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I listened to a lecture several years ago in which the speaker claimed that there is a theorem that shows that violation of charge conservation under classical electrodynamics is impossible in the strictest imaginable sense. I didn't think much of it at the time, but now I'm curious about it and cannot find references to this theorem anywhere.

The argument went something like this:

Assume that there is a region of spacetime in which the conservation of charge may be violated. Outside of this spacetime region, everything is normal, but inside all bets are off. Clearly, this special region must conserve charge as a whole since its temporal boundaries start and end with the traditional rules, but there might be events inside the region which violate charge conservation locally.

At this point, as I recall, he invoked conservation of energy (especially as it relates to electromagnetic radiation propagating out from this special region into the rest of the universe) and asserted that with all other laws of classical physics held in place the boundary of this special spacetime region would either have to be infinite or infinitesimal.

In other words, either the entire universe throughout all of time may violate charge conservation (and therefore violate all manner of other laws) or no part of it ever does. This is the theorem I am curious about. He called this special spacetime region on which the theorem is based an "Electromagnetic Miracle."

Can anyone say conclusively whether or not such a theorem exists?

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    $\begingroup$ Related? Feynman's layman proof of local charge conservation $\endgroup$ – Hal Hollis Jun 27 '18 at 21:06
  • $\begingroup$ @HalHollis It is certainly related, but the theorem I'm looking for would be an even stronger statement than that since the "miraculous" region would merely transform in spatial and temporal extent under Lorentz boosts. $\endgroup$ – Geoffrey Jun 27 '18 at 21:19
  • $\begingroup$ If charge is not locally conserved, some of the Maxwell equations is violated. Then the work-energy theorem and EM equations will not imply Poynting formulae for EM energy, but some other formulae. Did he indicate how the Maxwell equations are modified or what EM energy formulae are assumed? It is not meaningful to use the old Poynting formulae when Maxwell's equations are violated. $\endgroup$ – Ján Lalinský Jun 28 '18 at 8:09
  • $\begingroup$ @JanLalinsky Unfortunately, I don't remember much else about the talk. I don't think he mentioned any specific model for the "miraculous" region. The way I recall it, his argument sounded like it would be model independent - meaning that it shouldn't matter how charge conservation was violated only that it was. I think he may have made an argument that the violations could take place on the boundary of the region without loss of generality which would lead to some sort of discontinuity in the boundary conditions for Maxwell's equations outside the region, but honestly, it's really foggy. $\endgroup$ – Geoffrey Jun 28 '18 at 8:40

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