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In the context of classical physics,is there any renormalization method to avoid infinite energy of point charges?

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Related: physics.stackexchange.com/q/34437 –  Ben Crowell May 2 '13 at 1:04

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People spent a lot of time trying to do this kind of thing ca. 1910, i.e., after SR but before quantum mechanics. To make the electrostatic self-energy no greater than the observed mass of the electron, you have to create some kind of model of an electron as an extended object, with a size that's at least on the order of the classical electron radius.

You would kind of like this to be a little sphere that's perfectly rigid. However, there are fundamental relativistic problems with that.

For one thing, relativity doesn't allow a perfectly incompressible substance to exist, since then the speed of sound would be infinite, but we can't have speeds greater than c. There is a notion of a perfectly rigid object in relativity, called Born rigidity, in which the distances between nearby points remain constant as measured by a comoving observer. But Born rigidity can't be a passive property of a material; it can only be achieved if some external set of forces is planned in advance to maintain rigid motion when applied everywhere throughout the object.

A more subtle issue is that there is a theorem called the Herglotz-Noether theorem, which says that a Born-rigid object has fewer degrees of freedom then a nonrelativistic rigid object. (This was Fritz Noether, Emmy's brother.) If it's rotating, its center of mass has to move inertially. If its center of mass is moving noninertially, it can't be rotating. A rotating electron would have infinite inertia! Of course they didn't know about electron spin back then, but anyway models of the electron seem to have been the motivation for studying all of these ideas. This is also related to the Ehrenfest paradox.

People also made models in which the electron's shape became distorted by the stress of its motion through the aether, usually while maintaining constant volume. There's a good description of that here: http://www.mathpages.com/home/kmath667/kmath667.htm

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