 Inversion of the gauge-fixed Maxwell kinetic operator

Gauge fixing removes the noninvertible longitudinal direction of the Maxwell quadratic action. In a gauge with parameter $\xi$, the Fourier kernel is $-k^2{\eta }^{\mu \nu }+(1-{\xi }^{-1})k^{\mu }{k}^{\nu }$. Multiplying its inverse by $i$ gives the photon propagator, with the Feynman i-epsilon prescription. At $\xi =1$ the answer is $-i{\eta }_{\mu \nu }/(k^2+i0)$.