 Photon polarization sum

The sum over the two physical transverse photon polarization vectors is not the full tensor $-{\eta }_{\mu \nu }$. It differs by gauge-dependent terms proportional to the photon four-momentum. When the amplitude satisfies the Ward identity, those terms vanish on contraction and one may use $-{\eta }_{\mu \nu }$ as a replacement inside the squared amplitude. This replacement must not be mistaken for a free-standing equality of physical polarization tensors.