Definition of spinor

Question

In most physics books, spinors are sometimes defined in an adhoc way. From my understanding, a spinor transforms under the spin group. But I'd like to know the precise detail of the definition. Can someone point me in the right direction to understand this topic in the context of mathematical details of spinors used in physics?

Answer 1

Using clifford algebra, rotations of vectors can be represented using double-sided multiplication by a spinor*. That is, let $\psi$ be a spinor and $a$ be a vector, then $a'$ is the rotated vector, given by

$$a' = \psi a \psi^{-1}$$

The spinors in any given space can be constructed by a pair of sequential reflections. And you can see that if you apply another rotation that is associated with a spinor $\phi$, then we have

$$a'' = \phi \psi a \psi^{-1} \phi^{-1}$$

So while $a$ continues to transform in a double-sided (bilinear) fashion, $\psi \to \psi' = \phi \psi$, transforming in a single-sided way.

Odds are you're already familiar with spinors on a practical level. The clifford algebra for a 2d real vector space has spinors that are isomorphic to complex numbers. For 3d, the spinors are isomorphic to quaternions. In applications to physics, this gives an interpretation of the Pauli algebra as merely that of rotations in 3d space--in this sense, the Pauli matrices represent basis vectors, and their products help form spinors. The same can be said of the Dirac algebra with respect to spacetime.

(*If you like, I can elaborate on the construction with proves this. It involves taking a sequence of two reflections, so I assure you this double-sided stuff for rotations doesn't come out of thin air.)