The problems of damping force or Abraham-Lorentz force should, according to wikipedia, be solved in quantum mechanics or its relativistic counterpart, the quantum field theory. My question is, since wikipedia does not link any article to support this claim and answer my following question, what exactly is the quantum mechanical approach and quantum field theoretical approach to this problem?

I always thought of the problems of Abraham-Lorentz approach as a problem of oversimplification - we are simplifying the problem of a particle interacting with the magnetic field created by currents of charged particles - de facto an interaction between many-body system an a singular particle - as an interaction between a particle and an universal, infinite and absolute field(a field without a source, or a field generated in an infinity) which is clearly not what happens. In some sense, this is also how quantum mechanics works - there are no forces in quantum mechanics, only potentials(generated by interacting particles) and interactions.

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    $\begingroup$ As far as I know, the acausal solutions to the classical equations of motion are invisible to perturbation theory. Therefore, we cannot really analyse the problem with QED: we don't know how to solve the dynamics non-perturbatively, which means that we don't know if the quantum theory fixes the problems of the classical theory. $\endgroup$ – AccidentalFourierTransform Dec 30 '16 at 14:36

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