aa r X i v : . [ m a t h . G R ] F e b FINITE SYMMETRY GROUPS IN PHYSICS
ROBERT ARNOTT WILSON
Abstract.
Finite symmetries abound in particle physics, from the weak dou-blets and generation triplets to the baryon octet and many others. These areusually studied by starting from a Lie group, and breaking the symmetry bychoosing a particular copy of the Weyl group. I investigate the possibilityof instead taking the finite symmetries as fundamental, and building the Liegroups from them by means of a group algebra construction. Introduction
Historical motivation.
Einstein made a number of very pertinent remarksabout the foundations of quantum mechanics, often summarised in the famousphrase “God does not play dice”. For example, in 1935 he wrote [1]:“ In any case one does not have the right today to maintain that thefoundation must consist in a field theory in the sense of Maxwell.The other possibility, however, leads in my opinion to a renunci-ation of the time-space continuum and a purely algebraic physics.Logically this is quite possible [...] Such a theory doesn’t have tobe based on the probability concept.”In 1954, he went further, and opined [2]:“I consider it quite possible that physics cannot be based on the fieldconcept, i.e., continuous structure. In that case, nothing remainsof my entire castle in the air, gravitation theory included.”Nevertheless, modern physics has stuck with the field concept, and his ‘castle inthe air’ remains in general use.In 1935, there simply was not enough experimental evidence to provide muchof a clue as to the possible structure of a ‘purely algebraic physics’, beyond theobvious fact that it must contain a copy of the quaternion group. But today, thereis an enormous amount of experimental evidence, that essentially shows there is aunique possibility for a purely algebraic physics [3]. Whether it actually works ornot, is a separate question. But we should at least try it. I begin by consideringhow much of physics, including continuous approximations interpreted as spacetimeand fields in spacetime, can be obtained from the quaternion group alone.1.2.
The spin group and the quaternion group.
Group theory [4] first enteredinto quantum mechanics with the discovery of the spin of an electron, which has adirection relative to the ambient space, and therefore requires a group SU (2) forits description. On a macroscopic scale, the sum of the spins of a large number ofelectrons, protons and neutrons gives rise to the phenomenon of magnetism, whenthere is a sufficiently large bias in the directions of spin. Date : 12th February 2021.
But it is experimentally impossible to measure the direction of spin of an individ-ual elementary particle. All one can do is choose a direction, and measure whetherthe spin is ‘up’ or ‘down’ in that direction. Indeed, even for larger particles, suchas silver atoms, the Stern–Gerlach experiment [5] demonstrates that the directionof spin is quantised. Therefore we must assume that the elementary particles have,intrinsically, only finite symmetries. The smallest finite group that is available todescribe spin symmetries is the quaternion group Q , with 8 elements ± ± i , ± j and ± k satisfying the rules ij = k, jk = i, ki = j,i = j = k = − . (1)Now if we take a large number of copies of Q , and add them together, we obtain[6] the integral group ring Z Q , in which all the copies of Q are identical, meaningthat all electrons are identical, and so on. In macroscopic physics, the individualquanta disappear from view, and it becomes reasonable to approximate the scaledcopy of the integers ε Z by the real numbers R , and study the group algebra R Q instead. There is, up to equivalence, only one faithful representation of Q , inthe quaternion algebra H , and there are four 1-dimensional real representations, ofwhich one is trivial and three are representations of the three quotients Q /Z ∼ = Z .Therefore the group algebra has a canonical (Wedderburn) decomposition as R Q ∼ = 4 R + H . (2)Taking out the five real scale factors from this algebra, we are left with the group SU (2), which is exactly what we require in order to describe magnetic spin inordinary macroscopic space.1.3. Analysis of the group algebra.
In the representation theory of finite groups,the usual convention is for the finite group to act on the algebra by right multiplica-tion, and the continuous groups to act by left-multiplication. This allows us to treatboth the discrete and continuous symmetries simultaneously. The macroscopic be-haviour, on the other hand, requires the continuous group to act on both sides, byconjugation, so that SU (2) acts on H as SO (3), that is, fixing the real part, androtating the imaginary part as a Euclidean 3-space. It is standard to interpret this3-space as real physical space, in the case when SU (2) represents magnetic spin.It would be reasonable, then, to interpret the real part of H as a (non-relativistic)time. This allows the ‘spin’ to be modelled as something that takes place in spaceand time, rather than in isolation.But the finite group has given us something in addition, namely four copies ofthe real numbers. These presumably represent four physical particles, at least threeof which we can detect magnetically. These four particles are acted on by the finitegroup on the right, in three cases to change the sign, so that these three appearin a pair of spin states. They are invariant under conjugation by SU (2), so undersymmetries of macroscopic space. But they are not invariant under the left actionof the finite group. Since we started out modelling spin of electrons and protons,these had better be two of the particles, and the most plausible interpretation ofthe other two is as the neutron and (electron) neutrino. The first three appear intwo spin states, the last in only one, as we know also from the Wu experiment [7].Thus this action describes the weak interaction, in the form of beta decay: e + p ↔ n + ν. (3) INITE SYMMETRY GROUPS IN PHYSICS 3
Note that this is an action of the finite group, not of the spin group. In thestandard model [8], the weak interaction is described instead by a second copy of SU (2), commuting with the spin group. It is of course perfectly possible to definean action of SU (2) on this 4-dimensional real space, and have this action commutewith the spin SU (2), and try to build a model of physics on top. But this actionis not compatible with the group algebra. The standard model therefore puts theweak SU (2) elsewhere, and doubles up the Weyl spinor into a Dirac spinor so thatthere will be no conflict between the finite action and the continuous action. Bydoing so, however, it breaks the connection between the weak interaction and thespin group, which then has to be put back in by hand, in the form of a mixingbetween the weak force and electromagnetism. In the toy model discussed here,the mixing between magnetism and the weak force happens automatically.It is worth analysing the modelling of these particles in a little more detail.In the group algebra, the four 1-dimensional representations contain the followingelements: 1 a : 1 + ( −
1) + i + ( − i ) + j + ( − j ) + k + ( − k );1 b : 1 + ( −
1) + i + ( − i ) − j − ( − j ) − k − ( − k );1 c : 1 + ( − − i − ( − i ) + j + ( − j ) − k − ( − k );1 d : 1 + ( − − i − ( − i ) − j − ( − j ) + k + ( − k ) . (4)We then see that the elements of the group can be interpreted, not as particles,but as quantum numbers of the particles. Two quantum numbers are sufficientto distinguish these four particles, so that for example, if λ i and λ j denote thecoefficients of i and j respectively, then we can take the charge to be ( λ j − λ i ) / λ j /
2. We do not need to specify the coefficient of k ,that is λ k = λ i λ j .In a similar way, the other half of the group algebra has a basis consisting of t := 1 − ( − ,x := i − ( − i ) ,y := j − ( − j ) ,z := k − ( − k ) . (5)The names are chosen with the intention of suggesting a potential quantisation ofspacetime, with z representing the direction of spin, and the x, y plane relatingto the charge and isospin. The (speculative) idea, which I shall explore in moredetail below, is that the combination of spin, charge and weak isospin for a (large)collection of interacting elementary particles may be sufficient to define the ambientphysical spacetime.This example illustrates the general principle of how a macroscopic force, in thiscase magnetism, emerges from individual quanta. In particular, it illustrates how amacroscopic continuous symmetry arises from a microscopic discrete symmetry. Italso illustrates how a breaking of the symmetry at the microscopic level arises froman interaction between elementary particles, in this case the weak interaction, andgives rise to a breaking of symmetry also at the macroscopic level. The quaterniongroup Q on its own is therefore sufficient to model a united magneto-weak force.It is not sufficient for a complete electro-magneto-weak unification, however, forwhich we need a larger group. This larger group must be capable of dealing withthe fact that the electron comes in three different mass states, that is in threedifferent generations. ROBERT ARNOTT WILSON
The automorphism group.
The symmetries of this finite model of spin aredescribed by the automorphism group of Q , that is isomorphic to Sym (4), thesymmetric group on 4 letters. The automorphisms are of five types, where I denotethe letters
W, X, Y, Z : • the identity element, of order 1. • W, X )( Y, Z ); these are inner au-tomorphisms, that is conjugation by i , j and k . • X, Y, Z ); these can be representedas conjugation by unit quaternions ( − ± i ± j ± k ) / • W, X ); represented as conjugationby i ± j , j ± k and k ± i . • W, X, Y, Z ); represented as conju-gation by 1 ± i , 1 ± j and 1 ± k .These symmetries have many different interpretations, depending on what theoriginal copy of Q represents. The kind of symmetry that it might be useful to lookfor is an extension of the neutrino/electron/proton/neutron symmetry discussed inthe previous section, to three generations of electron, plus the baryon octet [9]. Inthis way the baryons would exhibit a 3-fold symmetry represented by the 3-cycles,that could perhaps be interpreted as a colour symmetry of the three constituentquarks [10]. Similarly, the electrons would divide into two transpositions, possiblyrepresenting the left-handed and right-handed spins.Alternatively, or in addition, one might want to consider the odd permutationsas representing leptons and quarks. The transpositions could then represent ei-ther the left-handed and right-handed electrons, or the left-handed parts of bothneutrinos and electrons. Similarly, the 4-cycles could represent quarks, either inup/down pairs or in left/right pairs. Additional interpretations might arise fromconsideration of particle interactions, so that the identity element represents elec-tromagnetic interactions, the 3 elements of order 2 represent the weak interaction,and the 8 elements of order 3 the strong interaction. The strong interaction man-ifests itself both in massless form (as gluons) and in massive form (as the mesonoctet, including pions and kaons).It may be worth mentioning in passing that if one extends the S action byconjugation of the quaternions listed above, to include both left and right multipli-cations, then one obtains the Weyl group of type F . This may be the fundamentalreason why the Lie group of type F , and various related types such as D and E ,are so attractive as potential ways to extend the standard model [11, 12, 13, 14, 15].2. The binary tetrahedral group
Overview.
There is only one non-trivial way to adjoin a triplet symmetry to Q , and that is to extend to the binary tetrahedral group, G say, of order 24, byadjoining the quaternion w := ( − i + j + k ) / , (6)and therefore also its quaternion conjugate v := ( − − i − j − k ) / . (7)This group and its representation theory are described in some detail in [3], but weshall require yet more detail here. I begin with a summary. INITE SYMMETRY GROUPS IN PHYSICS 5
The most important thing to note is the structure of the real group algebra: R G ∼ = R + C + H + M ( C ) + M ( R ) . (8)Taking out four real scalars and one complex scalar leaves us with the group U (1) × SU (2) × SL (2 , C ) × SL (3 , R ) , (9)which contains all the groups we need for the standard model of particle physics,apart from the fact that we have the split real form SL (3 , R ) rather than thecompact real form SU (3) of the Lie group of type A .This difference may reflect the fact that we are looking at finite (generation)symmetries that can be observed in experiments, rather than continuous (colour)symmetries that are not observable. It may be possible to resolve this issue byextending to the complex group algebra, but this process obscures a number ofimportant features of the real group algebra, so I will only do this if absolutelynecessary. In any case, we must bear in mind that this difference must be resolvedat some point, since it may be the difference between a viable model and an unviablemodel.2.2. Irreducible representations.
Representation theory is usually presentedfirst over the complex numbers, since this is the simplest theory, but we shallrequire the extra subtlety of the representation theory over real numbers. Firstnote that the conjugation action of the group on itself divides the group into 7conjugacy classes, of sizes 1 , , , , , ,
4, as listed in this table:Size Elements Order1 1 = e − i ± i, ± j, ± k w, wi, wj, wk v, − vi, − vj, − vk − w, − wi, − wj, − wk − v, vi, vj, vk ω and ¯ ω are the complex cube roots of unity.1 − i w − w v − v ω ω ¯ ω ¯ ω ω ¯ ω ω ω − − − − − − ω ω − ¯ ω ¯ ω − − ¯ ω ¯ ω − ω ω (11)The three 2-dimensional representations will eventually play the role of Weyl spinors,but there are three of them, not just the familiar left-handed and right-handed Weylspinors. Of course, the standard model for electro-weak interactions really has threeWeyl spinors, two left-handed and one right-handed, so the question is whether itis possible, and if so, how, to match up the three spinors in the two models. ROBERT ARNOTT WILSON
The structure of the complex group algebra can be read off from the dimensionsof the representations, and is3 C + M ( C ) + 3 M ( C ) . (12)In addition to the problem of mixing and matching the three copies of SU (2) or SL (2 , C ) inside the three copies of M ( C ), there is the problem of mixing andmatching the three copies of U (1) inside the three copies of C . There is too muchchoice at this stage, and the problem becomes rather more tractable if we restrictto the real group algebra. Of course, this restriction means that we lose the group SU (3) from M ( C ), but we have SL ( R ) instead. This different real form may ormay not be a satisfactory replacement for the gauge group of the strong force.The irreducible real representations can be described by taking the sum of acomplex representation with its complex conjugate, and at the same time takingthe union of each conjugacy class with its inverse class. This reduces the table tofive rows and five columns, as follows:1 − i w − w − −
13 3 − − − − − H , todenote a Hamiltonian quaternionic representation, and the second one 4 C , to denotea classical complex representation. Then the group algebra, as a representation ofthe finite group G , has the structure1 + 2 + 3 + 3 + 3 + 4 H + 4 C + 4 C . (14)2.3. Tensor products.
So far, I have associated the various Lie groups (gaugegroups and spin groups) with a single representation of the finite group. The finitegroup links them together in various ways, described in part by the decompositionsof tensor products of representations. These decompositions can be calculated easilyfrom the character table, and are as follows, with + signs omitted to save space:1 2 3 4 H C C C H C H C C H C C H C C H C C C H C H C C ⊗ H = 4 H + 4 C + 4 C meansthat the whole of the fermionic part of the algebra can be obtained from a real plusa complex ‘version’ of the spin representation 4 H . INITE SYMMETRY GROUPS IN PHYSICS 7
If one distinguishes the two copies of the real numbers by labelling one of themwith γ , then there is some prospect of being able to match up the finite model withthe standard model. Similarly, all of the 4 ⊗ iγ )into a real 2-space, and a similar mixing in the odd part of the Clifford algebra. Allthis suggests that a careful distinction between 4 H and 4 C may be able to throwsome light on the mixing of quantum electrodynamics and the weak interaction inthe standard model.A similar picture emerges from the bosonic part of the algebra, in which oneobtains the whole algebra from the tensor product (1 + 3) ⊗ ⊗ (1 + 3) ∼ = 1 + 1 + 2 + 3 + 3 + 3 + 3 ∼ = 4 C ⊗ C . (16)However one tries to interpret this equivalence, it seems to imply a mixing betweenthe strong force, with a gauge group acting on 1 + 3, and the electroweak forces,with a gauge group acting on 4 C .Indeed, there are a number of other suggestive isomorphisms between differenttensor product representations. For example, for any representation R that doesnot involve the trivial representation 1, we have(1 + 2) ⊗ R ∼ = 3 ⊗ R, (17)which gives ample scope for mixing a broken 1 + 2 symmetry with an unbroken 3symmetry. Another example is that3 ⊗ H ∼ = 3 ⊗ C , (18)which gives plenty of scope for mixing a spinor of type 4 H with a spinor of type 4 C .2.4. Explicit matrices.
For the purposes of explicit calculation, it will be usefulto have explicit matrix copies of all the irreducible representations. It is sufficientto specify matrices representing the generators i and w . In the 1-dimensional rep-resentation, they both act trivially. In the 2-dimensional representation, w is arotation of order 3, and i acts trivially, so we may take the matrices i (cid:18) (cid:19) , w (cid:18) − √ −√ − (cid:19) (19)The 3-dimensional representation is the representation as symmetries of a regulartetrahedron, which can be embedded as alternate vertices of a cube, so that thematrices can be taken as i − − , w . (20) ROBERT ARNOTT WILSON
The representation 4 H is the representation by right-multiplication on the quater-nions, so that the matrices can be taken as i − −
10 0 1 0 , w − − − − − − − − − − (21)In the standard model the quaternionic symmetry is broken, and a particular com-plex basis is chosen. The following matrices give an example, which may or maynot be similar to what is done in the standard model: i (cid:18) i − i (cid:19) , w (cid:18) − i i − i − − i (cid:19) (22)Finally, the representation 4 C can be written as i − −
10 0 1 0 , w α − β − β − αβ α α − βα − β β αβ α − α β (23)where α = 1 + √ β = 1 − √ . (24)It is unfortunately not possible to get rid of the annoying square roots of 3 in thisreal representation. On the positive side, they are what permits the extension ofthe standard model from one generation of fermions to three.2.5. Projections.
The theory of finite group representations contains a canonicalset of projections from the group algebra onto the various matrix subalgebras.First there is the basic division into bosons and fermions, obtained via projectionwith the idempotents ( e ± i ) /
2. Within the fermions, there are two projectionsobtained from the sum of all the elements of order 3, that is s := w + iw + jw + kw + v − iv − jv − kv. (25)These projections are defined by the elements (2 e − s ) / e + s ) / / / R ( e + i )( e + i )( e + j )( e + w + v ) / C ( e + i )( e + i )( e + j )(2 e − w − v ) / M ( R ) ( e + i )(3 e − i − j − k ) / H ( e − i )(2 e − s ) / M ( C ) ( e − i )(4 e + s ) / M ( R ) from this description we obtain the 15-dimensionalsubalgebra R + C + H + M ( C )(27)in which there is a close parallel between the splitting R + C using ( e + w + v ) / e − v − w ) /
3, and the splitting H + M ( C ) using (2 e − s ) / e + s ) / INITE SYMMETRY GROUPS IN PHYSICS 9
These two pairs of projections appear to be the finite analogues of the pair ofprojections (1 ± γ ) / M ( R ), to obtain a 16-dimensional algebra that fulfils the function ofthe Dirac algebra, but with a more subtle structure derived from the action of thefinite group.It is worth remarking here that because these projections involve dividing by 2and 3, they can be implemented in the real group algebra R G or the rational groupalgebra Q G , but not in the integral group ring Z G . Indeed, the structure of Z G ismuch more subtle than that of R G . Ultimately, to implement the finite model infull we will need to grapple with this structure in detail. For the purposes of thepresent paper, however, it is enough consider only the real group algebra.Just to give a flavour of the implications of using the integral group ring, I’lldescribe a toy model of the proton using the integral group ring of the group Z oforder 3. If we take e, v, w as the elements of Z , then the idempotents in R Z are( e + v + w ) / e − v − w ) / Z , and give a perfectlygood description of the internal structure of a proton.But they are not integral representations of Z . The integral group algebra Z Z does not support any projections, and is indecomposable. This I interpret as sayingthat the proton itself cannot be decomposed into smaller particles, so that, in thereal universe, protons never decay. Thus the quarks are ‘real’ particles, but theyare not ‘whole’ (integer) particles.It is also possible to use the action of Z by conjugation to describe the corre-sponding bosons. These bosons consist of one ‘left-handed’ and one ‘right-handed’quark, that correspond to ‘quark’ and ‘anti-quark’ in the standard model. So Iget three pions this way, consisting of two charged pions, π + = u ¯ d and π − = d ¯ u ,and a neutral pion π = u ¯ u . While this description of the neutral pion is not thesame as in the standard model, it is the only possibility in a discrete model, inwhich quantum superposition cannot be implemented as an intrinsic property of anelementary particle, but only as a property of the experiment or the environment.3. Implementing the standard model
The Dirac equation.
I have made various tentative suggestions for partsof the group algebra that could be used for various parts of the standard model.The time has come to make these suggestions more definite, and to begin to im-plement the standard model in this new mathematical framework. The first andmost fundamental requirement is to implement the Dirac equation. The most suit-able place to do this is surely the matrix algebra M ( C ). Thus we must use these2 × Actions on the Dirac spinors can then be obtained from both left-multiplicationsand right-multiplications, which together generate a 16-dimensional complex alge-bra that we must identify with the complex Clifford algebra in the standard model.For this purpose, we need a choice of Dirac gamma matrices. There are manypossible choices, but since each of the four Dirac matrices swaps the left-handedand right-handed parts of the spinor, the following looks to be a good choice. EachDirac matrix is written as a pair of a left-multiplication and a right-multiplicationby Pauli matrices. γ = (1 , σ ) = (cid:18)(cid:18) (cid:19) L , (cid:18) (cid:19) R (cid:19) ,γ = ( σ , iσ ) = (cid:18)(cid:18) (cid:19) L , (cid:18) − (cid:19) R (cid:19) ,γ = ( σ , iσ ) = (cid:18)(cid:18) − ii (cid:19) L , (cid:18) − (cid:19) R (cid:19) ,γ = ( σ , iσ ) = (cid:18)(cid:18) − (cid:19) L , (cid:18) − (cid:19) R (cid:19) . (28)With these definitions, we can write down the Dirac equation in the usual way.The group algebra model, however, contains more structure than the standardmodel, and in particular contains an important distinction between the Lie groupacting on the left, and the finite group acting on the right. Separating out the left-multiplications and the right-multiplications in the above, we have the followingright-multiplications: iγ = (1 , iσ ) ,iγ γ γ = (1 , iσ ) , (29)which together generate the quaternion group Q . Hence this Dirac equation canbe used to study four particles, as in the toy model discussed earlier. The sameequation applies to a variety of different particles, but only to four at a time. It ispossible to use this equation to study one generation of elementary fermions, butnot to study all three simultaneously. In particular, the standard model has toimplement the generation structure outside the Dirac algebra.On the left, we have i = ( i, γ γ = ( iσ , γ γ = ( iσ , , (30)which on exponentiation generate the Lie group U (2), that is, a group isomorphic tothe electro-weak gauge group in the standard model. Alternatively, we can obtain SL (2 , C ) from iγ γ and iγ γ . Note, however, that this copy of SL (2 , C ) is distinctfrom Dirac’s relativistic spin group generated by γ γ , γ γ and γ γ . In particular,these two copies of SL (2 , C ) have different physical meanings. At some point it willbe worth thinking carefully about the desired interpretations of these groups.3.2. Electro-weak mixing.
In the standard model, electro-weak mixing [17] isdescribed as a mixing of the gauge groups U (1) and SU (2), but it is fairly clearthat it is fundamentally a discrete phenomenon rather than a continuous one. Inpractice, there appears to be a small experimental variation of the Weinberg anglefrom a theoretical maximum of 30 ◦ . INITE SYMMETRY GROUPS IN PHYSICS 11
If we assume for the moment that the underlying discrete property is describedby an angle of exactly 30 ◦ , then this angle appears naturally as the angle betweenthe complex numbers i and ω . Since the group algebra model contains both i and ω , where the standard model only contains i , there is every prospect that thegroup algebra model can explain electro-weak mixing, at least to first order. Thesmall deviation of the Weinberg angle from 30 ◦ will of course require more detailedinvestigation.More specifically, the finite versions of U (1) and SU (2) in the standard model arethe scalar group of order 4 generated by i , and the quaternion group Q generatedby iσ and iσ . But the scalars that appear in the finite model at this point arethe scalars ω and ¯ ω that are used to convert from the representation 4 H to 4 C , andfrom one complex version of 4 C to the other, by multiplying with the elements oforder 3 in the group G . Thus the pair of Weyl spinors defined by i and − i in thestandard model becomes a triplet of Weyl spinors defined by 1, ω and ¯ ω .3.3. The three generations.
I have shown how the fermionic part of the algebra, H + M ( C ), contains all the structure of the Dirac spinor, the Dirac equation, theDirac algebra and the basic principles of electro-weak mixing. These parts of thestandard model use essentially all of the structure of this part of the group algebra.The rest of the standard model must therefore lie in the bosonic part of the algebra R + C + M ( R ). I have suggested ways of constructing this part of the algebra viaa finite analogue of the construction of the Dirac algebra.As far as the Lie group SL (2 , C ) is concerned, the Dirac algebra is the complextensor product of two copies of the Dirac spinor representation. But if this structurearises from the structure of the finite group, we must also be able to construct theDirac algebra from the (complex) tensor product of two copies of the 4 C represen-tation of G . Moreover, the complex structure adds nothing of physical significanceto the algebra, so that we might as well work with the real tensor product, whichhas structure 1+1+2+3+3+3+3. This contains the whole of the bosonic part ofthe algebra, together with a spare copy of 1 + 3.There are two ways of looking at this tensor product. One way is to look atthe finite group acting on the spinors on both sides. This gives a description ofthe internal symmetries of the elementary particles, without any Lie groups andtherefore without any measurements or observations. The other way is to look atthe finite group acting on the vectors on the right. This gives a description ofthings that can be measured by experiments that act on the vectors on the left.The representation theory of the finite group links the two viewpoints.In particular, the spare copy of 1 + 3 must represent things that cannot bemeasured, for example colours, or direction of spin. The corresponding part of theDirac algebra consists of two 4-vectors, one of which is usually interpreted as 4-momentum, so that we might interpret the other as spacetime position, and allocatethe non-measurability to the Heisenberg uncertainty principle. That leaves us with1 + 2 + 3 + 3 + 3 things that can be measured, with a macroscopic group U (1) actingon the 2, and SL ( R ) acting on 3 + 3 + 3.Now in removing the fourth copy of 3 we may have removed either momentum orposition, but not both. Since position is irrelevant in the standard model, we havepresumably kept the momentum. Moreover, we have a real scalar in the remainingcopy of 1, which must similarly be the energy. We then have a macroscopic group GL (3 , R ) acting on the momentum. This is a little disconcerting, since our everyday experience is that momentum has only an SO (3) symmetry. But we know from thetheory of special relativity [18] that momentum mixes with mass, and we know fromthe weak interaction that mass can be converted into momentum, so we should notbe too surprised.In any case, 3 + 3 + 3 contains a discrete set of 9 ‘things’ to measure, and acontinuous group of dimension 9 to do the measuring with. Hence we can measure9 masses, for example of 3 generations of electrons and up and down quarks. Whatwe observe is that three of these masses, the electron masses, are masses of particleswhose momentum we can also measure, and they are therefore well-defined andprecisely measured masses. The other six, the quark masses, are masses of particleswhose momentum we cannot measure, so that the masses are quite ill-defined,and vary from one experiment to another. Most of all, they vary from one typeof experiment to another, for example between baryon experiments and mesonexperiments.These 8 mass ratios are closely related in the group algebra to 8 parameters inthe subalgebra M ( C ), but the standard model does not include a finite group re-lationship between these two parts of the algebra, so has an entirely different set of8 parameters here. These are conventionally written in the symmetric square rep-resentation as 3 × Field theories
Principles.
There are three (or four) ways of looking at the groups actingon the group algebra. One can look at the finite groups acting on both left andright, or the Lie groups acting on both left and right, or one of each. The generalprinciple must surely be that if the Lie groups act on both sides, then we see noquantisation, and we must recover a reasonable approximation to a description ofclassical physics.Similarly, if the finite group acts on both sides, then all we see are quantumnumbers, and no macroscopic variables like position, momentum, energy, mass andso on. In between, we see a right-multiplication by a finite group, which flips thequantum numbers, and a left-multiplication by some Lie groups, that permits us tomeasure the properties of mass, momentum, energy, and so on, that are associatedto particular sets of quantum numbers in particular types of experiment that wemight contemplate performing. This, at least, must be the general set-up in anyhypothetical discrete theory of the type envisaged by Einstein.The standard model interprets things differently, and regards the Lie groups asbeing intrinsic to the elementary particles, despite being unable to provide anyplausible physical process by which this could happen. (Although in practice, thede Broglie–Bohm pilot wave scenario is rather close to the approach I am taking.)
INITE SYMMETRY GROUPS IN PHYSICS 13
What is clear, of course, is that the calculations that are done in the standardmodel are essentially correct. It is only the interpretation that must be somewhatdifferent in a discrete model. The discrete model must in fact be capable of ex-plaining not only the standard model, but also classical physics. This is a majorchallenge for any model, and success is hardly to be expected. Nevertheless, let ussee how the proposed finite model deals with this challenge. The first and mostbasic issue is to determine how macroscopic spacetime emerges from the discreteproperties of elementary particles and their interactions. This must be some kindof generalisation of the toy example already given in Section 1.3.4.2.
Classical forces and relativity.
Looking at the real group algebra firstfrom a macroscopic perspective, with the matrix groups acting on themselves byconjugation, we see three real and two complex scalars, that act trivially, plus threenon-trivial symmetry groups: SO (3) × SO (3 , × SL (3 , R ) . (31)Therefore, in addition to the magneto-weak ‘spacetime’ H with symmetry group SO (3), discussed in Section 1.3, we see an electromagnetic (special relativistic)‘spacetime’ with symmetry group SO (3 , SL (3 , R ). The issue then is to decide how the macro-scopic spacetime that underlies classical physics relates to these three different ver-sions of spacetime that seem to emerge in some way from different parts of particlephysics.There is nothing in classical physics directly corresponding to the group SL (3 , R ),which allows stretching and shearing of space. But combining it with the Lorentzgroup SO (3 ,
1) acting on the same spacetime gives the group SL (4 , R ) of all unimod-ular coordinate changes, that describes the general covariance of general relativity[23, 24, 25] as a theory of gravity. So SL (3 , R ) must in some sense describe a ‘fluid’gravitational ‘space’.Classically, of course, there is only one spacetime, with symmetry group SO (3)defined by the observer. The groups SO (3 ,
1) and SL (3 , R ) are then interpreted,not as different types of spacetime, but as forces. Clearly SO (3 ,
1) is the symmetrygroup of electromagnetism, as most elegantly expressed by Einstein’s interpretationof Maxwell’s equations in the theory of Special Relativity. To be more precise, theelectromagnetic field is an association of a (trace 0) element of M ( C ) with eachpoint in spacetime H . It is therefore expressed mathematically as a function f : H → M ( C ) . (32)Maxwell’s equations then describe how the values of the electromagnetic field, inthe adjoint representation of SO (3 , M ( C ), relate to the ambient physical space, in the adjointrepresentation of SO (3), and to time, in the scalar part of H . The theory ofspecial relativity explains how the equations are invariant under coordinate changeson spacetime described by the Lorentz group SO (3 , SO (3 ,
1) can be compensated forby the corresponding change in coordinates for the values of the electromagneticfield, by the corresponding element of SL (2 , C ) acting by conjugation on the trace0 matrices. The Lorentz group can in turn be interpreted as a change of spacetime coor-dinates between different observers. But the model under discussion is a modelappropriate to a single observer, with a fixed notion of space and time. Thereforethe appropriate interpretation of the Lorentz group in this context is as a gaugegroup, that can be used to choose coordinates for the (electromagnetic) gauge field.The content of the theory of special relativity is then that the theory of electro-magnetism does not depend on the choice of coordinates.Analogously, we must seek a gravitational field in the adjoint representation of SL (3 , R ). From the point of view of an individual observer, with SO (3) symme-try, this representation splits up as the sum of a spin 1 field and a spin 2 field.Newton’s universal theory of gravitation already includes both: the spin 1 fieldis the gravitational field, and the spin 2 field describes the tidal forces that arisefrom rotations of matter within a gravitational field (or, equivalently, rotations ofthe gravitational field around the matter). Newton’s theory is very good indeed inmost circumstances, but has two main drawbacks that have become apparent inthe ensuing centuries. One is that it does not distinguish between the gravitationalforce due to a rotating body, and that due to a stationary body. The other is thatit has a static gravitational field, rather than a dynamic field that propagates at afinite speed (presumably the speed of light).The former drawback is (at least partly) addressed by Einstein’s theory of generalrelativity, in the sense that rotations of the observer (or test particle) are taken intoaccount. But it is not clear that rotations of the gravitating body are fully accountedfor, nor the finite speed of propagation. It is therefore not required for a new modelto reproduce general relativity exactly, but merely to reproduce general relativityin the limit where the effects of the finite speed of light can be ignored. There areindeed two circumstances in which such effects might already have been observedin practice. One is in the effects on fast-moving satellites as the rotate very fastaround the Earth. The other is at the edges of galaxies, in which the gravitationalattraction of the galactic centre can take hundreds of thousands of years to reachthe outer edges.The former is known as the flyby anomaly [26], and it is claimed by Hafele [27]that this anomaly can be entirely explained by the finite speed of propagation ofthe gravitational field of a massive rotating object. The latter goes by many names,and many proposals have been made for theoretical explanations for the observedeffect, most notably the hypotheses of (a) dark matter, or (b) modified Newtoniandynamics, or MOND [28, 29, 30, 31, 32]. As far as I am aware, however, there is noproposed explanation that takes into account either the finite speed of propagationof the gravitational field, or the very fast rotation of the super-massive black hole(or other objects) at the centre of the galaxy.The proposed finite model permits a separation of the SL (3 , R ) symmetries ofgravitational space from the SO (3 ,
1) symmetries of electromagnetic spacetime, andtherefore permits a macroscopic theory of gravity that is independent of time, andtherefore independent of the finite speed of light. But to do that it requires thesymmetry group of the gravitating body to be extended from SO (3) to SL (3 , R ).In other words, the centre of the galaxy must be regarded as a rapidly rotatingfluid rather than a solid object. Given that a typical galaxy has many billions ofstars, this is surely a reasonable hypothesis to make. INITE SYMMETRY GROUPS IN PHYSICS 15
The general principle of relativity says that it makes no difference to the theorywhether the group SL (3 , R ) is attached to the galactic centre or to the periphery,so that we can interpret the theory either way. The latter interpretation seemsto be closely related to the dark matter hypothesis: that is the galactic centre isregarded as a Newtonian point mass, and all the otherwise unexplained behaviourof the outer stars must then be due to some ‘dark matter halo’ surrounding thegalaxy. But perhaps the assumption of a Newtonian point mass at the centre ofthe galaxy is unrealistic?The former interpretation is more closely related to the MOND hypothesis: thatthere is some new unexplained gravitational force, that can only be observed whenthe Newtonian gravitational field is extremely weak. The proposed finite modelfavours the MOND interpretation, and suggests (very speculatively) that the newforce might be explained in terms of the very fast rotation of the extremely massivegalactic centre. Most importantly, the group SL (3 , R ) is now interpreted as thegauge group of the theory of gravity, and is completely separate from the gaugegroup SO (3 ,
1) of classical electromagnetism. In the theory of general relativity,these two gauge groups are combined into a group SL (4 , R ), which has no mean-ing in the proposed model. This might explain why attempts to quantise generalrelativity with a gauge group SL (4 , R ) or GL (4 , R ) have failed [33, 34].4.3. Quantum field theory.
The above discussion suggests interpreting the 24dimensions of the group algebra as classical fields, divided into 7 dimensions ofscalar fields, 3 dimensions of space, 6 dimensions of electromagnetic field, and 8dimensions of gravitational field. These fields are mathematical constructs, ratherthan physically real objects. For example, we do not want to interpret the 3 dimen-sions of the ‘space’ field as an ‘aether’ in the 19th century sense. What is physicallyreal, however, is the propagation of the fields through space, or to be more precise,the propagation in spacetime.I have not precisely identified all of the scalar fields, though four of them appearto be time, inertial mass, charge and gravitational mass. Energy would seem tobe separate from both types of mass, and there appears to be a second type of(neutral) charge. This leaves one more scalar, which might be identified with theHiggs field in the standard model. Whatever identification is eventually decided on,these scalar fields propagate by being attached to matter particles. In other words,it is the movement of matter that defines the propagation of the scalar fields.Just as in the classical case, the fields must be modelled as functions from space-time to the appropriate subspace of the group algebra. In the quantum case, thisfunction decomposes as a sum of ‘infinitesimal’ quanta, that individually are lin-ear functions. These linear functions themselves lie in representations of the finitegroup G , so that we can use the representation theory to analyse the structure ofthe quantum fields. Since the spacetime representation H is self-dual, the quantumfields can be regarded as lying in the tensor product of H with the appropriaterepresentation.These tensor products then describe the particles that mediate the correspondingforces. Since the spacetime representation is fermionic in this model, the bosonicfields are carried by fermionic particles, and the fermionic fields are carried bybosonic particles. In particular, four of the scalars are bosonic, so carried byfermions, and three of the scalars are fermionic, so carried by bosons. The electromagnetic field values are fermionic in this model, and the field istherefore propagated by bosons. This agrees with the standard model, in whichthe propagation is effected by photons, which are parametrised by 3 dimensions ofmomentum in 2 distinct polarisations. The three fermionic scalars, which might beinterpreted as inertial mass (dual to Euclidean time, as opposed to energy, whichis dual to Lorentzian time), electric and neutral charge, are carried by the weakbosons, that is the Z and W bosons. Conversely, in this model the gravitationalfield values are bosonic, and the field is therefore propagated by fermions.No such process exists in the standard model, in which all fields are assumedto be carried by bosons. But it would appear to be consistent with experimentto suppose that these propagators are neutrinos, parametrised by 3 dimensionsof momentum in 3 distinct generations. This parametrisation would result in a9-dimensional gravitational field, however, and the model (as well as observation)supports only an 8-dimensional field. In other words, the 3 generations of neutrinoscannot be linearly independent. This means that macroscopic rotations act not onlyon the momentum coordinates, but also on the generation coordinates. The modeltherefore predicts that the generation of a neutrino is not an invariant. Indeed, thenon-invariance of neutrino generation is well-attested experimentally, and goes bythe name of neutrino oscillation [35, 36, 37].At this point, the neutrinos appear to have taken over the group SL (3 , R ), thatwas originally supposed to be allocated to the strong force, and in particular tothe 8 gluons. Indeed, since the group SL (3 , R ) in the proposed model acts on abosonic field, the corresponding mediators must be fermions. We can reconcilethe two viewpoints by interpreting the gluons as representing the values of thequantum field, rather than the field itself, which is a function . Then the mediatorscan be interpreted as virtual neutrinos, and the gluons as pairs of virtual neutrinos.However, the model suggests that it may be better not to interpret the gluons asparticles at all, but only as symmetries.At the same time, we need to address the distinction between the group SL (3 , R )used here and the group SU (3) used in the standard model. The latter is a compactgroup, and fixes a complex inner product, so describes rigid symmetries of a complex3-space. The phenomenon of asymptotic freedom [38, 39] suggests that such rigiditydoes not in fact characterise the strong force. The use of SL (3 , R ), on the otherhand, suggests complete freedom to change scale in one direction, providing thisis compensated for in another direction. In other words, replacing the (confined)gluons by the (free) neutrinos seems to require a split group, just as the (free)photons are described by the split group SL (2 , C ).4.4. Elementary particles and the standard model.
Let us now turn ourattention to the mixed case, with the finite group acting on the right and theLie groups acting on the left. This is the domain of the standard model, wheremeasurements of macroscopic variables such as mass, momentum, energy, angularmomentum, magnetic moments and so on are made on individual quanta or ‘el-ementary particles’. We then have 24 discrete objects on the right that we canmeasure, and 24 degrees of freedom for the operators on the left that define whatwe can measure. The standard model has a kind of duality between things thatcan be measured and things that can’t, which is sometimes an absolute distinction,and sometimes an uncertainty principle.
INITE SYMMETRY GROUPS IN PHYSICS 17
For example, the Heisenberg uncertainty principle says you can measure positionor momentum, but not both at the same time. Quantum chromodynamics, on theother hand, assigns (unobservable) colours to quarks, and puts the correspondingobservable property into the notion of particle ‘flavour’ or generation. In some cases,if the associated representation is a real or complex scalar, the appropriate conceptis its own dual, and there is no duality or uncertainty involved. This appears to bethe case, for example, with the electric charge. The details are not so important.What is important is, how many distinct measurements can we make?The finite model says that we can measure exactly 24 independent things. If wemake a specific choice of the 24 independent things we want to measure, we obtain24 (dimensionless) real numbers that describe everything there is to know abouthow the elementary particles behave. The standard model has made a particularchoice, and has measured precisely 24 independent things. These 24 things areusually described as 12 fermion masses (3 generations each of neutrino, electron,up and down quark), 3 boson masses (the Z , W and Higgs bosons), (hence 14mass ratios), 2 coupling constants (the finite-structure constant and the strongcoupling constant) and 8 mixing angles (4 each in the CKM and PMNS matrices)[19, 20, 21, 22], and are regarded as the fundamental parameters.The standard model therefore has exactly the right number of parameters tocontain a complete description of quantum reality. It produces the right answers,because it contains enough variables and enough equations to calculate everythingthat can be calculated. There is nothing wrong with the standard model. It isa mathematically correct, and complete, model of everything that can possiblyhappen. So why are people still looking for a ‘theory of everything’, if the standardmodel already is a theory of everything?The main reason is that we don’t understand where the 24 parameters come from.We understand perfectly well that the Lorentz group can be interpreted either asa change of coordinates on spacetime between two observers, or as a gauge groupfor electromagnetism, and that these two interpretations are equivalent, so that thetheory of electromagnetism is the same for all observers. We understand perhapsa little less well that the same applies to SL (3 , R ) and the theory of gravity.So why don’t we understand that the same applies to the gauge groups of thestandard model? The groups describe the relationship between the observer andthe observed, and we have a choice between regarding the groups as acting on theobserved (as in the standard model) or the observer (as in relativity). But we haveto make a choice: we cannot both have our cake and eat it. The incompatibility ofthe standard model with general relativity may be nothing more than the fact thatthe two disciplines have made incompatible choices of interpretation.So to resolve the issue we need to make a consistent choice of interpretation. Forpractical purposes, it makes no difference which choice we make. Either way wedo the same calculations, get the same answers, and reconcile them with experi-ment. But philosophically, there is no contest. The general principle of relativity issuch a powerful and obvious philosophical principle that we should not under anycircumstances contemplate abandoning it. The consequence of this philosophicalviewpoint is that the gauge groups of the standard model have to be transferredto act on the observer, so that (most of) the 24 parameters become parameters as-sociated with the experiment, the environment and the observer, rather than withthe elementary particles themselves. I re-iterate that both points of view are mathematically valid, but only one ofthem is philosophically valid. Moreover, since many of the 24 parameters are knownto vary with the experiment, and in particular to ‘run’ with the energy scale, itis also the case that only one of the two viewpoints is physically valid. It is notphysically reasonable to treat a parameter as a universal constant, if experimentshows that it is not. In the model I am proposing, 7 of the 24 parameters arescalars, and can therefore be taken as universal constants. The other 17 must beregarded as properties of the experiment.If we look carefully at the experimental evidence, 6 or 7 of the 14 mass ratiosshow evidence of not being constant, namely the 6 quark masses and the
Z/W mass ratio. All 8 of the mixing angles are similarly suspect. Finally, there is nopositive, model-independent, evidence of different masses for the three generationsof neutrinos, or indeed any mass different from zero. Moreover, the phenomenon ofneutrino oscillation suggests that there is no intrinsic difference between the threegenerations of neutrino anyway, which add another 2 or 3 parameters to make up atotal of between 16 and 18. In other words, the proposed finite model is consistentwith experiment on this point.4.5.
The wave-function.
The one thing that one might really hope for from afinite model of elementary particles is an insight into the measurement problem.This was really the focus of Einstein’s objections to quantum mechanics throughouthis life, and although his attacks, most notably the EPR paradox [40], were neverenough to sink the ship, the fundamental problem has not gone away. To simplifythe problem almost to the point of caricature, we may ask, what is the wave-function, and how does it ‘collapse’ ?In the group algebra model, the wave-function is implemented at the middlelevel, with the finite group acting on the right and the Lie groups on the left. Thetypical example is equation (32), which describes a function from spacetime to theDirac spinors. The ‘collapse’ is some kind of operation that moves down to thediscrete level, with the finite group on both sides. In practical terms, the wave-function describes the quantum field in the experiment, and the collapse describesthe result of the experiment.From this point of view, the measurement problem arises from the assumptionthat the wave-function is intrinsic to the elementary particle under investigation.The finite group model, however, does not allow any continuous variables to beassociated with an elementary particle. The continuous variables are always associ-ated with the macroscopic measuring apparatus. The model, in other words, doesnot solve the measurement problem. It does not find the measurement problem.Philosophically, this is always the best way to solve problems, namely, to realisethat, if looked at in the right way, the problem does not exist.4.6. The real universe.
The structure of the model indicates that of the 24 un-explained dimensionless parameters of the standard model, exactly 7 are universalconstants, and the rest are dependent in some way on the experiment, the environ-ment and/or the observer. There is nothing like enough detail in the model as so fardeveloped to indicate which parameters should be regarded as universal constants,nor how the other parameters vary. Indeed, there may be a certain amount of choiceas to which parameters can be defined as constant, so that the other parameterscan be calibrated against them.
INITE SYMMETRY GROUPS IN PHYSICS 19
What is clear, however, is that we must be prepared for the possibility thatcertain parameters that we are certain must be universal constants, may not be so.At this stage, we can do little more than speculate on these matters, and applysome educated guesswork. There is no real need to use the same fundamentalparameters as the standard model, but other mass ratios such as those betweenelectron, proton and neutron might also be worth looking at. The model suggeststhat approximately 8 of the fundamental parameters are dependent in some wayon the gravitational field, including tidal forces.At first glance, the very idea is preposterous. But on closer inspection, onerealises that while the standard model is in theory based in an inertial frame, theexperiments that measure the fundamental parameters are not done in an inertialframe, but in a frame which moves with the Earth. It is therefore very hard torule out, on experimental grounds, the possibility that some of the measurementsdepend crucially on some dimensionless property of the tides that is constant overall experiments done on the Earth.There are four basic dimensionless parameters of the tides on the Earth, ofwhich two are obvious spatial angles: the angle of tilt of the Earth’s axis, and theinclination of the Moon’s orbit to the ecliptic. The other two are the ratios of theperiods of rotation/revolution, so are angles in Euclidean spacetime. Of course,these parameters are not precisely constant, and there are other effects that mightbe expected to contaminate the results, such as the gravitational pull of Jupiter. Itwill therefore be somewhat tricky to distinguish a real correlation from an unwantedcoincidence.Some 8 such correlations/coincidences were presented in [41], without any verysolid justification, but with the suggestion that, while some of them may indeed bepure coincidences, it is very unlikely that they are all coincidences. The parametersdiscussed in that paper include the electron/proton/neutron mass ratios, the pionmass ratio, the kaon mass ratio, the kaon/eta mass ratio, the Cabibbo angle, theWeinberg angle and the CP-violating phase in the CKM matrix. In addition, thepaper discusses a number of other equations that seem to suggest that some ofthe fundamental parameters may not be independent of each other. This may bea somewhat less unpalatable suggestion than the suggestion, made originally byEinstein [42] more than a century ago, that they might depend on the gravitationalfield! On the other hand, the paper [41] also shows that the observed CP-violatingbehaviour of neutral kaons [43] is quantitatively consistent with the hypothesis thatthe effect is caused by the small difference in the direction of the gravitational fieldbetween the two ends of the experiment.5.
Conclusion
In this paper, I have examined what I believe to be the unique possible mathe-matical model of a discrete algebraic universe, in the hope that it will have some-thing useful to say about the seemingly intractable problems in the foundations ofphysics. I have shown how all the essential ingredients of the standard theories ofboth classical and quantum physics arise from this finite model, and discussed ata general level the relationships between them. I have traced the conflict betweenquantum mechanics and relativity to a conflict in interpretations, that does notaffect the mathematics of either theory, and sketched a possible way to resolve thisconflict.
I have not done all the necessary detailed calculations to show that the proposedmodel reproduces the standard models exactly, so there is still room for doubt asto whether the model I propose is viable. Nevertheless, I have shown that theproposed model has enough complexity to incorporate all of the subtleties of thestandard model of particle physics, including the 24 dimensionless parameters. Theproposed algebraic model seems to be consistent with the experimental fact thatthe standard model is essentially a complete and correct theory of everything. Asfor physics ‘beyond the standard model’, the model explains what experiment hasdemonstrated, namely that, essentially, there is none.Similarly, it appears to be consistent with general relativity as a theory of grav-ity, but suggests that the simplifying assumptions made in practical calculationsbreak down in extreme circumstances. The model suggests that incorporating acontribution to gravity from the rotation of a (not spherically symmetric) gravitat-ing body, and taking into account the finite speed of propagation of gravitationalwaves, may be sufficient to account for the anomalous rotation of stars in the outerregions of galaxies, that was the original reason for the hypothesis of dark matter. Ifthis is not enough, then the model has room to distinguish gravitational mass frominertial mass, and to incorporate a scale factor between the two that is dependenton properties of the motion of the observer.So what remains of Einstein’s ‘castle in the air’ ? The finite model has, if any-thing, given it slightly firmer foundations. Special relativity has always been builton solid rock, but the foundations of general relativity are more fluid. I see generalrelativity, therefore, not as a castle in the air, but as an aeroplane, that stays updespite having no visible means of support. My proposed model provides somesupport, in the form of a neutrino wind, to keep it flying.
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