I am prepping for my QFT2 exam tomorrow, and in one of the mock exams I found the following question (and I'm not quite sure how to go about this). Given the following Lagrangian:
$$ L = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} - (D_{\mu}\phi*)D^{\mu}\phi - m^2\phi\phi* $$
where
$$F_{\mu\nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}$$ and $D$ the covariant derivative:
$$D_{\mu}\phi = (\partial_{\mu} - ieA_{\mu})\phi $$ The question being asked is the following: Consider a plane-wave solution to the vacuum Maxwell equations of motion with a transverse polarization vector $\epsilon_{\mu}^{(\lambda)}(p)$:
$$A_{\mu}(x) = \epsilon_{\mu}^{(\lambda)}(p)e^{ipx}$$
Show that you can obtain another plane-wave solution with a polarization vector:
$$\epsilon_{\mu}^{(\lambda)}(p) + f(p)p_{\mu}$$
by performing a gauge transformation $A_{\mu}(x) \rightarrow A_{\mu}(x) + \partial_{\mu}\alpha(x)$ with a suitably chosen $\alpha(x)$.
I started by plugging in the given polarization vector in terms of A in the equation of motion to see if this indeed results in a solution, which indeed it does. However, I am not quite sure on how to proceed. Do I try and plug in the new plane wave solution into the equation of motion and see if this is also a solution? Or do I just show that the Lagrangian is gauge invariant and that the $f(p)p$ term has to coincide with the partial alpha term? Any help is greatly appreciated!
