Duals, contragredients and opposites

I tried to continue the discussion of Peter Woit’s talk on “Spacetime is right-handed” by talking about the second section, where he describes what he means by this phrase. However, I messed it up badly, so thought it best to start again from scratch. The real problem is that different interpretations of the standard model appear to use the terms “left-handed” and “right-handed” differently. There are in fact two meanings, which are quite different. The first meaning is attached to the two Weyl spinor representations of SL(2,C), whereby the left-handed spinors are complex conjugates of the right-handed spinors. The second meaning is attached to groups, in particular the weak gauge group SU(2)_L, but also in some interpretations also to two copies of SL(2,C), called SL(2,C)_L and SL(2,C)_R.

Problems start to arise if you confuse left-handed groups with left-handed representations. If you think the left-handed representations are representations of the left-handed groups, then you will get into trouble. What Woit points out in slides 13-15 is that it is possible to treat the left-handed Weyl spinor as the left-handed representation of the right-handed SL(2,C). What I would like to point out is that it is not only possible, but it is necessary. There are two reasons for this. The first reason is that this is what is actually done in practical calculations (the Feynman calculus): the left-handed Weyl spinor is not a representation of SU(2)_L. The second reason is that it is forbidden by the Coleman-Mandula theorem: if SU(2)_L did act on the left-handed Weyl spinor, then it would not commute with the Lorentz group.

So, as far as I can see, what Woit does in section 2 of the talk is to describe exactly how the calculations in the standard model actually work. He contrasts this with section 1, in which he describes how a great many theorists think the standard model works. His assertion that “Spacetime is right-handed” is therefore not so much a major breakthrough in the understanding of spacetime, as he seems to think, but more like a tautology. I therefore revise my opinion of his talk. I no longer consider it to be wrong, but I consider it to be devoid of content.

Now let’s get technical, and try to see what is really going on in the mathematics. There are three interesting things you can do to the complex 2×2 matrices: you can take the complex conjugate (C), the transpose (T) and the inverse (I). Of these, T and I reverse the order of multiplication, while C does not. If you reverse the order of multiplication, you take a right-handed group SL(2,C)_R to the opposite group, that is the left-handed group SL(2,C)_L. This is a technical meaning of “opposite”, which I mistakenly called “dual” before. The standard meaning of “dual” is TI. For finite groups, and compact Lie groups, TI is equivalent to C, and if TI=C on a particular representation, then that representation is called unitary. But SL(2,C) is not compact, and on the Weyl spinors TI is equivalent to 1, not C. Thus C needs a new name: I call it the contragredient, but this word seems to be widely confused with dual, so let’s just call it the complex conjugate, and hope we can avoid this confusion.

Before we go on to talk about the opposite, it is important to note that the property that TI is equivalent to 1 is a very special property of SL(2,C), and does not apply to any other Lie group. In particular TI is not equal to 1. The duality TI swaps the spin up particles with the spin down particles. Moreover, it depends on the basis chosen for the spinors, which means that in the standard interpretation the “direction” of spin is always the z direction. However, this “direction” can only be interpreted as a direction in spacetime once bases have been chosen for both Weyl spinors.

This opens up a huge can of worms, because if we choose bases independently then there is a symmetry group SL(2,C)_L x SL(2,C)_R acting to change bases, at which point we need to go back to treating the left-handed Weyl spinor as a representation of the left-handed SL(2,C)_L. BUT, this procedure produces a well-defined “direction of spin” for every particle, which is known to be unphysical. The experimental properties of entangled particles prove beyond a shadow of doubt that elementary particles do not have a well-defined “direction of spin”: this is the kind of (continuous) hidden variable that is disallowed by Bell’s theorem, the Free Will Theorem, and numerous related results. In other words, using the group SL(2,C)_L x SL(2,C)_R is disallowed by the fundamental properties of quantum mechanics. This adds a third reason to the two already given, as to why we must not under any circumstances treat left-handed representations as representations of left-handed groups.

So, let us return to the discussion of what Woit means by L and R. I think he means that they are “opposite”, because this is the only way he can treat the L group(s) as internal symmetries, and the R group(s) as spacetime symmetries. The Coleman-Mandula theorem is true in this interpretation, because left-multiplication commutes with right-multiplication: this is the associative law of matrix multiplication. Then he restricts the spacetime symmetries from SL(2,C)_R to SU(2)_R, which means restricting from SO(3,1) to SO(3) space rotations. But then I lose the thread of his argument when he uses SU(2)_L x SU(2)_R to define SO(4) acting on a “Euclidean spacetime”, because SU(2)_R then acts as SO(3) on MInkowski spacetime, fixing a time direction, but as SU(2)_R on Euclidean spacetime, fixing no directions at all. This doesn’t make any sense to me.

What does make sense to me is to embed the Lorentz group SL(2,C)_R in the 2×2 matrix algebra M_2(C), acting on itself by right-multiplication, and embed the corresponding internal symmetries in M_2(C) acting on itself by left-multiplication. Compactness of the gauge groups implies that the internal symmetry group here is U(2)_L (not only SU(2)_L), so that altogether we get U(2)_L x SL(2,C)_R as the complete symmetry group for the electro-weak sector. Surely this is what he wants to do? So why doesn’t he do this? I could try asking him, but he will ignore me, so I don’t think I will bother.

After that, he wants to discretise the internal symmetries, which he admits he doesn’t know how to do. I do it by taking a particular finite subgroup of order 24 in U(2)_L. He doesn’t appear to be interested in listening to this idea, even though it solves his problem. There are two different things you can do at this point: (1) tensor the natural representation of U(2) with its complex conjugate, to get a complex non-relativistic spacetime, from which the complex structure cancels out, to reveal a classical space+time; (2) tensor the natural representation of U(2) with itself, to get a complex non-relativistic “spacetime”, from which the complex structure on “time” does not cancel out, to reveal the three-generation structure of matter. If you want more, you can read my 50-page paper https://arxiv.org/abs/2102.02817 where I give lots of details.