 Solution

The Maxwell quadratic action has a null longitudinal direction: its momentum-space kinetic operator annihilates $k_{\mu }$, reflecting gauge invariance under addition of a gradient to the potential. It cannot be inverted on all four vector components before gauge fixing.

Add the covariant gauge density ${\mathcal{L}}_{\mathrm{gf}}=-({\partial }_{\mu }{A}^{\mu }{)}^2/(2\xi )$. Away from the pole, introduce the transverse and longitudinal momentum projectorsWith one index raised they obey $P^{\mathrm{T}}+P^{\mathrm{L}}=I$, $(P^{\mathrm{T}}{)}^2=P^{\mathrm{T}}$, $(P^{\mathrm{L}}{)}^2=P^{\mathrm{L}}$ and $P^{\mathrm{T}}{P}^{\mathrm{L}}=0$. The gauge-fixed kinetic operator is $K=-k^2(P^{\mathrm{T}}+{\xi }^{-1}{P}^{\mathrm{L}})$. Its inverse times $i$ gives the photon propagatorThe pole factors are interpreted with the Feynman boundary prescription; the algebraic projector inversion is performed off the pole. The choice $\xi =1$ is Feynman gauge.

Changing $\xi$ changes only a longitudinal term proportional to $k_{\mu }{k}_{\nu }$. Between conserved currents it vanishes because $k_{\mu }{J}^{\mu }=0$. In full QED amplitudes, the Ward-Takahashi identity supplies the corresponding cancellations when all diagrams at an order are combined. Thus physical on-shell scattering is gauge-parameter independent, whereas off-shell Green functions and the propagator itself can depend on the gauge choice. For the charged scalar, derivative vertices and contact terms must likewise be included for this cancellation.