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-Gauss' Law is observed experimentally, shows us there's this thing E

-Biot-Savart's Law is observed experimentally, shows us there's this thing B

-Ampere's Law (after fixed by Maxwell) observed experimentally, along with Faraday's law define the interrelationship between B and E

-Maxwell's equations condense these experimentally observed behaviors of these B and E things.

-Maxwell's equations are not Galilean invariant but Lorentz invariant.

-By assuming Lorentz Invariance, Linear, Temporal and Angular Homogeneity and Extremal Action we derive Relativist Mechanics which in the small energy limit gives us Classical Mechanics.

My question is this: Is there ANY WAY to divorce Maxwell's equation from those 4 experimentally observed facts (i.e. there exists something called E and it behaves this way). Is there any appeal to symmetry or some such that allows us to say "If we assume our universe has a symmetry of the form blah and blah we see that there MUST be some quantity E attached to each point in space and it MUST have these properties)?

Is there any more abstract way to motivate and derive Maxwells equations like we do for Classical Mechanics (like in the first chapter of Landau's mechanics) or MUST they be taken as 100% the result of experiment?

This seems to me to be the ultimate linchpin in the derivation of classical mechanics through the eyes of Euler-Lagrange and Extremal Action. We can motivate and derive everything else from some assumptions about the homogeneity of space and such but then we ALWAYS just tack on Maxwell's Equations and the existence of E and B as a matter of experimental fact.

In parallel in Quantum Mechanics we do the same things with Spin (we simply tack it on because experiment say it is there), however, when we generalize to quantum field theory, spin actually becomes a PREDICTION and not an experimental artifact. Can the same be done for Maxwell's equations?