I promised to tell you about gravity, from the point of view of the group Sp(1) x Sp(3,1). The first point to make is that we need to factorise Sp(3,1) as SO(3,1) x Sp(1). That is, we separate out the quaternionic scalars Sp(1) from the real Lorentz group SO(3,1). These are right-handed scalars, and the left-handed scalars are in the other Sp(1), commuting with Sp(3,1). Or you could swap the left-handed and right-handed copies – it’s only a mathematical convention, after all. Anyway, the group you get after all that is in fact SO(3,1) x SO(4), not SO(3,1) x Spin(4) that you might expect, or Spin(3,1) x Spin(4) that you might want, or Spin(3,1) x Spin(3,1) that Dirac and Einstein appeared to want.
So let’s look at SO(3,1) x SO(4) in the cold light of day, and try not to pay too much attention to the dreams, visions and hallucinations of the past 100 years of theory. Quantum theory arose from the canonical factorisation of SO(4) into SU(2) x SU(2), which was then interpreted as a factorisation of Spin(4) instead of SO(4), so that SU(2) x SU(2) becomes SU(2) + SU(2). This mistake has never been understood, because 2×2=2+2=4, so no-one ever figured out that there was anything wrong. You cannot add the left-handed and right-handed spinors, you can only multiply them. Anyway, inside SU(2) + SU(2) you can create a diagonal copy of SU(2), which is a double cover of SO(3) sitting inside SO(4). But this copy of SU(2) is an illusion – it doesn’t really exist. So that’s the second mistake. The third mistake is to confuse SO(3) in SO(4) with SO(3) in SO(3,1), and hence convert the left-handed and right-handed spinors of SO(4) (which do exist) into left-handed and right-handed spinors of Spin(3,1) (which don’t exist).
OK, are you with me so far? Dirac’s QM has three fundamental mistakes in it. One is replacing SO(4) by Spin(4), the second is replacing SO(3,1) by Spin(3,1), and the third is pretending that Spin(3,1) and Spin(4) are the same thing. Einstein, by contrast, only made one of these mistakes. In his theory of gravity, he pretended that SO(3,1) and SO(4) are the same thing. So it is actually easier to explain how to correct Einstein’s theory of gravity than it is to explain how to correct Dirac’s theory of quantum mechanics.
Einstein then worked with the tensor product SO(3,1) x SO(3,1) instead of SO(3,1) x SO(4), which meant he was able to separate out the symmetric part (Ricci tensor) from the anti-symmetric part (field-strength tensor). Then he separated off the Ricci scalar, which is basically the Lorentzian metric on (local) spacetime, to create the 9-dimensional Einstein tensor. That is all very well, but it only works when the identification of SO(3,1) with SO(4) doesn’t stretch too much, in other words when the timescales are relatively short. In this situation the corrections to Newtonian gravity are linear, so you can treat time and space separately, and not worry too much about how they mix together. You just have to stick in a negative sign here and there to switch between real and imaginary time (squared).
But over long timescales, it no longer works. If you have to wait tens of thousands of years for the gravity of the rest of the galaxy to reach you, everything has moved a long way from where it was. So you need to take this into account, as Yahalom does in his theory of “retarded gravity”. But that may not be enough, because Yahalom still uses a Minkowski SO(3,1) spacetime, with Einstein gravity acting on it locally. He does not convert one of the tensor factors from SO(3,1) to SO(4). So it is likely that, while Yahalom’s theory explains a lot, it does not explain everything.
In the SO(3,1) x SO(4) model there is no splitting of the 16 dimensions of the tensor into 6+10, and no splitting of the 10 into 9+1 (Einstein tensor plus cosmological constant). It is an irreducible representation. The SO(4) factor splits into SU(2) x SU(2), where we see all the Lorentz-invariant properties of matter. We have, therefore, a copy of the quaternions, with left and right multiplications defined on it. On this structure we have to find all the (discrete) quantum numbers for quantum gravity, and all the (continuous) masses associated to those quantum numbers. I’ve analysed this structure in quite some detail, and discovered how one can use these quantum numbers to describe the masses of the proton, neutron, electron, muon and tau particle, with a linear relation between them.
If we analyse the 16 coordinates in full, we have four energy/momentum coordinates, plus these four mass coordinates, but they overlap, so we really only have 7 coordinates for properties of matter, leaving 9 coordinates for properties of the gravitational field. These factorise into 3 directions of space (parameterised by SO(3) in SO(3,1)) times 3 generations (parametrised by SO(3) in SO(4)), and are therefore represented by neutrinos. So we still have a splitting of 16 into 6+10, or more precisely into (3+3)+(1+9), and we still have the Einstein tensor (9) , field-strength tensor (3+3) and Ricci scalar (1), but the interpretations are different. The (1) is still an energy (density), but it is the energy density of the gravitational field.
The (9) is still generated by matter, but it is a product of three colours (directions of spin) and three generations. The sum of the three generations of electrons has mass almost indistinguishable from that of electron+proton+neutron, or two neutrons, so that the latter is a very good approximation to the gravitational mass of ordinary matter. But it does not split up correctly to the elementary particles, and it does not work correctly when the proportion of hydrogen (no neutrons) to helium (half neutrons) varies too much, on a galactic scale. The (9) describes not only the matter itself, but also the propagation of the gravitational field using three generations of neutrinos and three directions of momentum. The equivalence of these two descriptions of the (9) is what Einstein’s field equations are trying (but failing) to express.
And the (3+3)? Three generations of electrons, plus 3 colours of protons. This is electromagnetism, of course. So we get the same general description that Einstein gets in his rank 2 tensors over spacetime. But significant differences in the details. What about the rank 4 tensors? This is where Einstein gets the Riemann curvature tensor (20 dimensions), and a splitting thereof into 1+9+10. He gets this by first taking the adjoint representation of SO(3,1), then taking the symmetric square (dimension 21) and splitting off a scalar. We have to take instead the adjoint representations of both SO(3,1) and SO(4), and tensoring them together. So we get 36 dimensions, and an irreducible representation. Where is the Riemann curvature tensor in here? Well, that really isn’t a meaningful question. Curvature of spacetime is an illusion caused by a failure to appreciate the physical and mathematical differences between SO(3,1) and SO(4).
Next time, I’ll tell you more about Dirac’s mistake – adding spinors instead of multiplying them. This is what introduces unphysical (i.e. meaningless) variables into quantum mechanics, and is the cause of the measurement problem – the problem is, basically, how can experiment measure things that don’t exist? Or perhaps it is more accurate to say that this is the solution to the measurement problem.