I may have asked this question before, but it’s like the story of economics exams – the questions are the same every year, but the answers are different. The question about the Dirac algebra is whether the complex structure is physically important, or just a kludge to get the answers to come out right. For some time now, I have been building models that use the complex structure in a way that seems physically important, but earlier on I used to assume it was just a kludge. Now I’ve come back to the viewpoint that it is just a kludge. So let me explain.
The one thing we definitely need in the Dirac algebra is a copy of so(3,1) for infinitesimal Lorentz transformations. This would normally be made from the Clifford algebra Cl(3,1) or Cl(1,3) – it doesn’t matter which one you use, we can either take the standard gamma matrices (which I will call G0, G1, G2, G3 so I don’t have to keep saying “gamma”), which generate Cl(1,3), or i times them, which generate Cl(3,1). In both cases, the Lorentz transformations are the products of pairs G0.G1, …, G2.G3. The ones with G0 in them are supposed to be boosts (to give Lorentz contraction, time dilation etc), and the others are supposed to be rotations. When you do the calculations, you find that G0.G1 is hermitian and G2.G3 is anti-hermitian. This means we have to use the mathematicians convention, that hermitian matrices represent boosts and anti-hermitian matrices represent rotations.
Physicists unfortunately have adopted the convention that rotations are represented by hermitian matrices, and therefore boosts are represented by anti-hermitian matrices. I have no idea why they do this, but in particle physics at least it is completely universal. But it is completely unnatural, because the one thing you need to know about hermitian matrices is that they have real eigenvalues, and the one thing you need to know about rotations is that they do not have real eigenvalues. Hence, in order for particle physicists to interpret G2.G3 as a rotation, they have to multiply by i. But mathematicians don’t need to do this, because G2.G3 is already a rotation , because it has no real eigenvalues.
In other words, physicists need the Dirac algebra to be complex purely in order to switch hermitian and anti-hermitian matrices. There is nothing physical about it, it is purely a kludge to deal with the fact that Pauli made a very bad choice (hermitian) for his matrices in 1927, and nobody has seen fit to replace this with a better choice in the last 97 years. Unfortunately, this choice is not just a convention, it actually messes up a whole heap of things, as I shall explain.
Grand unified theories of the modern type try to put everything, including all the gauge groups and the Lorentz group, into one big Lie algebra. That means that we must treat the Dirac algebra as a Lie algebra. If we take G0, G1, G2, G3 as generators we get the Lie algebra so(4,1), if we use the maths convention, or so(3,2) if we use the physics convention. If we multiply by i first, it’s the other way round. This is a rather important distinction, and it’s important to get it right! For the sake of argument, let’s assume that physicists made their choices for a reason, and that so(3,2) is the one that correctly describes the reality of physics, independently of the unfortunate choice of notation.
Then we have one more decision to make, concerning G5 = G0.G1.G2.G3 in the math convention, or i times this in the physics convention. The question is whether G5 represents a rotation or a boost. It seems fairly clear from the way it is used that it is a rotation, but I’m not 100% sure about that. Certainly in the maths convention G0.G1.G2.G3 is hermitian, so represents a rotation, and the extra factor of i is required to convert to the physics convention, while still representing a rotation. Now for simplicity let us stick to the math convention, so we don’t have to introduce extra factors of i in unpredictable ways, but keep the standard Dirac matrices for our notation. This really means that the natural generators are iG0, iG1, iG2 and iG3, generating a Lie algebra so(3,2), with a Minkowski signature -+++.
To get the full Dirac algebra we need the products of 3 or 4 of the Gs, as well as the products of 2 obtained from the Lie algebra. That means we need to add G0.G1.G2.G3, which is already in the real Clifford algebra, so does not require the physicists’ complexification. Now we can work out the signature, which is -+++ for the single Gs, —+++ for the pairs (Lorentz group), -+++ for the triples, and – for the quadruple, making (9,6) altogether, which implies the Lie algebra is in fact so(3,3).
This conclusion comes from analysing from a pure mathematical point of view what it is that physicists actually write down in their calculations. We put it on firm mathematical foundations, so that the kludges (complex Dirac algebra, hermitian rotations, etc) are no longer necessary. The Clifford algebra is the real Clifford algebra Cl(3,1), no messing, and the Lie algebra we get from it is so(3,3), no messing. Of course, so(3,3) = sl(4,R), which means that the infinitesimals in the Dirac Lie algebra are the same as the infinitesimals in the Einstein Lie algebra of local transformations of spacetime. The Dirac Lie algebra and the Einstein Lie algebra, done correctly, are the same Lie algebra.
So where does the contradiction between quantum mechanics and gravity come from? Is it a real contradiction, or is it a failure to translate correctly between the physics convention and the maths convention for Lie algebras? I believe it is the latter. The reason I believe that is because I have myself failed to do this translation correctly. I assumed that G5 is a boost, because it is hermitian, and hence I extended so(3,1) to so(4,2). That seems to be wrong. G5 seems to be intended as a rotation, and hence the extension is either to so(3,3) or so(5,1). Then we also need to distinguish between signatures -+++ and +— for spacetime in order to distinguish between so(3,3) and so(5,1). Previously I just guessed which one is correct, as other people seem to do also. Now I’ve analysed what the Dirac algebra actually does, I don’t have to guess any more, I know it is so(3,3).
Woit guesses so(4,2), Chester, Marrani and Rios guess so(3,3), Manogue, Dray and Wilson guess so(5,1). I’ve tried all three, starting with so(3,3), then so(5,1) and most recently so(4,2). Now I know which is right I can go back to the beginning, throw away everything I have done for the past few years, and start again from scratch. Then I will have a model of particle physics that is generally covariant, because it is built with general covariance in the Dirac algebra rom the very beginning. Indeed, it is even better than that, because the Dirac algebra is the same thing as general covariance.