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The point charge concept is clearly a very useful mathematical fiction, but it is also problematic from the point of view of "physical intuition". Even a layperson would feel that an explanation is required as to why a point charge is not split asunder by self-repulsion. A friend of mine who knows far more physics than me said that Maxwell's equations would require a point electric charge to rapidly attenuate (on a time-scale of something like $10^{-18}$ sec!). She refused to be drawn into discussing the following question, however:

How did classical physicists soothe their anxieties on this matter, and is the conundrum really dissipated, or merely sidelined, in quantum theories of elementary particles?

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Yes, they did, and the problem persists in quantum field theory to the extent that electrons can be called "point charges" (they are neither waves nor particles despite common terminology, and technically each one is smeared all over the universe). The classical electron theory of Lorentz-Abraham had multiple issues, some related to electron's point like nature, like infinite self-energy and self-stress, some not, like pre-acceleration (charge 'feels' application of force from the future) and internal instability. Lorentz tried to 'dissolve' electron in ether reducing it to an artifact of electromagnetic field, Abraham attempted to build a theory where electron was a small sphere of finite radius in Principles of the Dynamics of the Electron (1902), and Poincaré suggested some attractive non-electromagnetic force to hold it together.

Here is from Rohrlich's lecture:

"... when the radius shrinks to zero, the electron mass becomes infinitely large... This is the famous problem of the electron self-energy. It exists in the classical theory of the electron as well as in the quantum theory. A satisfactory solution of it is not known. An at best temporary solution is provided by the renormalization procedure... It is therefore not surprising that many attempts were made to keep the electron of finite extension and carry along throughout the development of the theory an explicit electron structure."

And that's not all:

"Obviously, an accumulation of negative charge is not a stable configuration: Any finite charge distribution of only one sign would explode... A solution to it was proposed by Poincaré who showed that an attractive and consequently non-electromagnetic force can always be added to the theory so as to just balance the stresses and establish stability. It is a very ad hoc solution and not at all fitting a "fundamental" theory. On the other hand, in the point electron theory the electromagnetic self-stress is infinite and so must be the Poincaré's cohesive force."

Electrons in quantum field theory are still "point like" in the sense that they technically collapse into points when positions are measured. In perturbative expansions their paths are represented by lines in Feynman diagrams, which leads to divergences. Renormalization is roughly a consistent way of "canceling" infinities coming from all relevant diagrams. This depends on a "renormalization scale", and to fix it measured quantities, like electron mass, then have to be substituted into theoretical formulas. With gravity the trick doesn't work, it is not renormalizable, which is a big reason why we still do not have a theory of quantum gravity.

String theory is an attempt to deal with this issue in a principled way. Points are replaced with segments or circles, and line diagrams with smooth two dimensional "fat graphs", whose contributions are expected to be finite. If string theory works out perhaps electrons will no longer be point like but have an internal stringy structure.

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  • $\begingroup$ Doesn't relativity (Dirac's equation) account for the observed mass of the electron? $\endgroup$ – hjhjhj57 May 8 '15 at 1:33
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    $\begingroup$ @Javier Not exactly. Dirac's ground state energy is still infinite, but he "absorbed" it into "redefined" mass of the electron. That was a prototype of renormalization. Under single particle interpretation Dirac's equation is ruled out because of negative energy levels, jumps to which are not observed. So he filled them all with extra particles introducing infinite charge so that Pauli principle precludes jumps. To do it properly his equation needed to be "second quantized" into multi-particle one, and when all is said and done we get a quantum field theory with modern style renormalization. $\endgroup$ – Conifold May 10 '15 at 18:24

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