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. ;)