Feynman-weight de Donder gauge fixing

Strictly, ${\mathsf{P}}_{\mathrm{gr}}$ is a constant invertible bundle factor times a normally hyperbolic operator. Its Green theory follows from that of $\square$; its principal symbol need not be called the scalar identity symbol.

Gauge fixing specifies the free operator. It does not impose $C[l]=0$ on every configuration in its unrestricted covariant free-field carrier. Indeed a retarded source with nonzero divergence generally gives $C[{{\Delta }_{\mathrm{gr}}}^{\mathrm{R}}A]\ne 0$.