# Projective representations of the spin group

To define spinors in QM, we consider the projective representations of SO(n) that lift to linear representations of the double cover Spin(n). Why don't we consider projective representations of Spin?

## Answers and Replies

I answered my own question (I think).

Modulo redefinition of the phases of operators, projective representations are in correspondence with central extensions (as both are built out of nontrivial algebraic 2-cocycles). For n>2, Spin(n) is a universal cover, so the phase of any of its projective representations is a coboundary, which is to say that operators can be redefined to make the representation linear.

tl;dr: There are no nontrivial projective representations of Spin.

What happens in n=1,2? I haven't thought about it.

Ah, these must be anyons.

dextercioby