Show that the given Green Function is the propagator of a certain Lagrangian

[Markus Kahn]

Homework Statement

The Lagrangian density for the electromagnetic potential with gauge-fixing term reads
$$\mathcal{L}=-\frac{1}{4} F_{\mu \nu}(x) F^{\mu \nu}(x)-\frac{1}{2} \xi^{-1}\left(\partial_{\mu} A^{\mu}(x)\right)^{2}.$$

Show that the photon propagator (Green function) with arbitrary gauge parameter $\xi$ is
given
$$G_{\mu \nu}^{\mathrm{V}}(x-y)=\int \frac{\mathrm{d} p^{4}}{(2 \pi)^{4}} \frac{\mathrm{e}^{i p(x-y)}}{p^{2}}\left(\eta_{\mu \nu}-(1-\xi) \frac{p_{\mu} p_{\nu}}{p^{2}}\right).$$
Note: we will not care about on-shell contributions to the propagator.

Relevant Equations

All given above

My fundamental issue with this exercise is that I don't really know what it means to "show that X is a propagator".. Up until know I encountered only propagators of the from $\langle 0\vert [\phi(x),\phi(y)] \vert 0\rangle$, which in the end is a transition amplitude and can be interpreted as a probability.
Green Functions on the other hand I only know as mathematical objects for which we have $D G(x,y)=\delta(x-y)$, where $D$ is some kind of differential operator. But what exactly am I supposed to do now with the given "Propagator" $G_{\mu\nu}^V$ to show that it actually is one?

 

[Markus Kahn]

After some trying out I was able to obtain a partial result I think.

First derive the EoM for the given Lagrangian, which results in
$$\partial^2 A^\rho -\partial^\rho\partial^\lambda A_\lambda + \xi^{-1}\partial^\rho\partial^\lambda A_\lambda=0.$$
Now we can apply the following trick to get the EoM into a nicer form for our purposes:
$$D^{\rho\lambda}A_\lambda\equiv(\partial^2 \eta^{\rho\lambda} -\partial^\rho\partial^\lambda(1-\xi^{-1})) A_\lambda=0.$$

My claim now would be that we have $D^{\rho\lambda}G^V_{\mu\nu}(x-y)=\delta^\rho_\mu\delta^\lambda_\nu\delta(x-y).$ I can show that we have $D^{\mu\nu}G^V_{\mu\nu}(x-y)=\delta(x-y)$, but for $\mu,\nu\neq\rho,\lambda$ respectively I just can't show that we have $D^{\rho\lambda}G^V_{\mu\nu}(x-y)=0$. So is my claim wrong, or can it be shown? (and obviously, does what I have done here even make sense in the context of the exercise?)