[hansen]

> Given the trajectories of charges it will tell you what the electromagnetic field will be, and given the electromagnetic field it will tell you how charges will move. Unfortunately, these two parts of the theory seem to be incompatible, and the theory will not tell you how fields + charges will evolve in time.

The Problem is not coupling charged matter fields to the EM field. You get a well defined set of coupled and (now) non-linear PDEs. The problems arise when you try to model point charges. Then the theory is plagued by infinities that originate in the infinite charge and current densities.

The infinities in QED are actually far less problematic than those in the classic theory. They just seem to be more problematic because you cant (approximately) ignore the backreaction of EM and matter fields.

[weedwarrior]

what you expounding upon is [1] of the indicators that we dont completely understand physics at the "point charge" scale, and very possible there is no such thing as a point charge, rather there is a centroid of field intensty/probability I.E. a wave function. point charges are likely an overly simplified view, and artefactual convienience of extrapolation.

[hansen]

The problems in the classical theory are easily understood. The charge and current densities of point particles are not smooth functions but distributions (think of the Dirac δ-“function”). If they act as the sources of the EM field the EM field itself becomes singular. Now if you try to solve the full Maxwell equations including the backreaction of matter & radiation fields you would have to multiply distributions which is ill defined.

There are similar problems in the quantum theory but the divergences are less severe and can be dealt with in a systematic way. Most physicist believe they will totally disappear in some more fundamental underlying theory. From a mathematicians point of view there is the hope that at least some QFTs are finite and the divergences are just an artifact of the construction & pertubation theory.