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.