In classical electromagnetism we have two space vector fields E and H for the electric and magnetic field and a four vector made with spatial charge density and current [ρ, J] with determines EM field together with external sources and whose dynamic is affected by those E and H fields according to the Lorentz force. Classical electromagnetism assumes that each point of the charge density interacts with fields E and H and moves independently on how the rest of the charge moves (except, of course because the dynamics of the rest of the charge will affect the future EM field).
However, in nature there are not infinitesimal charges that move freely, but discrete particles such as electrons and protons. In quantum mechanics at first order a particle EM field does not interact on itself. An example of this is the Hydrogen's electron Hamiltonian, the potential we see is the one created by the proton and there is no contribution from the electron itself. At higher levels, we may find interactions due to the vacuum but they are much weaker, for example, the Lamb Shift in the Hydrogen atom (4*10^-6 eV) is much smaller that the difference between radial levels 1 and 2 (10.2 ev), so I consider a good approach to say that in general, the EM field generated by discrete a particle does not affect itself.
So, can we assume that a particle only interacts with the field created by other particles? In that case (still in the classic non-quantic theory), we would have N charged particles that require N Hilbert spaces to define the charge and current distribution for each of them. Each particle “sees” a different EM field because it has to exclude its own contribution to the EM field, so we would have N spaces with EM fields plus an overall field.
As far as I can see, we can have this way a theory with discrete particles that respects Maxwell equations, however, the field energy density should be reviewed because in absence of other charges, there is no work done to “bring together” the density of charge needed to build a particle since the own field does not affect particle. Explained from another point of view, we can have a charged sphere which generates an electric field, which should have energy, but we don’t have spent any energy to bring together its charge.
Let the formula for the EM field density of energy be:
u = (ε/2)E2 + (μ/2)H2
Therefore, some correction must be done to remove the term of the energy needed to build isolated particles and once this is done, I consider this approach would be consistent.