Consider a Chern-Simons Lagrangian $\mathscr{L}=\mathbf{e}^2-b^2+g\epsilon^{\mu \nu \lambda} a_\mu\partial _\nu a_\lambda$ in 2+1 dimensions, where the 'electromagnetic' fields are $e_i=\partial _0a_i-\partial _ia_0$ and $b=\partial _1a_2-\partial _2a_1$.
From the Lagrangian, we obtain the classical equations of motion for $a_\mu$ fields ($\mu=0,1,2$): $(\partial _0^2-\partial _1^2-\partial _2^2)a_\mu+\partial _\mu(\partial _1a_1+\partial _2a_2-\partial _0a_0)\pm g\epsilon^{\mu \nu \lambda}\partial _\nu a_\lambda=0$; or the 'Maxwell' equations for the 'electromagnetic' fields: $\partial _0b-\partial _1e_2+\partial _2e_1=0, \partial _1e_1+\partial _2e_2+gb=0, \partial _0e_1+\partial _2b+ge_2=0, \text{and } \partial _0e_2-\partial _1b-ge_1=0.$
It's easy to solve the four 'Maxwell' equations with a gapped dispersion relation $\omega ^2=\mathbf{k}^2+g^2$. My questions are:
(1) Is it possible to solve the equations of motion for $a_\mu$ fields (maybe under some gauge fixing) where we get the same dispersion relation $\omega ^2=\mathbf{k}^2+g^2$? As we know, when $g=0$, it's easy to solve the equations (under Lorenz gauge) to show that each $a_\mu$ field and each $e_i,b$ field fulfil the same wave-equation $(\partial _0^2-\partial _1^2-\partial _2^2)\phi=0$ with the same gapless dispersion relation $\omega ^2=\mathbf{k}^2$.
(2)If we further modify the Lagrangian to $\mathscr{L}=\mathbf{e}^2-b^2+g\epsilon^{\mu \nu \lambda} a_\mu\partial _\nu a_\lambda+m(a_0^2-a_1^2-a_2^2)$, $m\geqslant 0$, then the resulting equations of motion become $(\partial _0^2-\partial _1^2-\partial _2^2)a_\mu+\partial _\mu(\partial _1a_1+\partial _2a_2-\partial _0a_0)\pm g\epsilon^{\mu \nu \lambda}\partial _\nu a_\lambda\pm ma_\mu=0$, and it seems impossible to write down some 'Maxwell' equations which are ONLY in terms of the 'electromagnetic' fields. Thus, for this case, how to solve the equations for $a_\mu$ fields and get the dispersion relation?
(3)Following Q(2), do ONLY the gauge-invariant Lagrangians (up to a total derivative term) can produce the classical equations of motion containing ONLY the 'electromagnetic' fields?
Thank you very much.
