Spinors in 2+1 dimensions

Question

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!

Answer 1

The spin-statistic theorem holds in all dimensions of spacetime greater than two. The proper definitions of "half-integer" or "integer" spin in general dimension is simply how the rotation operator of a full rotation, $R(2\pi)$ is represented - "integer spin" or "bosonic" representations will have it as the identity, while "half-integer" or "fermionic representations will have it as minus the identity. More formally, fermionic representations are those that are representations of the double cover of the Lorentz group that do not descend to representations of the actual Lorentz group. If you examine the proof of the spin-statistic theorem you will see that it is indeed the behaviour of the $2\pi$ rotation that is crucial, not that spin is "half-integer".

Talking about "spin up/down" in other than 3+1 dimensions is indeed difficult in general representations. We are used to talk about spin in 3+1 because the Lorentz algebra $\mathfrak{so}(1,3)$ has an accidental equivalence for its finite-dimensional representations to $\mathfrak{su}(2)\times\mathfrak{su}(2)$, i.e. two copies of the rotation algebra. The algebra of actual physical rotations embeds diagonally into these, and we label a finite-dimensional representation of the Lorentz algebra thus by two spins $(s_1,s_2)$ where the total physical spin is $s_1+s_2$. The Dirac spinor in 4D is $(0,1/2)\oplus(1/2,0)$, the sum of the left- and right-handed Weyl spinors.

However, the Dirac representation does have a notion of spin up/down - you can group the Clifford algebra of the gamma matrices into pairs of raising/lowering operators (with the last one remaining unpaired in odd dimensions and being akin to a parity operator), and the Dirac representation is the unique (or, in odd dimensions, one of the two unique) $2^{\lfloor d/2\rfloor}$-dimensional irreducible representation of this algebra, in which the basis states are labeled by "spins" $s_1,\dots,s_{\lfloor d/2\rfloor}\in\{\pm 1/2\}$ and the $i$-th apir of raising/lowering operators raises/lowers $s_i$. For more on constructing the Dirac representation in arbitrary dimensions see this question and this question.

Answer 2

For a physical interpretation, spinors are like vectors with a phase. Steane, in his An Introduction to Spinors, describes spinors as flag poles with little stiff flags attached to them. Otherwise spinors have a lot of similarities with vectors, having a direction and magnitude and not inherently quantum mechanical, and there is a spinor formulation of electrodynamics.