Is spacetime left-handed?

Comparing my model with Peter Woit’s, we basically agree that SU(2)_L acts on quaternions H (that I called Hamilton spinors last time), and that SL(2,C)_R acts on a Weyl spinor W and its complex conjugate W*, say. The standard way to construct spacetime is as the complex tensor product W x W*, but strictly speaking this is a complex spacetime, and not a real spacetime. Woit worries about this, but I don’t: the reason is that it is mathematically straightforward to factor out the complex numbers, and just throw them away. The complex numbers don’t do anything here, and don’t interact with any of the symmetry groups, so they are superfluous.

This standard spacetime is “right-handed” in Woit’s sense, since it is a representation of the “right-handed” Lorentz group SL(2,C)_R. In my model, however, this representation is equivalent to the quaternionic tensor product H x H, which is a representation of SU(2)_L. So shouldn’t we say that spacetime is “left-handed”? Well, not really. It is both and it is neither. In my model, both ways of factorising spacetime as a product of spinors make sense, so you can treat spacetime as left-handed if you want to, or you can treat it as right-handed if you want to.

What you cannot do is treat it as a product of a “left-handed” (Hamilton) spinor with a “right-handed” (Weyl) spinor (in Woit’s sense, which is different from the standard sense, of course). What Woit does is restrict from SL(2,C)_R to SU(2)_R in order to force a quaternionic structure onto the Weyl spinor. He then takes a quaternionic tensor product H x W to define spacetime. This does not work, for several reasons, which are probably all the same fundamental reason. First of all, the restriction to SU(2)_R makes his model non-relativistic, and involves a choice of splitting of spacetime into space and time. This means that he destroys the natural complex structure on H x W, which is an 8-dimensional complex representation, required for the adjoint representation of SU(3) that describes the strong force.

Second, and perhaps even more important, the imposition of a quaternion structure on W mixes the spin-direction (defined by W) with the generation (defined by H). Well, hang on a minute: is this a bug or a feature? Surely we need this feature in order to explain neutrino oscillations? As a neutrino travels through spacetime (defined by W x W*) it changes generation (defined by H x H), while keeping W x H (representing the strong force) unchanged. Yes, I believe this is a useful feature. But Woit interprets W x H as Euclidean spacetime, which doesn’t really work.

The reason this doesn’t work is because W x H, calculated by choosing a complex subalgebra of H, rather than imposing a quaternion structure on W, does not have a scalar as the time coordinate, but an amplitude. Hence it is impossible to factor out the complex numbers to get a real spacetime this way. Again, this is a feature, not a bug, and it is only the interpretation as spacetime that is wrong here. It is “spacetime” (or its dual) multiplied by an amplitude. The amplitude, as I have explained several times, is used to select the electron generation. But it also acts on space to select a direction of spin, and a direction of momentum. That is why, when a cobalt 60 nucleus ejects an electron (first generation), the direction of spin of the nucleus and the direction of momentum of the electron are correlated (Wu experiment, 1957).

At least, that is how my model explains the Wu experiment. Have you got a better explanation? The critical factor is to distinguish the cases when the product of two spinors contains an amplitude, from the cases when it contains a true scalar. In my model, H x H and W x W* contain scalars, so can be used to define spacetime, while W x W, W* x W*, H x W and H x W* contain amplitudes, so can be used to define the three generations. In the usual interpretation of SU(2)_L as internal symmetries and SL(2,C)_R as external symmetries, we may distinguish between an internal spacetime H x H and an external spacetime W x W*, but my model implies that these concepts are interchangeable.

The crucial point is to distinguish correctly between H and W, because H x H is a real spacetime, but W x W is not. The Standard Model makes the distinction here between spin (W and W*) and weak isospin (H), and further splits the spin into left-handed (which we may take to be W) and right-handed (that is then W*). Woit makes the same distinction in different words. Both attempt to unify electromagnetism (described by tensors of W and W*) with the weak interaction (described by H) in terms of H x W. But both destroy the fundamental structure of H x W: the Standard Model destroys the quaternion structure, and hence destroys the three generations; and Woit destroys the complex structure, and hence destroys Minkowski spacetime. If you want both, my model is the only game in town. And in my game you can win the jackpot, because H x W is also the strong force.