Wow, wasn’t SU(3,2) a sensation? Theory Of Totally Everything, Man! TOTEM, for short. To recap, the Georgi-Glashow model of 1974 proposed embedding the gauge group of the Standard Model into SU(5), but then ended up with 12 extra gauge bosons, that caused protons to decay, whereas experiment says they don’t decay. There is exactly one other real form of SU(5) that contains the SM gauge group, and that is SU(3,2), which does not contain any extra gauge bosons, because the 12 extra dimensions turn out to be fermions. So why don’t we use SU(3,2) instead? Remarkably, it actually works! The fermions have the 2x2x3 structure that they have in the SM, the bosons have the SM structure 1+3+8, and by breaking literally all the symmetry, we get 24 parameters, which is (roughly) the number in the SM, once neutrino oscillations are included.
It contains two different types of Lorentz group, which I want to describe to you. The first one is SL(2,C), contained in SU(2,2), inside U(2,2) fixing (say) the first of the five coordinates. The 9 dimensions of SU(2,2) outside SL(2,C) split 1+4+4 into a real scalar and two Lorentz 4-vectors. The 8 dimensions of SU(3,2) outside U(2,2) form an irreducible complex 4-dimensional representation of U(2,2), in other words, a Dirac spinor. The 16-dimensional group U(2,2) plays the same role as the 16-dimensional Dirac algebra, and the two transform in the same way under the action of the Lorentz group.
The only difference is that the Clifford algebra has signature either (10,6) or (6,10), depending on whether we start with a spacetime metric of signature (3,1) or (1,3), whereas the group has signature (8,8). This change of signature explains why the Dirac algebra needs to be a complex Clifford algebra – both of the possible real Clifford algebras have the wrong signature. In any case, the model now contains all of the standard formalism of quantum mechanics, including the Dirac equation for mass. Moreover, it explains why there are three independent choices of Dirac algebra U(2,2) within SU(3,2), obtained by fixing one of the first three coordinates. Hence we get all three generations into the model at once.
The other Lorentz group is obtained in a similar way, by fixing one of the last two coordinates instead. There are of course two independent possibilities here, obtained by splitting the weak doublets. For example, we can distinguish the charged electron/proton pair from the neutral neutrino/neutron pair, and hence distinguish electromagnetism from gravity. Anyway, restricting from SU(3,2) to U(3,1) we see a way to fit in the Lorentz group in its SO(3,1) form, by restricting from complex to real numbers in each coordinate. The 10 dimensions of U(3,1) outside SO(3,1) transform in the same way as the stress-energy tensor and the Ricci tensor in General Relativity, so they describe the way that SO(3,1) changes under the influence of gravity. And they do it in exactly the same way that GR does it. Then the 8 dimensions of SU(3,2) outside U(3,1) form a complex 4-dimensional representation of U(3,1), which restricts to SO(3,1) as the sum of two (real) Lorentz 4-vectors. One of these represents spacetime, and the other is 4-momentum.
What this means is that SU(3,2) has three entirely different interpretations, depending on which subgroup we take as the important symmetry group.
- Symmetry group SO(3,1) contained in U(3,1): Special and General Relativity.
- Symmetry group SL(2,C) contained in U(2,2): Quantum Mechanics and the Dirac Equation.
- Symmetry group S(U(3) x U(2)): the Standard Model of Particle Physics.
In other words, SU(3,2) actually does unify all three theories. How cool is that?! Everything in the universe is encoded by a 5×5 complex matrix, and existing theories work with 4×4 submatrices, or 3×3 and 2×2 and 1×1. But what I really want to point out is that the Dirac Equation puts the mass into the top-left entry in the matrices, and the Einstein Equations put the mass in the bottom-right entry in the matrices. The Weak Equivalence Principle says these are the same. The mathematics says they are not. That on its own wouldn’t matter too much, were it not for the fact that the real world also says they are not the same. The experiments were done in the 1950s and 1960s, and provided a 3 sigma signal that gravitational mass and inertial mass are not the same thing. It is a pity that since 1969 these experiments have been completely ignored, and never repeated.
The fundamental assumption of quantum mechanics is that the group SL(2,C) is the double cover of SO(3,1). This is a mathematical fact that is completely irrelevant to physics. The sooner physicists forget about it the better. The group SL(2,C) is isomorphic to the double cover of SO(3,1), but is not equal to it. In the model, SL(2,C) and SO(3,1) are completely different subgroups of SU(3,2), and the absurd assumption that they are essentially equal is responsible for breaking literally all of the symmetry of all the models of fundamental physics. In other words, the 24 unexplained parameters of the SM arise purely and simply from equating two incompatible definitions of mass. One is a scalar representation of SL(2,C) – the Dirac mass – and the other is a scalar representation of SO(3,1) – the Einstein mass.
The 24 parameters did not arise from the Big Bang, or from natural selection of our universe from a multiverse of possibilities, or from the anthropic principle, or any other such nonsense. They arose purely and simply from generalising the concept of “mass” from its useful range of perhaps 20 orders of magnitude to a meaningless range of up to 80, or even more. Einstein mass and Dirac mass can be regarded as approximately equivalent in limited circumstances provided we do not get too much bigger or smaller than weighing potatoes in the market. If we try to generalise further than that, the equivalence fails, in a big way. Dark matter is a fictitious concept invented to try to weigh galaxies in the same way that you weigh potatoes. Neutrino mass is a fictitious concept at the other end of the scale. Quark mass sits in the grey area where the concept of mass starts to break down, because Einstein mass of quarks doesn’t make sense – although one could try to argue that Dirac mass of quarks still makes sense. But whichever way you look at it, you cannot escape the inevitable conclusion that the 24 parameters arise from the interference of gravity in particle physics. There is simply no other reasonable mechanism.
I don’t know about you, but I’m starting to look forward to the Year of the Snake.