Chern-Simons and massive A^{\mu}

[IRobot]

Hi, I am struggling with a problem in field theory:
We are looking at a Chern-Simons Lagrangian describing a massive A field:
$L = -\frac{1}{4} F^{\mu\nu}F_{\mu\nu}+\frac{m}{4}\epsilon^{\mu\nu \rho}F_{\mu\nu}A_{\rho}$
I find those field equations:
$\partial_{\mu}F^{\mu\lambda}=-\frac{m}{2} \epsilon^{\lambda\mu\nu}F_{\mu\nu}$ and now I need to show that F satisfies the Klein-Gordon equation: $(\Box+m^2)F_{\mu\nu}=0$ using the EL equations, but after a time playing with both equations, I still can't prove that.

 

[fzero]

It's normal here to introduce the dual of the field strength

$$\tilde{F}^\lambda = \frac{1}{2} \epsilon^{\lambda\mu\nu}F_{\mu\nu}.$$

Then taking the divergence of your equation of motion above yields $\partial_\lambda \tilde{F}^\lambda = 0,$ while multiplying by an appropriate epsilon yields the KG equation for $\tilde{F}^\lambda$.

 

[IRobot]

It's normal here to introduce the dual of the field strength

$$\tilde{F}^\lambda = \frac{1}{2} \epsilon^{\lambda\mu\nu}F_{\mu\nu}.$$

Then taking the divergence of your equation of motion above yields $\partial_\lambda \tilde{F}^\lambda = 0,$ while multiplying by an appropriate epsilon yields the KG equation for $\tilde{F}^\lambda$.

Thanks a lot. I did manage to find the Klein Gordon equation for the dual, then it's just a matter of applying another LeviCivita to find it for the "regular" form. ;)