Around New Year 2015 I found my first clue to the link between gravity and particle physics. I had come to the conclusion that this link must manifest itself in some coincidence of dimensionless numbers in astronomy (representing curvature of spacetime) and particle physics (representing the mass that supposedly gives rise to curvature of spacetime), and at that time I found the first suspicious coincidence of that kind. By 5th January, I had found enough coincidences that I was completely convinced that they were not just coincidences. Eventually I wrote a paper analysing the probabilities as best I could, essentially proving that they could not possibly be mere coincidences, and that there must be real physical meaning to them. The paper appears in https://arxiv.org/abs/2205.05443 but has not been published in a journal.
Nevertheless, certified physicists insist that they are indeed meaningless coincidences, despite the fact that this is statistically untenable. So I am essentially the only person who is in a position to use these coincidences as real clues as to how to unify gravity with quantum mechanics. But the clues tell us very little, and the Holy Grail is a complete mathematical model that explains them. So I’ve tried every model I can think of, and given up on all of them, sometimes more than once. The only model I am left with at the moment is Sp(1) x Sp(3,1). Strictly speaking, this is a tensor product, rather than a direct product, since both Sp(1) and Sp(3,1) contain the element -1.
First looking at this group from the gravitational point of view, we fix a subgroup SO(3,1), together with its centralizer SO(4). Thus we have a tensor product SO(3,1) x SO(4), or equivalently a direct product SO^+(3,1) x SO(4). Here the + just means we are not allowed to reverse the direction of time. Now SO(4) splits as the tensor product of two copies of SU(2)=Sp(1), one of which is the scalar Sp(1), and the other of which is inside Sp(3,1). So now we can decompose the 36-dimensional adjoint representation of Sp(3,1) into representations of SO(3,1) x Sp(1). The calculations are not difficult, and we find the decomposition 6×1 + 1×3 + 9×3. Here 6×1 is the adjoint representation of SO(3,1), called the field strength tensor in GR but also used for the electromagnetic field in SR, 1×3 is the adjoint representation of Sp(1), and 1+9 is the representation that supports the various tensors: stress-energy tensor and Ricci tensor, the latter breaking into Ricci scalar plus Einstein tensor. GR does not have the group Sp(1) in it, so we reduce to three copies of 1+9. GR uses two of them, to write down the Einstein field equations, and mixes in the third copy of 1 as a cosmological constant. But the third copy of 9 is not used. Thus the proposed model can (at least in principle) extend GR to a more general theory of gravity.
Since GR does not have Sp(1) in it, it cannot build 1×3 + 9×3, but has to build each copy of 1+9 separately. In fact, it builds 6+1+9 as 4×4, the tensor product of two Lorentz 4-vectors. Now the adjoint representation of Sp(3,1) is the symmetric square of the sum of these two Lorentz 4-vectors, so it contains the symmetric squares of each of them, in addition to their tensor product. In GR, a symmetric square is calculated in order to make the Riemann tensor, but it is not the symmetric square of two vectors, it is the symmetric square of the adjoint representation of the Lorentz group. By coincidence, however, the Riemann tensor is 20-dimensional, so it fits the remaining 20 dimensions of adjoint Sp(3,1), and provides the right number of degrees of freedom. But it splits over SO(3,1) to 1+9+10, not 1+9+1+9. Which one is correct? Well, 1+9+10 cannot be unified with quantum mechanics, but 1+9+1+9 can, so my money is on the latter. In standard GR, however, the 10 (which is actually a complex 5, or spin 2 representation) is the Weyl tensor, and is supposed to represent gravitational waves, consisting of spin 2 gravitons. I believe this is an error, and the spin 2 gravitons do not exist.
The reason I believe this is because particle physics requires the existence of three copies of 1+9 in order to have three generations of fermions. In order to explain this, I need to break the symmetry of Sp(3,1) in a different way. For particle physics, we must have a Dirac equation, and the Dirac gamma matrices must generate the Lie algebra so(4,1)=sp(1,1), and the Lie group Spin(4,1)=Sp(1,1). This contains a copy of the Lorentz group in the form Spin(3,1)=SL(2,C), which centralizes Sp(2). In other words, the breaking of Sp(1) x Sp(3,1) is now as Sp(1) x Sp(2) x Sp(1,1). Each of the three factors contains a scalar -1, but now they are not all equal: the product of any two is equal to the third. The scalar in Sp(1) is charge conjugation C, and the other two combine into PT, but we really have to distinguish between P_1 (reflecting one direction of space) and P_3 (reflecting all three), so that the scalar in Sp(2) is P_1P_3 and the scalar in Sp(1,1) is P_1T. In any case, the consistency condition in the model is the same as the standard consistency condition CP_3T=1.
So now we find a gauge group Sp(1) x Sp(2) of dimension 13, in place of the Standard Model gauge group Sp(1) x U(3) of dimension 12. We need to ask which is correct. Certified physicists will automatically say that Sp(1) x U(3) is correct – we’ve “known” that for fifty years. No, not really, we’ve assumed that for fifty years, and nothing has gone wrong yet. Apart from loads and loads of anomalies of course. And loads and loads of unexplained symmetry-breaking, and unexplained parameters. So we cannot just assume that U(3) is correct – why couldn’t it be Sp(2)? And why couldn’t all the symmetry-breaking be a result of trying to force real Sp(2) physics into a fictitious U(3) mould? And let’s not forget that the Standard Model has a Higgs boson as well as the 12 gauge bosons, so the count of 13 rather than 12 appears to be correct.
Of course, it is not just a matter of replacing U(3) with Sp(2), because there’s no guarantee that the Sp(1) facto can be identified with the weak gauge group SU(2) in the Standard Model. In fact, I can guarantee that it can’t. So we need to consider the whole gauge group at once, and compare the two options: first option, Standard Model complex tensor product U(1) x SU(2) x SU(3), dimension 12; second option, Crackpot Model quaternionic direct product Sp(1) x Sp(2), dimension 13. Of course, the SM was developed over a period of 50 years, and has had 50 years of consolidation since, while I’ve been developing the CM for about 50 days, without any period of consolidation, so it’s not really a fair fight. But David did kill Goliath, so all I need is a good sling and a well-aimed stone.
Remember that Sp(1) x Sp(2) = Spin(3) x Spin(5). This splitting of 8 into 3+5 appears in SU(3) as well, on restricting to the real subgroup SO(3). It splits the gluons into 5 symmetric and 3 anti-symmetric (under colour/anticolour symmetry). In the SM this splitting isn’t used properly, which results in a failure to understand how neutral kaons really work – lots of anomalies there, not just CP violation. In the CM, it is absolutely fundamental. It provides a meson octet with three pions and five kaons, as experiment confirms, not the four kaons plus quantum superpositions that the SM insists on, in defiance of all logic. It provides a 3-fold degeneracy for protons, but a 5-fold degeneracy for neutrons, as is required to explain the mass formula e+mu+tau+3p=5n.
The SU(3) illusion probably comes from trying to split off the charge U(1)=Spin(2), which gives us Spin(2) x Spin(2) x Spin(3), with the two copies of U(1) looking after pion charge and kaon charge respectively. Since only one of these copies can be scalars, the other one is mixed with Spin(3) to act on a complex 3-space, which is assumed to be unitary. But where is the experimental evidence that it really is unitary? I don’t see any. I have just slung the 3+5 (experimental) stone and killed two birds at once. What is Goliath’s answer to that?
I could go on. I almost certainly will. But that’s enough for now. Have I found the Holy Grail? Only time will tell. Time for a New Year’s Revolution or two, I think. Here’s to (New Year’s) Resolutions of Auld (Lang Syne) problems!
