= Coulomb-gauge propagator between conserved currents
{c}
{title2=$J^\mu D_{\mu\nu}^{\rm C}K^\nu=-iJ\cdot K/(q^2+i0)$}
For $q\cdot J=q\cdot K=0$, the instantaneous term in the <Coulomb-gauge photon propagator> combines with its spatial longitudinal subtraction:
$$
J^\mu D_{\mu\nu}^{\rm C}(q)K^\nu=-\frac{i}{q^2+i0}J\cdot K.
$$
The identity follows from $\mathbf q\cdot\mathbf J=q^0J^0$ 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.
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