I may have been a bit hasty to dismiss GR in my most recent paper, posted here a couple of weeks ago at https://robwilson1.files.wordpress.com/2024/04/camps3.pdf. What happens in that paper is that I work with 3×3 matrices over the split quaternions H’, in order to implement phase space, and I find that they split as 1+14+21, which is compatible with QM, and not as 1+15+20, as would be required for GR. However, when I tried to embed the model into E8, in order to try and get the E8 crowd interested, I found that the embedding I wanted was A1+A2+A5 (which I’ve written about before, but not managed to get onto the arxiv), and that I had got the wrong real form of A5.
I’m not sure exactly how many real forms A5 has, but it is at least seven, of which two occur in the semisplit magic square that underlies most of the E8 models that people consider, including our “octions” model https://arxiv.org/abs/2204.05310. The real form I was considering occurs in the split magic square only. The 3×3 matrices over ordinary (non-split) quaternions were studied in some detail in the octions paper, as they formed one of the main stepping stones to get to E8. But I am now convinced that the octions paper got it wrong, and should have transposed the magic square, and used the other real form, which is SU(3,3). And I also got it wrong, and should have used SU(3,3).
The reason is that SU(3,3) not only contains Sp_6(R), which is the symmetry group of phase space in Hamiltonian mechanics, and is, as I have recently shown, the formalism that underlies the Dirac algebra in (relativistic) quantum mechanics, but also contains SO(3,3), which is how the group of general covariance in GR acts on phase space. In other words, we can combine the two provided we complexify phase space. This is not even a crazy idea, it was already done by Maxwell in the 19th century. Complex phase space allows one to treat momentum and current at the same time. If you can’t treat momentum and current at the same time, then you can’t do electrodynamics at all! And it was done by Dirac in 1928 in the context of QM. So it is known to be necessary, and you may well ask, why I thought I could get away without it? Good question, to which I do not have an answer.
Anyway, now that I know it is necessary, I know that I have to study the action of SU(3,3) on complex phase space. This group has got everything in it that is needed for physics. It’s got the Dirac algebra for QM, it’s got general covariance and the Riemann curvature tensor for GR, it’s got twistors if you want to follow Penrose, it’s got Hamiltonian symmetries of phase space if you want to follow Bohm/Hiley/de Gosson, and its centralizer in E8 is SU(3) x SL(2,R), which is the correct real form for the gauge group of the Standard Model of Particle Physics. What more could you want? The Moon?
Maybe you want to see the Riemann curvature tensor, because that’s the bit that no-one believes can be there in quantum mechanics. As a representation of SO(3,3), the RCT is usually constructed by first converting to compact SO(6), then taking the symmetric 6×6 matrices and subtracting off the identity matrix. If you want to do this with SU(3,3), then what you get is a 21-dimensional irreducible complex representation, in place of the real 1+20. In other words, we get a significant generalisation of the RCT, and a corresponding extension or correction to GR.
On the other hand, it may be worth noting that SU(3,3) has got a real 20-dimensional irreducible representation, that is constructed in quite a different way, as the anti-symmetric cube of the natural 6-dimensional representation (instead of the symmetric square). Is this what the RCT is really trying to be? I have no idea. BUT this representation plays a prominent role in the E8 model, and represents (among other things) the left-handed leptons. There are two copies of this representation, one for neutrinos, one for electrons. So does this mean that the RCT is trying to describe neutrinos? Can this possibly make sense? Instead of working in the fourth power of spacetime, we work in the cube of phase space? The cube of phase space breaks up as 1+9+9+1 if we separate position from momentum, whereas the RCT breaks up as 1+9+10 if we restrict to SO(3,1). So we get the Einstein tensor coming out, but not the (spin 2) Weyl tensor. So we get something like the field equations, but not the spin 2 graviton. Instead of the spin 2 graviton we get three directions of momentum times three directions of distance-squared. Well, if that isn’t quantum gravity travelling at the speed of light then I’ll eat my hat!
But, did you notice that there are also 3 dimensions of momentum squared times 3 dimensions of distance? That means that, in addition to the 1/r^2 ordinary Newtonian gravity, there is a 1/r type of gravity, which is what is needed for MOND. Moreover, there is a transition between the two that occurs when the momentum/distance relation swaps over. Why does this occur empirically at a particular acceleration scale? I have no idea. But one thing is for sure, this isn’t caused by Dark Matter. It is caused by neutrinos interacting with each other, because it has a momentum-squared term – one momentum for each neutrino.
But, did you also notice that there is a momentum-cubed term? Independent of distance? Is that what Dark Energy really is? Interactions of three neutrinos at once?! No wonder the cosmological constant is so small! No wonder it is 120 orders of magnitude smaller than particle physicists think it is!
So, are we any nearer to a resolution of this conundrum? Is Einstein’s GR like Eric Morecambe’s piano concerto? All the right notes, but not necessarily in the right order?