November 2008

Spinors are somewhat subtle objects in field theory. They are our mathematical representation of fermions, which are spin-1/2 objects, and hence have the unintuitive property that a 2\pi rotation does not return them to their initial state, but a 4\pi relation does. (For a classical analogue, see Bolker’s spinor spanner.) Any quantum field theory text will teach how to manipulate spinors… but it’s not always made clear where spinors come from in the first place.

Here I’d like to say a few introductory words on the spin representation. I’ll assume a background in representations of Lie groups but will try to be very qualitative. For a proper introduction, see Weinberg Vol. I section 2.7.

Even before learning group theory physics students have an intuition for vector and tensor representations of the Lorentz algebra, SO(3,1). These are just the usual objects with indices in special and general relativity. These correspond to the usual fundamental and tensor reps that one constructs for a general Lie algebra. Classically, those are all the reps that we would expect nature could choose from.

But alas, our universe is not filled with only vectors and scalars. We also observe fermions, which are not spin-1 or spin-0, but rather spin-1/2. The spin-1/2 representation is inherently quantum in origin (and this is the part that I think is really neat).

In quantum mechanics an object’s state is given by its wavefunction, \psi(x,t). This is a complex number that can be decomposed into an magnitude and a phase. Physical observables, however, are given only by the magnitude of the wavefunction and are independent of the phase. Relative phases can, of course, produce physical effects; but we’re focusing on one-particle states.

This independence on the phase allows us to relax our restrictions on the representation of a group on quantum states. Usually we require that elements of a lie group/algebra g_1,g_2 are represented by matrices U(g_1), U(g_2) with the property (by the definition of a representation) that

U(g_1) U(g_2)=U(g_1g_2).

In quantum mechanics, however, we have the freedom to allow the product of representations to introduce a phase. That is to say, acting on a wavefunction \psi, our representation permits a \theta such that

U(g_1) U(g_2) \psi = U(g_1g_2)e^{i\theta(g_1,g_2)}\psi.

These representations “up to a phase” are called projective representations. Neat. But so what?

It turns out that it’s actually rather difficult to construct projective representations of a group/algebra. In fact, most groups don’t even permit projective representations — attempts to write a projective representation can be rewritten in terms of `normal’ representations.

One sufficient condition for a group to furnish a projective rep is that the group is not simply connected. We’ll leave it at this with no further proof, but it is rather cute that the quantum properties of a group’s representation can depend on its topology.

The point is that the Lorentz group is not simply connected, and hence it permits projective representations. This projective representation corresponds to the spinor rep. One can get a flavor of this by noting that the Loretnz group is doubly connected. This is the source of the rotation-by-4\pi property of spinors.

To complete the story, we note that it also turns out that instead of working with projective representations of a group, one can equivalently work with regular representations of the universal covering of that group. Practically, this means that instead of working with the Lorentz group, SO(3,1) = SL(2,C)/Z_2 we work with the simply connected group SL(2,C). The fundamental representations are the very Weyl spinors that we know and love.

Further Reading

For a more rigorous treatment of the above discussion see Weinberg, Vol I section 2.7. There are many more interesting things to learn about spinors. Spinors in higher dimensions are treated in Polchinski, Vol II appendix B. There’s a very nice relation between the Clifford algebra in arbitrary dimension and differential forms. See appendix A of hep-th/0510176 and the references therein. A proper mathematical treatment require deciphering the statement “spinor representations are the square root of a principle fiber bundle,” I suppose a good place to start is Nakahara, but probably the place to go is Kobayashi and Nomizu. [This is something I’d like to learn more about… maybe I could toss around the idea of an informal reading group with some of the students next semester.]

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