Effect of a charge's own field on itself (Feynman Lec. Vol. [I]-28 )

[Swamp Thing]

From Feynman's Lectures, Part I , Ch. 28

There was a problem that was not quite solved at the end of the 19th century. When we try to calculate the field from all the charges including the charge itself that we want the field to act on, we get into trouble trying to find the distance, for example, of a charge from itself, and dividing something by that distance, which is zero. The problem of how to handle the part of this field which is generated by the very charge on which we want the field to act is not yet solved today. So we leave it there; we do not have a complete solution to that puzzle yet, and so we shall avoid the puzzle for as long as we can.

Purely in terms of predictive success and useful applications, what kind of physical / practical problems are we not able to calculate because of this gap in our understanding? Have things become clearer in any way, as of 2019?

 

[vanhees71]

FAPP there's nothing we can't handle with appropriate approximations. The point is that classical point particles don't exist, i.e., they are at best an approximation. The resolution FAPP is to use not the Abraham-Lorentz-Dirac equation but the modification by Landau and Lifshitz of it, which is accurate to the same order of approximation but doesn't suffer from all the trouble.

Another way out, known for more than 100 years, is not to use point-particle mechanics in relativity at all but only continuum mechanics, where no problems of this kind exist.

For a nice discussion, see

J. Rafelski, Relativity Matters, Springer International
Publishing AG, Cham (2017).
https://dx.doi.org/1007/978-3-319-51231-0