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Back in my undergrad I had a course on classical electrodynamics where the fields had values in the space of tempered distributions. In this way one could correctly treat self-interaction and effectively solve the differential equations involved in the distribution sense.

Unfortunately the few notes that I have are in Italian, and I am looking for some resources in English, but don't seem to find anything, could someone point me in the right direction?

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  • $\begingroup$ Related: physics.stackexchange.com/q/127376/2451 $\endgroup$
    – Qmechanic ♦
    Commented Jul 22, 2014 at 11:31
  • $\begingroup$ "In this way one could correctly treat self-interaction and effectively solve the differential equations involved in the distribution sense." You mean self-interaction of particles? Which differential equations can you solve? $\endgroup$
    – Ján Lalinský
    Commented Jul 22, 2014 at 11:52
  • $\begingroup$ Exactly, I mean the self-interaction of particles. In particular, the field is singular along the worldline of a particle, so all the derived quantities (such as the EM tensor) also are. But in the 'distribution sense', one can take derivatives that are meaningful also where the field is singular (because, for instance, the Dirac delta is derivable). $\endgroup$
    – Ziofil
    Commented Jul 30, 2014 at 21:18

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There is an excellent paper of 15 pages that contains all the electromagnetism distributional approach fundamentals from Skinner and Weil. Its title is "An introduction to generalized functions and their application to static electromagnetism point dipoles, including hyperfine interactions". Let me know if this is useful for you.

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I found a compact version of the course notes, re-written in english by the same professor who gave the original lectures. You can find them at this link.

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