Massive Photon in $2+1$ Dimensions

Question

I have been reading a little on the Maxwell-Chern-Simons Lagrangian, an attemp to create a massive photon in $2+1$ dimensions. \begin{align}\mathcal{L} &= -\frac14 F_{\mu\nu}F^{\mu\nu} + \frac{m}{4} \epsilon_{\sigma\mu\nu}F^{\mu\nu}A^{\sigma}\\ &= -\frac14 F_{\mu\nu}F^{\mu\nu} + \frac{m}{2} \epsilon_{\sigma\mu\nu} \partial^{~\mu}A^{\nu}A^{\sigma} \end{align} The action but not the Lagrangian is gauge invariant (changes by a total derivative). I derived the equations of motion, \begin{align} \partial_{\mu}F^{\nu\mu}-m\epsilon^{\nu\mu\sigma}\partial_{\mu}A_{\sigma}=0 \end{align} If we combine this with the Bianchi identity and define a new vector $~f^{\mu} = \frac12\epsilon^{\mu\nu\sigma}F_{\nu\sigma}$, it takes a bit of algebra to show, $$ (\partial^{~\mu}\partial_{\mu}+m^2)f^{\nu} = 0$$

I'm really interested in the number of polarizations this "photon" can have. The little group should be $SO(2)$ right? So a $J_z$ should generate this? So one polarization? I wonder if there is a nice argument for this? Maybe something similar to how it's done in the normal case from using the Lorenz gauge? Note that the Lorenz gauge condition holds trivially from the Bianchi identity.

Answer 1

This system has a primary constraint: $$\Pi_0=0$$ and a secondary constraint: $$-\partial_i \Pi_i+m\epsilon_{0ij} \partial^i A^j=0$$ The Poisson bracket of these constraints is zero, so they are first class constraints. Using the formula for counting the degrees of freedom of a constrained system:$$DOF=\frac{N-2M-S}{2}$$

see, for example:

Counting degrees of freedom in presence of constraints

Counting number of degrees of freedom in constrained system

where $N$ is the dimension of the phase space, $M$ is the number of first class constraints and $S$ is the number of second class constraints, the degree of freedom is $\frac{6-2\times2-0}{2}=1$.

The little group for a massive particle is indeed $SO(2)$,but the number of independent polarizations is not equal to the number of generators of the little group; it equals to the dimension of the irreducible representation of the little group that the particle carries.