx ) − ( δx ) with respect to the local coordinates (albeit with a walkingvelocity not too small compared with light speed needed for a significant δx contribu-tion). On turning left or right the δx coordinate will similarly augment this expressionand upon entering a building and climbing the stairs the vertical δx component willcomplete the full 4-dimensional quadratic spacetime expression of equation 1. Thecentral feature is that the watch itself continues untroubled in measuring the invari-ant ‘tick-tock’ of proper time, with the same observation applying hypothetically forthe addition of extra spatial coordinates in equation 6 – along a trajectory no longerconfined to 4-dimensional spacetime as represented in figure 1(b).Alternatively, from the original stationary position of the pedestrian, record-ing the linear progression of proper time intervals δs , for a passive Lorentz boost in4-dimensional spacetime to the perspective of another local frame in uniform relativemotion (which can readily approach the speed of light), and with the local coordinates( x , x , x , x ) now assigned to the new frame, the same time interval δs will againbe expressed in the quadratic form of equation 1 from the new perspective. The fourcomponents { δx a } for a = 0 , , , + (1 ,
3) Lorentz transformations. Similarlyall n local components on the right-hand side of equation 6 are unphysical in that theydepend upon the choice of arbitrary SO + (1 , n −
1) transformations. These transforma-tions however leave the left-hand side invariant. We might then consider equation 6to represent a possible generalisation of equation 1 with both interpreted as possibleexpressions for a proper time interval, that is the one objectively measurable quantityin these equations, which can be arithmetically expressed in such a quadratic form,and hence accorded a corresponding geometric spatial interpretation.15iven then that we can equate proper time with a non-linear quadratic struc-ture for the 4-dimensional external spacetime arena that we do perceive, we might alsoconsider augmentations to more general higher-order homogeneous polynomial formsthat may be utilised by ‘extra dimensions’ that we do not observe in a geometricalsense. This can be achieved by exploiting the basic arithmetic properties of the realnumbers to obtain expressions for δs ∈ R , with this infinitesimal proper time intervalinvariant under a full symmetry group ˆ G that generalises the local Lorentz transfor-mations. Indeed expressions can be written down for ( δs ), ( δs ) , ( δs ) , . . . , or ( δs ) p in general for any power p = 1 , , , . . . , of which equation 6 represents a particularcase for p = 2. Expressions of quadratic order with p = 2 are of significance for di-rectly identifying components with ‘spatial’ properties, as noted for equation 10 and as needed for, and only for, external 4-dimensional spacetime. Hence from the perspectiveof local proper time on the left-hand side, and the extra components on the right-handside, equation 6 can be generalised to a p th -order homogeneous polynomial expression,for p = 1 , , , . . . , in n components { δx a } with each of a, b, c, . . . = 0 , . . . , n − δs ) p = α abc... δx a δx b δx c . . . with each α abc... ∈ {− , , } (11) provided we can extract a specific 4-dimensional quadratic substructure in four compo-nents ( δx , δx , δx , δx ), in the form of the right-hand side of equation 1, as requiredto represent the local geometric structure of the external spacetime M . That is, werequire that equation 11 can in general be written in the form:( δs ) p = h η ab δx a δx b i ( δx , . . . , δx n − ) p − + ( δx , . . . , δx n − ) p (12)where here in the first term a, b = 0 , , , p − th -order polynomial in the remaining ( n −
4) components, whilethe second term, in all components, represents the further p th -order polynomial con-tributions to equation 11. This expression clearly generalises the 4-dimensional formfor proper time in equation 1 and also reduces to the quadratic form of equation 6 asa special case, now interpreted as a possible form of proper time itself.The sense of a linear ‘one-dimensional’ progression in proper time is somethingwe are intimately familiar with. With regards to spatial constructions we can alsoreadily conceive in our mind’s eye of a one-dimensional straight line. In this case wecan picture a second dimension adjoined by a right angle to the first, and in turn athird spatial dimension adjoined at right angles to each of the first two, with eachpair forming a basis for the two quadratically added components of the Pythagoreantheorem. Here the progression ends in terms of our ability to picture such a geometricstructure with a fourth or more spatial dimension, as does our ability to physicallyperceive or explore such a space given the 3-dimensional world we inhabit, as describedabove for the pedestrian exploring the neighbourhood streets.However we can gain some handle on the properties of a fourth dimensionof space and beyond through a purely mathematical augmentation, by incorporatingfurther components into the Pythagorean theorem as for the ( δx ) term and beyondin equations 6 and 10. This is clearly a mathematical possibility, however since ingeneralising beyond a 3-dimensional space we are compelled to employ a mathematicalextrapolation we should consider what the limits are in a purely algebraic , rather thangeometric, sense. For the case of generalising the 4-dimensional spacetime structure16f equation 1 this leads beyond the extra spatial dimensions in the quadratic form ofequation 6 to the more general algebraic expression of equation 11 which is then opento mathematical exploration. In this case we can still extract a local 3-dimensionalspatial structure, as an integral part of the external 4-dimensional spacetime factor inequation 12, forming the basis of the locally Euclidean world that we do physicallyperceive.Essentially we have abstracted the arithmetic composition of equation 1 awayfrom the context of a local inertial reference frame and temporarily neglected thePythagorean geometric significance of the quadratic expression on the right-hand side.This initial arithmetic argument in focussing upon the possible mathematical formsfor a proper time interval δs ∈ R as the chief guide is somewhat disorienting in that theprominence of the geometric structure of the spacetime background has melted away.From the point of view of the flow of time, which is generally conceived of as a linearprogression, the cubic and higher-order homogeneous polynomial expressions for δs implied in equation 11 are just as mathematically permissible and no stranger than thequadratic forms of equations 1 and 6. From this more abstract perspective equation 1 isconsidered to represent a possible arithmetic composition for an infinitesimal interval ofproper time δs ∈ R on the left-hand side that directly generalises to equation 11, with( δs ) p invariant under a full symmetry group ˆ G as a generalisation of the Lorentz groupSO + (1 ,
3) acting on the right-hand side components. We then regain our spacetimeorientation by extracting out from equation 11 a 4-dimensional quadratic substructure,as described for equation 12, with a Lorentz ⊂ ˆ G symmetry as a necessary geometricbasis for the required external 4-dimensional spacetime arena.The underlying shift in focus is towards the continuum of proper time as theobjectively measurable quantity in these expressions. The form of equation 11, poten-tially involving cubic or higher-order homogeneous compositions, is not problematicfor the extra dimensional structures provided that we can extract the 4-dimensionalquadratic spacetime form of equation 1, which underlies the visible external geom-etry of physical 3-dimensional space. However, as described in equation 12 and aswill be explicitly demonstrated in the following subsection, we can readily embed thequadratic spacetime structure of equation 1 within specific higher-order cases for equa-tion 11, just as legitimately as we can within equation 6. Hence the assumption thatgeneralisations from the metric structure of equation 1 should be limited to quadraticforms can be dropped.This generalisation to equation 11, involving the relaxing of an assumptionin augmenting the 4-dimensional spacetime metric form, is in this sense proposed ina similar spirit as for the earliest unified field theories reviewed in subsection 1.2.In the present case the basis is even simpler in that we focus upon generalising theexpression for a proper time interval δs in a local inertial reference frame with the localmetric η ab in equation 1 to that with the coefficients α abc... in equation 11, and hencebegin with a more elementary structure than the global metric g µν ( x ) of equation 2 ofgeneral relativity in the extended 4-dimensional spacetime manifold as incorporatedinto equations 3–5. While the metric g µν ( x ) within equations 3–5 locally reducesto the Lorentz metric η ab in appropriate local coordinates, the Lorentz metric η ab extracted here via equation 12 will be locally equivalent to the metric g µν ( x ) in suchlocal inertial reference frames in 4-dimensional spacetime. This contrasting perspectivewill be discussed further in subsection 5.1 in particular in relation to figure 2.17n order to establish a convenient notation and avoid expressions with infinites-imal elements, and similarly as equation 6 generalises to equation 11, we can in turngeneralise equation 7 by again defining an n -vector v n with the generally finite com-ponents v a = δx a δs (cid:12)(cid:12) δs → , and on dividing both sides of equation 11 by ( δs ) p we define: L p ( v n ) ˆ G := α abc... δx a δx b δx c . . .δs δs δs . . . (cid:12)(cid:12)(cid:12) δs → = α abc... v a v b v c . . . = 1 (13)with each of a, b, c, . . . = 0 , . . . , n − α abc... ∈ {− , , } , while the equalitywith unity on the right-hand side, via equation 11, is simply from ( δs ) p ( δs ) p = 1. In thisequation L p for p = 1 , , , . . . denotes a p th -order homogeneous polynomial expressionin the n components of v n with full symmetry group ˆ G . (Any of the subscripts inthis expression may be dropped if their value is implied from the context, see also thediscussion in [39] between equations 11 and 13 there, although generally this notationwill be manifestly unambiguous in this paper). While the underlying simple conceptualbasis for this theory in terms of generalised proper time is readily made explicit inequation 11, the equivalent expression in equation 13 provides a convenient notation asa basis for the explicit mathematical analysis and physical interpretation of the theory.The kernel symbol ‘ L ’ in equation 13 originates from a consideration of p th -ordermulti L inear forms that might generalise the bilinear metric forms of equations 1, 6and 7, while also having a connection with the role of a conventional L agrangian infield theory as will be described in the following subsection.The symmetry breaking identification of the subcomponents ( x , x , x , x )of equation 12 with a set of local coordinates and the local geometric structure ofthe external spacetime M now corresponds to the projection of the subcomponents v = ( v , v , v , v ) ∈ TM out of equation 13 onto the external tangent space, simi-larly as described for equations 7–9. Indeed equation 7 represents a special case ofequation 13 with: L ( v n ) SO + (1 ,n − = | v n | = ˆ η ab v a v b = 1 (14)while equation 13 allows generalisation for p > n -dimensional vector v n over 4-dimensional spacetime M for this pseudo-Euclidean case depicted as a projec-tion from a 3-dimensional Euclidean space over an embedded 2-dimensional plane), thegeneral form of equation 13 cannot be pictured at all in such a geometrical manner.In fact the necessary extraction of a quadratic substructure, to match the ge-ometry of the locally Euclidean 3-dimensional spatial arena incorporated within thelocally pseudo-Euclidean 4-dimensional external spacetime background against whichall physical phenomena are observed, from a cubic or higher-order form for equation 13might also be interpreted as a central feature of the mechanism for the symmetry break-ing itself, unlike for the uniformly quadratic expression of equation 7 or 14. Howeverthe explicit connection with non-Abelian Kaluza-Klein theories for models with extraspatial dimensions, as alluded to after figure 1 with reference to [40], remains the sameand hinges upon the limit of the local structure in which equation 7 is a particular caseof equation 13, as will also be discussed for equation 25 in the following subsection.The expression in equation 13, equivalent to equation 11, represents the ‘gen-eral form of proper time’, as distinct from a ‘spacetime form’, emphasising the simple18nterpretation of this theory as deriving directly from the basic arithmetic substructureof an infinitesimal interval δs of the continuum of proper time alone. The adjective‘proper’ essentially refers to the invariance of the time interval δs under symmetrytransformations that can be applied to the subcomponents in equation 11 or 13. Viaequations 11–13 expressions for proper time can incorporate the geometric structureof 4-dimensional spacetime as well as the physical structures of matter in spacetimeassociated with the residual components. While this perspective may be unfamiliarthe new theory has a very simple and conservative interpretation in being foundedupon the underlying flow of time which we do intimately perceive rather than uponthe fashionable hypothesis of extra spatial dimensions, over and above a 4-dimensionalspacetime background, which we do not discern at all. For the present theory thereare no extra spatial dimensions of a ‘bulk space’ to be compactified or otherwise hid-den from direct observation, as alluded to in the opening of this subsection, ratherthe properties of the additional components in equation 13 over and above those of4-dimensional spacetime are interpreted directly as matter fields.Despite this underlying simplicity, in generalising from equation 7 to equa-tion 13 on dropping the assumption of a local quadratic p = 2 spatial form for theextra components, we now have a seemingly more complicated structure with the po-tential in principle for both p → ∞ and n → ∞ , while subsuming the p = 2 and n = 4 case for equations 13 and 14 for the external 4-dimensional spacetime. Forthe p = 2 case of equations 7 and 14 any number of dimensions through to n → ∞ can be considered, as discussed following equation 10, although particular structuresfor n -dimensional spacetime are singled out in the context of sophisticated theoreticalframeworks that employ extra spatial dimensions, such as with n = 11 for supergravityand n = 26 or n = 10 for string theory as reviewed in the previous subsection.However for p >
2, as we consider for example possible cubic and quartic formsfor equation 13, particular values for p and n will be intrinsically preferred as uniquemathematical structures which possess a high degree of symmetry, while supplantingequation 1, will be highlighted. In this sense the progression from ‘spacetime forms’to ‘forms of proper time’ is both more general and yet more restrictive, and in amanner that will lead to well-known unification symmetry groups as we shall describe insection 3. By comparison with the elementary analysis for the extra spatial dimensionsin equations 7–9 now applied for the generalised form of proper time of equation 13 thequestion can then be addressed regarding the form of matter fields over 4-dimensionalspacetime that can be deduced for this theory in practice. In the following subsectionwe first consider the features and consequences of a minimal non-trivial generalisationfrom the form of proper time of equation 1 in the manner of equations 11 and 13. A source of homogeneous p th -order polynomial forms for equations 11 and 13 which ex-hibit a high degree of symmetry between the contributions of each component is foundin the determinant function for p × p matrices. With the matrix composition propertydet( AB ) = det( A ) det( B ), for any such square matrices A, B of the same size, thesestructures are also naturally suited for the description of symmetry transformations,via the determinant-preserving multiplication of B by any such A with det( A ) = 1.19s a means of explicitly embedding the 4-dimensional quadratic spacetime form ofequation 1 inside a higher-order homogeneous polynomial form for the proper timeinterval δs we hence first note that there is a standard way of expressing the norm ofa Lorentz 4-vector such as ( δx , δx , δx , δx ) ∈ R in terms of the determinant of a2 × δs ) = η ab δx a δx b = det δx + δx δx − δx iδx + δx i δx − δx (15)Here δs is invariant under the actions of the symmetry group SL(2 , C ) througha 2 × + (1 , × cubic expression in nine components for a propertime interval δs , consistent with equation 11 and now with an augmented SL(3 , C )symmetry, which we can write as:( δs ) = det δx + δx δx − δx i δx + δx iδx + δx i δx − δx δx + δx iδx − δx i δx − δx i δx (16)= h η ab δx a δx b i δx + ( δx , . . . , δx ) (17)In the construction of this cubic form for proper time in equation 16 we em-phasise the deviation from the quadratic structure of extra spatial dimensions, suchas in equation 6, while noting that this minimal augmentation from the 4-dimensionalspacetime form of equations 1 and 15 maintains a balanced contribution from the newcomponents. In equation 17 the same expression of equation 16 is written in the formof equation 12, where here the first term, with a, b = 0 , , ,
3, corresponds to partof a standard cofactor expansion for a 3 × δx , . . . , δx = 0 and δx = δs , similarly as equation 6 reduces to equation 1 on settingeach of δx , . . . , δx n − = 0.On rearranging equation 1 in the form of equation 13 the matrix expression inequation 15 can be written more conveniently as: L ( v ) SL(2 , C ) = η ab v a v b = det( h ) = det v + v v − v iv + v i v − v = 1 (18)with the components of v = ( v , v , v , v ) ∈ R embedded in the 2 × h ∈ h C . As indicated this determinant form is again invariant underthe actions of the symmetry group SL(2 , C ) as the double cover of SO + (1 ,
3) (see for20xample [40] equations 16 and 17). With all four components of equation 18 projectedlocally onto the external spacetime tangent space, with v ∈ TM and no residualstructure, this effectively represents the ‘matterless vacuum’ case ([40] subsections 2.1and 2.2). This 4-dimensional form can be embedded within a 3 × L ( v ) SL(3 , C ) = det( v ) = det v + v v − v i v + v iv + v i v − v v + v iv − v i v − v i v = det h ψψ † n = 1(19)= h η ab v a v b i v − h · ( ψψ † ) = 1 (20)with v ∈ h C , h ∈ h C , ψ ∈ C and here n = v ∈ R (consistent with the notationof [40] equation 19) while a, b = 0 , , ,
3. The second term in equation 20 is the Lorentzinner product h · ( ψψ † ) = tr( h )tr( ψψ † ) − tr( h ψψ † ) between the Lorentz 4-vectorsassociated with h , ψψ † ∈ h C (see for example [53] equations 23 and 70). This cubicexpression in equations 19 and 20 is a specific example of the general form of propertime in equation 13 which, via the first term in equation 20, can be seen explicitlyas an extension from the 4-dimensional spacetime form of equation 18 via a naturalminimal symmetric augmentation from a 2 × × η in a higher-dimensional spacetime metric ˆ η , through the first fourcomponents of the quadratic form in equation 6 or 7, is immediately evident. While thecase here is a slightly more obscured such a 4-dimensional quadratic metric structurecan also be readily embedded in a cubic or higher-order expression in a less obvious,but nevertheless direct, manner as seen for equation 20. In this form equations 19 and20 reduce to equation 18 on setting each of v , . . . , v = 0 and v = 1, similarly asequations 7 and 14 reduce to the form of equation 18 on setting each of v , . . . , v n − = 0.Hence we have no reason to suppose that extra components should not be incorporatedthrough the more general form for proper time in equation 13 with the restriction tothe quadratic form of equation 7 being unnecessary. In either case in augmentingfrom the basic matterless vacuum of equation 18 the symmetry of the generalised formwill be broken through a projection of the local 4-dimensional spacetime substructure.We might then consider the properties of the residual components deriving from thissymmetry breaking for equations 19 and 20, interpreted as a basis for matter fields in4-dimensional spacetime, for comparison with equation 8–9 for the restricted case ofextra spatial dimensions.In beginning this analysis with the fully SL(3 , C )-symmetric 9-dimensional cu-bic form L ( v ) SL(3 , C ) = det( v ) = 1 there are a number of ways that a 4-dimensionalLorentzian substructure could be extracted. However, without loss of generality, fromequations 19 and 20 we can choose the four components originating from equation 18that we have effectively extended about – indeed equations 19 and 20 were constructedin this way in order to explicitly demonstrate that such an extraction is possible. Theseextracted components v = ( v , v , v , v ) ∈ TM are then aligned with the local coor-dinates ( x , x , x , x ) of a local inertial reference frame of the external spacetime M .21n turn a preferred external SL(2 , C ) ⊂ SL(3 , C ) symmetry will act upon these subcom-ponents v ∈ TM of v ∈ h C projected onto the external spacetime from equations 19and 20, which we can then write as: L (cid:30) ( v ) SL(2 , C ) × U(1) = det( h ) n − h · ( ψψ † ) = 1 (21)The extraction of the necessarily quadratic substructure for v ≡ h ∈ h C to describethe geometry of the external 4-dimensional spacetime results more fully in the brokensymmetry SL(2 , C ) × U(1) ⊂ SL(3 , C ), with the kernel symbol L (cid:30) in equation 21 denotingthe broken form.While equation 18 has been subsumed into equation 21 the latter contains theexternal SL(2 , C )-invariant Lorentz 4-vector norm | v | = | h | of the projected fragment v ∈ TM with: | v | = η ab v a v b = h (22)where h = | h | = p det( h ) ∈ R (23)which, unlike equation 18 for the matterless vacuum, is not equal to 1 in general. Beingcentral to the symmetry breaking, and now taking a ‘vacuum value’ h = | h | in theprojection onto TM , the four components of the vector field v ( x ) ≡ h ( x ) ∈ h C of equations 21–23 are associated with a non-standard Higgs in this theory (also forthe further reasons reviewed in [53] after figure 4, as also discussed in the followingsection).In the context of the present theory the components h ( x ) ∈ h C play a piv-otal role in relating the Standard Model of particle physics and the general relativistictheory of gravitation by connecting the ‘origin of mass’ in these two frameworks. Vari-ations in the value of h ( x ) in equation 22 in the projection out of equation 13, for thegeneral case, are associated directly with a local warping of the external 4-dimensionalspacetime geometry as can be expressed by the Einstein tensor G µν ( x ) (see discussionof [54] figure 13.1 and equations 13.2–13.4). For the present theory this is proposed tounderlie the physical property of mass through a contribution to the energy-momentumtensor T µν ( x ) via Einstein’s field equation 43 (discussed in section 5 here), with ([54]equation 13.4) written for G µν ( x ) as a function of h ( x ): G µν = − h − ∂ ρ h ∂ ρ h g µν − h − ∂ µ ∂ ν h + 2 h − (cid:3) h g µν =: − κT µν (24)with ρ, µ, ν = 0 , , , δh ( x ), are on ascale set by the vacuum value for h ( x ) in equation 22.For the case of equation 21 the full U(1) group manifold is incorporated ([40]subsection 2.3 in particular figure 3(b)) in place of the group SO( n −
4) as describedfor equation 8–9 after figure 1. This structure will further generalise for full symmetrygroups larger than ˆ G = SL(3 , C ) in equation 13, with a residual internal gauge sym-metry G , in general larger than U(1), related to the geometry of the base space M in a principal fibre bundle structure analogous to that of non-Abelian Kaluza-Kleintheories ([40] subsection 4.1 in particular points ‘a)–e)’). Specifically, the Einstein ten-sor G µν ( x ) can be related to the gauge curvature components F ρσα ( x ), where ρ, σ arespacetime indices and α is a Lie algebra index for the gauge group G , and considered22s a further source of energy-momentum with ([40] subsection 4.2 equation 93, with χ considered a normalisation constant): G µν = 2 χ ( − F αµρ F ρνα − g µν F αρσ F ρσα ) =: − κT µν (25)again explicitly describing a direct warping of the 4-dimensional spacetime manifoldand a corresponding form of energy-momentum. The dynamics of the gauge fields areproposed to be determined in turn by geometric constraints such as Bianchi identities(see discussion of [40] equations 93 and 94 and the accompanying references).For equation 21 with SL(2 , C ) being the external symmetry the residual in-ternal symmetry U(1) can be interpreted as a gauge group underlying a theory ofelectromagnetism alongside gravitation ([40] subsection 4.2), with equation 25 thendescribing the energy-momentum of the electromagnetic field. In this sense this min-imal extension from equation 18 to the cubic form of equation 19 is analogous to theearly unified field theories reviewed here in subsection 1.2. There we described howWeyl’s original geometric ‘gauge theory’, with the scaling factor λ for the metric g µν ( x )in equation 3, was superseded by a U(1) gauge theory for electromagnetism indepen-dent of the external metric structure. Here we have described how a U(1) gauge theory can be incorporated through an augmentation of the local spacetime metric η ab viathe structure of equations 19 and 20, considered as cubic form for proper time.In addition to the fragment of equations 22 and 23 at the elementary local levelthe broken symmetry reduces the full 9-dimensional vector space h C to three partswith the Lorentz SL(2 , C ) and U(1) factors acting upon these subcomponent partsintroduced in equation 19 as ([40] subsection 2.3, [53] subsection 4.1):SL(3 , C ) → SL(2 , C ) × U(1) matter: (26) h ∈ h C : vector 0 : ‘Higgs-like’ role in TM v ∈ h C → ψ ∈ C : L -spinor 1 : charged spinor over M n ∈ R : scalar 0 : neutral scalar over M (27)with the 2-component Weyl spinor ψ taken to be left-handed by convention as denotedby the prefix ‘ L -’ above. A distinct feature of this unification scheme is the directand natural manner in which spinor components such as ψ arise in the local symmetrybreaking structure, unlike the typical case for non-Abelian Kaluza-Klein theories whichrequire a specific additional extension – for example via supersymmetry as alluded toin subsection 2.1. Hence here not only can the symmetry breaking pattern be linkedwith a gauge field A µ ( x ) for electromagnetism via the internal symmetry U(1), butthis gauge group also acts non-trivially upon the spin- field ψ ( x ) in spacetime, asindicated by the normalised unit charge ‘1’ in equation 26–27.These structures deriving from the residual components and symmetry of equa-tion 19 as projected over 4-dimensional spacetime to the broken form of equation 21then provide a framework for electrodynamics incorporating a charged Weyl spinor.Given the ambition to ultimately account for properties of the Standard Model, witha range of spinor states for the charged leptons and quarks and also neutral spinors inthe form of neutrinos, there is here then the potential for accommodating such statesthrough further augmentations of the form of proper time. Hence equation 26–27clearly provides a better starting point for this goal than the equivalent analysis of23quation 8–9 as applied for the restricted quadratic case of extra spatial componentsin equation 7. We shall return to a possible interpretation for the neutral scalar field n ( x ) ∈ R of equation 26–27 in the next subsection and in particular in subsection 4.2In general the local symmetry breaking projection of v ∈ TM ( ≡ h ∈ h C )in equations 22 and 23 out of the full set of components for the n -dimensional formof equation 13 partitions the components of v n ∈ R n into subsets of subcomponentpieces that transform under irreducible representations of the subgroup:Lorentz × G ⊂ ˆ G (28)where the external local Lorentz symmetry group for 4-dimensional spacetime can beSO + (1 ,
3) or its double cover SL(2 , C ), the group G is the internal gauge symmetry andˆ G is the original full symmetry, as listed for the ˆ G = SL(3 , C ) case in equation 26–27(and also discussed in [40] subsection 2.3 for equation 23 there). At the same timethe corresponding form L p ( v n ) ˆ G = 1 of equation 13, which is invariant under ˆ G , canbe expanded and partitioned into subsets of terms with each part invariant under theLorentz × G broken symmetry of equation 28 as: L (cid:30) p ( v n ) Lorentz × G = X (invariant parts) = 1 (29)The individually invariant parts in equation 29 which contain a factor of h ,or a scalar combination of components such as | h | in equation 23, are proposed tobe associated with ‘mass terms’ in an effective Lagrangian deriving from the the-ory, in part motivating the kernel symbol ‘ L ’ in equations 13 and 29. For examplewhile L ( v ) SL(3 , C ) = det( v ) = 1 of equation 19 is invariant under the full symme-try ˆ G = SL(3 , C ), each of the two terms in equation 21 is invariant under the brokensymmetry SL(2 , C ) × U(1). In this case the two terms each contain a factor in thecomponents of h ( x ) and might ultimately be interpreted as mass terms for the fields n ( x ) and ψ ( x ) in spacetime. That is, such terms provide a source of field interactionssuch as δψ ( x ) ↔ δ h ( x ) that can perturb the external spacetime geometry in a mannerthat is proposed to generate the property of mass as described for equation 24.While the components of h ≡ v ∈ TM are composed with other fields inthe terms of equation 21 in manner that begins to resemble Lagrangian mass terms acloser correspondence will require a more complete theory with more components in ahigher-order form of proper time for equations 13 and 29, as will be discussed furtherin section 4. Such a construction is possible here for equation 13 unlike the case for therestricted quadratic forms of equations 7 and 14, similarly as spinor states are also nowreadily identified as described above. Indeed for higher-order forms for equation 29there is the potential for spinors to be composed in terms incorporating not only aneffective Higgs but also further factors that might act as a source of Yukawa couplingsfor possible mass terms, as will be described in subsection 4.2.Equation 29 will also act as a constraint on dynamical expressions for matterfields in 4-dimensional spacetime, yielding equations of motion with explicit interac-tions between gauge fields and spinor fields for example (see discussion of [54] equa-tions 5.51 and 11.33). As described earlier in this subsection the symmetry breakingprojection of equation 13 over 4-dimensional spacetime also has physical consequencesthrough the relations of both equation 24 and 25. Collectively the set of constraintsfor the full structure of the theory will subsume the role of effective Lagrangian terms24n determining the detailed empirical properties of matter at the most elementarylevel (see discussion of [54] equation 11.29 and table 15.1), as will be described furthertowards the end of subsection 5.2.For this theory there is no ( n > x , x , x , x ) underlying v ∈ TM as theprojected quadratic 4-dimensional part of equation 13 are utilized as implicit localcoordinates in defining the local structure of an extended spacetime manifold M .The additional components in an expression for a proper time interval, such as equa-tion 19, are directly associated with matter fields in spacetime, as explicitly seen forthe ψ ( x ) ∈ C components of equation 26–27. The identification of such matter fields,which might be described in terms of an ‘associated fibre bundle’ related to the prin-cipal fibre bundle constructed for the internal symmetry G of equation 28, followsdirectly from the identification of the distinguished external 4-dimensional spacetimebase space M . The only physical space is this external spacetime M , upon whichextended geometric structure and energy-momentum can be defined as for examplethrough equations 24 and 25.The symmetry breaking hence revolves around the necessary choice of a pre-ferred Lorentz ⊂ ˆ G subgroup symmetry in equation 28 that acts upon a 4-dimensionalquadratic substructure of equation 13 that is identified with the local external space-time geometry. This necessary identification and extraction of the geometric structureof the spacetime manifold M itself, for example via the local ( x , x , x , x ) compo-nents of equations 16 and 17, is inextricably linked with a complete distinction betweenthe external and internal components that hence applies for all physics that can bedefined in spacetime. In turn the full symmetry ˆ G of equation 13, with which we beginin the mathematics of the theory as for example with ˆ G = SL(3 , C ) in equation 19, isbroken absolutely to the product of the external Lorentz, or SL(2 , C ), symmetry andinternal G symmetry, as for G = U(1) in equation 26 or for the general case of equa-tion 28, as a basis for the analysis of physical structures in 4-dimensional spacetime.Hence while the mathematics of the theory begins with equation 13 the physics beginswith equation 29. There are no surviving symmetries that mix subcomponents of v n transforming under different representations of the external Lorentz group.The group product structure for the external and internal symmetries in equa-tions 28 and 29 is consistent with the demands of the Coleman-Mandula theorem [55]ultimately for the relativistic quantum theory limit ([40] subsection 5.3). That is, sim-ilarly as a physical model that begins with 4-dimensional spacetime M and posits theSL(2 , C ) × U(1) symmetry and field structure of equation 26–27, without reference toSL(3 , C ), would be compatible with the Coleman-Mandula theorem, the same conclu-sion applies for the present theory in which these structures, as the starting point forphysics, derive from the fundamental origin of the mathematical form of proper timein equation 19 through the necessary absolute symmetry breaking in the identificationof the base space M itself.These observations apply for the general case of equation 13 resulting in equa-tion 28 and also for the restricted quadratic case of equation 14 resulting in the brokensymmetry of equation 8, and is hence similar to an argument that could be made forsome unification schemes based upon extra spatial dimensions – through the necessary25xtraction of a distinguished 4-dimensional base space M from a more uniform higher-dimensional structure, as depicted for the case of equation 14 in figure 1(b). Howeverthere are also significant differences, with for example additional assumptions neededto incorporate spinor fields in models with extra spatial dimensions as we have notedfollowing equation 26–27.The Higgs, U(1) gauge theory and spinor physics alluded to above for equa-tions 22–24, 25 and 26–27 respectively was unknown in 1918 when unified field theoriesbased on extending general relativity were first proposed. The Higgs mechanism forsymmetry breaking was developed much later in the 1960s (see for example [24] sec-tion 21(e) part 4). Even particle spin was not discovered until 1925, with the Paulimatrices introduced in 1927 and the relativistic Dirac equation for 4-component spinorsfollowing in 1928 ([24] chapter 13). As described in subsection 1.2 an understandingof the gauge principle to obtain a theory of electromagnetism culminated in 1929 [28],incorporating an application of 2-component Weyl spinors. Hence the structures ofequation 26–27 would not have been natural to consider as a possible extension fromgeneral relativity in the years immediately following 1915. Also from 1925 the de-velopment of the principles of spin and of gauge theory were inextricably linked withdevelopments in quantum mechanics led by Heisenberg, Schr¨odinger and others – thecomprehension of which itself became the focus for theoretical activity from that time,as also noted in subsection 1.2.Although founded at an elementary level the nature of the generalisation fromthe form of the local 4-dimensional spacetime geometry for a proper time interval inequation 1 according to that proposed in equations 11 and 13 might also have seemedinappropriate during the period straight after the publication of general relativity. Atthat time, around one hundred years ago, relatively minimal extensions to generalrelativity were sought, with only electromagnetism to incorporate then alongside grav-itation, while the general form of equation 13 allows for much broader and more openpossibilities. Now with the benefit of hindsight, afforded not only by the empiricalknowledge accumulated in the rich properties of the Standard Model but also by themodern-day understanding of particular mathematical symmetry structures that ex-emplify equation 13 in a manner naturally subsuming equation 18, direct progress canbe made.We can then explore the possibilities for a generalised proper time intervalbeyond the initial step of equation 19 to determine and assess the nature of furtherphysical structures beyond those of equation 26–27 that might be uncovered for thistheory. The properties of the new matter fields, again obtained through a symmetrybreaking projection of the subcomponents v ∈ TM locally onto the 4-dimensionalexternal spacetime, will be reviewed systematically in the following section. Of par-ticular interest will be to observe the extent to which these natural mathematicalstructures dovetail with the empirical features of the Standard Model within the con-text of the conceptual scheme of the theory presented here based upon generalisedproper time, which might then provide a firm foundation for the deduction of newphysical phenomena as will be explored in section 4.26 Exceptional Lie Groups and the Standard Model and E While the Lie algebras, including the five exceptional cases of G , F , E , E and E ,were classified by Killing and Cartan in the late 19 th century (see for example [56]section 4 opening) an understanding of explicit expressions for certain representationsof these algebras and their corresponding Lie groups developed from the mid-20 th and continues into the 21 st century. Elements of the space of 3 × O , which with an octonion incorporating eight real components is27-dimensional over R , comprise the ‘exceptional Jordan algebra’ as first described in1934 ([56] section 3). The smallest non-trivial representation of the Lie algebra E is27-dimensional and was described in terms of transformations on the 27-dimensionalspace h O in 1950 [57]. These actions correspond to E − , one of the non-compactcases of the five real forms for E , and preserve a cubic norm, or determinant, definedon h O . The corresponding explicit E − ≡ SL(3 , O ) group action on elements ofh O has been described in detail more recently (see for example [58] and referencestherein, as reviewed in [54] chapter 6), making intrinsic use of the octonion algebracomposition properties.In the context of the present theory, while we obtained equation 19 from equa-tion 18 by a natural minimal symmetric extension from the 2 × × C to the octonions O to obtain the cubic 27-dimensionalnorm: L ( v ) E = det( v ) = det X θθ † n = 1 (30)= det( X ) n − X · ( θθ † ) = 1 (31)with v ∈ h O , X ∈ h O , θ ∈ O and again here n ∈ R (in line with conventions inthe main references as for equations 19–21, see also [53] equations 25 and 72). TheSL(3 , C ) symmetry of L ( v ) SL(3 , C ) = 1 in equation 19 is augmented correspondinglyto SL(3 , O ) ≡ E − . Here equation 31, while of a similar form to equation 21,will break down further, for the broken form of equation 29 at this ˆ G = E level, toan augmented set of parts each invariant under the broken symmetry. The symmetrybreaking now results from the extraction of an external Lorentz 4-vector v ≡ h ∈ h C from a set of subcomponents of X ∈ h O in equations 30 and 31 as projected ontothe external local tangent space of the 4-dimensional spacetime M . On identifyingan external Lorentz symmetry SL(2 , C ) ⊂ E − acting upon the external 4-vector v ∈ TM a symmetry breaking pattern can also be identified for the components of v ∈ h O , as an augmentation from equation 26–27, as we shall summarise below.As a normed division algebra with the octonion norm compatible with octo-nion multiplication, in that | ab | = | a || b | for any a, b ∈ O ([53] equation 1) similarly asfor the composition of matrix determinants described in the opening of subsection 2.3,27he octonions naturally describe symmetry transformations, for example via the norm-preserving multiplication of b by any a with | a | = 1. Being non-associative octonioncomposition can in fact be employed to express a high degree of symmetry with rela-tively few elements ([54] section 6.2). However since the non-associativity property iscounter to the basic axioms of group theory, unlike the case for matrix algebras, careis needed in applying a standard analysis for symmetry breaking where octonions areinvolved since there may be some deviation from a direct computation via the technicaltools developed to study Lie groups generally.Indeed there are subtle differences between the explicit analysis for the realform Lie group E − that here intrinsically involves the octonion composition asan instantiation of equation 13 and the parallel Dynkin analysis for the complex Liealgebra E (as described for [53] tables 1 and 2 respectively). The full explicit analysisis presented in detail in ([54] chapters 6 and 8, [39] sections 4 and 5, [53] subsection 4.2)with the resulting branching properties for the subcomponents of equation 30 obtainedas: E − → SL(2 , C ) × SU(3) c × U(1) Q matter: (32) X ∈ h O : vector ν -lepton / h scalar : u -quark v ∈ h O → θ ∈ O : L -spinor e -lepton L -spinor : d -quark n ∈ R : scalar n (33)With the complex numbers C and octonions O being respectively 2-dimensionaland 8-dimensional over R a consequence of this fourfold increase is that there are nowfour spinor states identified in equation 32–33 compared with the single spinor ofequation 26–27. In contrast to the parallel Dynkin analysis here as a feature arisingfrom explicit use of the octonion algebra structure in the symmetry transformationsall four spinors have the same chirality, taken to be left -handed by convention, asdemonstrated explicitly in ([54] equations 8.10–8.13).Through this natural augmentation from equation 26–27 we also find an in-ternal non-Abelian symmetry, which is proposed to correspond to the colour gaugegroup SU(3) c of quantum chromodynamics (QCD), alongside the original Abeliangauge group of electromagnetism, now denoted U(1) Q . The three SU(3) c singlet parts for the E − symmetry breaking in equation 32–33 are subsumed from theSL(3 , C ) case of equation 26–27 while the SU(3) c triplet parts are new. The patternof U(1) Q fractional relative charge magnitudes listed in equation 32–33, deriving di-rectly from the intrinsic mathematical structure of the E − group action, as alignedwith the SU(3) c singlets and triplets leads to the provisional matter field interpreta-tion of the subcomponent decomposition of v ∈ h O under the broken symmetryas representing a generation of Standard Model leptons and quarks, as listed in thefinal column of equation 32–33. The appropriate features are here determined by thealgebraic structure without needing to be introduced by hand.28hile a full electroweak SU(2) L × U(1) Y symmetry, with L signifying a non-trivial action on left-handed spinors and Y denoting hypercharge, is not identified atthis stage there are SU(2) ⊂ E − symmetries that act between the real subcom-ponents of the three octonion elements of h O in equation 30 in a manner resemblingproperties of the Standard Model SU(2) L doublets (cid:0) νe (cid:1) L and (cid:0) ud (cid:1) L which, along withthe SU(3) c × U(1) Q transformation properties in equation 32–33, further justifies theassociation of the ν -lepton and u -quark states with particular components ([54] sub-section 8.3.1). Given that with respect to the external Lorentz SL(2 , C ) symmetry the e -lepton and d -quark states in equation 32–33 transform uniformly as a set of four2-component left-handed Weyl spinors the identification of complete doublets of aninternal SU(2) L symmetry would require the ν -lepton and u -quark states also to trans-form as left-handed spinors. However at this stage for equation 32–33 the ‘ u -quark’components transform as Lorentz scalars while the neutral components most directlyassociated with the neutrino, with respect to the internal symmetries, are incorporatedinto a Lorentz 4-vector.Compounding these discrepant features, this natural slot for the ‘ ν -lepton’ inequation 32–33 is in fact already occupied specifically by the Lorentz 4-vector h ∈ h C subcomponent of X ∈ h O , which is projected onto the tangent space of the externalspacetime with h ≡ v ∈ TM and associated with the Higgs as discussed after equa-tion 23 and similarly as listed in equation 26–27. In fact the partial identification ofelements of an SU(2) L × U(1) Y electroweak symmetry breaking structure as alludedto above further motivates the association of the h ∈ h C components with the Higgs([54] section 8.3), as will be discussed further in subsection 4.2. This ambiguity inthe assignment of the corresponding components, indicated by the ‘ ν -lepton/ h ’ entryin equation 32–33, already at this E − stage suggests an intimate link betweenneutrino and Higgs physics in the context of this theory.Despite these caveats the above overall observations with the identification nowof a set of four Weyl spinors and the overall SU(3) c × U(1) Q representation patternin equation 32–33 are encouraging for this direct development from the simple under-lying basis of the theory in generalised proper time that motivated equations 11 and13. Through this E − symmetry a connection has also been established with theexceptional Lie groups, which are known to be of interest for unification as noted insubsection 1.2. This leads in turn to the potential for further natural extension beyondthe E symmetry as we describe below.The smallest non-trivial representation of the next largest exceptional Lie groupE , specifically E − of the four real forms, can be described by an action on the56-dimensional space of the ‘Freudenthal triple system’ F (h O ), investigations of whichhad begun by 1954 ([59] section 4.11), as again has been studied in detail more recently(see for example [60] and references therein). The E − action preserves a homoge-neous quartic norm q defined on the space F (h O ) which, while not being expressed asa matrix determinant function itself, contains an E − ⊂ E − subgroup action onthe 27-dimensional cubic norm of equation 30 and hence can be considered a furthernatural augmentation consistent with equation 13. This quartic 56-dimensional formfor proper time, identified with the norm q , can be written explicitly as: L ( v ) E = q ( v ) = − αβ − ( X , Y )] − α det( X )+ β det( Y ) − ( X ♯ , Y ♯ )] = 1 (34)where v ≡ ( X , Y , α, β ) ∈ F (h O ), the bilinear form ( X , Y ) is the trace of the Jordan29roduct of X , Y ∈ h O and the quadratic adjoint map X ♯ is also defined in [60], while α, β ∈ R (see also [53] equations 30 and 63). The components of the 27-dimensionalrepresentation of E − in equation 30 can be associated explicitly with subcompo-nents in equation 34 via v ≡ X ∈ h O , here with det( X ) = 1 in general, whilethe 27-dimensional complex conjugate representation of E − corresponds to the Y ∈ h O subcomponents of F (h O ). Given this straightforward embedding of thesubgroup E − ⊂ E − action the main consequences of this further augmentationcan be inferred directly from equation 32–33. However, there is still only one set offour real subcomponents to project onto the external 4-dimensional spacetime with v ∈ TM in equation 22 which, without loss of generality given an arbitrary choice,we now extract as v ≡ h ∈ h C from the Y ∈ h O subcomponents. The resultingbreaking pattern for the E − symmetry of equation 34 is then determined ([54]section 9.2, [39] section 6, [53] subsection 4.3) as summarised here:E − → SL(2 , C ) × SU(3) c × U(1) Q matter: (35) X ∈ h O : vector ν L ’scalar : ‘ u L ’ L -spinor e L L -spinor : d L scalar n v ∈ F (h O ) → Y ∈ h O : vector h scalar : ‘ u R ’ R -spinor e R R -spinor : d R scalar Nα, β ∈ R : scalar α, β (36)where the upper sector for X ∈ h O is essentially a copy of equation 32–33. Theimmediate consequence of this augmentation in the symmetry of equation 13 throughthe exceptional Lie groups from ˆ G = E to ˆ G = E is the incorporation of right -handedspinor states as well as the original left -handed states, via the inclusion of componentscorresponding to the complex conjugate 27-dimensional representation of E . Thatis, in addition to the four left-handed spinors of equation 32–33 a corresponding setof four right-handed spinors is identified in the Y ∈ h O subcomponents, with the X and Y components of equation 35–36 hence referred to respectively as the ‘left-handed’and ‘right-handed’ sectors of the theory. With the internal symmetry transformationsbeing the same for both sectors, the 2-component Weyl spinors for the e L and d L statesin equation 32–33 have been augmented to 4-component Dirac spinors (cid:0) e L e R (cid:1) and (cid:0) d L d R (cid:1) in equation 35–36. Provisionally anticipated through the study of SU(2) ⊂ E − doublet actions alluded to in the discussion of equation 32–33 corresponding L and30 subscripts are also added to the ‘ ν -lepton’ and ‘ u -quark’ states in equation 35–36,albeit within quotation marks since the need to identify an explicit Lorentz spinorstructure for these states remains and will require yet further augmentation beyondthis E stage.Nevertheless the branching patterns obtained for this elementary symmetrybreaking for natural augmentations of the form of time L p ( v n ) ˆ G = 1 of equation 13,through equations 19, 30 and 34, over the local structure of 4-dimensional spacetime viathe projected fragment of equation 22, leading to equations 26–27, 32–33 and 35–36respectively, achieve far more success in terms of the direct emergence of StandardModel properties than the equivalent case of extra spatial dimensions described forequations 7–9. In all cases we are uniformly applying the simplest symmetry breakingscheme around the extraction of the external v ∈ TM subcomponent part, as wasdepicted for equations 7–9 in figure 1(b). It is particularly striking that these resultshave been obtained by dropping the unnecessary assumption that extra componentsshould have the ‘quadratic spatial’ form of the Pythagorean theorem at an elemen-tary level, rather than by adding new structures specifically tailored to accommodateStandard Model features, as are required for models with extra spatial dimensions asdiscussed in subsection 2.1.In addition, while there is some overlap between the Standard Model propertiesidentified in equation 35–36 and features known to arise from the abstract mathemati-cal analysis of the symmetry breaking patterns for certain candidate unification groupssuch as E and E as alluded to in subsection 1.2, here we do have a clear underly-ing conceptual origin for the significant role played by these exceptional Lie groupsas symmetries of generalised forms of proper time. As a distinct feature, comparedwith a unification of the internal symmetry alone for the GUT models of [32, 33] forexample, here we are generalising from the local structure of 4-dimensional spacetimeand begin by identifying the external Lorentz symmetry as a subgroup of E and E ,with residual structures then identified as matter fields in spacetime.While additional structures beyond those motivated directly by the conceptualbasis of the theory have not been added by hand, we also find that Standard Modelfeatures are identified very efficiently in equation 35–36 with very little redundancy ofcomponents. Indeed of the 56 real components in equation 34 only four, { n, N, α, β } ,have not been utilised for the above correspondence with Standard Model structures –with N a subcomponent of Y ∈ h O corresponding to the n subcomponent of X ∈ h O as incorporated from equation 30, while α and β are two further new components.Potentially looking beyond the Standard Model we note that at this stage these fourcomponents, as the scalar invariants { n, N, α, β } listed in equation 35–36, providecandidates for dark matter or even a source for ‘dark energy’ phenomena in this theory.These augment the original single scalar invariant n ∈ R of equations 26–27 and 32–33and will be discussed further in subsection 4.2 where an alternative interpretation ofsuch components as Yukawa couplings will also be considered.Indeed further structure is still needed to describe the complete Standard Modelitself. In addition to the required spinor structure for the ν -lepton and u -quark statesin equation 35–36 the principal Standard Model symmetry and particle multiplet fea-tures that remain to be accounted for are that of a full electroweak theory, with anSU(2) L × U(1) Y symmetry that breaks to U(1) Q , and a full three generations of lep-tons and quarks. We note however that while a larger unification group beyond E may incorporate appropriate SU(2) and U(1) internal factors in the symmetrybreaking we do already possess a natural explanation for a left-right asymmetry, whichwill be needed for the SU(2) L factor of a complete electroweak theory. The left-rightasymmetry is a significant feature of particle physics that in general is very difficult toaccount for in an uncontrived manner in model building (see for example [61] as dis-cussed for [53] equation 65). The empirical asymmetry between left and right-handedstates is particularly conspicuous in the neutrino sector, with such properties againusually imposed by hand in neutrino models as described in subsection 1.1.Here an intrinsic left-right asymmetry is implied in the full symmetry break-ing through the necessary choice of a preferred set of subcomponents v ∈ TM projected onto the local 4-dimensional external spacetime, upon which a unique ex-ternal SL(2 , C ) ⊂ E − subgroup acts. The asymmetry arises as v ∈ TM ( ≡ h ∈ h C ) is necessarily projected out from either the left-handed sector X ∈ h O or the right-handed sector Y ∈ h O components of v ∈ F (h O ), as described beforeequation 35–36. We can observe at this stage for equation 35–36 that this left-rightasymmetry is indeed particularly marked for the neutrino states since the embeddingof the external 4-vector v ≡ h ∈ h C , associated with the Higgs, within the Y ∈ h O components prohibits the accommodation of a neutrino state ‘ ν R ’ in the right-handedsector while implying that the corresponding slot is now available for a neutrino state‘ ν L ’ in the left-handed sector associated with the components of X ∈ h O , withoutthe conflict described for equation 32–33. The theory is hence in principle able to pro-vide a natural framework for left-right asymmetric properties in the neutrino sectoras an intrinsic feature of the symmetry breaking structure, as we explore further insubsection 4.1.Although the analysis of equation 35–36 strongly suggests that the theory isprogressing in a favourable direction, since none of this structure has been pragmati-cally tailored for this purpose a perfect fit to Standard Model structures is not neces-sarily to be expected until the full picture has been established. While additional ele-ments of electroweak theory and Higgs physics have been partially identified as notedearlier in this subsection (with reference to [54] section 8.3, see also [39] section 5)and the required spinor structures for equation 35–36 might be contrived through theintroduction of further components ([54] section 9.1) the necessary threefold augmen-tation to account for two additional generations of leptons and quarks clearly requiresa substantial extension, and in all cases ideally via a further natural mathematicalaugmentation. Such a further possible extension for the generalised form for propertime of equation 13 beyond the E and E symmetries described in this subsection willbe considered in that to follow. The intrinsic preference for certain values for the polynomial order p and the numberof components n for the general form of proper time in equation 13, as suggested atthe end of subsection 2.2, has been explicitly demonstrated in subsections 2.3 and 3.1.The augmentation from the matterless vacuum case with p = 2 , n = 4 in equation 18with the local Lorentz symmetry of 4-dimensional spacetime to the p = 3 , n = 9 case ofequation 19 with SL(3 , C ) symmetry was obtained by a minimal symmetric extension32rom a 2 × × × p = 3 , n = 9 form of proper time to the case of p = 3 , n = 27 in equation 30 with an E symmetry at the 3 × C to theoctonions O , with a compact internal symmetry obtained as presented in equation 32.This means of extension in turn terminates here since the octonions are uniquely thelargest normed division algebra ([56] sections 1 and 1.1).However, while leaving behind matrix and division algebra extensions, from theproperties of exceptional Lie groups a further natural augmentation from p = 3 , n = 27to p = 4 , n = 56 in equation 34 has been identified through the embedding of E withinthe E symmetry action on the components of this quartic norm. In considering thepossibilities for further extension a further progression from the E symmetry to E as uniquely the largest exceptional Lie group is naturally suggested, potentially termi-nating the series of augmented forms for equation 13 with a high degree of symmetrythat might be of most significance for physics.This line of argument hence converges with the latter end of the well-knownsequence of unification groups SU(5) → SO(10) → E → E → E , alluded to in sub-section 1.2, which are linked by a progression of augmented Dynkin diagrams that alsoterminates uniquely in E (see for example [54] figures 7.2(c,b,a) and 9.1(a,b,c)). Whilein most cases these groups are employed in Grand Unified Theories, accommodatingonly the Standard Model gauge symmetry SU(3) c × SU(2) L × U(1) Y , for the presenttheory we initially incorporate the external Lorentz or SL(2 , C ) symmetry and then in-clude an internal gauge symmetry in this unification. Indeed here a natural connectionhas been made with the unique high-symmetry structures of the exceptional Lie groupsE and E through a generalisation from the quadratic form of 4-dimensional space-time and extra spatial dimensions by exploring higher-order homogeneous polynomialforms of proper time as described in the previous subsection.The case for an extension to an E symmetry is further strengthened on notingthat the three largest exceptional Lie groups E , E and E are also known to describea sequence of symmetries acting on structures that can be interpreted as ‘generalisedspacetimes’, in particular based on the space h O ([62] equations 64, 66 and 67),and with those same E − and E − actions interpreted here as symmetries of‘generalised proper time’ for the cubic and quartic expressions of equations 30 and 34respectively.Hence with E as uniquely the largest exceptional Lie group the above obser-vations lead to the proposal of a homogeneous polynomial form: L ( v ) E = 1 (37)as the ultimate instantiation for equation 13, as originally suggested in ([54] section 9.3,[39] section 7) and considered in detail in [53]. The progression through E − andE − suggests the symmetry action of E − , one of the three real forms of E , inequation 37. This provisional form is potentially of octic order with p = 8 (see for33xample [63]), and a close connection with the smallest non-trivial E representationwith n = 248 dimensions is here assumed; although other values for p and n mightbe conceivable. The nature of this structure and the plausibility of encompassing theprincipal remaining Standard Model features required (as summarised at the end ofthe previous subsection) in a correlated manner is the main focus of [53].As a unification symmetry the Lie group E itself is comfortably able to incor-porate a broken symmetry corresponding to a product of the external Lorentz SL(2 , C )and internal Standard Model gauge groups in the form of equation 28 with:SL(2 , C ) × SU(3) c × SU(2) L × U(1) Y ⊂ E (38)While the external and internal symmetries derive from the same single unifying math-ematical source in E the absolute nature of the above symmetry breaking prior to thederivation of any physics in 4-dimensional spacetime is again compatible with the re-quirements of the Coleman-Mandula theorem for the QFT limit, as also for the E andE levels of the previous subsection and as discussed towards the end of subsection 2.3.On the other hand as an extension from the representation of E underlyingequation 32–33, as combined with the complex conjugate for equation 35–36, inbroad terms a possible factor of three for three generations of leptons and quarksis suggested by the factors of ( – ) in the subgroup embedding of E ⊂ E with therepresentation branching pattern ([53] equation 68):E ⊃ E × SU(3) : → ( , ) + ( , ) + ( , ) + ( , ) (39)However, as also explained in [53], unlike the case for the direct embedding ofthe subgroup E ⊂ E action described for equation 34 the embedding of E and E in the E action for the form L ( v ) E = 1 that we are seeking is expected to be lessstraightforward than that suggested by equation 39 if the needed spinor structures for ν -leptons and u -quarks under SL(2 , C ) together mutually with a complete electroweaksymmetry action under SU(2) L × U(1) Y are to also be identified compatible with thesymmetry breaking pattern of equation 38. It is possible for example that features ofthe full Standard Model may be more closely aligned with a different maximal subgroupembedding such as E ⊃ F × G , in terms of the two other exceptional Lie groups(as discussed for [53] equation 81), or another possible algebraic decompositions of E (such as reviewed in [56] section 4.6). The central ambition is to identify an explicitstructure for equation 37 and supplant equation 35–36 with a full matter field listingunder equation 38 for the E case.It is also known that the structure of the 248-dimensional E Lie algebra itselfexhibits some correlation with the full symmetry structure of the Standard Model [34],albeit with seemingly prohibitive flaws, as alluded to in subsection 1.2. While a fullthree generations of ‘leptons’ and ‘quarks’ are identified in the E Lie algebra struc-ture within that analysis there is an irreconcilable inconsistency in the representationsunder the Lorentz and electroweak symmetry subgroups as explained in [34, 64]. Thatdifficulty may be related to a key issue for the present theory regarding the needto identify further Lorentz spinors and a full electroweak theory in augmenting fromequations 32–33 and 35–36 for the proposed E level.However rather than reading off particle states directly from the abstract com-position of the complex E Lie algebra structure and its representations, which may be34iagrammatically illustrated in an aesthetically appealing manner ([34] figures 2–4),here we seek an explicit E symmetry action on a homogeneous polynomial form asdescribed for equation 37 as a unique expression for equation 13 with a high degree ofsymmetry and consistent with the simple underlying motivation of this theory, basedupon the generalisation of proper time as described for equation 11. In particular asan augmentation from the E − action for equation 30 and the E − action forequation 34 the proposed E − symmetry action on the components of v ∈ R for equation 37 is expected to incorporate octonion composition in an essential way.As noted before and after equation 32–33 the application of the octonion algebra inthis way implies subtle but significant differences compared with a standard analysisfor the abstract structures of the corresponding complex Lie algebra, which hence can-not be fully relied upon for a rigorous assessment here. This implies that the presenttheory, while pointing towards a full unification based on E or a very closely relatedstructure, is not explicitly constrained by the prohibitive conclusions of [64].The octonion composition, exhibiting properties such as ‘triality’, is anticipatedto be at the heart of the unravelling of the full Standard Model spinor structure fora full three generations of leptons and quarks ([53] section 5, [54] discussion of equa-tions 9.9–9.12). As noted for equations 26–27, 32–33 and 35–36 spinor states do heredirectly arise for a natural progression in extending the form of proper time (unlikefor the restricted case of extra spatial dimensions in equation 8–9). The nature of theaugmentation from the one spinor identified in equation 26–27 to four spinors withthe property of the same handedness in equation 32–33 was already intrinsically deter-mined by properties of the octonion algebra. With both left-handed and right-handedcomponents then identified in the extension to equation 35–36 the spinor states exhibitan accumulation of properties that increasingly resemble structures of the StandardModel, and in particular can be associated with e -leptons and d -quarks. The idealsituation would be to continue this progression by identifying a spinor structure alsofor ν -leptons and u -quarks through a subsequent E stage without needing to contrivethe final Standard Model symmetry features in any way. That this might in princi-ple be achieved through an underlying octonion structure for an E − action for ahomogeneous form L ( v ) E = 1 for proper time in a natural mathematical mannerconstitutes a non-trivial prediction of the theory [53].Central to this aim is an understanding of the interconnections between vari-ous algebraic structures related to E (including those reviewed in [53] section 2). Forexample with the proposed E − octic invariant in 248 real variables of equation 37pursued as a generalisation from the E − cubic invariant defined on h O for equa-tion 30 and the E − quartic invariant on F (h O ) for equation 34 a close relation tothe properties of the exceptional Jordan algebra h O is implied. As well as the centralrole as a ‘generalised spacetime’ [62], alluded to before equation 37, the properties ofh O are also inextricably linked with the E , E and E entries of the 4 × invariant as proposed for equation 37 – in the pursuit of uncovering threegenerations of Standard Model spinor states for SL(2 , C ) ⊂ E through a symmetry35reaking structure under equation 38. A link between the role of octonion triality inthe magic square and that for the proposed E octic invariant might for example beforged via the shared association of the space h O with these applications.With the properties of the octonions, uniquely the largest division algebra,central to these connections the mathematical structure required for equation 37 inobtaining the appropriate spinor properties for ν -lepton and u -quark states might alsobe related to that employed in some studies of supersymmetry [67, 68, 69, 70, 71, 72,73]. These studies also relate explicitly to the above discussion of the magic squareand triality (see for example [71] chapter 9, [72, 73]). Such structures in mathematicalphysics might then provide a pivotal guide for the further advancement of the presenttheory. Here however we have constructed and developed the theory from the elemen-tary first principles described for equations 11 and 13. While significant elements of theStandard Model have already been obtained through the E level of equations 30–33and E level of equations 34–36 a more complete structure is predicted to emergethrough an explicit analysis of the E level based on equation 37, as provisionally de-noting this ultimate form for proper time. In the meantime, guided by the successesof the E and E levels and general considerations such as equations 38 and 39, wecan already anticipate possible features of new physics that may emerge beyond theStandard Model for the E level, as we describe in the following section. In general terms, while keeping in mind the caveats regarding the need to incorporatefurther spinor structures and a full electroweak theory, given the potential of incor-porating three generations of charged leptons and quarks at the E level as suggestedby equation 39 we might anticipate some of the implications for neutrino physics ofan explicit expression for equation 37 with a symmetry breaking decomposition underequation 38. Extrapolating from the ambiguity of neutrino and Higgs components forthe E level in equation 32–33 and the resolution of that ambiguity at the E level inequation 35–36 the assumption of the simplest further augmentation for the neutrinosector within a three generation symmetry breaking pattern at the E level pointstowards the progression: L ( v ) E = 1 : ν L / h (equation 32–33) L ( v ) E = 1 : ν L and h (equation 35–36) L ( v ) E = 1 : ν L ν L ν L and h ν R ν R (40)Here the necessary projection of the original external components h ≡ v ∈ TM istaken, without loss of generality, from the right-handed sector of components at thisE level, as for the case of the E level in equation 35–36. This schematic augmentationthen suggests that the accommodation of a full three generations of both left and36ight-handed charged leptons and quarks at the E level may be accompanied by threegenerations of left-handed neutrinos but only two right-handed neutrinos, with theexternal components of h associated with the Standard Model Higgs now prohibitingthe identification of a third ν R state (developing the comment made in [53] section 7,second bullet point).In this manner the present theory, developed from the first principles underly-ing equation 13 upon generalising the form of proper time, could provide a unifyingtheoretical basis for phenomenological models based on two right-handed neutrinostates, including [13, 14, 15, 16] as reviewed in subsection 1.1. Given the firm con-ceptual foundation the further mathematical development of this theory then offersthe potential to greatly narrow down the specific features of such models and lead torobust predictions.As a further implication of equation 40 the Higgs sector, associated with thecomponents of h ≡ v ∈ TM , will be intimately connected with the neutrino sector,as had already been suggested by the E level of equation 32–33, and we can nowconsider at the E level how some of the Higgs and neutrino properties may be closelycorrelated. For example the need to identify a spinor structure for both the ν L and ν R states under the SL(2 , C ) ⊂ E action of equation 38 on the components underlying L ( v ) E = 1 for the symmetry breaking pattern of equation 40 suggests that h mayalso have an underlying spinor composition, rather than being directly extracted as h ∈ h C vector subcomponents as was the case for equations 26–27, 32–33 and 35–36.Indeed in general spinor components can be combined to form vector objectsas for ψψ † ∈ h C with ψ ∈ C in equations 20 and 21, suggesting that h ∈ h C inthese equations might have a similar decomposition, or a generalisation of it, withina higher-dimensional form for proper time such as at the full E level. On the otherhand spinors can also be combined to form scalar objects as is the case for Dirac orMajorana mass terms in a Lagrangian and also for | h | in equation 23 if h is composedof spinors. This latter case, with the scalar | h | associated with the Higgs, might thenhave some similarities with composite models in which the Higgs field is analogous topion fields for a scaled up adaptation of QCD (see for example [74, 75] and referencestherein). While pions are composed of quark spinor states the possibility of Higgs-like composites constructed specifically from right-handed neutrino states is describedin [76, 77, 78, 79], a connection which is here for equation 40 proposed to leave adistinct residual of two free ν R states. Here the components of the would-be third ν R state are proposed to be fused into an effective Higgs field through the necessaryidentification of the 4-vector projection h ≡ v ∈ TM of equation 22, representingthe local external spacetime structure.In the Higgs sector itself the implications of a composite structure can beinvestigated and constrained at the Large Hadron Collider (LHC; see for example [75]section 4, [80, 81]). For composite Higgs models the coupling of the Higgs to fermionpairs can deviate from the Standard Model expectation by of order 10%, sufficient forthis new physics to also be observable at a 250 GeV e + e − linear collider ([82] section 5).Here for a detailed analysis and specific predictions an explicit expression forequation 37 and a full symmetry breaking structure for the components of v ∈ R under equation 38 will be needed. As a progression from the broken SL(3 , C ) symmetrycase of equation 21, via the E and E levels, this will also involve a full expansion of37nvariant terms for the broken E form: L (cid:30) ( v ) SL(2 , C ) × SU(3) × SU(2) × U(1) = X (invariant parts) = 1 (41)This expansion at the E level is proposed to yield an explicit form for massterms in a corresponding effective Lagrangian as described for equation 29. In partic-ular for the neutrino sector such terms will depend upon the nature of the embeddingof the neutrino spinor structures, under the external SL(2 , C ) ∈ E symmetry of equa-tions 38 and 41, and their composition with a factor of the vacuum value for a Lorentzinvariant scalar combination of the components of h such as h = | h | in equation 23.These structures relating to spinor compositions will be intimately connected with thetriality properties of the octonion composition that is anticipated to play a centralrole in the construction of the homogeneous polynomial form L ( v ) E = 1 itself, asdescribed towards the end of the previous subsection.When partitioned into invariant pieces under the broken symmetry in the ex-pansion of equation 41 Dirac mass terms containing a factor of the form ψ † L,R ψ R,L can be sought for the e -lepton and d -quark spinor states ψ L,R subsumed from the E level of equation 35–36. The ‘ u -quark’ scalar components in equation 35–36 will alsoneed to be identified as spinors in the components of v at the E level, through theoctonion composition triality properties, and similarly incorporated into such effectivemass terms, and again for a full three generations. For the case of the neutrino sectorthe nature of this spinor embedding at the E level may determine both the numberof ν L and ν R states as well as the form and combination of Dirac and Majorana, orother, mass terms for this sector.In general a Majorana mass term contains a factor of the form ψ TL,R σ ψ L,R ,where T is the transpose and σ = (cid:0) − ii (cid:1) is a Pauli matrix, which is numerically equalto zero for any spinor values ψ L,R ∈ C . For a Lagrangian field theory with such aterm the component ψ L,R ( x ) hence needs to be treated as an anticommuting field evenat the classical level, anticipating the statistical fermionic properties of such a spinorin the corresponding quantum theory (see for example [83] problem 3.4). Similarlyhere the expansion of terms in equation 41 may also require an understanding of therole of the ‘spin-statistics theorem’ in the QFT limit in order to fully interpret andassess the nature of the particle properties that might be deduced (as alluded to in[40] subsection 5.3). Indeed the connection with QFT will be needed to assess theconception of particle states generally for the present theory, as will be discussed forequation 44 in subsection 5.1.While the above possibilities are hence provisional one distinctive feature con-cerning the lightest left-handed neutrino can already be surmised. With the compo-nents of h ≡ v ∈ TM being associated with the Higgs and the origin of mass (asdiscussed for equations 22–24) for this preliminary structure a clear basis for a massasymmetry in the neutrino sector is also implied at the E level in equation 40, whichhints at a form of ‘seesaw’ imbalance between the left and right-handed states. Asreviewed in subsection 1.1 in a standard neutrino model seesaw mechanism each ν R state generates one ν L state mass. Hence with only two ν R states available in the pro-jected structure outlined in equation 40 there is a strong hint that the lightest activeneutrino mass state may be massless, that is m min = 0 eV.38s also discussed in subsection 1.1, while ongoing and future experiments ontritium β -decay and neutrinoless double- β decay will improve the corresponding con-straints on m min , the most sensitive measurement is currently provided by the cos-mological observations limiting the total mass of the active neutrino states, currentlywith m tot < .
12 eV implying an upper bound of around m min < .
03 eV (for thenormal hierarchy) or m min < .
02 eV (for the inverted hierarchy). The prospects forfurther observations, alluded to in subsection 1.1, suggest that within the ΛCDM cos-mological model the consequences of even the most challenging case with m min = 0 eVin the normal hierarchy (hence with m tot = 0 .
06 eV) could be detectable within theforeseeable future. If the value of m tot could be determined to be greater than 0 .
06 eVand different from 0 .
10 eV (allowing for the inverted hierarchy case) with statisticalsignificance then the case for m min = 0 eV would be disfavoured and is hence testable.As suggested in subsection 1.1 with reference to ([2] figure 62.1, [3] figure 5) the moststringent test of a prediction for m min = 0 eV might be provided in the coming yearsby a combination of cosmological and neutrinoless double- β decay data.The property of a lightest active neutrino mass m min = 0 eV is naturally sharedwith models that assume there are only two right-handed neutrinos [13, 14, 15, 16],as noted in subsection 1.1. Such models are ‘minimal’ in the quantitative sense ofaccounting for the well-established neutrino oscillation data with the smallest numberof additional states over and above those of the Standard Model. On the other hand the ν MSM [7, 8], for which m min is non-zero but still far too small for the deviation from m min = 0 eV to ever be detected by any known means, is ‘minimal’ in the symmetricsense that a uniform three-generation pattern of lepton and quark states is maintainedwithout needing to assume an exception for the case of ν R states – although two ofthe ν R states in the ν MSM are distinguished in being much heavier than the third. Inall cases the above models incorporate two right-handed neutrinos that can accountfor the observed solar and atmospheric oscillations between left-handed neutrinos, viaan appropriate structure of mass terms, while also providing a mechanism for thebaryon asymmetry originating in the early universe, via CP -violating properties ofthese two heavy ν R states. The additional state of the ν MSM, the third and lighter ν R component, provides a dark matter candidate, as also alluded to in subsection 1.1. Inthe event of confirmation of any of the anomalous neutrino observations, discussed insubsection 1.1 in connection with reference [12], a further extension from these modelsor an alternative approach would be required.In the absence of decisive observations, or the guide of building up a theoryfrom first principles, it may prove difficult to empirically distinguish between the aboveneutrino models. For all models the stark contrast between the empirical properties ofleft and right-handed neutrinos is essentially built in by hand, with the ν R states beingtypically far heavier, sterile to Standard Model gauge interactions, and in some casesfewer in number, compared with the ν L states. The present theory however does derivefrom the first principles of a clear underlying conceptual motivation in a generalisationof proper time as described for equations 11 and 13. Through equations 32–33 and35–36 we are led to a proposed symmetry breaking pattern for the neutrino sectorat the E level as described for equation 40 in which a stark contrast between theleft and right-handed states is indeed implied . Hence we do have a clear origin for asignificant asymmetry between the properties of the ν L and ν R states, regardless of anyfurther specific features. However, until further developed there is an open question39oncerning how much new physics in the neutrino sector might be accommodated,with the need to account for the present and future requirements in empirical neutrinophysics providing a possible means to test the theory. More generally the challengewill be to address a range of outstanding questions in neutrino physics including thoselisted in the penultimate paragraph of subsection 1.1.In summary, the provisional structure of equation 40 favours the case for theaccommodation of only two right-handed neutrino states. In light of the correspondingneutrino models, if these two ν R states are found to have the appropriate properties,this may be sufficient to account for both the compelling neutrino oscillation phenom-ena observed and in principle the baryon asymmetry of the universe. On the otherhand in equation 40 there is no room to accommodate a third ν R state, suggesting that m min = 0 eV. The third ν R is not needed to account for dark matter here since suchcandidates may be provided by another sector of the symmetry breaking or throughan alternative form for proper time itself, as we describe in the following subsection. The four scalar components { n, N, α, β } at the E level in equation 35–36, in beinginvariant under the internal symmetry gauge group, provide a prototype set of darkmatter candidates as noted in subsection 3.1. As a progression from the lone scalarinvariant n of equations 26–27 and 32–33 these components may generalise further intoa broader range of possible dark matter states at the full E symmetry level, invariantunder the full Standard Model gauge group in equation 38, and offer the possibilityto test this new physics and explore the corresponding cosmological phenomena. Suchcomponents could in fact relate to an extended ‘dark sector’ more generally, incorpo-rating also the dark energy impact on the universe at the present epoch and in principlethe nature of any ‘inflationary’ period during the very early stages of cosmic evolution(as considered in [54] chapter 13).While in the discussion of equation 40 in the previous subsection we describeda close link between the neutrino and Higgs sectors, here we note that a dark sectorassociated with a set of scalar invariants is also anticipated to be closely related to theHiggs. With the Higgs associated with the projected components of h ≡ v ∈ TM in equations 22 and 23 onto the external spacetime this latter connection is madein particular through several dilation transformations identified in the symmetries ofequations 19, 30 and 34 as described for ([53] equation 90), with their possible role inthe very early universe considered in more detail in ([54] section 13.2).High energy physics experiments might also probe the link between the Higgsand scalar invariant components based on a generalisation of { n, N, α, β } through thepossibility of ‘invisible Higgs decays’. Limits can be set on the branching ratio of theHiggs to such a dark or ‘hidden’ sector at the LHC via missing energy in events withother features observed that might accompany standard Higgs production [84, 85].Similarly invisible Higgs decay events might also be detectable via a visible recoiling Z boson decay at a future e + e − collider ([82] section 6).The connection between the Higgs and dark matter here may be similar to thatof a ‘portal interaction’ model, in particular for such a model incorporating more thanone scalar invariant state (see for example [86]). Portal interactions with dark matter40an also be utilised in studies of a composite Higgs involving the neutrino sector [78],hence also connecting with the discussion of equation 40 in the previous subsection.The nature of any new physics in the Higgs, dark and neutrino sectors would needto be mutually consistent within the context of the present theory, as well as withempirical observations in particle physics and cosmology more generally, broadeningthe scope for making predictions that might then be tested.A pivotal element in analysing the physics will be the role of the externalcomponents h ≡ v ∈ TM as extracted from the full form for proper time in equa-tion 37. Through the simple conceptual underpinning for this theory there is only onefundamental mass scale as associated with the magnitude of this projected 4-vector h = | h | = | v | in equations 22 and 23. That is, mass terms will be identified in theexpansion of equation 41 via the composition of matter fields ψ ( x ) with componentsof h ( x ) such that their mutual interaction will imply δh ( x ) variations that generatemass via the impact on the local external geometry through equation 24, in which theoverall scale is set by the vacuum value for h = | h | . Through the dilation transforma-tions a very different value for h might have been attained in the very early universe,with for example h → t → h = h in [54] section 13.2). At this earlyepoch of cosmic evolution the properties of the Standard Model emerged, in principlewith any ‘hierarchy problem’ avoided since only a single stable basic mass scale remains(as discussed in [40] towards the end of subsection 5.3).While we have considered a unique sequence of mathematical structures forequation 13 in augmenting from the Lorentz symmetry of equation 18 to the proposedE case of L ( v ) E = 1 in equation 37 the possibility remains of an alternativeextension from the local 4-dimensional spacetime form L ( v ) Lorentz = 1 to a full formfor proper time that might be denoted: L p ′ ( v n ′ ) ˆ G ′ = 1 (42)and hence also subsuming equation 1. Indeed the original ‘extra spatial dimensions’form of equations 6, 7 and 14 presents such a possibility. Variations δh ( x ) in equa-tion 23 for the magnitude | v | in equation 22 can be associated with the projection ofthe common fragment v ∈ TM onto the local external spacetime out of both v n ′ ∈ R n ′ of equation 42 as well as v ∈ R of equation 37, impacting the geometry of ouruniverse and exhibiting gravitational effects as described for equation 24. Howeverthe symmetry breaking down to Lorentz × G ′ ⊂ ˆ G ′ , corresponding to equation 28 forthis new projection over the local structure of M , would yield internal gauge symme-tries G ′ and v n ′ fragments from equation 42 independently of the symmetry breakingpattern of ˆ G = E on the original v components of equation 37.Hence the ‘ordinary matter’ deriving from the breaking of L ( v ) E = 1 wouldnot interact with matter fields deriving from the parallel form L p ′ ( v n ′ ) ˆ G ′ = 1 via anygauge forces, and hence the latter could also be considered as a potential source of‘dark matter’. The gravitational link between the two forms of proper time for v and v n ′ via the common projection h ≡ v ∈ TM is perhaps even more reminiscentof a kind of ‘Higgs portal interaction’ as alluded to above [86]. For the example inwhich such a parallel dark or hidden sector derives from the breaking of equations 741nd 14, as pictured in figure 1(b), the fragments of equation 8–9 are obtained, andhence the matter field v n − ( x ) associated with ‘extra spatial dimensions’ might thenyet play a role in the present theory as a candidate source of dark matter that caninteract gravitationally with ordinary matter. The nature of this quadratic form, witha potential for n → ∞ and a role for ‘fractal-like’ or ‘Bott periodicity’ properties (seealso discussion in [54] towards end of section 13.3), and the possibility of several parallelhidden sectors each describing a form for equation 42, might also account for the excessof dark matter over ordinary matter by around a factor of five ([2] sections 2 and 26).While hence identifying a possible role for alternative expressions for equation 13 herewe focus on the properties of matter fields deriving from the E form of equation 37,proposed to be responsible for the properties of our visible universe.With the physics beyond the Standard Model deriving from equation 37 poten-tially involving an extended Higgs sector closely related to the neutrino sector as wellas elements of an extended dark sector an explicit branching of the components of v in equation 37 under the broken symmetry of equation 38, as well as a full expansionof the invariant terms for equation 41, may be needed to untangle the various physicscomponents according to their transformation properties under the external and in-ternal symmetries. As noted in the discussion following equation 41 the full physicalpicture will require an understanding of the QFT limit and the nature of field quanti-sation itself, the origin of which for the present theory will be reviewed for equation 44in the following section. However, as for the analysis described in section 3, a numberof features can be directly deduced from the elementary symmetry breaking structurefor comparison with the Standard Model.In the Standard Model Yukawa couplings are added to mass terms in the La-grangian by hand alongside the Higgs vacuum value to determine the fermion massmatrices (as described for [54] equations 7.69 and 7.70). Given the potential source ofdark matter through alternative forms for proper time, as suggested for equation 42above, the specific composition of stable vacuum values emerging from the Big Bang forscalar invariants, such as { n, N, α, β } at the E level, in the expansion of equation 29could play the role of Yukawa couplings and complete the ‘mass term’ interpretation.Indeed under augmentation to the full E case, with terms in the expansion of equa-tion 41 being of octic order, a combination of several such factors into an effectiveYukawa coupling might be needed, alongside the Higgs represented by a vacuum valuescalar combination of components of h ≡ v ∈ TM and the components of spinorstates, if such terms are to closely resemble Lagrangian mass terms and account forthe wide range of observed fermion masses.An understanding of these factors may hence be central in particular in ulti-mately calculating specific masses for the left and right-handed neutrinos while specif-ically incorporating any corresponding ‘seesaw’ imbalance, as provisionally suggestedby equation 40, with potentially a large Majorana mass sought for the ν R states con-sistent with the models described in subsection 1.1. As well as accounting for thehierarchy of neutrino masses and ‘textures’ of the neutrino mass matrix (see for exam-ple [14, 87]) the aim would be to explain the Standard Model lepton and quark massspectrum more generally.From equation 38 at the E level the further breaking of the electroweak sym-metry down to U(1) Q ⊂ SU(2) L × U(1) Y is proposed to involve a non-trivial actionof a subset of the SU(2) L × U(1) Y transformations upon components of the 4-vector42 ≡ v ∈ TM , itself projected onto the local external spacetime tangent space fromsubcomponents of the full set of v ∈ R components in equation 37. This is con-sidered to account for the masses of the W ± and Z gauge bosons (as suggested by thestudy of SU(2) × U(1) ⊂ E subgroups in [54] section 8.3, alluded to here in subsec-tion 3.1). For a ‘composite’ Higgs the set of spinors acted upon by the SU(2) L × U(1) Y symmetry could include the components of a spinor decomposition of h ≡ v ∈ TM .While such a composition of the 4-vector h ≡ v ∈ TM in terms of spinor subcompo-nents under the external SL(2 , C ) symmetry was suggested in the previous subsection,the physical Higgs itself might be associated with a more extended set of componentsin the decomposition of v ∈ R under equation 38 and need to be disentangledcollectively from the neutrino states, elements of the dark sector and also Yukawafactors as considered in this subsection.The Higgs coupling to the W ± and Z gauge bosons, as well as to the twoheaviest quarks (top and bottom) and two heaviest leptons (tau and muon, with anupper limit in the latter case) have been determined at the LHC [88, 89, 90]. TheseHiggs couplings are seen to be directly proportional to the mass of the particle coupledto ([88] figure 5, [89] figure 10, [90] figure 15), consistent with the Standard Modelprediction and allowing constraints to be placed on new Higgs physics (see for exam-ple [89] section 9). For the present theory the underlying origin of this high degreeof uniformity arises from the direct connection between the couplings to the identifiedHiggs components in the symmetry breaking of equation 13, in particular in the formof equation 37, and the conception of mass in general relativity as discussed for equa-tions 22–24. Particle masses are here to be determined from terms in equation 41 by acombination of specific scalar invariant ‘Yukawa’ coupling factors in composition withHiggs components together with the uniform vacuum value h = | h | for the Higgs thatunderpins the overall mass scale. Such properties, and deviations that might arise ina full development of the theory, might be tested as measurements improve with moredata at the LHC, while further tests of Higgs couplings would be possible at a futureILC experiment [82] as noted in the previous subsection.For the present theory as well as the Higgs couplings and particle mass spectruma full understanding of electroweak theory is to be sought at the E level, in particularwith an internal SU(2) L not only impacting upon the Higgs subcomponents in anappropriate way but also acting on doublets of left-handed leptons and doublets of left-handed quarks. In augmenting from the E level of equation 35–36 this will require theidentification of Lorentz spinor properties for both ν -lepton and u -quark states, andfor a full three generations of leptons and quarks. This ambition is expected to dependupon the intrinsic incorporation of the properties of octonion triality as discussed insubsection 3.2.That all of these required features of the Standard Model in augmenting fromthe E level to the E level are correlated is further emphasised by noting that theSU(2) L doublet actions are anticipated to meld with the structure of the fermionmass terms and ‘textures’, from the partitioning of equation 37 under the breakingof equation 38 to the invariant terms of equation 41, in a manner relating to inter-generation mixing. With the symmetry breaking pattern, including the internal SU(2) L transformations, oriented around the external SL(2 , C ) action on the external 4-vector h ≡ v ∈ TM the fermion mass and flavour generation mechanisms will be intimatelyrelated to this projection. In particular with h projected out of the ‘neutrino sector’43s described for equation 40 the very different PMNS and CKM weak mixing matricesfor the lepton and quark sectors respectively might in principle be accounted for. Thiswill relate to the nature of CP -violating effects in both sectors, and in particular for ν L phenomenology in the leptonic case. While in turn a link can be made between CP -violation for ν L states and that for ν R states in a model-dependent manner, asreviewed in subsection 1.1, here the aim will be to determine this link from first prin-ciples to establish whether CP -violating effects associated with ν R phenomenology inthe very early universe might be on an appropriate scale to act as a potential sourceof the baryon asymmetry as proposed by models described in subsection 1.1.Since the form of equation 37 for the E level involves a much larger spaceof components in v ∈ R , beyond the 27-dimensional E and 56-dimensional E levels of subsection 3.1, in addition to accommodating the three generations of knownleptons and quarks together with two right-handed neutrinos as well as a non-standardHiggs with associated Yukawa couplings and potential dark sector candidates furthernew particles beyond the Standard Model might also in principle be predicted, with thepotential for further laboratory tests. In addition, for the full symmetry group ˆ G = E ,the complete breaking pattern might also accommodate further internal gauge groupsbeyond that of the Standard Model in equation 38 (see for example [54] equation 9.51),in principle implying new gauge interactions that might also have observable conse-quences in high energy physics experiments. While a range of additional states beyondthe Standard Model might hence be accommodated a large multiplicity of new statesis not anticipated, unlike for example the case in general for supersymmetric modelsand in particular for extended supersymmetry.With the further required symmetry and representation features of the Stan-dard Model beyond equation 35–36 being mutually closely correlated, as noted in thediscussion above, it is plausible that they may all be uncovered together in one furtheraugmentation from the E form of equation 34 to the proposed E form described forequation 37 (as considered in detail in [53]). If these required features do emerge at theE level this will provide a very firm basis for precise predictions of a wealth of physicsbeyond the Standard Model. In this section we have described the nature of the newphysics, including for the Higgs, neutrino and dark sectors, that can already be an-ticipated. In the previous subsection we have considered the particular significance ofthe intrinsic left-right asymmetry for the neutrino sector associated with equation 40,and emphasised the manner in which the theory favours models with two, and onlytwo, right-handed neutrino states with a close link to the Higgs sector. In this subsec-tion we have described in particular how these interconnections might extend into thespecific structure of mass terms for particle states in equation 41 and into a ‘portal’interaction with a dark sector of the theory as described for equation 42, augmentingthe potential scope for empirical tests in the particle physics laboratory or throughcosmological observations.While the analysis for the E level has been of a provisional nature we notethat here we are not building a model pragmatically, for example by adding termsto a Lagrangian by hand, but rather the theory has been developed from elementaryfirst principles , as we discuss further in the remaining two sections. Precise empiricalpredictions will require a full understanding of the structure of the theoretically pre-dicted E − symmetry action for the full form of time L ( v ) E = 1 in equation 37and a symmetry breaking pattern. However given the simple unifying basis underlying44quation 13 in terms of a generalisation of proper time an opportunity to uncover theultimate origin of the physics of the Standard Model and beyond at the most funda-mental level is in principle provided by this theory. In the following section we returnto consider the elementary foundations of the theory from an historical perspective. In 1922, seven years after the generalisation from special relativity had yielded a theoryof gravitation, concerning the ambition for a geometric unification with electromag-netism, and in light of the first attempts by Weyl and Kaluza, Einstein wrote in aletter to Weyl: ‘I believe that in order to make real progress one must again ferret outsome general principle from nature’ ([22] section 17(b)). More specifically, in the early1930s and having embarked upon his own endeavour to find such a unified field theory,Einstein summed up the nature of the task faced with the question [91]:Is there a theory of the continuum in which a new structural elementappears side by side with the metric such that it forms a single wholetogether with the metric?This principle, closely associated with the sentiment expressed by Einsteinthat the quest for unification should be guided by simplicity, applies to the unifiedfield theories of Weyl, Einstein himself and Kaluza/Klein, for which the metric g µν ( x )of equation 2 is augmented as reviewed here for equations 3, 4 and 5 respectively insubsection 1.2. The aim of each of those early unification schemes was to incorporatea theory of electromagnetism alongside the theory of gravity via a minimal extensionfrom the latter theory. In each case the corresponding generalisation was one thatcould be interpreted as dropping a further geometric assumption from the frameworkof general relativity, as also described in subsection 1.2.For general relativity itself a theory of gravity had been obtained on droppingthe assumption that spacetime should be globally flat, as noted before equation 1, withspecial relativity applying only locally. This was a somewhat counter-intuitive startingpoint since the external 3-dimensional space with which we are very familiar appears topossess extended Euclidean properties, and it had seemed perfectly natural to assumesuch properties as a basis for all science for centuries. As well as requiring a significantlymore complicated mathematical description taking away the assumption of a stableflat background arena was hence somewhat disorienting, even though justified by theexplanatory power and empirical success in accounting for gravitational phenomena.Since, like gravity, electromagnetism is a long range force it was reasonable tosearch for a generalisation from general relativity by dropping a further assumptionconcerning the global geometry of 4-dimensional spacetime. Ideally the aim for sucha unified field theory was not only to incorporate classical electromagnetism alongsidegravitation but also to account for particle states in terms of classical field solutions andto provide an underlying explanation for quantum theory without relying on seeminglyad hoc postulates ([22] chapters 17 and 26). Somewhat like the limitations of Newto-nian mechanics, for Einstein the apparent incompleteness of quantum mechanics, in45articular in terms of its probabilistic nature, could not be addressed by incrementalinternal refinements or interpretations, but rather demonstrated the need for an ex-planation from a new foundation, such as might be achieved through a generalisationof general relativity. Indeed for Einstein this incompleteness of quantum theory was initself a significant motivation for the need of a unified field theory, from which quantummechanics might emerge as a limiting case. This initially applied to the ‘old quantumtheory’ associated with Planck’s radiation law of 1900 and the Bohr atom of 1913, andlater for the ‘new quantum theory’ of Heisenberg and Schr¨odinger from around 1925.An early conception proposed that particle-like properties might be associatedwith ‘energy-knots’ of very high classical field values ([26] chapter III section 25).The original paragraph containing the above quote from Einstein goes on to considerwhether simple field laws might be obtained to describe the properties of both gravi-tational and electromagnetic fields, and continues [91]:Then there is the further question whether the corpuscles (electrons andprotons) can be regarded as locations of particularly dense fields, whosemovements are determined by the field equations.A similar hypothesis is alluded to in the ‘Concluding Remarks’ of Klein’s pa-per [36] where it is also suggested that the properties of quantum phenomena may orig-inate out of a projection from a 5-dimensional spacetime. That the indeterminacy ofquantum mechanics might arise through an inherent ambiguity of 4-dimensional phys-ical laws obtained from an embedding within the 5-dimensional spacetime of Kaluzaand Klein is described in ‘Part III: Unified Field Theories’ of Bergmann’s 1942 text-book on relativity (as quoted and discussed in [22] section 17(c)). While that quest forsuch a unification was ultimately unsuccessful many modern-day models build uponthe elegant idea of Kaluza-Klein theory, as noted in subsection 2.1, featuring furtherextra spatial dimensions in aiming to address the wider challenge of accommodatingthe Standard Model of particle physics. However the postulates of quantum theoryand the quantum particle description are generally adopted, or adapted, as a basis forsuch unification schemes without any further underlying explanation.Upon reconsidering the motivation for such a unified framework in subsec-tion 2.2 we noted that since we do not perceive any ‘extra dimensions’ there seems littlejustification for assuming such components, if they exist at all, to possess any form of‘spatial properties’ even in the purely local sense of consistency with the Pythagoreantheorem for infinitesimal intervals as described for equation 10 and as might be de-picted in figure 1(b). Dropping this assumption is in some sense more natural thanrelaxing the assumption of a flat external spacetime since we do not even see the extradimensions, and hence there is no compelling argument to restrict the generalisationof a local proper time interval from the quadratic expression of equation 1 to that ofequation 6, leading to the more general expression of equation 11. Given that the no-tion of extra spatial dimensions is now very familiar loosening that preconception andgeneralising to the local form of equation 11 might itself seem disorienting, with forexample the ‘visualisation’ of figure 1(b) no longer applying. However a mathematicalpath can be followed and affirmation sought through empirical success, as was the casefor the extended non-flat geometry of general relativity.With reference to the first displayed quote from Einstein in the opening of thissubsection here equation 11, equivalent to equation 13, is proposed as a ‘single whole’,46eriving from the continuum of proper time, which can incorporate the 4-dimensionallocal spacetime metric η ab of equation 1 as described for equation 12 and exemplifiedby the cubic form of equations 19 and 20, ‘side by side’ with additional structuresthat are interpreted as a basis for matter fields. For this theory the minimal sym-metric extension from the matterless vacuum case of equation 18 to equation 19 alsoincorporates a framework for electromagnetism together with gravitation as describedfor equations 25–27, similarly as for the ambition of early unified field theories. Herethe potential for further natural extension described in section 3 leads, via a uniquesequence of mathematical structures utilised for equations 30–36, directly and effi-ciently to a series of explicit structures of the Standard Model, hence vindicating thisapproach, and with the direction of further extrapolation providing insight into newphysics beyond as described in section 4.A fundamental dependence upon the continuum properties of time and in-finitesimal intervals can in fact be traced back somewhat further to the earliest equa-tions of physics in the 17 th century as developed by Newton. The methods of calculusinvented by Newton for this purpose incorporate at the most elementary level the no-tion of a ‘fluxion’ ˙ x for the rate of change of a quantity x with respect to progressionin time [92]. The definition of such a fluxion, or derivative, involved taking the limitof a ratio of two quantities both decreasing without any finite bound. (Such a con-ception was not without controversy for many years, with fluxions famously criticisedas a ‘Ghosts of departed Quantities’ by Berkeley in ‘The Analyst’ in 1734). The samecontinuum property of time is central to the present theory on taking infinitesimalintervals approaching the limit δs → v a definedfor equation 13 the quantities δx a and δs composing this limiting ratio are intimatelylinked through equation 11.Hence the historical roots concerning the conception of time for the presenttheory date back before relativity to the Newtonian worldview of the 18 th and 19 th centuries. Throughout that era the purely linear progression in absolute time provideda fundamental independent parameter for the recording of events in a 3-dimensionalEuclidean arena of absolute space with global coordinates ( x , x , x ). Spatial lengthsof arbitrary extent ∆Σ could be determined by the simple Pythagorean relationship(∆Σ) = (∆ x ) + (∆ x ) + (∆ x ) invariant under 3-dimensional rotations. In theearly 20 th century for special relativity the time coordinate ( x ) was also introducedinto this quadratic structure, now augmented to a 4-dimensional spacetime form forarbitrary ‘proper time’ intervals ∆ s with the form (∆ s ) = (∆ x ) − (∆Σ) invari-ant under global Lorentz transformations between inertial reference frames in thisMinkowski spacetime. The space and time components of the corresponding globallydefined coordinates ( x , x , x , x ) are hence connected through the Lorentz invariantform of equation 1 expressed for arbitrary finite proper time and coordinate intervals,as described after equation 1. This form for proper time was determined by the pos-tulates of special relativity, and in particular the constancy of the speed of light with∆ s = 0 in any inertial frame, and implies that each such rest frame carries its ownproper time parameter, abandoning universal Newtonian time.In general relativity the Lorentz metric η ab is necessarily replaced by the generalsymmetric metric function g µν ( x ) of equation 2 with respect to global coordinateson the extended scale, in a form for the infinitesimal proper time interval δs now47nvariant under general coordinate transformations. Through this more general globalframework a unification is achieved between the geometry of 4-dimensional spacetimeitself and gravitation, with the structure of the former inextricably accommodating atheory of the latter. For this progression from special relativity to the more generalgeometry of general relativity inertial frames are strictly local, with Lorentz invarianceonly holding for infinitesimal intervals of proper time as introduced here for δs in thequadratic form of equation 1.While incorporating the linear flow of time within such quadratic expressionsmight have originally been considered somewhat eccentric, the present theory repre-sents a further natural progression in the conception of time through an arithmeticgeneralisation of invariant expressions for infinitesimal proper time intervals to formsof greater-than-quadratic order through equation 11. As a progression from the cen-tral role played by the invariance of proper time in special and general relativity, andagain noting the quote from Einstein in the opening of this subsection [91], it is theelementary properties of this local continuum of time that we exploit for the theoryproposed here in permitting this generalisation of the infinitesimal proper time interval δs described for equation 11, which can be written as equation 13. With the resid-ual components, over and above a local 4-dimensional spacetime structure, and theirproperties providing the source of matter fields this elementary local generalisation ofproper time leads to a unification of 4-dimensional spacetime with structures resem-bling the Standard Model at the elementary particle level of matter as reviewed insection 3. The conceptual comparison and contrast with general relativity as depictedin figure 2.Figure 2: Complementary generalisations of expressions for a proper time interval fromthe Lorentzian form ( δs ) = η ab δx a δx b : globally to the metric field g µν ( x ) underlyinga theory gravity and locally to non-quadratic forms with coefficients α abc... providinga basis for matter fields through a local projection over 4-dimensional spacetime M .Essentially figure 2 describes two complementary generalisations from specialrelativity: globally for general relativity and locally for the present theory. In generalrelativity the source of gravitation in the form of spacetime curvature, described bythe Einstein tensor G µν ( x ) which is a function of the first and second derivatives of48he metric g µν ( x ), is identified with posited forms of matter, described by the energy-momentum tensor T µν ( x ), through Einstein’s field equation: G µν = − κT µν (43)with κ a normalisation constant. For the present theory matter itself derives fromthe complementary local generalisation of a local proper time interval depicted infigure 2. More precisely the properties of the residual components in the projectionof the generalised local form of proper time over M directly shape the spacetimegeometry on the extended scale through which the energy-momentum is in turn defined([54] equation 15.1, [53] equation 85): G µν = f ( A, v n ) =: − κT µν (44)The structure of the spacetime geometry, represented by the function f ( A, v n ),includes a contribution from gauge fields A ( x ) via the corresponding field strength ten-sor F ( x ), such as the electromagnetic field, in a manner similar to modern Kaluza-Kleintheories as reviewed here for equation 25, as well as from subcomponents of v n ( x ) of thefull form of proper time for equation 13 via the impact of the projected 4-dimensionalspacetime component v ∈ TM , as described here for equations 22–24 and 29. Thequantum properties of matter are proposed to arise as an intrinsic feature through adegeneracy of spacetime solutions for equation 44 ([54] section 10.1, chapter 11 andsection 15.2), with quantum particle states exhibiting the properties of equation 35–36as generalised further for the full form of time proposed for equation 37. In the limitof a flat spacetime approximation, with field dynamics constrained by equation 41, thelocal interactions of a QFT and quantum particle effects exhibiting the properties ofthe Standard Model are envisaged to arise from the symmetry breaking projection ofthe full form for proper time over the local 4-dimensional spacetime. In this mannerthe properties of ‘corpuscles’ such as electron states, alluding to the second of theabove displayed quotes from Einstein, are proposed to be identified from this furthergeneralisation of relativity.For the Standard Model itself, as usually presented, fields are added to a flat4-dimensional spacetime background and then quantisation rules applied, completelydetached from any consideration of general relativity. This standard theoretical frame-work reflects the negligible impact of gravity on phenomena recorded in the particlephysics laboratory. A more technically challenging approach is needed for any attemptto incorporate gravitation within a theoretical scheme in which quantisation rules areafforded precedence, as for the ambition of string theory – which also aims to ac-commodate the Standard Model as discussed in subsection 2.1. Here we begin withthe very local structure of general relativity and make the generalisation for a propertime interval as depicted in figure 2. Through this unrestricted augmentation, beyondquadratic spacetime forms, matter fields are obtained with features both resemblingthe Standard Model and also in principle accommodating probabilistic quantum prop-erties through equation 44. Here the nature of gravity is inextricably linked with theseproperties of matter, without needing to be subsequently appended.Hence the complete theory is not anticipated to involve classical fields in space-time which are then ‘quantised’ by applying the postulates of a quantum field theory(such as reviewed in [54] chapter 10), rather the aim is to account for and explain quan-tum phenomena through this generalisation of the local spacetime structure. Matter,49ogether with its quantum properties, is directly correlated with the spacetime ge-ometry through equation 44, which also describes gravity by incorporating Einstein’sfield equation 43 along with an account of the elementary composition of the energy-momentum tensor. The gravitational field g µν ( x ) remains an essentially classical fieldon a smooth spacetime manifold, with no ‘quantisation’ of the spacetime geometryitself. This perspective on quantum theory and particle states is then in a similarspirit to that of the ambition of the early unified field theories as described earlier inthis subsection.As described in subsection 2.2 and earlier in this subsection the present the-ory can also be motivated in a similar spirit as for the earliest attempts at a unifiedfield theory in that in all cases an assumption concerning the 4-dimensional spacetimemetric is dropped , as a possible further progression from the relaxing of the flat space-time assumption of special relativity that underlies general relativity. For the Weyl,Einstein and Kaluza-Klein theories of equations 3, 4 and 5 for the case of A µ ( x ) = 0and F µν ( x ) = 0, with vanishing electromagnetic field, the original metric g µν ( x ) stilldescribes the curved spacetime of general relativity when related to other matter fieldsthrough equation 43. However for the present theory the trivial case of equation 18corresponds to the matterless vacuum of a flat spacetime, with all matter arising in amanner intrinsically related to the spacetime curvature under augmentations to equa-tion 18 in the form of equation 13 as reviewed above for equation 44.Although in a similar spirit this approach also differs from the early unifiedfield theories reviewed in subsection 1.2, which were based on a generalisation of thegeometry of the global metric g µν ( x ) of equation 2 on an extended spacetime, in thathere we focus upon the local metric η ab of equation 1 of a local inertial reference framewhich is generalised to the expression for proper time in equation 11. The embeddingof the general metric g µν ( x ) in the generalised geometric structures of equations 3, 4and 5 is analogous to the embedding of the local metric η ab within the generalisedproper time interval of equation 11 in the manner of equation 12 as described beforeequation 13. With η ab extracted from equation 11 to be locally incorporated into themetric g µν ( x ) of general relativity in the extended external 4-dimensional spacetime,matter fields derive from the residual components of the generalised form of time ofequation 11 in this projection over the local spacetime geometry as depicted in figure 2.In a sense the present theory is hence based upon a more elementary general-isation than that of the early unified field theories. Compared with the unified fieldtheories that immediately followed general relativity the shift in focus here towardsthe local spacetime structure also seems reasonable given the aim of accounting for themicroscopic properties of matter. As figure 2 implies we can zoom into an infinitesimallocal inertial reference frame anywhere on the spacetime manifold M and generalisefrom the form of proper time in equation 1 to explore the microscopic structure ofmatter that arises. As alluded to in subsection 1.2 the spirit of Einstein’s approach toa unified theory, for example via the generalisation of the global metric components g µν → ˜ g µν in equation 4, remains enlightening as can be seen here in the augmenta-tion from the local metric components of equation 1 to the coefficients of equation 11with η ab → α abc... . This approach, which can hence be considered a further generali-sation from special and general relativity, has been shown to lead to properties of theStandard Model and beyond as presented in the previous sections. In the followingsubsection we further emphasise the underlying simplicity of the theory.50 .2 One Simple Equation As implied in figure 2 the underlying order and structure of matter in spacetime es-sentially arises from the composition of the continuous flow of time. Through anelementary and direct analysis, from an abstract mathematical perspective, any finiteinterval of proper time ∆ s ∈ R can be decomposed down to a limit of infinitesimalintervals: ∆ s = δs + δs + δs + . . . (45)with substructure: δs = p p α abc... δx a δx b δx c . . . (46)for δs → p th -root of a homogeneous polynomial of p th -order in n infinitesimalcomponents { δx a } ∈ R n labelled by a, b, c, . . . with each coefficient α abc... ∈ {− , , } ,maintaining a consistent order of infinitesimals. Here we are simply exploiting the basicarithmetic properties of the continuum of real numbers, representing the continuum oftime with δs ∈ R , which together with addition include the operations of multiplicationand extracting roots. Equation 46 is equivalent to equation 11, which as a directgeneralisation from equation 1 is the starting point for the whole theory.There is a simplicity in the foundation of this theory in dropping the assump-tion that a proper time interval should be expressed through a quadratic form, asfor equations 1 and 6, and hence generalised to higher-order homogeneous polynomialstructures, as described for equation 11. Here we see that this simplification is furtheremphasised by observing that rather than adding extra spatial dimensions that wedo not see, with the components ( δx , . . . , δx n − ) in equation 6, here equation 11 canbe interpreted as expressing a basic general arithmetic form inherent in an infinites-imal interval δs of the continuum of proper time as described for equations 45 and46. Hence the theory is essentially grounded conservatively in this ‘one dimension’ oftime alone , the passage of which we are intimately familiar with (prompting the titleof [54] to emphasise the contrast with theories based on extra spatial dimensions). Inequations 11 and 45–46 we are not adding anything to time, nor replacing time withanything, but simply expressing an intrinsic arithmetic substructure that is carried simultaneously with time.Generally in the equations of physics a lot of attention is naturally paid to theobjects on the left-hand side and the right-hand side but the ‘equals sign’ in the middleoften carries a significant meaning or interpretation in itself. For example in generalrelativity the equals sign in Einstein’s field equation 43 is often interpreted to implythat matter (energy-momentum T µν on the right-hand side) ‘bends’ spacetime (thegeometry G µν on the left-hand side). This description conforms with the historicaldevelopment since the notion of matter very much preceded the conception of space-time curvature. However for the present theory the form of the spacetime solution G µν ≡ f ( A, v n ) in equation 44 takes priority, with T µν being defined in terms of thisstructure in a manner proposed to incorporate the particle and quantum propertiesof matter consistently with the field equation of general relativity as reviewed in theprevious subsection.Similarly, given the historical background to equation 1 the initial prioritymight be given to the components of space ( δx , δx , δx ) and the interval of coordinatetime ( δx ) on the right-hand side, since these are effectively assimilated from the earlierNewtonian worldview as alluded to in the previous subsection. In relativity these51omponents are collectively combined in a 4-dimensional spacetime form which can then be identified with the square of the Lorentz invariant proper time interval δs onthe left-hand side of equation 1. For the present theory, however, we place all theinitial emphasis on the left-hand side interval δs itself, which can be arithmeticallyexpressed in the quadratic form on the right-hand side of equation 1 and interpreted in geometric terms as a basis for 4-dimensional spacetime.Consideration of possible arithmetic expressions for the proper time interval δs then leads to the general form of equation 11, which is equivalent to the expression for δs in equation 46, taken to subsume the 4-dimensional form of equation 1 as describedfor equation 12. Physical structures arise through the breaking of the symmetry ofthe full form of time when projected over the 4-dimensional spacetime substructure asdepicted in figure 2, leading to properties of matter fields that resemble the StandardModel of particle physics as described in section 3. Hence the pivotal role of the equalssign in equation 1 is central to the interpretation of the theory as deriving entirely from generalising and analysing the single simple entity of proper time.Attempts to account for all physical phenomena through a single simple entitydate back to the pre-Socratic philosopher Thales of Miletus (circa 600 B.C. ) on identi-fying water as the basis for all matter ([93] book 1 chapter 2). As arguably the first‘unified theory’ this proposal was perhaps motivated on observing that water could betransformed into three known forms, as a solid, liquid or vapour, from which furtherextrapolation might account for all types of material substances. In light of the appar-ent empirical implausibility of that theory by the time of Aristotle (circa 350
B.C. ) thebasic elements had grown to four in number: earth, water, air and fire with a furtheraugmentation by a fifth element, or ‘ether’, originally associated with celestial phenom-ena. The properties of each element could in turn be ascribed to tiny indestructibleatoms of matter, according to the theory of Democritus (circa 400
B.C. ), which mightbe attributed to the unique geometric structures of the five Platonic solids.In 1808 John Dalton proposed the modern atomic theory of matter with chemi-cal combinations of originally around twenty elements, including hydrogen, carbon andoxygen, in simple numerical ratios accounting for the wide variety of compound sub-stances. By 1869 the list had grown to over sixty basic elements which when orderedaccording to their atomic weights were seen to exhibit recurring physical propertiesat a regular pattern of intervals in Mendeleev’s Periodic Table of chemical elements.By that time the ‘ether’ had been reinvented as the ‘luminiferous ether’ proposed topermeate all space, with elastic mechanical properties permitting the transmission oflight similarly as vibrations in the air allow the propagation of sound. However sucha hypothetical ethereal medium substratum for the electromagnetic field in Maxwell’stheory [20], discussed here in subsection 1.2, or an ‘ether drift’, was never detectedand that concept along with the notion of a Newtonian absolute space with which toassociate such an ether was ultimately discarded as superfluous with the advent specialrelativity in 1905 ([23] introduction).The regular pattern of Mendeleev’s Periodic Table of chemical elements hintedat an inner structure for atoms that foreshadowed the discovery of their compositionand quantum properties, as alluded to in the opening of section 1. The 1911 Rutherfordmodel of subatomic structure with a central massive nucleus surrounded by a sphericalcloud of electrons was followed by the 1913 Bohr atom with the electrons confined tocircular orbits in discrete steps of angular momentum in multiples of Planck’s constant52s an application of the ‘old quantum theory’ ([24] chapters 9 and 10), subsequentlysupplanted by the atomic picture of the ‘new quantum theory’ in the mid-1920s ([24]chapter 12). By 1925 in addition to the two established fundamental forces of elec-tromagnetism and gravitation only the electron, proton (identified with the hydrogennucleus) and photon were believed to be needed to account for elementary materialphenomena, albeit with a third force seemingly required to hold the atomic nucleustogether ([24] section 12(a)). The empirical study of the properties of those and furthercomponents discovered over the following half a century led to the Standard Model ofparticle physics as established by 1975. With the Standard Model incorporating a col-lection of regular symmetry patterns associated with multiplets of elementary particlesthe focus then turned to the empirical and theoretical investigation into the underlyingsource of these structures.In the late 19 th and early 20 th century gaps were filled in and the Periodic Ta-ble grew in size with the discovery of further chemical elements, before an underlyingexplanation in terms of atomic structure was uncovered. Similarly since 1975 the Stan-dard Model has been both confirmed with new empirical discoveries and augmentedwith an extended neutrino sector and other models of physics beyond, while the searchcontinues for an explanation of the origin of these properties.Rather than conceiving of the composition of the world as an indefinite sequenceof ‘onion layers’ of comparable complexity the intuitive idea is often expressed thatthe underlying structure of matter is expected to become simpler as deeper layers areexplored. On the macroscopic scale a huge variety of elaborate material structurescan be observed. While a great deal of complexity remains for structures that can beobserved with the most powerful optical microscopes all such matter is composed froma wide range of molecules and compounds in turn constructed from combinations of lessthan a hundred chemical elements. As arranged in the Periodic Table these elementsare associated with a series of basic atomic structures built from a common set of asmall number of component types. Analysis of these components led to the discovery ofa range of elementary particles, somewhat smaller in number than the variety of atoms,as arranged in the Standard Model. The thread of ever greater simplicity is proposedhere to culminate in equations 45 and 46, with the basic arithmetic decomposition oftime itself providing the template for the elementary particle multiplet structures. Thedistinctive patterns of the Standard Model arise from the symmetry of the generalisedmulticomponent form of time as this symmetry is broken in the projection over thelocal geometric form of 4-dimensional spacetime as described for figure 2.Atoms listed in the Periodic Table can be broken down into constituent pieceswhile in other experiments the particle states of the Standard Model can mutate intoeach other through interactions. Seemingly no such experiment can be performed for‘time’ as the basic element of the present theory. However time can be ‘broken down’and ‘mutated’ through the simple mathematical analysis and identities of equations 45and 46. Indeed it is precisely through this analysis that substructures of time canbe identified as a basis for both the geometry of 4-dimensional spacetime and thematter content within this arena as studied in the laboratory as described explicitlyin subsection 3.1. In this sense particle physics experiments could be considered asan investigation into the elementary substructure of time itself. Since it is difficult toconceive of a simpler basis for a theory than time alone, and since the ‘fragmentation’of this entity in equations 45 and 46 is already employed at the heart of this theory,53here is a sense of reaching the ultimate ‘bedrock’ in accounting for the properties ofmatter, with no further ‘onion layers’ to be sought at a yet deeper level.On the other hand given equation 46, which is equivalent to equation 11 andcan be conveniently written in the form of equation 13, specific mathematical possibil-ities for this generalisation of proper time and their empirical consequences are verymuch open to further exploration. This has been described through to the proposedlevel of an E symmetry in subsection 3.2 with tentative consequences for physics be-yond the Standard Model considered in section 4, demonstrating the potential of thistheory for making testable predictions. The possibility of a yet higher-order fragmen-tation for equation 46, with potentially fractal-like properties, or a role for alternativemulticomponent forms for proper time as suggested for equation 42 as a candidatesource of a dark sector in subsection 4.2, can also be considered.The flow of time pervades the entire spacetime manifold in figure 2, sharingthis property with the hypothetical ether. However, while the ether was abandonedwith relativity theory the notion of proper time, and its local invariance, is central tothe progression through special and general relativity to the theory presented here, asdescribed in the previous subsection. For Maxwell the luminiferous ether, while neverdetected, was a postulated material substratum underlying the observed phenomenaof the electromagnetic field. Here on the other hand there is no question of perform-ing an experiment to search for ‘time’, rather we are directly and intimately familiarwith the flow of time as an irreducible element infusing all experiments and observa-tions, including in the high energy physics laboratory, with all particle and materialphenomena proposed to arise through the substructure of time.Indeed the usual basis for most physical models and theories is to begin bypositing a basic entity or entities in space and time, whether for example water, anether, atoms, particles or fields. Here the left-hand sketch in figure 2 is not ultimately tobe similarly interpreted with the basic entity time s flowing through a pre-existing spaceand time. Rather the geometric spacetime structure itself , as well as the matter withinit, is extracted through the composition of the continuum of time as the sole basicentity. While conceptually very different to earlier unification schemes, the simplicityof this perspective then provides a further motivation for the theory. The historicaland philosophical aspects associated with this simple interpretation of the theory asderiving from the intrinsic arithmetic substructure of proper time alone are elaboratedin [94]. In particular this change in perspective with time promoted to the prior role, viathe substructure of equations 45 and 46, as the progenitor for both spacetime structureand matter in spacetime, subject to laws of physics determined by the constraintsimplied by this underlying simplicity, is further described for ([94] figure 1).The properties of elementary particles are uncovered at the most elementarylevel of the theory through a simple symmetry breaking analysis for equation 46, writ-ten as equation 13, deriving directly from the extraction of an external Lorentz ⊂ ˆ G symmetry acting on the projected v ∈ TM subcomponents of v n ∈ R n that underliethe basis of the local external 4-dimensional spacetime itself, without adding anythingelse to the theory. This symmetry breaking is implied through the necessity of per-ceiving a physical world in space as well as through time, as also discussed in ([94]section 4), with the properties of matter entirely determined by the residual compo-nents in the form for proper time of equation 13 over the 4-dimensional spacetimemanifold M . 54s noted above this contrasts with the usual approach of formulating a theorythrough the introduction of entities with particular properties into a pre-existing spaceand time. Such is the case for example in Maxwell’s theory with the introduction ofelectric and magnetic fields, whether or not supported by an ethereal substratum, to-gether with a set of equations to describe empirical observations. In many modern-daytheoretical frameworks particle properties are typically built in by hand in a similarspirit as for Maxwell’s theory by introducing the corresponding fields and interactionsinto the proposed Lagrangian for the theory, as is the case for the ‘Standard Model’ it-self as well as typically for many models beyond. While masses and charges are assignedto physical bodies in the equations of gravitation and electromagnetism of Newton andMaxwell, similarly in the Lagrangian of the Standard Model masses, charges and cou-plings generally for elementary particles are typically assigned according to empiricalobservations.There are a number of constraints on the type of fields and terms that canbe included in a Lagrangian. This is particularly the case in the context of a quan-tum field theory (the postulates of which are also generally imposed pragmatically)for consistency with unitarity and causality. Typically the Lagrangian will be a realfunction consisting of a series of terms in the fields and their first or second orderderivatives in a Lorentz and gauge invariant form, with the renormalisability of theQFT placing further restrictions such as avoiding coupling constants with negativemass dimension. Nevertheless considerable arbitrariness is still permitted, particularlyfor the construction of models beyond the Standard Model. Indeed on occasions whena provisional hint of new physics is seen in the high energy physics laboratory, in dataotherwise implying a significant statistical fluctuation above the expected background,a large number of new models may be prompted as there will generally be many waysto accommodate the apparent observations through augmenting the Standard ModelLagrangian. Even for the Standard Model, with a minimal extension to incorporatethe current phenomenology of left-handed neutrinos, at least 25 free parameters alsoneed to be introduced and determined or constrained from the data (eighteen for theStandard Model together with seven neutrino mass and mixing parameters).In the light of these observations the sentiment is sometimes expressed that akey ambition for a unification scheme would be to replace the lengthy and complicatedLagrangian of the Standard Model by ‘one simple equation’ from which all of theproperties of particle physics might be derived. A number of questions could be raisedabout any candidate for such a primordial equation, with key issues regarding thenature of the basic entity described, whether for example a particle, field, string etc.,concerning why it should exist itself at all, why it should be the basic entity and whyit should be subject to the ‘one simple equation’.For the present theory ‘time’ is the basic entity. As well as being fundamental toempirical and theoretical physics, as discussed in this section, time is also a necessary,intrinsic and inherent element of any subjective experience we can have, unlike the casefor water, an ether, atoms, particles, fields or other proposed basic physical entities.These fundamental objective and subjective features of time make it an appropriatebasic element for any theory. For the present theory time is the sole basic entity.Further, here we are not proposing or positing an equation to be imposed on time,rather we are simply utilising a direct arithmetic property innate in the concept of thecontinuum of time as expressed for equations 45 and 46.55he continuum properties of time have been recognised as central to the devel-opment of the equations of physics in describing the behaviour of matter since the daysof Newton, as noted in leading up to figure 2 in the previous subsection. Here howeverwe do not introduce any material or other entity with a particular ‘time dependence’,rather material phenomena derive from the ‘subcomponents of time’ itself throughequation 46, which also accommodates the basis for the local Lorentz metric structureof the background spacetime arena via the substructure of equation 12. In this man-ner all of physics is proposed to follow from the ‘one simple equation’ for the invariantproper time interval δs in equation 46, which is equivalent to equation 11 and can bewritten as equation 13 as described in subsection 2.2, as the basis of the theory. Aslabelled in equation 13 the symmetry transformations applied to the subcomponentsof time belong to a group ˆ G that generalises the Lorentz group of relativity theory.A connection with the Lagrangian approach is anticipated to arise throughterms in the breaking of the full symmetry ˆ G of equation 13 in the projection overthe local substructure of 4-dimensional spacetime M . Matter fields deriving from thebroken fragments of v n ∈ R n will feature in ‘mass terms’ in the resulting expressionof equation 29 if composed with a factor derived from the projected subcomponent h ≡ v ∈ TM , which is here central to the ‘origin of mass’ itself and associated withthe Higgs as described for equations 22–24, with Yukawa coupling factors to be iden-tified as proposed in subsection 4.2. The broken symmetry expression in equation 29also places constraints on the dynamics of the matter fields, coupling them with gaugefields, as discussed in subsection 2.3. The role of Lagrangian kinetic terms for eachgauge field A ( x ) associated with the internal symmetry is proposed to be appropriatedby the quadratic terms in the internal gauge curvature F ( x ) in the relation with theexternal spacetime geometry of equation 25, which is similar to that in many Kaluza-Klein theories, with the dynamics of the gauge fields constrained by geometric identitiesas also reviewed in subsection 2.3. Further, rather than applying the postulates of aquantum theory, a local degeneracy of field solutions for describing the external space-time geometry is proposed to underlie the ‘quantisation’ of the matter and gauge fieldsand the corresponding particle phenomena as reviewed for equation 44.That these particle phenomena will exhibit properties closely resembling theStandard Model has been demonstrated for the analysis through to the ˆ G = E levelof equation 35–36 in subsection 3.1. The full picture is predicted to emerge for theproposed ˆ G = E level of equation 37 as discussed in subsection 3.2, for which physicsbeyond the Standard Model in the neutrino, Higgs and dark sectors can be anticipatedas described in section 4. It is striking to observe how properties of the StandardModel, and contemporary physics beyond, can be uncovered in this manner from sucha simple underlying basis. In particular the above analysis illustrates how a rangeof relatively complex phenomena, matching empirical observations, can be obtainedthrough the ‘one simple equation’ for an infinitesimal interval of time in equation 46,or the equivalent expressions of equations 11 and 13. The ultimate ambition would beto determine not only the elementary particle multiplet structure but also the masses,charges and couplings of elementary particle states as far as possible from the intrinsicconstraints of the theory. In deriving from the natural generalisation for a proper timeinterval in equation 46 as a simple elementary basis for the theory, there is then thepotential for a genuine understanding of the underlying origin of the Standard Modeland physics beyond, all accounted for by a fundamental unified theory.56 Discussion and Conclusions
One of the main aims of experiments in particle physics, observations in cosmologyand the construction of phenomenological models, typically via a proposed Lagrangianfunction, is to point the way to a unified theory incorporating a more elementaryunderstanding of the workings of the universe. Given a set of empirical data and anarray of associated models an intuitive leap, rather than a series of small iterativesteps, is likely to be needed to uncover the fundamental theory. Having posited sucha theory, based largely on an internal motivation relating to its conception, the aimwould then be to work forwards developing the theory and seeking correspondence withmodels that have been devised on the basis of empirical observations. In this paper weare not building a model pragmatically, for example by adding terms to a Lagrangianby hand, but rather we have posited such a fundamental theory and developed theconsequences from underlying elementary first principles.In place of directly adding matter fields independently on top of 4-dimensionalspacetime to match empirical observations, as for the models alluded to above, theearly approaches to a unified field theory reviewed in subsection 1.2 aimed to accountfor the empirical properties of matter in spacetime within a single unified framework byfurther generalising the structure of spacetime itself. This principal goal was presentfrom the beginning, as alluded to in the title of Kaluza’s paper [35], and continuesto motivate modern-day unification schemes with extra spatial dimensions aiming toaccount for the Standard Model through an augmented spacetime structure. Via sucha conception the aim has been to achieve a unification in which the field and particlestates we observe in experiments reside within the internal ‘mathematical space’ ofextra spatial dimensions, which is more akin to the external ‘geometrical space’ withinwhich macroscopic structures such as ourselves reside and through which we perceivethe physical world.Extra spatial dimensions have then provided the potential to connect thephysics of the external world of 4-dimensional spacetime and the internal world ofmaterial phenomena within a more unified structure (see for example [37] section 9first two of final three paragraphs). However even the most sophisticated attemptsto account for the Standard Model via the structures of extra spatial dimensions, asdiscussed in subsection 2.1, have if anything been less successful than the original5-dimensional spacetime theory of Kaluza and Klein in accommodating electromag-netism, as reviewed for equation 5.From a local perspective the addition of extra spatial dimensions augments aproper time interval δs from the form of equation 1 to that of equations 6 and 7, asdepicted by the generalisation from figure 1(a) to 1(b). An elementary analysis forthe implied matter content in 4-dimensional spacetime is described for equation 8–9.The present theory is based on a direct further generalisation of a proper time intervalover and above the 4-dimensional form of equation 1 for the local external spacetimeto the form of equation 11. Compared with augmentations restricted to extra spatialdimensions with the quadratic form of equation 6 the more general homogeneous poly-nomial form of equation 11 demands fewer assumptions, as explained in subsection 2.2.As described explicitly for equation 12 this more generalised form for proper time canstill readily accommodate a substructure matching the local external spacetime metricgeometry. 57dopting the generalisation of equation 11, which can be expressed as equa-tion 13, matter fields are then associated with the properties of the residual componentsresulting from the symmetry breaking of the generalised form of proper time when pro-jected over the local 4-dimensional geometric spacetime substructure, as depicted infigure 2. The elementary physical structures identified in spacetime for a minimal caseare described in subsection 2.3. Leading to the structure of equation 35–36 the resid-ual components are found to exhibit a significant correlation with the Standard Modelof particle physics as reviewed in section 3. This direct analysis for the general formof proper time then provides a far better template for connecting with the StandardModel, and with features beyond as described in section 4, than the equivalent analysisapplied for the restricted case of extra quadratic spatial components as described forequation 8–9, and in this respect also compares favourably with the more sophisticatedapproaches to extra spatial dimensions discussed in subsection 2.1.Similarly as for extra spatial dimensions this theory based on generalised propertime can be seen as a natural further development from special and general relativ-ity as described in subsection 5.1. Compared with the employment of extra spatialdimensions the present theory is however more unifying in reducing the fundamen-tal entity of theory to the more elementary starting point of the single parameter ofproper time alone. The greater simplicity of the theory is manifested in its expressionthrough ‘one simple equation’ for infinitesimal intervals of time as discussed in sub-section 5.2 for equation 46, which is equivalent to equations 11 and 13. The theoryis also more conservative in that the flow of time is something that we are intimatelyfamiliar with while, essentially by definition, we do not perceive any ‘extra’ spatialdimensions. Further the theory is more unique in terms of having an unambiguousstarting point in proper time as parametrised by s ∈ R with a trivial topology, ratherthan having a potentially arbitrary number of extra spatial dimensions with a largerange of possible topological properties. This uniqueness is further seen in the mathe-matical development of the theory in leading to equations 34–36, as summarised in theopening of subsection 3.2. Each of the above points applies for the present theory bothin comparison with the direct approach to extra spatial dimensions of equations 6–9and figure 1 as well as in comparison with the most technically sophisticated approachvia string and M-theory which, for example with regards to uniqueness, is faced withthe landscape problem as also discussed in subsection 2.1.The internal simplicity and uniqueness of the present theory, which neverthelessyields a far more direct and efficient connection with empirical structures of the Stan-dard Model at the most elementary level of matter compared with the models basedon extra spatial dimensions, strongly suggests that equation 13 rather than equation 7provides a more appropriate core basis for a unified theory. The extraction of a nec-essarily quadratic substructure to represent the local external spacetime also underliesa mechanism for the symmetry breaking itself for equation 13 but not for equation 7,as described after equation 14. While models with extra spatial dimensions essentiallypropose a ‘materialisation of space’ the present theory goes further by describing a‘materialisation and spatialisation of time’. The nature of our necessary perception ofthe world in space as well as through time is central to the symmetry breaking as alsoalluded to in subsection 5.2 with reference to [94] where further historical backgroundto this conceptual picture is also discussed. Here we briefly summarise the historicaldevelopments in mathematics and physics underlying this theory.58he first particle of the Standard Model to be discovered was the electron at theend of the 19 th century, around the same time that the exceptional Lie algebras werefirst classified as noted in the opening of subsection 3.1. The Standard Model itself wasfully established in the 1970s, representing an amalgamation of the inferences drawnfrom many experiments over a number of decades, and has been tested subsequentlyand largely confirmed by further decades of empirical data through to the discovery andanalysis of the Higgs in recent years. Mathematical developments from the mid-20 th to early 21 st century, including [57, 58, 59, 60] as cited in subsection 3.1, have enabledthe development of the present theory, as expressed for equation 13, through the 9,27 and 56-dimensional forms of equations 19, 30 and 34. These have led respectivelyto the identification of the symmetry breaking structures of equations 26–27, 32–33and 35–36, closely resembling the features of a generation of Standard Model leptonsand quarks – including a set of spinor states, colour SU(3) c singlets and triplets withthe appropriate electromagnetic U(1) Q fractional charges and an intrinsic left-rightasymmetry.This progression then represents an intimate coming together of mathematicaland physical structures which, while contemporary with each other and studied inparallel, developed largely independently over the past hundred or more years, andwith very little redundancy as seen for equation 35–36, drawn together through naturalmathematical expressions for equation 13 as the central equation of this unified theory.While the theory is rooted in the firm conceptual basis of generalised proper time thesubsequent employment of mathematical structures relating to the exceptional Liegroups has advanced the theory to the point of making connections with empiricalphenomena. Mutually, the application of these mathematical structures, which havebeen developed in a largely abstract manner, within the context of the present theory,presents an explicit means of relating the exceptional Lie groups to the physical world.That is, the expressions in equations 30 and 34 provide their respective symmetrygroups E and E with a clear and simple conceptual basis within this theory, assymmetries of generalised proper time, without needing to introduce them in an adhoc fashion as might be the case for model building (dating back to [32, 33] as alludedto in subsection 1.2), here making their relevance for the Standard Model explicitthrough equations 32–33 and 35–36. Such an interplay of mathematical developmentand physical application is analogous to the incorporation of fibre bundle structures inthe framework of gauge theories and non-Abelian Kaluza-Klein theories as discussedin subsection 1.2 with reference to [37, 38].While properties associated with the Standard Model emerge much more read-ily for the present theory, compared to models with extra spatial dimensions, a com-plete correspondence is not to be expected until a mathematically complete under-standing of the full form of proper time, and its corresponding symmetry breakingpattern, has been established. The unique development of the theory through to theE quartic form of equation 34 with the implications of equation 35–36 together withproperties of the exceptional Lie groups naturally lead to the prediction of a furtherunique progression to an E symmetry of a full form L ( v ) E = 1, as described forequation 37 in subsection 3.2, proposed to result in the uncovering of the full particlemultiplet structure of the Standard Model and beyond. While elements of the Stan-dard Model have been accounted for through a rigorous analysis of the E and E levelsunderlying equations 32–33 and 35–36, the investigation at the E level, including that59or equation 40, has to date been of a more provisional nature. An explicit descriptionof a real E action constructed on the, currently hypothetical, homogeneous form ofequation 37 with a symmetry breaking pattern of equation 38 applied to the subcom-ponents of v ∈ R , subsuming equation 35–36, will be needed to attempt a fullreconstruction of the Standard Model structure and to make firm predictions beyond.The need to unfold the full Standard Model structure, including three genera-tions of leptons and quarks with a complete set of spinor states also for the ν -leptonsand u -quarks and with a full electroweak symmetry, implies that there are signif-icant requirements on the desired structure of the homogeneous polynomial form L ( v ) E = 1 which may also be utilised as a guide in identifying the E symmetrystructure and breaking pattern itself. However the main goal for the predicted E structure, as achieved for the intermediate E and E levels, is principally to deducethe properties of the Standard Model, and beyond, as constrained by the natural math-ematical expressions for equation 13. That it is not obvious that this can be achievedfor E , or a closely related symmetry structure consistent with equation 13, provides anon-trivial test for the theory. The technical mathematical details of the connectionswith the Standard Model made through to the E level and support for the predictedrole for E are described extensively in [53] and the references therein. Here a signif-icant direction for further progress on the mathematical side, through investigating apossible role for octonion triality in the construction of an octic E invariant in 248components, has been described in subsection 3.2 – in particular with the guide of thereferences cited in the last three paragraphs of that subsection.The attempt to advance the status of the theory beyond the conceptual ba-sis and corresponding mathematical formalism by making connections with empiricalphenomenology has also been extended beyond the Standard Model. In this paper wehave focussed upon the opportunity for a mutual development of this theory along withan understanding of new physics, in particular in the neutrino sector. Building uponthe progression through the E and E levels general consideration of the possibility ofa significant role for E leads, via the schematic extrapolation from the relevant partsof equations 32–33 and 35–36 to equation 40, to the preliminary prediction for only two right-handed neutrino states to be accommodated in a manner compatible witha structure of three generations for the other lepton and quark states. As also notedin subsection 4.1 this suggests that one of the active left-handed neutrino states willbe correspondingly massless, with forthcoming improving limits in the laboratory andfrom cosmology able to test this tentative but specific empirical feature of the theory.By making a connection with models featuring two right-handed neutrinos (for exam-ple [13, 14, 15, 16]) in this paper the focus has been to demonstrate a specific bridgefrom the forefront in developing this theory, based on a very simple underlying origin,to a key area in recent developments in particle physics.If this theory had reached the present stage of development in the 1980s, withthe Standard Model established but still with relatively limited empirical input inneutrino physics, then the argument for two right-handed neutrinos would not havebeen a compelling prediction of the theory at this point. Rather we have aimed todemonstrate how the theory might in principle accommodate two right-handed neu-trinos naturally within a structure of three generations of Standard Model fermions,consistent with a series of recent neutrino models that have been motivated by contem-porary neutrino phenomenology. While at this stage lacking in more explicit detail for60he neutrinos, since the present theory is based on elementary first principles ratherthan being explicitly tailored to fit the data as a pragmatic model, through furtherdevelopment there is here an opportunity for a deeper understanding of the nature ofneutrino physics generally, as discussed in subsection 4.1.While the mathematical pursuit of the full symmetry action of E for theform L ( v ) E = 1 and the resulting breaking pattern may elucidate the originof the esoteric properties of the neutrino sector, building upon the basic generationstructure of equation 40, existing empirical knowledge of neutrino phenomenology asestablished in recent decades, as expressed by models with two right-handed neutrinos(or even other models such as for example the ν MSM in which two right-handedneutrinos have distinct properties and a distinct role from a third ν R state [7, 8] as alsoreviewed in subsection 1.1), might itself provide a pertinent clue, on top of the StandardModel itself, towards deciphering the detailed structure of the specific mathematicalapplication for E that is predicted. In particular the packing of the neutrino spinorstructure, under an SL(2 , C ) ⊂ E action of equation 38 on the subcomponents of v in equation 37 utilising octonion triality as alluded to above, and their Diracor Majorana nature in mass terms identified in the expansion of equation 41, willbe central to this ambition. A connection might then be made between Lorentz andgauge invariant terms in the expansion of equation 41 and Lagrangian mass terms inmodels with two ν R states. More generally there is an open opportunity to explore themutual development and close interplay of the underlying mathematical structure ofthe theory together with an understanding of Standard Model physics and empiricalconsequences beyond.The main approach of this theory is ‘deductive’ in starting with the hypothesisof the central role for proper time in equations 11 and 13 and then developing themathematics, for example through to equations 35–36 and 40, to then examine theempirical consequences in a manner ideally leading to testable predictions. As alludedto in the opening of this section this contracts with the ‘inductive’ approach employedin the construction of a model, as inferred from patterns in the data, for example byadding fields and terms to a Lagrangian. While that data hence provides evidence forsuch a model the inductive argument is not in general unique and typically leads to arange of plausible models, as is generally the case for neutrino phenomenology. Here,while an interplay with inductive elements might be utilised to aid in the mathematicalconstruction of the E action and the explicit form of equation 37 sought, the ultimateaim is for a clear, unique and deductive argument unambiguously covering a wide rangeof physical phenomena, and for example homing in on the features of a particular modelin the neutrino sector.A degree of confidence in any predictions of the theory can be gained throughthe intrinsic simplicity of the underlying basis of the theory, as described in the pre-vious section, and the significant foothold in features of the Standard Model that hasalready been efficiently attained as summarised here for equations 26–27, 32–33 and35–36. Indeed most of the subcomponents of v in equation 35–36 are associatedwith elements of the Standard Model with the only apparent redundancy being theset of four scalar invariants { n, N, α, β } , which provisionally form candidates for darkmatter as described in the opening of subsection 4.2. The prospects for insights intonew physics include the nature of the Higgs and electroweak symmetry breaking, forwhich empirical predictions might be sought for the laboratory. The close connection61ere between the Higgs sector, associated with the h ≡ v ∈ TM projection of equa-tions 22 and 23, and neutrino physics, as implied by equation 40, suggests the Higgsmay have a composite structure as described in subsection 4.1. This feature, togetherwith the possible connections between this Higgs and a dark sector as explained in sub-section 4.2, may impact precision electroweak measurements in a manner accessible atthe LHC or at a future e + e − linear collider as also discussed in section 4.In subsection 4.2 we described for equation 42 how the impact of possiblealternative forms for proper time on the 4-dimensional spacetime geometry might itselfprovide a quintessential source for the dark sector, in only being linked with the visiblesector deriving from the E form of equation 37 through purely gravitational means.Correspondingly the generalisation from { n, N, α, β } at the E level of equation 35–36could yield a broader set of scalar invariants under the full E broken symmetry which,on taking vacuum values and being composed in terms of the expansion of equation 41,may form a range of Yukawa couplings and hence in principle augment the predictivepower of the theory, as also noted in subsection 4.2. More generally analysis of theproperties of the full set of subcomponents of v ∈ R from the breakdown ofequation 37 offers the opportunity to explore further new physics, as does the possibilityto fit a further gauge group alongside that of the Standard Model in the E symmetrybreaking pattern of equation 38.While previous papers (including [39, 53]) have described in detail the connec-tions made with the Standard Model (summarised here in section 3), in this paper wehave emphasised the potential connections with new physics (as assessed in section 4),while always keeping in sight the simple origins of the theory based on an elementarygeneralisation of a proper time interval as described for equations 11 and 13. Withreference to subsection 1.2, in section 2 and subsection 5.1 we have described how thebasis for this theory connects with the early unification schemes from the immediatepost-general relativity era based on generalisations of the 4-dimensional spacetime ge-ometry. Hence in developing the present theory we connect the original conceptionof a unified field theory dating from a hundred years ago with the current quest tounderstand the empirical data from recent and ongoing laboratory experiments andcosmological observations. In particular the modern-day empirical status of neutrinophysics, as reviewed in subsection 1.1, is hence connected with the historical proposalsof subsection 1.2 through this theory, ultimately making contact with models featuringtwo right-handed neutrinos as explained in subsection 4.1 (with the central argumentsummarised in [1]).As described for equations 45 and 46 it is difficult to conceive of a more ele-mentary basis for a unified theory than this simple direct generalisation of proper time.On the other hand the natural mathematical possibilities for the invariant infinitesimalproper time interval in equation 46, which is equivalent to the ‘one simple equation’ ofequation 11 and also equation 13, are open to further exploration. Also open to investi-gation are the wider implications of the theory in relation to the geometric structuresof Kaluza-Klein models [40], quantisation and the nature of physical particle statesas studied in laboratory experiments ([54] chapters 10, 11 and section 15.2, as alsodiscussed for [94] equations 9 and 10) and the cosmological questions regarding thedark sector and early universe ([54] chapters 12 and 13) as well as the connectionswith the Standard Model itself ([54] chapters 6–9, [39, 53]). With the aim of makinga range of testable predictions there is an opportunity for mutual advances in these62heoretical and related experimental investigations. 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