> 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.
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.