vanhees71 said:
... A "classical electron" can in some cases be treated as a classical point particle, but it's not structureless, because it has electric charge and a magnetic moment.
Classical point electron does not have magnetic moment, I believe. Magnetic moment was introduced for extended models of electron (charged rotating sphere) and in quantum theory.
This radiation in turn acts back on the electrons accelerated motion, and at this point all hell breaks loose. In my opinion, there's not a satisfactory exact solution to this problem but only approximate ones with the Landau-Lifshitz modification of the old Abraham-Lorentz treatment, avoiding artificial effects like self-acceleration.
The only satisfactory solution I know is modifying the premise; radiation of point particle does not act back on the point particle. Assuming it does has lead to contradictions nobody was able to resolve.
The reason for this is that the point-particle "idealization" cannot be made rigorous in classical physics but introduces problems.
The idea of limiting charged sphere to point introduces problems. If we begin with point particles right from the start and do not allow self-interaction in the first place, none of those problems arise. Self-interaction is not necessary to explain known experiments and it only brings problems.
Consistent theories of charged point particles were described many times in the past. The first case I know of is the paper by Frenkel:
J. Frenkel, Zur Elektrodynamik punktfoermiger Elektronen, Zeits. f. Phys., 32, (1925), p. 518-534.
http://dx.doi.org/10.1007/BF01331692
In English, this article also explains it concisely:
R. C. Stabler, A Possible Modification of Classical Electrodynamics, Physics Let- ters, 8, 3, (1964), p. 185-187. http://dx.doi.org/10.1016/S0031-9163(64)91989-4
Quantum field theory is in somewhat better shape than the classical point-particle model, because with ("soft-photon resummed" renormalized) perturbation theory you can at least define a clear scheme to find approximate solutions. As far as I know also here we still do not have a fully selfconsistent non-perturbative solution nor a mathematical proof that one exists.
Could you please give some references which discuss these issues in quantum field theory?