In Feynman's paper "Mathematical Formulation of the Quantum Theory of Electromagnetic Interaction" he writes that the semiclassical Coulomb interaction can be written by eliminating longitudinal photons. I believe this was done by Fermi and Bethe in 1932, however there paper was written in German so is hard to read. Does anyone have an explanation or reference for how to eliminate longitudinal photons (in the Coulomb gauge I presume) in QED?
The noncovariant quantization of electrodynamics is treated in great deal in Advanced Quantum Mechanics by Sakurai. It actually begins with a nonlocal theory containing an explicit Coulomb interaction between the sources, as well as transverse excitations of the electromagnetic field, later demonstrating the equivalence (at least at the lowest orders in perturbation theory) to a covariant theory including fictitious longitudinal and timelike photons.