= Solution
Let $G_{\mu\nu}(x,y)=\langle\Omega|T A_\mu(x)A_\nu(y)|\Omega\rangle$. Invariance of the normalized path integral under $A\mapsto A+\delta A$ gives the <Schwinger-Dyson equation>
$$
D_x^{\mu\rho}G_{\rho\nu}(x,y)
=i\delta^\mu_\nu\delta^{(4)}(x-y)
+e j^\mu(x)\langle A_\nu(y)\rangle_j,
$$
together with $D^{\mu\rho}\langle A_\rho\rangle_j=e j^\mu$. This assumes the functional measure is translation invariant, boundary terms in field space vanish, and vacuum bubbles are removed by normalization. The current is a fixed c-number and conserved; conservation removes dependence on the longitudinal, gauge-parameter part of the <photon propagator>. An $i\epsilon$ prescription and adiabatic switching select the interacting vacuum.
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