Subgroups of Clifford algebras

Clifford algebras are used for constructing spin groups, and are therefore of particular importance in the theory of quantum mechanics. But the spin group is not the only subgroup of the Clifford algebra. An algebraist's perspective on these groups and algebras may suggest ways in which they might be applied more widely to describe the fundamental properties of matter. I do not claim to build a physical theory on top of the fundamental algebra, and my suggestions for possible physical interpretations are indicative only, and may not work. Nevertheless, both the existence of three generations of fermions and the symmetry-breaking of the weak interaction seem to emerge naturally from an extension of the Dirac algebra from complex numbers to quaternions.

The Dirac algebra, as used in the Feynman calculus in the standard model of particle physics, is a complex version of the real algebra Cl(1, 3) generated by the Dirac gamma matrices γ 1 , γ 2 , γ 3 and γ 0 , whose essential properties are that they anti-commute with each other, and square to −1, −1, −1 and +1 respectively. Instead of using the scalar i as a fifth generator, it is conventional (or at least instructive) to use the matrix γ 5 := iγ 0 γ 1 γ 2 γ 3 , which anti-commutes with the original four gamma matrices and squares to +1. With respect to these five gamma matrices as generators, therefore, the algebra has the structure of Cl (2,3).
Another reasonable set of generators is which gives a structure Cl(4, 1), and a third possibility is which gives a structure of Cl(0, 5). All these algebras are isomorphic (as algebras) to the algebra of all 4×4 complex matrices. The same physical information can be packaged into slightly different mathematics by reversing the signature of the Clifford algebra. We obtain Cl (3,2), which consists of two copies of the algebra of 4 × 4 real matrices, and Cl(5, 0) and Cl (1,4), both of which consist of two copies of 2 × 2 quaternion matrices.

Clifford algebras beyond the standard model
Larger Clifford algebras have also been suggested [8,9] for purposes such as including the strong force [10] gauge group SU (3), although SU (3) is not itself a spin group. But SU (3) is a subgroup of SO (6), and by the Klein correspondence Spin(6) ∼ = SU (4), which contains SU (3) as a subgroup. Hence this approach may also be useful in extending from the standard model to a 'four-colour' model such as the Pati-Salam model [11,12]. Similarly, Penrose's twistor theory [13,14] is based on Spin(4, 2) ∼ = SU (2, 2), which can be constructed via the Clifford algebra Cl (4,2) or Cl (2,4). In fact, the algebras Cl (3,3), Cl(0, 6) and Cl (4,2) are all isomorphic to the algebra of 8 × 8 real matrices, so all three approaches can be pursued simultaneously if desired. The reversed signatures Cl(6, 0) and Cl (2,4) are both isomorphic to the algebra of 4 × 4 quaternion matrices, as are Cl(5, 1) and Cl (1,5), so that these provide an alternative approach for those who appreciate the virtues of using quaternions [15]. By now, I have mentioned 19 distinct Clifford algebras, and it is obvious that some clear mathematical principles are going to be necessary in order to make good choices for which ones to use in which physical contexts. The main principle I want to discuss in this paper is the principle of choosing a particular Clifford algebra structure from the many that are in general available in a suitable abstract algebra. This is a mathematical symmetry-breaking principle, which may or may not have anything to do with the physical process of 'spontaneous symmetry-breaking' that is usually invoked to explain the structure of the fundamental forces.

.1 Examples
Every Clifford algebra is either a full matrix algebra over the real numbers, complex numbers or quaternions, or the direct sum of two such algebras, isomorphic to each other. Therefore the algebra contains much larger groups than the spin group it was designed to construct. Restricting from the large group of all invertible elements of the Clifford algebra, to the spin group, is a process of breaking a large symmetry group down to a small one. For example, Cl(0, 2) is a quaternion algebra, so contains not only Spin(2) ∼ = U (1), but also a copy of SU (2) ∼ = Spin(3). Another example that I hope to show is relevant to physics is Cl (3,1), that is a 4 × 4 real matrix algebra, so contains not only Spin(3, 1) ∼ = SL(2, C) but also a copy of SL(4, R) ∼ = Spin (3,3).
In both these cases, the Clifford algebra breaks the symmetry. Thus Cl(0, 2) breaks the symmetry of SU (2) by choosing a particular subgroup U (1) to call Spin(2). This example may be relevant to the symmetry-breaking of the weak gauge group SU (2), that distinguishes between the neutral Z boson and the two charged W bosons. Similarly, Cl(3, 1) breaks the symmetry of SL(4, R) by choosing a particular copy of SL(2, C) to call Spin(3, 1). This example is not currently used in physics, as far as I am aware, but I hope to show that it is relevant to the breaking of symmetry between the three generations of fermions.
If further symmetry-breaking is required, one can use Cl(0, 2) to break SU (2) down to U (1), and Cl(2, 0) to break SL(2, R) down to U (1), or even use Cl(1, 1) to break SL(2, R) down to GL(1, R). These chains of symmetry-breaking should be sufficient to deal with the main examples of symmetry-breaking in the standard model. The chain down through Cl(3, 1), Cl(3, 0) and Cl(2, 0) looks particularly promising.

A quaternionic Dirac algebra
One of many ways to construct Cl(3, 3) is to extend the complex numbers in the Dirac algebra to the quaternions. This is not just a matter of extending the 4×4 complex matrices to 4×4 quaternion matrices, but of adjoining quaternions i, j, k that commute with the four Dirac matrices γ 1 , γ 2 , γ 3 and γ 0 . This process requires 8 × 8 real matrices rather than 4 × 4 quaternion matrices, since j and k must anti-commute with γ 5 . To obtain anti-commuting generators required for a Clifford algebra, we can take which exhibits the structure of Cl (3,3). Some alternative sets of generators for Cl (3,3) are It is not immediately obvious which of these choices will be most useful for physical applications. But the mathematical structure of Cl(3, 0) is quite different from that of Cl(0, 3), so the choice between γ 1 , γ 2 , γ 3 generating Cl(0, 3) and iγ 1 , iγ 2 , iγ 3 generating Cl(3, 0) is likely to be significant.
As already noted, the signatures (0, 6) and (4,2) are available in the same abstract algebra, for example in the two sets of generators: We therefore have a choice of three distinct spin groups but only the first of these is suitable for further symmetry-breaking with a smaller Clifford algebra. Both Cl(3, 1) and Cl(2, 2) are available for this purpose, and although the former is the 'obvious' one, the latter may also be useful. The remaining spin group can similarly be used as input to further symmetry-breaking using any of Cl(4, 0), Cl (1,3) or Cl(0, 4). For the moment I will stick as closely as possible to the standard model, and work with Cl(3, 3) as an algebra of 8 × 8 real matrices, with Cl(3, 1) used for symmetry-breaking of Spin(3, 3). I have suggested three possible generating sets, two of which have γ 5 in the top (so-called pseudoscalar) degree, while the third has γ 0 in the top degree. The choice between these options must be made on the basis of subgroups other than the spin group itself.

Electroweak gauge groups
In the standard model Dirac algebra Cl(2, 3), the modelling of electro-weak interactions uses the 2-dimensional complex subalgebra spanned by 1 and γ 5 . This permits the implementation of two copies of U (1), generated as Lie groups by i and iγ 5 . In other words, only a subgroup U (1) of the weak gauge group SU (2) appears in the Dirac algebra. The full SU (2) can only be implemented by extending to the quaternionic Dirac algebra Cl(3, 3) described above. The Coleman-Mandula theorem [16] implies that the gauge group must lie in the centralizer algebra of the relativistic spin group SL(2, C), generated by even products of the original four gamma matrices. This centralizer algebra is generated by j, k and γ 5 , and is therefore isomorphic to Cl(1, 2), and thereby to the algebra of 2 × 2 complex matrices.
By choosing instead the generators jγ 5 , kγ 5 and γ 5 we can identify this algebra also as Cl (3,0). Which generators should we choose? The four rotations are i, j, k and iγ 5 , and the three boosts are γ 5 , jγ 5 and kγ 5 . Whatever happens, the symmetry is visibly broken, but we have a choice of which bits of symmetry to keep, and which to abandon. This is largely a matter of physical interpretation, rather than mathematics, so it is not appropriate to try to make such a decision here. What is clear, is that the four rotations give a group U (1) generated by iγ 5 , and a group SU (2) generated by i, j, k, and that these groups commute with each other, and generate a group U (2). This is exactly the formalism that is normally used to describe the unification of quantum electrodynamics (QED) with the weak force.
Thus consideration of the electro-weak interactions suggests that the third of the three suggested sets of generators for Cl (3,3) is likely to be the most useful. There appear to be some minor technical differences between what we see here and what appears in the Feynman calculus, in that the roles of i and iγ 5 appear to have been interchanged. Since this can be adjusted by applying complex conjugation on half of the spinor, I suspect this is not a serious issue. Indeed, this difference may be more apparent than real: it may well be possible to take the Feynman calculus for one generation exactly as it is, and then to use jγ 5 or kγ 5 for the second and third generations, in place of γ 5 .
There is complete symmetry amongst these three elements of the Clifford algebra, that is slightly obscured by the conventional notation. If we write them instead as iγ 0 γ 1 γ 2 γ 3 , −kγ 0 γ 1 γ 2 γ 3 and jγ 0 γ 1 γ 2 γ 3 , the symmetry becomes obvious. There is also no clearcut answer to the question whether we should use compact SU (2) or split SL(2, R) or complex SL(2, C) as the gauge group. This ambiguity also exists in the standard model, so again, there should not be a problem with this. Indeed, the groups themselves play a rather minor role in the theory: it is the Clifford algebra where all the calculations take place.
It is clear, I think, that electro-weak mixing can be expressed mathematically, not as a process of 'mixing' groups together, or mixing Lie algebras together, but as a process of choosing bases for the matrix algebra to turn it into one or more Clifford algebras. This viewpoint has some consequences for how to think about gauge groups in the general case. Although they may sometimes appear in the formalism as groups or Lie algebras, it is certainly possible that these mathematical objects are secondary to a primary manifestation as a Clifford algebra.

Spin groups
The ordinary spin group Spin(3) ∼ = SU (2) is generated as a Lie group by γ 1 γ 2 , γ 2 γ 3 and γ 3 γ 1 . It can be regarded either as the even part of Cl(3, 0), generated by iγ 1 , iγ 2 , iγ 3 , or as the even part of Cl(0, 3), generated by γ 1 , γ 2 , γ 3 . The algebra Cl(3, 0) is isomorphic to the algebra of all 2 × 2 complex matrices, and therefore contains a subgroup SL(2, C) that is isomorphic to, but not equal to, the relativistic spin group. I do not know if this copy of the group SL(2, C) has any reasonable physical interpretation.
The algebra Cl(0, 3), on the other hand, is the direct sum of two copies of the quaternion algebra. Generators for these two copies may be taken as: They therefore have a fairly obvious interpretation as left-handed and righthanded spins. Exponentiating these generators in the usual way we obtain two copies of SU (2), one left-handed and one right-handed. These two copies of SU (2) intersect trivially, and therefore they generate a copy of Spin(4).
The same effect is achieved in the standard model using Cl(4, 0) generated by γ 0 , iγ 1 , iγ 2 , iγ 3 , with pseudoscalar element γ 5 . Here the full Clifford algebra is a 2 × 2 quaternion matrix algebra, and the even part is the sum of two quaternion algebras, which can be separated by the projections with the idempotents (1 ± γ 5 )/2. The generators can be taken as The obvious question that occurs to an algebraist [17] at this point is whether the extra complications in the standard model are actually necessary, if the same effect can be achieved with the simpler mathematical machinery presented above? That is, can we take the same spin terms as always, γ 1 γ 2 and so on, but project with (1 ± γ 1 γ 2 γ 3 )/2 without the extra factor of iγ 0 ? I do not pretend to provide an answer to this, I merely ask the question. The relativistic spin group Spin(3, 1) ∼ = SL(2, C) can be obtained either from the even part of Cl(3, 1), generated by iγ µ for µ = 0, 1, 2, 3, or from the even part of Cl(1, 3), generated by the γ µ themselves. The algebra Cl(3, 1) is isomorphic to the 4 × 4 real matrix algebra, while Cl(1, 3) is isomorphic to the 2×2 quaternion matrix algebra, and the even part in both cases is isomorphic to the 2 × 2 complex matrix algebra. In both cases the pseudoscalar is γ 0 γ 1 γ 2 γ 3 , which squares to −1. Hence there are no projections onto left-handed and right-handed spins as such, but in the standard model this distinction is made instead by complex conjugation. This is done by extending the Clifford algebra either from Cl(3, 1) to Cl(4, 1), or from Cl(1, 3) to Cl(2, 3). Or, equivalently, complexifying the Clifford algebra.
So again, the question that occurs to an algebraist is, is this extension really necessary, or is there already enough information in the odd part of Cl(3, 1) or Cl(1, 3)? In both cases, the Clifford algebra provides the necessary symmetrybreaking, either by choosing a copy of the complex numbers inside the quaternions, or by imposing a complex 2-space structure onto a real 4-space. The discussion in this section suggests that the algebra Cl(1, 3) provides a closer link to experimental properties of left-handed and right-handed spin, while the standard model is somewhat ambivalent.

The mass term in the Dirac equation
In the standard model Clifford algebra, whether it is regarded as Cl(4, 1) or Cl(2, 3) or Cl(0, 5), the pseudoscalar term is always i. This pseudoscalar is conventionally used for the mass term in the Dirac equation. But it may not be the most natural choice. The Dirac equation was modelled on a factorisation, into two linear factors, of Einstein's equation in that particular form, rather than, for example, If the mass term is a scalar, then any re-arrangement of the equation introduces unphysical cross-terms in the squaring process, so is not possible. But the symmetrical form suggests that we should really be looking at a group SO(4, 1), and therefore the Clifford algebra Cl(4, 1), with generators iγ µ for the energy and momentum terms, and therefore γ 5 for the mass term. This is consistent with the earlier observation that extending to three generations of fermions seems to force the roles of i and iγ 5 to be interchanged. Indeed, if we want to extend to a larger Clifford algebra in order to incorporate the three generations of fermions, then i is no longer a natural choice for pseudoscalar. The discussion so far has suggested either γ 0 or γ 5 as pseudoscalar, with perhaps a preference for γ 0 . The question really is whether (rest) mass (γ 5 ) or (total) energy (γ 0 ) is a better choice for pseudoscalar. One has to make a choice on physical, not mathematical, grounds. Experiment makes it clear that mass is not conserved in the weak interaction, while the whole of physics relies on the principle of conservation of energy. This suggests that taking γ 0 as the top degree may ultimately result in a more fundamental theory.
The current suggestion for generators for the quaternionic Dirac algebra is as follows, in two different notations, the second, more cumbersome, notation exhibiting the symmetry between the three generations of fermions: This version has pseudoscalar γ 0 , and therefore does not have an explicit energy term. One can alternatively include the energy term as one of the generators, in order to obtain something closer to the standard model: The latter breaks the 3+3 symmetry to the 3+1+2 symmetry that is a prominent feature of the standard model. But it has pseudoscalar γ 5 , and therefore treats rest mass as a more fundamental concept than energy. At this point, we have a proposal for embedding the electro-weak gauge groups and the spin groups into the quaternionic Dirac algebra Cl (3,3), and the stage is set for attempting to reproduce the rest of the standard model, with three generations of fermions, and including the mixing angles and other parameters. Obtaining the values of the parameters is not a matter of algebra, but obtaining a classification of the parameters is. 4 The structure of the Clifford algebra 4

.1 Undetermined parameters
There are some parameters within the algebra Cl(3, 0), that describes electroweak unification. These parameters include one that describes the breaking of the symmetry within SU (2), and three that describe how the pseudoscalar U (1) mixes with the three dimensions of SU (2).
The remaining parameters describe how Cl(3, 0) relates to Cl(0, 3). The latter has 8 dimensions in total, but under rotation symmetries of space these reduce to 4 different types, in degrees 0, 1, 2 and 3. The degree 0 is the identity element of the algebra, so cannot mix with anything. Similarly, Cl(3, 0) has 8 dimensions, and since the symmetry is broken, all 7 non-identity elements participate individually in the mixing. Hence there are 3 × 7 = 21 more parameters, making a total of 4 + 21, exactly the number required for the standard model, according to some counts at least. The 25 parameters consist of 5 × 3 mass terms, for 12 fermions and 3 bosons, 3 + 1 mixing angles each from the Cabibbo-Kobayashi-Maskawa (CKM) matrix [18,19] and the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [20,21], plus the fine-structure constant and the strong coupling constant.
The main group of 21 parameters splits under electroweak symmetries, according to my suggestion, into 3 sets of 3 + 3 + 1 of which one may contain lepton masses, and one quark masses, leaving the other to contain 3 + 3 + 1 of the 3 + 1 + 3 + 1 mixing angles. The other two scalars here, I suggest, might be the fine-structure constant (to go with the lepton masses) and the strong coupling constant (to go with the quark masses). That leaves the electroweak group of four parameters as the three boson masses (Higgs, Z and W ), together with the last mixing angle, which may be the CP-violating phase in the CKM matrix. This proposed allocation of parameters may not be exactly correct, of course, but it shows that the overall pattern of parameters matches closely to the standard model.
In a little more detail, we can label the 3 × 7 parameters with products of γ 3 , γ 1 γ 2 , γ 1 γ 2 γ 3 with i, j, k, γ 5 , iγ 5 , jγ 5 , kγ 5 , and the remaining four with the products of γ 5 with 1, i, j, k. In the main block of 3 × 7 parameters, the i, j, k symmetry is a generation symmetry for fermions, while in the remaining four it represents the broken symmetry 1, k, j of the weak interaction, since there is a hidden multiplication by i in the theory.

A quaternionic notation
Since the notation inherited from Dirac obscures some of the symmetries, in particular the generation symmetry, I suggest an alternative notation. The algebra Cl(1, 3) is the algebra of 2 × 2 quaternion matrices, so is the tensor product of quaternions with 2 × 2 real matrices. Let us take i ′ , j ′ , k ′ for these quaternions, to distinguish them from i, j, k that have already been used for a different quaternion algebra, and 2 × 2 real matrices Then we can take Dirac matrices as generators, defined as follows: Our canonical copy of Cl(0, 3) is then generated by Ji ′ , Jj ′ , Jk ′ , and our canonical copy of Cl(3, 0) by Ii, Ij, Ik. Space symmetry is described by i ′ , j ′ , k ′ , and generation symmetry by i, j, k.
The pseudoscalar elements of these two subalgebras and the whole algebra are respectively. One then sees the breaking of symmetry between the three generations, in the choice of γ 5 = −Ii in the standard model. The even part of the algebra corresponds to the diagonal matrices 1 and K, while the odd part corresponds to the off-diagonal matrices I and J. The ordinary spin group SU (2) is generated by i ′ , j ′ , k ′ , and extends to the relativistic spin group generated by Ii ′ , Ij ′ , Ik ′ . The weak gauge group appears either as SU (2) generated by i, j, k, or as SL(2, R) generated by i, Ij, Ik, with a commuting copy of U (1) generated by I. In particular, we have separated two concepts of complexification: one with I and one with i. These two are mixed together in the standard model, in a way that depends on the choice of the first generation of fermions as special.
To describe the 25 parameters, we must choose an arbitrary direction in space, say the k ′ direction, corresponding to the conventional choice of the z direction in which to measure spin. Then we must mix the three elements J, k ′ , Jk ′ in Cl(0, 3) with the seven elements i, j, k, I, Ii, Ij, Ik in Cl(3, 0). Hence the corresponding 21 parameters lie in the elements The remaining four parameters are perhaps best thought of as lying in i, Ii, Ij, Ik, and the whole system of 25 parameters can then be arranged as follows: The second row of parameters does not have a momentum direction k ′ , and the first six are rotations, and contain generation labels i, j, k, so could reasonably be interpreted as the mixing angles between the three generations in the CKM and PMNS matrices. At a guess, Ji, Jj, Jk lie in the CKM matrix, and Ki, Kj, Kk in the PMNS matrix. The first six elements in the bottom two rows have momentum and generation labels, and 9 of them are boosts, so could reasonably be interpreted as 9 of the 12 fundamental fermion masses. The remaining 3 are rotations, so perhaps represent the so far undetected neutrino masses. Of the remaining 7 parameters, two are rotations, and I suggest i to hold the CP-violating phase of the CKM matrix, and Kk ′ to hold the CP-violating phase of the PMNS matrix, although it is perhaps equally plausible to allocate these the other way round.
This leaves five boosts, K, Ik ′ , Ii, Ij and Ik, to hold the remaining three masses and two coupling constants. A reasonable guess might be that those involving I are electroweak parameters, and K is a strong force parameter, therefore the strong coupling constant. Perhaps Ik ′ is the fine structure constant, and the remaining three the masses of the Higgs, Z and W bosons. But electroweak symmetry-breaking may significantly complicate this picture.

Even and odd
The even part of Cl(3, 3) consists of degrees 0, 2, 4 and 6, with dimensions 1, 15, 15 and 1 respectively. Bases are as follows, arranged with degrees 0 and 4 in the first block, and degrees 2 and 6 in the second: Neither block is closed under the Clifford multiplication, but the first block is closed under Jordan multiplication AB + BA, and the second is closed under Lie multiplication AB − BA. Indeed, the Lie bracket converts the degree 2 part of the algebra into the Lie algebra of Spin (3,3). Exponentiating the elements of degree 2 therefore gives (the canonical copy of) Spin (3,3). This is the only copy of Spin(3, 3) that preserves the Clifford algebra structure under its action by conjugation.
The pseudoscalar K defines two orthogonal central idempotents (1 ± K)/2 in the even part of the Clifford algebra, and the corresponding projections map onto two 16-dimensional subalgebras, each of which is isomorphic to the full 4 × 4 real matrix algebra. Since each subalgebra annihilates the other. In particular, each algebra contains a copy of SL(4, R) that acts on only half of the 8-dimensional real spinor. These subgroups may be useful for physics, but it must be stressed that they do not preserve the Clifford algebra structure, and therefore they unavoidably break a great deal of symmetry. They do, however, preserve the distinction between the even and odd parts of the Clifford algebra. The odd part of the algebra splits into degrees 1, 3 and 5, with dimensions 6, 20 and 6 respectively. Degrees 1 and 5 have bases respectively. As a representation of Spin(3, 3), the degree 3 part splits into a dual pair of 10-dimensional real representations. These are obtained by multiplying the 10 elements so that each of these two 10-spaces has Clifford product identically zero. Their product in one order gives one projection of the even part of the algebra, and their product in the other order gives the other projection.

A change of signature?
The unification of space and time in special relativity mixes momentum (Ji ′ , Jj ′ , Jk ′ , in the odd part of the Clifford algebra) with energy (K, in the even part). Similarly, the unification of the electroweak interactions mixes i, j, k in the even part with I in the odd part. In other words, both these unifications require groups that extend into the odd part of the Clifford algebra. This might mean that the Clifford algebra that I have suggested is irrelevant for physics. Or it might mean that the relationship between different Clifford algebra structures may throw some light on the process of symmetry-breaking. In particular, the necessity for using the odd part of the algebra for part of the Lorentz group, or relativistic spin group, and for part of the electro-weak gauge group, suggests the possibility that the strong gauge group SU (3) may also lie partly in the odd part of the algebra. Alternatively, we may prefer to change our choice of Clifford algebra structure to try to bring the standard model symmetries into the even part of the algebra. Essentially, we have to make I even and K odd. One way to do this is to choose the following generators, given in both notations: This has the effect of changing the signature to (0, 6), and the pseudoscalar to I or iγ 5 , and changing the spin group to Spin(6) ∼ = SU (4).
There are now two obvious ways to collapse the Clifford algebra from a 6space to a 4-space, by multiplying together either the first three generators or the last three. Both give copies of Cl(1, 3). One of these has generators equal to the original gamma matrices, and the corresponding spin group Spin(1, 3) is the usual relativistic spin group. The other one has generators iγ 0 γ 5 , iγ 0 , jγ 0 , kγ 0 , and the corresponding spin group is generated by i, j, k, γ 5 , jγ 5 , kγ 5 .
Hence it describes electro-weak mixing in exactly the same way as I have already suggested. Both collapsed Clifford algebras also have pseudoscalar iγ 5 . There is then an obvious copy of U (3) inside the spin group, generated as a Lie group by the 9 elements The scalar copy of U (1) is generated by I(i ′ i+j ′ j +k ′ k), and SU (3) is generated by the orthogonal complement of this. There are other choices of SU (3), but this is the most symmetrical. Now comes the problem of interpreting the group SU (3) constructed here. It is hard to support an interpretation as the gauge group of the strong interaction, since the mathematical properties of this copy of SU (3) are not the same as those of the gauge group in the standard model. In particular, it does not commute with the relativistic spin group, generated by and thereby violates the Coleman-Mandula theorem. Moreover, it does not commute with the weak gauge group generated by i, j, k. The only things it does commute with in the even part of the algebra are two copies of the complex numbers, generated by I and I(i ′ i + j ′ j + k ′ k). Nevertheless, this copy of SU (3) may repay further study, so let us investigate its action on the Clifford algebra in more detail.
We have already seen the adjoint representation and a scalar in degree 2, and the rest of the degree 2 part is a 6-dimensional real representation on the generators The latter can be given the structure of a complex 3-space, using the natural scalar I(i ′ i + j ′ j + k ′ k). The degree 4 part consists of equivalent pieces 8 + 1 + 6, just multiplied by the pseudoscalar I. In degrees 1 and 5 we again see copies of the 6-dimensonal real representation, and again there is a natural complex structure defined by In degree 3, however, there are two natural complex structures, defined by I and I(i ′ i + j ′ j + k ′ k). Both give this 20-dimensional space the structure of a complex 10-space, breaking into irreducibles as 1 + 3 + 6. The complex scalars are contained in J and K times (1 + i ′ i + j ′ j + k ′ k). The 3-space consists of the anti-symmetric part and the 6-space consists of the symmetric part All of this structure is manifest in the quaternionic Dirac algebra, and therefore must surely have a sensible interpretation in physics. But it seems quite clear that this copy of SU (3) cannot be sensibly interpreted as the gauge group of quantum chromodynamics (QCD). So what is it? 5 Prospects for unification 5

.1 Mass versus energy
Mathematically, the distinction between Cl(3, 3) and Cl(0, 6) is most evident in the distinction between using I or K as pseudoscalar, that is, in physics notation, the distinction between iγ 5 and γ 0 . Physically this corresponds to a distinction between symmetry groups for which mass is scalar, and those for which energy is a scalar. Roughly speaking, mass is a scalar in the theories of special relativity and (classical) electromagnetism, as well as quantum electrodynamics and quantum chromodynamics, while energy is a scalar in general relativity, and in the theory of the weak interaction.
The former (QED and QCD) are the quantum interactions for which the gauge bosons are massless. The gauge groups U (1) and SU (3) therefore embed in the spin group Spin(0, 6) ∼ = SU (4) which is the subgroup of the Clifford algebra that fixes the mass coordinate. But these groups do not necessarily commute with each other. The group U (1) maps onto a subgroup U (1) of SU (3), or more generally of U (3), and this subgroup is defined by a 3 × 3 unitary matrix, that appears in the standard model as the CKM matrix.
In the weak interaction, on the other hand, the gauge bosons are massive, so that the gauge group does not act on mass as a scalar, but instead acts on energy as a scalar. Hence the gauge group SU (2) embeds instead in the spin group Spin(3, 3) ∼ = SL(4, R) that fixes the energy coordinate. The Pati-Salam model extends SU (2) ∼ = Spin(3) acting on three coordinates to Spin(4) ∼ = SU (2) × SU (2) acting on all six. The Clifford algebra formalism identifies the two scalars here, so that the group Spin(4), which is not a subgroup of SL(4, R), is replaced by SO(4), which is. In other words, the Pati-Salam model in effect uses both Clifford algebra structures Cl(0, 6) supporting SU (4), and Cl(3, 3) supporting SO(4) inside SL(4, R). But the model does not address the issue of symmetry-breaking, which arises from embedding both groups into SL(8, R), by identifying the two Clifford algebras as being two different ways of looking at the same thing.
Similarly, in general relativity, in theory energy is a scalar (conserved, but not necessarily invariant), so that the relevant group is SL(4, R), but most formulations assume that mass is a scalar (both invariant and conserved), which breaks the symmetry down to SO(3, 1). Now, as I have shown, in a flat spacetime it is not possible for both mass and energy to be scalars, and therefore general relativity incorporates a curvature of spacetime in order to solve this problem. On the other hand, if one abandons the assumption that mass is a scalar, which we must do in the case of the weak interaction anyway, then it may be possible to construct a version of general relativity in a flat spacetime, with gauge group SL(4, R), in such a way that the curvature of spacetime is replaced by variations in rest mass between different observers, accelerating with respect to each other.
In this way, Cl(3, 3) and the associated group Spin(3, 3) ∼ = SL(4, R) can potentially describe all the theory that one sees if one assumes that energy is a scalar, including the weak interaction, and a theory of gravity that reduces to general relativity in the limit that rest masses can be treated as constants; while Cl(0, 6) and the associated group Spin(0, 6) ∼ = SU (4) can describe all the theory one sees if one assumes that mass is a scalar, including electrodynamics and the strong interaction.
Any unified theory that uses a Clifford algebra must reconcile this difference in viewpoint in some way, either by choosing one or the other of mass and energy to be a scalar and allowing the other to vary, or by providing a translation between the two. This fact is already evident in the unification of QED (with scalar mass) and the weak force (with scalar energy). The translation between the two is expressed in the standard model by a symmetry-breaking that arises from treating both mass and energy as scalars simultaneously.
In terms of both physical principles, and experimental reality, it surely makes more sense to treat energy as a scalar, rather than mass. This is already done very effectively in the theory of the weak interaction. But this can only be done by breaking the symmetry groups that rely on mass being a scalar. To do the same for the other forces requires breaking the Lorentz symmetry group SO(3, 1) down to SO(3), and breaking SU (3) down to essentially nothing using the CKM matrix.

The strong gauge group
If it is really possible to include all the unexplained parameters of the standard model inside the quaternionic Dirac algebra Cl (3,3), then this algebra must effectively contain the gauge group SU (3) of the strong force as well. But there are some serious mathematical problems with such a proposal, as this Clifford algebra is simply not big enough to contain a commuting product of all the gauge groups and the relativistic spin group.
The only copies of SU (3) that are available break the symmetry significantly. But we have already seen, in the case of electro-weak mixing, that the Clifford algebra imposes a symmetry-breaking that is not obvious in the standard model, so maybe the same is true for the mixing of the strong force with the electroweak forces. The triplet colour symmetry is unobservable, and the only available place to put it is in the triplet γ 1 γ 2 , γ 2 γ 3 , γ 3 γ 1 that is independent of the 25 significant parameters. This unavoidably identifies colour as being essentially the same thing as the direction of spin. The direction of a quark spin is surely not measurable, at least in practice, if not in theory. The possible directions, moreover, form a 2-parameter family, consistent with the property of colour confinement.
From a mathematical point of view, there is little prospect of linking the gauge group SU (3) of quantum chromodynamics (QCD) as a theory of the strong force to a subgroup SU (3) of the Clifford algebra. The alternatives therefore seem to be either to keep the gauge group outside the Clifford algebra completely, as in the standard model, or to try to re-write the formalism in terms of the 8-dimensional subalgebra Cl(0, 3), generated by γ 1 , γ 2 , γ 3 , rather than the 8-dimensional group SU (3). I have no idea whether this is possible, and therefore leave this as an open problem. It seems to require a rather radical reinterpretation of the strong force as being not so much a force as a quantisation of space itself. Such an interpretation is certainly unconventional, but is perhaps hinted at by the phenomenon of asymptotic freedom [22,23].
A more conventional approach would be to take the direct product of all the gauge groups and the spin group and embed it in the real group so that there is an obvious 16-dimensional real representation. Thus the group can be embedded in a Clifford algebra of an 8-dimensional space with signature (8, 0), (5,3), (4,4), (1,7) or (0, 8). A breaking of symmetry into 2+6 and 4+4 can be accommodated in Cl (4,3) or Cl(0, 7). Alternatively, a breaking of symmetry to incorporate the complex structure can be accommodated in Cl(7, 0), Cl(5, 2), Cl (3,4) or Cl(1, 6). All of these approaches, however, introduce a large number of extra parameters whose meanings and values still have to be explained.

Further remarks
The suggested splitting as 3 + 1 + 2 keeps Cl(0, 3) intact, but splits Cl(3, 0) into Cl(1, 0) and Cl(2, 0). A slightly closer match to the standard model splitting into SU (3), U (1) and SU (2) can be obtained by reversing the signatures, so that Cl(0, 3) splits into Cl(0, 1) and Cl(0, 2), which are the complex numbers and quaternions, respectively. Hence we have obvious groups U (1) and SU (2), as subgroups of the group Spin(4) that lies inside Cl(0, 3). The other half of the algebra is then Cl (3, 0), the algebra of all 2 × 2 complex matrices, as a real 8-dimensional algebra to replace the real 8-dimensional group SU (3). The group that we actually get is, however, GL(2, C), also 8-dimensional as a real Lie group, with a normal subgroup SL(2, C), that is isomorphic to, but distinct from, the relativistic spin group of Dirac. From a mathematical point of view, I cannot see any strong argument for adopting this reversal of signature, but it is possible that are good physical arguments for it. It is also interesting to consider the Georgi-Glashow model [24] in this context. They unified 3 + 2 into 5, and because they were working with unitary groups, they used SU (5) for unification of SU (2) and SU (3). But this group has dimension 24, and turned out to be too big, as it contains extra gauge bosons that cause proton decay, a phenomenon that has never been observed experimentally. If instead we unify Clifford algebras Cl(2, 0) and Cl(0, 3), then we obtain instead Cl(2, 3), that is isomorphic to the Dirac algebra. The largest group that could possibly be relevant here is the general linear group GL(4, C) of (complex) dimension 16. This group is small enough to avoid the problem of proton decay, since it contains 13 dimensions for the standard 13 bosons, plus 3 dimensions for the spin group itself.
A final remark on chirality is in order. The suggestion to use iγ 5 as a generator for the electromagnetic U (1), rather than i, is affected by the fact that iγ 5 anti-commutes with γ µ for µ = 0, 1, 2, 3. This implies that electromagnetism does not have a chirality, since the sign of γ 1 γ 2 γ 3 is not well-defined. On the other hand, the weak SU (2) generated by i, j, k commutes with these γ µ . This implies that the weak force does have a chirality, as is experimentally observed [25], since it distinguishes between γ 1 γ 2 γ 3 and its negative. Our choice of possible generators for the strong force algebra suggests that the strong force also should not have a chirality, since the γ µ anti-commute with each other.

Possible relationships with gravity
At a philosophical level, the suggestion that the strong force is essentially describing the quantum structure of space relates closely to Einstein's approach [26,27,28] to gravity, namely that gravity is not so much a force as a description of the shape of spacetime on a macroscopic scale. At a mathematical level, the use of Cl(3, 3) and the spin group Spin(3, 3) ∼ = SL(4, R) may possibly be related to the use of GL(4, R) in some modern approaches to general relativity [29,30]. But these superficial observations are a far cry from actually unifying the theories of general relativity and particle physics.
A first remark is that in Cl (3,3) there are many ways to find two copies of SL(4, R) that commute with each other. Therefore one could in principle use one copy of SL(4, R) for particle physics and the other for general relativity, simply placed side-by-side and not interacting with each other. Neither copy is equal to the spin group Spin (3,3), and neither of them respects the Clifford algebra structure, but they are obtained by suitable projections, defined by the pseudoscalar.
One could try to use the standard particle physics projections with 1±γ 5 , but the above discussion suggests that projections with 1 ± γ 0 are likely to be more useful in this context. This reflects the fact that general relativity describes a gravity that depends on total energy (γ 0 ) rather than on rest mass (γ 5 ). If one tries to use both pairs of projections together, one runs into problems because γ 0 and γ 5 do not commute with each other. Whatever choice of projections we make, the splitting of Cl(3, 3) into two copies of 4 × 4 real matrices is a symmetry-breaking of the kind I have been discussing, down to Cl (3,2). Further symmetry-breaking down to a single copy of 4 × 4 real matrices can be achieved by going to Cl(3, 1) or Cl(2, 2).
I have no idea whether it is actually possible to incorporate general relativity into Cl (3,3) in this way. If it is, then the algebra implies that there is a certain overlap between the parts of the algebra that are used for particle physics and those that are used for gravity. Such an overlap is not part of current mainstream thinking on fundamental physics, but has been considered at various times by Einstein [31], Sachs [32], Penrose [33] and others [34].
An alternative approach, more in line with mainstream ideas, is to go to a bigger Clifford algebra, so that this overlap can be avoided. For example, Cl(4, 3) is the sum of two copies of 8 × 8 real matrices, which permits one copy to be used for Cl(3, 3) to model particle physics, and the other to act on four dimensions of spacetime and four dimensions of energy-momentum, to describe classical physics and general relativity. Possibly it might be useful to go further, to Cl (4,4), in order to have enough room for the standard model strong gauge group SU (3), rather than reducing this to a smaller group as I have suggested.

Conclusion
In this paper I have explored the subgroups of many small Clifford algebras in sufficient detail to conclude that Cl(3, 3) is the closest match among the Clifford algebras to the algebraic foundations of the electro-weak part of the standard model of particle physics. I have shown that it incorporates the three generations of fermions in the minimal possible way, consistent with the experimental properties of electrodynamics and the weak interaction. Moreover, it contains exactly 25 free parameters. The main mathematical technique employed is to use the structure of the Clifford algebra to describe a symmetry-breaking within the underlying matrix algebra (or pair of matrix algebras).
I have suggested a possible way to incorporate a unification with the strong force, but this is somewhat speculative, in the sense that it requires adopting the point of view that the strong force is not so much a force, as a description of the quantisation of space itself. Moreover, it requires some modification to the gauge group SU (3), which may well be sufficient to rule out such an approach on experimental grounds. Further unification with general relativity is hinted at by the coincidence of two groups, both isomorphic to SL(4, R). This coincidence suggests the possibility of using either the Clifford algebra Cl (3,3) or perhaps a larger Clifford algebra such as Cl (4,3) or Cl(4, 4) to include models of both particle physics and gravity. I make no attempt to build such a model, and again there may be good physical reasons why such a model cannot exist.