A couple of posts ago I argued that there are essentially only three possibilities for a mathematical model to support a Theory of Everything: the Lie algebra must be of type B4, C4 or A5. The “Grand Unified Theory” part of this model is compact of type B3 or C3. Recently, I have been looking at the C3 case, but the B3 case is also interesting. The argument that the GUT must be 21-dimensional rests on the assumption that all the gauge groups and mixing angles must lie in the algebra. However, an argument has been made to me that only the broken symmetries should lie in this algebra, and the unbroken symmetries (u(1) and su(3)) should be somewhere else. I don’t entirely buy this argument, but I felt it was worth looking at anyway.
In this case, we need the broken symmetries su(2)_L of the weak interaction, and the broken symmetries so(3) of the three generations. That’s six degrees of freedom, plus 9 mixing angles, making 15. Hence the algebra is so(6)=su(4), since this is the only compact real simple Lie algebra of dimension 15. In other words, we have Lie type A3=D3 instead of B3 or C3. The first question is, how do the two symmetry groups fit together? The simplest assumption is that they commute with each other, and together generate a subalgebra so(4)=su(2)+su(2), before symmetry-breaking kicks in and moves things around. Then we need to know that there are two completely different ways of embedding so(4) in so(6).
One of them is the “obvious” way, splitting 6 into 4+2 to get so(4)+so(2). That is what the octions model does. That is what most people do. But I hope to be able to convince you that it is wrong. The other way to do it is to split 6 into 3+3, and use the fact that so(4) = so(3)+so(3), to get a different embedding of so(4) in so(6). Of the two possibilities, only 3+3 is compatible with General Relativity. Perhaps a better way to see the difference is to look at the algebra in the form su(4). The octions version splits this to su(2)_L + su(2)_R, whereas the other version splits it to su(2)_L x su(2)_R, where L and R denote the conventional “left-handed” and “right-handed” groups. The Standard Model is ambivalent about when and how it uses + (direct sum) and x (tensor product), and therefore confuses the two cases.
This confusion is something I have been writing about for years, and it is the essential reason why unification of the Standard Model with anything else is impossible. The SM pretends that these two copies of so(4) are equal, when they are not. So if a model uses one form of so(4), it can always be rejected on the grounds that it is incompatible with the other form of so(4). This is a “Catch 22” that completely prevents any solution to the problem. My argument is, and always has been, that the + version (i.e. the 4+2 version) is impossible, and therefore we must use the x version (i.e 3+3). Today I want to present a new argument that proves the same thing in a different way.
The argument proceeds by trying to put the mixing angles into both versions. The 4+2 version divides the mixing angles into 1+8 – the 8=4×2 consists of all the cross terms, and the 1 is the so(2) term. Then we have to separate the two 4’s, and split each into 1+3, to match the mixing angles in the CKM and PMNS matrices. By the time we’ve done that, we’re left with so(3) symmetry, which is a generation symmetry because it acts on the two triples of generation-mixing angles. So far, so good. But what we don’t see is the u(3) formalism for the CKM and PMNS matrices. We do not see the complex structure that breaks so(6) to su(3), or even to u(1) x so(3), unless we break the generation symmetry.
The 3+3 version has 3×3 cross terms for the 9 mixing angles. The SM breaks the symmetry of the weak so(3), first into 1+2, to separate the Z boson from the two W bosons, and then into 1+1+1 to separate the W+ from the W-. Hence we have a breaking of the mixing angles into 3+3+3. So far this looks worse, until we take into account that the generation symmetry acts on both terms in 3+3: this means that the actual breaking is into 3 diagonal terms (that do not mix the generations) plus 6 off-diagonal terms (that do mix generations. Hence weak symmetry-breaking acts differently on the diagonal and off-diagonal terms. The diagonal terms split 1+1+1 on the basis of charge/hypercharge/isospin, without changing the generation (i.e. mass), while the off-diagonal terms split first into 3+3 (again on the basis of hypercharge, to separate leptons from quarks), and then each 3 splits into 1+1+1 on the basis of mass, for three different generations.
So why is 3+3 better? First because it is compatible with symmetry-breaking, which the 4+2 version is not. Second, because the symmetry-breaking splits the CKM from PMNS matrices in a completely natural way, rather than by choosing arbitrary elements of so(2) to label the two. Third, the choice of so(3) for generation symmetry defines a copy of u(1) that acts 2+2+2 on the six coordinates, and hence defines a copy of u(3), so that the formalism for the CKM and PMNS matrices exactly matches the SM implementation in U(3). Fourth, this copy of u(1) is an unbroken version of electromagnetic u(1), with the individual terms of 2+2+2 representing the broken versions of u(1) that implement the CP-violating phases.
To summarise, the 3+3 version matches what is actually done in the Standard Model. The octions model, on the other hand, uses 4+2, and matches what physicists think they are doing. Unfortunately, what physicists think they are doing with the mathematics is not the same as what they are actually doing. They think they are using Spin(4) = SU(2)_L + SU(2)_R, but what they are actually using is SO(4) = SU(2)_L x SU(2)_R. The difference is only a single sign, but it is important. It is the difference between a model of particle physics that is compatible with General Relativity, and one that is not.
I rest my case.