I am trying to understand representations of the Poincare/Lorentz group, and in particular spinors, in 2+1 dimensions. I know some of the math, but I'm not sure about the physical interpretation of it all. For example, I know that Dirac spinors have two complex components in 2+1 D instead of four in 3+1 D. In 3+1 D the Dirac equation "projects out" two of the four components, and the remaining two complex ("on-shell") degrees of freedom corresponds (I think) to spin up and down. By the same reasoning I think spinors in 2+1 D have one on-shell degree of freedom. Does this mean that there is no analogue of spin up/down for a Dirac spinor in 2+1 D?
Going further, is there no such thing as spin at all in 2+1 D? If so, how do you classify one-particle states? And is there still something like the spin-statistics theorem?
I think the answers are related to the fact that the little group of the Poincare group in the massive case in 2+1 D is SO(2) instead of SO(3) but I'm not sure how to make this precise. (I'm not even sure how to make this question as precise as I'd like...) Any clarifications or sources to discussions in the literature are appreciated!