Landau gauge photon propagator ($D_{\mu \nu }^L=-i({\eta }_{\mu \nu }-k_{\mu }{k}_{\nu }/k^2)/(k^2+i0)$)

The covariant Landau gauge propagator is the transverse limit of the inversion of the gauge-fixed Maxwell kinetic operator. Contracting with $k^{\mu }$ gives zero away from the pole. The Lorenz gauge condition still has residual transformations satisfying $\square \chi =0$, so boundary conditions and the pole prescription complete the choice.