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Assuming the electron to be a classical point particle, if one calculates the self-energy one finds $$U=-\frac{e^2}{8\pi\epsilon_0r}$$ which diverges as $r\rightarrow 0$. Therefore, the measured mass of the electron should be $$m_{0e}+U/c^2=m_e.$$

Now, $m_{0e}$ is the mass of the electron in absence of its electric field (which is therefore unobservable because its electric field cannot be switched off) and $m_e$ is the measured electron mass.

Why is it a problem that in classical electrodynamics, the self-energy of a point electron diverge? The divergence in $U$ may be absorbed in $m_{0e}$ which is unobservable as often done in renormalization.

This is presented as a problem in textbooks because often one equates $m_ec^2=U$, and forget about $m_{e0}$. But I don't see a reason why $m_{e0}$ should be neglected. See page 28 here.

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  • $\begingroup$ Divergences are always a problem, whether we can reabsorb them into unobservable parameters or not. Just because we can hide the divergence into a counter-term doesn't mean that there is not a problem. In classical mechanics, and in quantum mechanics, a divergence always make the theory ill-defined. (note that "rigorous" treatments of QFT always study the regularised theory, say, in dimensional regulatisation, in which case there are no divergences at all, and the theory is "well" defined, at least in a perturbative sense). $\endgroup$ – AccidentalFourierTransform Dec 25 '16 at 19:43
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    $\begingroup$ @AccidentalFourierTransform Assuming a cut-off in length (by assuming new physics exists at short distance), one can take a similar regularization scheme here, too. Right? $\endgroup$ – SRS Dec 25 '16 at 20:08
  • $\begingroup$ @SRS Yes but that kind of regularization would not be Lorentz (ie Relativistic) invariant. $\endgroup$ – user139175 Dec 27 '16 at 9:03
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This is indeed not a fundamental problem, because classical electrodynamics is only an approximate description of electrodynamics valid in some appropriate scaling limit. If one then attempts to formulate such an approximate theory completely divorced from the real fundamental laws of Nature, then one typically encounters problems when naively taking limits to scales where the theory should not be valid.

A non-trivial problem in classical electrodynamics where renormalization is needed is the rigorous treatment of electromagnetic radiation. While one can calculate the emitted radiation of accelerated charges without problem, the backreaction of the emitted radiation on the charges is typically ignored. E.g. the OP's book contains a calculation of the emitted radiation by pulsars, it mentions that these pulsars slow down due to conservation of energy, but it's not pointed out that this implies that the Lorentz force equation as mentioned in the book must therefore be wrong as it doesn't contain any terms that could exert a torque on a magnet that is freely rotating in empty space.

What needs to be included, therefore, is the force exerted on a charge due to its own electromagnetic field, but this is divergent for point charges. A rigorous treatment of the self-force was only given recently in this article.

In general, analogues of the renormalization group methods as used in QFT will have to be invoked in all physics disciplines where the degrees of freedom reside in arbitrarily small length scales (e.g. fluid dynamics), as soon as one is rigorous enough. The fact that we typically don't see such methods applied in textbooks is because they typically don't treat the subject in a rigorous way.

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