Coulomb-gauge photon propagator ($D_{00}=i/|\mathbf{q}{|}^2$)

For $q=(q^0,\mathbf{q})$ with $\mathbf{q}\ne 0$, the Coulomb gauge propagator hasThe temporal component is instantaneous; the spatial part projects onto transverse photon polarizations. Although these components are not individually Lorentz covariant, their contraction with conserved external currents equals the covariant answer. Zero-spatial-momentum limits should be taken after combining the current contractions.

For $q\cdot J=q\cdot K=0$, the instantaneous term in the Coulomb-gauge photon propagator combines with its spatial longitudinal subtraction:The identity follows from $\mathbf{q}\cdot \mathbf{J}=q^0{J}^0$ and $q^2=(q^0{)}^2-|\mathbf{q}{|}^2$. It is equality after contraction with conserved currents, not equality of the uncontracted propagator tensors. It is especially useful in showing that tree scattering amplitudes computed in Coulomb gauge are Lorentz invariant.