I am somewhat used to simplified non-relativistic quantum mechanics (both canonically and grand canonically), describing a system by a Hamiltonian containing a kinetic part, an external potential as well as a (in my case mostly electrostatic) pair potential. Now when trying to get to work relativistically I take it one should be considering the coupled dynamics of the quantized electromagnetic field with the charged particles. Most of the stuff I have seen so far goes along the lines of quantizing an electromagnetic field neglecting any charged matter then describing charged matter, with static potentials and then introducing some coupling terms allowing the absorption of photons to create excited matter states and vise versa.
From my understanding these are just approximations and splitting electromagnetism up like that into one part described by photons and one part described by static interactions seems arbitrary.
Is there a way to describe interactions in a system of electrons entirely by coupling to a photon field with no electron-electron pair potential? If yes how would the Hamiltonian/Dirac-equation of such system look like? Is this even the right way to think about it?
