How to find Maxwell's equations approximations to more general set of equations?

Question

Are Maxwell's equations approximations of something more general? How does one recover Maxwell's equations by taking the appropriate limit of the more general set of equations? Has it been worked out in the literature? Where can I find it?

In this article (which is on a completely different topic), Weinberg says that "They are understood today to be an approximation that is valid in a limited context (that of weak slowly-varying electric and magnetic fields)". What might he have in mind?

You may take a look into its relativistic formulation, into QED or look at how other forces are treated in the Standard Model formalism — Mauricio, Mar 07 '23 at 14:42
If you mean like being nonlinear or something like that then I can't think of anything offhand, one can write Maxwell's equations even for nonabelian theories like QCD and Electroweak theory. Even gravity has a form of Maxwells equations. — Triatticus, Mar 07 '23 at 15:01
Maybe you can start here: https://physics.stackexchange.com/questions/169553/how-wrong-are-the-classical-maxwells-equations-as-compared-to-qed?rq=1 — Samuel, Mar 07 '23 at 15:11
https://fermatslibrary.com/s/feynmans-proof-of-the-maxwell-equations — Amit, Mar 07 '23 at 15:48
@Triatticus Somewhere, I have read that Maxwell's equations are valid only for weak enough electric and magnetic fields. — Solidification, Mar 07 '23 at 16:20
@Solidification Only if "weak" means up to the Schwinger limit and that would be a very strange definition of "weak". — John Rennie, Mar 07 '23 at 16:22
@JohnRennie Here is an article by Weinberg (but on a completely different issue) where he says that "They are understood today to be an approximation that is valid in a limited context (that of weak slowly-varying electric and magnetic fields)" See this: https://physics.nyu.edu/sokal/weinberg.html — Solidification, Mar 07 '23 at 16:24
I suppose you could regard Maxwell's equations as the low energy limit of the Euler-Heisenberg theory, but in this case "low energy limit" actually means "such a high energy that we haven't been able to experimentally create it". — John Rennie, Mar 07 '23 at 16:25

score 3 · Accepted Answer · answered Mar 07 '23 at 16:36

Maxwell's equations are not approximations, they are the exact equations of motion of classical electromagnetism.

What Weinberg is likely referring to is that the classical equations of motion are of course only valid in situations where the classical limit of quantum electrodynamics is appropriate. For instance, at (very) high energies (above the Schwinger limit), you get significant contributions from quantum two-photon physics, contradicting e.g. the classical statement that two light beams shot "through" each other don't interact with each other.

Note that Maxwell's equations of motion are still the correct, non-approximative equations of motion for the quantum field, the "approximation" that fails is simply the approximation of a quantum theory by its corresponding classical theory, not something specific to electromagnetism as such.

How to find Maxwell's equations approximations to more general set of equations?

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