A finite version of SU(3)

In the previous post I drew attention to the fact that M12 contains a finite version of SU(3), or more precisely of PSU(3), that contains the binary tetrahedral group that has been the main point of my models of physics in the past few months (or even years). You will have noticed that my efforts to get SU(3) out of the binary tetrahedral group, in order to model the strong force, might appear contrived, and might not be entirely convincing. There might therefore be some point in extending the model to this finite version of PSU(3), or even SU(3), in order to include the strong force in a more “natural” way.

I will start by considering PSU(3), as a subgroup of M12, so that it has an action on 12 fermions, grouped as 3×4 as explained in the previous post, and on 12 bosons, grouped as 3+9, breaking to 3+1+8 on restriction to the binary tetrahedral group. There may be some point in extending to the full SU(3), which can be found inside the Weyl group of E_6, and which may therefore relate to the various E_6 models in the literature. It all depends on how we can interpret the normal subgroup of order 9 in PSU(3), or the normal subgroup of order 27 in SU(3).

First of all, if we interpret the 12 fermions as above, then we do not appear to have any explicit “colours” for the quarks, but we have a group of order 9 that acts as generation symmetries. There is no single element that acts simultaneously as generation symmetries on all four types of particles, but each element acts on three types of particles, and not on the fourth. If you think about it, then something like this is essential, in order to cope with the fact that generations of quarks do not behave in the same way as generations of leptons. So let us take two specific generation symmetries, one that acts on electrons but not on neutrinos, and one that acts on neutrinos but not on electrons. The former changes the mass on the electrons, does nothing to neutrinos, and presumably therefore also changes the mass of the quarks. The latter, on the other hand, does nothing to the electrons (so does not change their mass), and changes the generation (flavour) of the neutrinos (but without a detectable change in mass), so presumably also has an undetectable effect on the quark masses.

Well now, as far as the quarks are concerned, what is the difference between this symmetry and the colour symmetry? Can you see any difference? I can’t. As far as the electrons are concerned, it does nothing, so it is again the same as the colour symmetry on electrons. Only on neutrinos do we see a “flavour” symmetry instead of a “colour” symmetry. Is this a deal-breaker for this model, or is it a hint that the “flavour” of a neutrino is really a “colour”, and that the labels have been accidentally switched at birth? Of course, the Standard Model insists that flavours of neutrinos are really generations and not colours, but where is the experimental evidence for this? I don’t think there is any, because this is just a theoretical assumption built into QCD.

Mathematically, the assumption that there is a “generation” symmetry that acts on all four types of fermions, and a “colour” symmetry that acts on just two, is inconsistent with the existence of the weak force, with an SU(2) gauge group containing the essential isospin symmetry group Q_8. The only way that group theory allows two triplet symmetries of “generation” and “colour” to work together is if each triplet symmetry acts on three of the four triples. Of course, this is not an argument I can win, against the accumulated “might is right” attitude of thousands of mainstream physicists. I know, because I tried it ten years ago, and gave up. But it is an argument that is correct, and strongly supported by experiment, which confirms that neutrinos and quarks have a “colour” symmetry that does not affect the mass, and that electrons and quarks have a “generation” symmetry that does affect the mass.

More importantly, it is essential that neutrinos have a colour rather than a generation, in order to allow the “colour” to leak out of nucleons, without which there can be no (quantum) force of gravity between nucleons. Yes, you heard me right, a quantum theory of gravity is impossible unless the neutrinos have a colour, in order to glue together nucleons across the vast expanses of space and time. The colour of the neutrinos is the gravitational field. I’ve said it before, and I’ll say it again. You can call me a crackpot all you like, and you can try and pretend that my mathematics is not relevant to your physics. But the lesson of history is that physicists who ignore the simple but powerful arguments of mathematics are consigned to history. Mathematics is an imperious queen, and does not brook dissent.