Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 49 / 4 / d / Solution

The covariant Landau gauge is the zero-parameter limit $\xi \to 0$ of the covariant gauge family. It imposes the Lorenz gauge condition ${\partial }_{\mu }{A}^{\mu }=0$ sharply. The Landau gauge photon propagator is the transverse part of the inverse from solution:Away from the pole, $k^{\mu }{D}_{\mu \nu }^{\mathrm{L}}=0$, matching the divergence-free condition. This is transversality to the four-momentum, rather than a projector onto only the two physical spatial photon polarizations. The Lorenz gauge equation alone still admits residual gauge transformations with $\square \chi =0$; boundary conditions and the pole prescription complete the propagator choice. The parameter limit specifies the Landau propagator without trying to invert the ungauged Maxwell operator on its longitudinal null space.