This is a homework problem that I am confused about because I thought I knew how to solve the problem, but I'm not getting the result I should. I'll simply write the problem verbatim:

"Consider QED with gauge fixing $\partial _\mu A^\mu=0$ and without dropping the Fadeev-Popov ghost fields. Thus the gauge fixed Lagrangian is
$$
\mathcal{L}=-\frac{1}{4}F_{\mu \nu}F^{\mu \nu}+\frac{1}{2}(\partial ^\mu A_\mu )^2+\overline{\psi}(i\gamma ^\mu D_\mu -m)\psi+\overline{c}(-\partial ^\mu \partial _\mu)c
$$
Verify that the Lagrangian is invaraint under the BRST transformation
$$
\delta A_\mu=\epsilon \partial _\mu c,\delta \psi =0,\delta c=0,\delta \overline{c}=\epsilon \partial _\mu A^\mu"
$$
Two questions. First of all, if we are working in the gauge where $\partial _\mu A^\mu =0$, then why has he left this term in the Lagrangian? Does he mean something different by "gauge fixing" than I am thinking? Second of all, I am not getting this Lagrangian to be invariant under the transformation listed. I find that the transformation of the gauge field gives an extra term of the form
$$
-\epsilon e\overline{\psi}(\gamma ^\mu \partial _\mu c)\psi
$$
that doesn't cancel. This term arises from the $A_\mu$ contained in the covariant derivative. Did I screw up the computation somewhere? What's going on?

This post imported from StackExchange Physics at 2014-04-13 14:07 (UCT), posted by SE-user Jonathan Gleason