Photon polarization completeness relation (${\sum }_r{ϵ}_i^{(r)}{ϵ}_j^{(r)\ast }={\delta }_{ij}-k_i{k}_j/|\mathbf{k}{|}^2$)

Two orthonormal transverse polarization vectors span the plane perpendicular to a nonzero momentum. Summing their dyadic products gives the spatial transverse projector. It has trace and rank two and acts as the identity on physical photon polarizations.