Let me start by saying that I don’t believe in GUT. Or Grand Unified Theory to give Him His full title. But I do believe in the Holy Trinity of three generations of elementary fermions. In particular, the electron, muon and tau particle. Part of the reason I don’t believe in GUTs is because they never seem to deal adequately with the Holy Trinity. But now that I am beginning to see how the three generations can fit into an E8 GUT, I can also see how they fit into an old-fashioned Georgi-Glashow (horse-drawn) SU(5) GUT, and into a Spin(10) (all spinning, all dancing) GUT. I can therefore see how a belief in GUT can take hold and become difficult to eradicate.
So let’s start with the GG model, in which the fermions are split into 5+10 complex dimensions, with 5 comprising left-handed leptons and right-handed (anti)down quarks, and 10 comprising right-handed electrons, left-handed down quarks and all the up quarks. My generation symmetry (as originally conceived) acts on the 10 but not the 5. This is interesting, but probably wrong, so I looked for modifications. But those look to be probably wrong too.
So let’s see if Spin(10) can help. This unifies 1+5+10 into 16, where the extra 1 is for the so-called “right-handed neutrino”. The 16 is a complex spin representation of Spin(10), in physicists’ usual interpretation, but it is “really” an 8-dimensional quaternionic representation. Well, what I mean is that the Clifford algebra Cl(0,10) is an algebra of 8×8 quaternionic matrices. This is important, because the quaternions unite the left-handed and right-handed (complex) parts of each particle into a whole particle. This is important, because the left-handed and right-handed parts are mathematical fictions, without physical reality. This is important, because we want to model physical reality, and we’re not interested in mathematical fictions.
Now what does my generation symmetry do? It acts on all particles, because in every case it acts on either the left-handed part or the right-handed part, or both. In fact it acts on both parts of the up quarks, and one part only of all the other particles. In particular it acts on the “right-handed” part of the neutrino only. This is important, because it means there is no such thing as a “right-handed neutrino”. What we have is a collection of eight particles, each of which is represented by a quaternion. These quaternions are not (Majorana-Weyl) spinors, they are labels for Weyl spinors. The 8 labels are electron, neutrino, and three colours of up and down quark. On each quaternionic label we have a generation symmetry, as I’ve already explained.
Hence the Spin(10) GUT from 1974 already contains a generation symmetry. But nobody appears to have noticed it before, because physicists refuse to use quaternions. They insist on treating the spin representation of Spin(10) as a 16-dimensional complex representation, when it should really be treated as an 8-dimensional quaternionic representation of the full Clifford algebra Cl(0,10). The quaternionic structure is important, because you can’t describe the generation symmetry properly without it.
So now that we have re-united the elementary particles into whole particles, by stitching their left-hand sides to their right-hand sides, we are in good shape for understanding what these particles do, and how the three generations behave. Interestingly, I already gave you all the details of the electron quaternion a year or two ago. At the time, I didn’t realise it was part of the Spin(10) GUT. But now, the GUT has spoken to me, and I realise I was speaking in tongues.
To recap, then, I wrote the three electrons as the quaternions -1+i+j, -1+j+k and -1+k+i, so that rotating i,j,k gives us the generation symmetry. Now the projection onto the “left-handed” part obliterates the generation distinction, so that all the left-handed electrons are -1+(i+j+k)2/3. Hence the right-handed electrons are (i+j-2k)/3, (j+k-2i)/3 and (k+i-2j)/3. But now you will notice there is something missing – we’ve only got three particles in a 4-dimensional space. Well, I’m glad you asked that question, because I have an answer that’s been ready and waiting for years: the fourth dimension is taken by the proton. Why? Because the negative charges on the electron don’t make any sense unless we’ve got a positive charge to compare it with.
So, what next? We need to decode the symmetry-breaking between the different masses of the three generations of electrons. I made a start on that, by finding the mass axis in the electron quaternion, and using it to predict or explain three of the mixing angles in the Standard Model. But the real question is, where does this mass axis come from? It defines a particular copy of the complex numbers inside the quaternions, but it seems completely arbitrary. If it is coming from E8, then it comes from Spin(12,4), so from the Clifford algebra Cl(12,4) or Cl(4,12), both of which are real 64×64 matrix algebras. So the complex structure must come from the complex structure of Cl(4,1), I suppose, i.e. from the Dirac equation. But that doesn’t answer the question, it only re-phrases it.
A possible approach is to split Spin(12,4) into Spin(11) x Spin(1,4), so that I have added a mass coordinate to both Spin(10) and Spin(1,3), and can interpret the former as an “internal” mass, and the latter as an “external” mass. Then the unification of the internal and external worlds hinges on a calibration between the internal and external masses, as I have suggested in various forms before. Or perhaps the E8 GUT is a false GUT, a mere graven image, and asking it for oracles is a waste of time. But you have to admit it is a very beautiful graven image. Mesmerising, in fact.