I know some of you are puzzled that I have put forward loads of different models, and keep on presenting new ones. Which one is right? Probably none of them, but the point is to explore all the possibilities, and try and find one that fits experiment better than the standard models. At the core of all these models is the problem of reconciling the Einstein picture of spacetime with the Dirac picture. If you just assume these pictures are the same, you get contradictions coming out of your ears, and then you start adding more and more epicycles to your theory to get rid of these contradictions, and you end up tying yourself in knots with string theory.
Both the Einstein and Dirac pictures are expressed in terms of 4×4 matrices, the Einstein real matrices and the Dirac complex matrices. But these matrices are doing different things, and you can’t just say they are the same 4×4 matrices. However, my first models tried to do just that, basically saying either Einstein or Dirac must be wrong. First I tried saying Dirac was wrong, and used the Einstein matrices for everything. I even managed to publish that paper, but it doesn’t match experiment very well, so I abandoned that approach. Then I realised that Einstein was wrong, because Einstein gravity is contradicted by observations of galaxies. The standard LCDM “fix” doesn’t actually work, although most physicists still think it does.
My favourite model is the binary icosahedral group algebra model. This contains both the Einstein matrices and the Dirac matrices, separately, but the finite group breaks up both sets of 16 matrices into 1+3+3+4+5 in the same way. So if you quantise everything with this finite group, you can reconcile Dirac with Einstein without changing either of them. Actually you get even more, because the Dirac algebra gets a quaternionic structure too, becoming 2×2 quaternion matrices. I thought that was pretty cool, but selling discrete models to physicists is very difficult. They’re just not in the market for discrete models, they want continuous models with go-faster stripes.
So I went back to continuous E8 models, and then I found that E8 gives you 4×4 complex matrices with SU(2,2) structure instead of either the real SL(4,R) Einstein structure or the quaternionic SL(2,H) of my discrete model. This gives spacetime a conformal SO(4,2) symmetry group instead of Einstein SO(3,3) or quaternionic SO(5,1). There are two different ways of putting SU(2,2) in E8, and I tried one and then I tried the other. The second one seems to work better than the first. But either way, it means Einstein was wrong, and Dirac might have been right, which is good for experimental reasons, because the evidence that Einstein was wrong is much stronger than the evidence that Dirac was wrong. Or perhaps I should say the evidence that Einstein was right is much weaker than the evidence that Dirac was right. If you treat the Dirac gamma matrices (including gamma_5) as real, without multiplying by complex numbers any time you feel like it, then they generate the Lie algebra of SU(2,2), so you don’t need to modify Dirac’s work in any way.
Then I realised I didn’t need all of E8, I only needed SU(7,2). Now I’m beginning to think I only need SU(4,2), so I’ll have a look at that possibility and see what happens. The smaller the group, the tighter the structure and the less freedom there is in the model. Hence the smaller the group, the greater the explanatory power of the model. I don’t think it is possible to get any smaller than SU(4,2), although I did try SU(3,2) at one point, and it might be worth looking at that again. It all depends what you think the gauge group of gravity is. Currently I am working under the assumption it is SU(1,2), but Woit and others seem to think it is SU(0,2), i.e the right-handed (spacetime) SU(2). If he is right, then SU(3,2) might work, with a splitting SU(3) x SU(2) for the strong force and gravity, and a splitting SU(2) x SU(1,2) for the weak force, and U(2,2) for electromagnetism, i.e. the Dirac algebra.
But I am inclined to think that I need SU(1,2) for gravity, because SU(2) only gives us three degrees of freedom, enough for Newtonian gravity but not enough for either Einstein or MOND. In that case I can split SU(4,2) into SU(3) x SU(1,2) for the strong force and gravity, and SU(2) x SU(2,2) for the weak force and electromagnetism. All the mixing then goes into the third coordinate, that swaps sides between the gravi-strong splitting and the electro-weak splitting. That gives us only 6 free parameters, nowhere near the 20+ needed in the standard model, and therefore gives us a vast amount of explanation of the standard model parameters. If it works, that is.
If it works. Did you say, if it works? It does work, and I’ve already told you how it works. You just look at the action on the third complex coordinate, including the scalars of order 3 in SU(3) and the scalars of order 4 in SU(2,2), and write down the obvious four equations between the nine mixing angles to reduce them to five, one for each of the off-diagonal entries in the third column. Then you calculate the Weinberg angle from a 2,3,sqrt(13) triangle, and calculate the other four mixing angles from the five masses of electron, proton, neutron, muon and tau in the way I told you. Ah no, that only gives three independent mixing angles, we need one more from somewhere. Or maybe we don’t? Maybe we only need SU(3,2) after all? Anyway, you get my drift – the idea basically works, it just needs a bit of finishing off and tidying up.
To summarize, the current proposal to reconcile Einstein with Dirac using SU(2,2) has huge explanatory power, that is consistent with experiment everywhere I have looked so far, so it is by far the best game in town that I know of. It means that Einstein was wrong, in quite a serious way. It means that the concept of general covariance is not useful for describing gravity, but the concept of conformal covariance (i.e. Lorentz covariance plus scale covariance) is. It means that the identification of Dirac’s SL(2,C) version of the Lorentz group with Einstein’s SO(3,1) for the description of quantum electrodynamics still works, because replacing general covariance with conformal covariance removes the fundamental contradiction from this identification. This is a huge relief, because it is the contradiction between quantum mechanics and general covariance that has prevented unification for the best part of 100 years. It means that Dirac was right all along, and Einstein was wrong.
Once you get rid of general covariance, you also get rid of the spin 2 graviton, which lives in the adjoint representation of SL(3,R), that splits as 3+5, i.e. spin 1 plus spin 2. Under conformal covariance, this changes to a different real form, SU(1,2), that splits as 3+4+1 under the chiral (right-handed) SU(2), that is spin 1 plus spin 1/2 plus spin 0. This is what experiment actually sees, when you look far enough out in the galaxy. But if you still believe in general covariance, you have to add in the 4+1 by hand. The 4 is a spinor, so it behaves like matter, so you call it “dark matter”, because you can’t see it. The reason you can’t see it is because it isn’t matter, it is part of the gravitational field. You can’t “see” it, any more than you can “see” a photon. The 1 is a scalar, so it behaves like energy, so you call it “dark energy”, because you can’t feel it. Actually you can feel it, because it is part of the gravitational field. But it is very weak, so you don’t notice.
In particular, there is another sacred cow that must join general covariance on the funeral pyre, that is the concept of gauge bosons. There are gauge fermions as well. Any unification of the gauge groups that adds in new gauge bosons is falsified by experiment, because they all predict new forces and new phenomena that have never been detected. If, however, you add in (essentially massless) gauge fermions, i.e. neutrinos and antineutrinos, then you do not predict anything new, except quantum gravity. Oh, and while we’re about it, we must slaughter the Coleman-Mandula Theorem. The gauge group of quantum gravity does not commute with the conformal group SU(2,2), as Coleman-Mandula would have it, it is a subgroup of the conformal group. The only thing it commutes with is the other gauge groups.
So this is another reason to believe the SU(2,2) version of Einstein-Dirac unification. Not only does it come down solidly on the side of Dirac against Einstein, but it distinguishes all the other correct and incorrect assumptions that have been made over the past 100 years. It sorts them into one pile of correct assumptions, and another pile of incorrect assumptions. The assumptions have nowhere to hide any more, they are hidden no longer. The only problem with that is, if there are no more Hidden Assumptions, what will I write about on this blog from now on?