We know that there are two ways to get charge conservation in electrodynamics by using the following action: $$S[A]~=~\int\! d^4x {\cal L},$$

$$ {\cal L} ~=~{\cal L}_{\rm Maxwell} + {\cal L}_{\rm matter} , $$

1) Using global gauge symmetry and matter field equations of motion, or

2) Using local gauge symmetry and gauge field equations of motion.

This has been discussed at many places but let me provide few links for those who want to check it more thoroughly: see here and here.

Now here is my first question:

Is there any way to proceed in the reverse direction? i.e. Using continuity equation and gauge invariance to derive field equations?.

Note: maybe extra symmetries (like Lorentz symmetry) are needed to build up the action itself, suppose we know them and we use them in both directions for the sake of simplicity.

Based on Noether's general identity presented here and here it seems that the answer to my question is yes. However, I'm not sure!.

In fact I'm trying to see the minimum number of independent assumptions from which we can construct the whole electrodynamics (Considering the lorentz force law as a separate assumption itself-see here) plus charge conservation. It seems to me taking gauge invariance (and probably some other symmetries) and conservation of charge as our independent postulate is a good candidate.

When I was searching I accidentally found very interesting paper in which the author has derived Maxwell's equations from continuity equation. He uses a few extra assumptions to do it though. Here are the extra assumptions as the author himself describes them in the conclusion part:

It can be argued that other physical assumptions like the use of the Minkowski spacetime with its speed of light c and its associated retarded Green function G are ingredients implicit in the covariant derivation of Maxwell’ equations presented here. However, if we first postulate the validity of the continuity equation in any four-space, the particular use of the Minkowski spacetime (together with its associated retarded Green function G and speed c) would not really be a new postulate, but merely an application of our initial postulate.

Then we reach my second question:

I guess that if the answer to my first question is yes, then the extra assumptions of the author are(should!) be equivalent to gauge invariance(plus other symmetries that we use in my first question). And it seems highly counter intuitive to me!. Why should anything related to Minkowski space be related to gauge invariance?!

  • $\begingroup$ "Using continuity equation and gauge invariance to derive field equations?" What do you mean by 'gauge invariance' here? Invariance of the equations for potential $A_\nu$ with respect to change of potential that preserves $\partial_\mu A_\nu - \partial_\nu A_\mu$? $\endgroup$ – Ján Lalinský Jul 19 '16 at 13:24
  • $\begingroup$ I mean invariance of action up to boundary terms with respect to $A_{\mu} \rightarrow A_{\mu}+ \partial_\mu \lambda$ and also corresponding change to matter fields. It is defined in the links that I have cited. Of course for global case $A_\mu$ doesn't change. $\endgroup$ – user121238 Jul 19 '16 at 13:35
  • $\begingroup$ Notice that Heras also assumes in his paper that EM fields reduce to Coulomb and Biot-Savart fields in static cases. He uses this to derive the Maxwell equations, but this is based on experience, not derived from gauge invariance. $\endgroup$ – Ján Lalinský Jul 19 '16 at 13:59

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