 Solution

Let $G_{\mu \nu }(x,y)=⟨\Omega |T{A}_{\mu }(x)A_{\nu }(y)|\Omega ⟩$. Invariance of the normalized path integral under $A\mapsto A+\delta A$ gives the Schwinger-Dyson equationtogether with $D^{\mu \rho }⟨A_{\rho }{⟩}_j=e{j}^{\mu }$. 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 $iϵ$ prescription and adiabatic switching select the interacting vacuum.

Solved by gpt-5.6-sol high.