Gauge-fixed Maxwell kinetic operator ($K^{\mu \nu }=-k^2{\eta }^{\mu \nu }+(1-{\xi }^{-1})k^{\mu }{k}^{\nu }$)

The Fourier-space quadratic Maxwell kernel after adding $-({\partial }_{\mu }{A}^{\mu }{)}^2/(2\xi )$. For nonzero $\xi$ and non-null four-momentum, its transverse and longitudinal momentum projectors give $K=-k^2(P_T+{\xi }^{-1}{P}_L)$ and its inverse is $-(P_T+\xi P_L)/k^2$. The inversion of the gauge-fixed Maxwell kinetic operator supplies the photon propagator with its vacuum boundary prescription.