Derivation: Maxwell's equation only from the Lorenz gauge

[greypilgrim]

Hi.

Here, somebody apparently derives Maxwell's equations using only symmetry of second derivatives and the Lorenz gauge condition. Unfortunately it's in German, but I think the basic ideas are clear from the maths only.

In this derivation, the magnetic field turns out to be divergence-free, even before the application of the gauge condition. Why? Shouldn't such a general derivation allow for magnetic monopoles?

Also, it seems a bit suspicious that this derivation only works by fixing a specific gauge. How can this be the same as usual ED which allows different gauges?

 

[vanhees71]

I think, in this derivation the author used already what he claims to derive. Usually you start from the Maxwell equations, which cannot be derived, because they are a fundamental law of Nature. The homogeneous Maxwell equations are constraints, involving only the field (with its electric an magnetic components) and thus can be substituted by using a four-vector potential (or in (3+1)D language a scalar and a vector potential), which however is only defined modulo a gauge transformation since electromagnetism turns out to be a gauge theory.

If, on the other hand, you make the postulate that the electromagnetic field is the gauge boson of an un-Higgsed U(1) local gauge symmetry you can almost without much further input guess the Lagrangian and thus Maxwell's equations. The only allowed terms in the Lagrangian are such that lead to gauge invariant terms in the action. Then by power counting the leading term must lead to the free Maxwell equations. Then you can also couple a conserved U(1) current to get the equation with sources.

 

[greypilgrim]

I think, in this derivation the author used already what he claims to derive.

Can you see where exactly they did that?

 

[vanhees71]

Well, I've no clue what the author is really aiming at, but obviously he assumes to know the Maxwell equations and then uses the potentials in the usual way to eliminate the homogeneous Maxwell equations. As I said, I think he simply used the Maxwell equations to derive the Maxwell equations in terms of the potentials in Lorenz gauge, but writes it down in an ununderstandable way. I've no clue, what this webpage is good for and to whom it may be aimed. I'd recommend to read a standard textbook on the subject. Since obviously you understand German, my recommendation for theoretical physics is

M. Bartelmann, B. Feuerbacher, T. Krüger, D. Lüst, A. Rebhan, and A. Wipf, Theoretische Physik, Springer-Verlag, Berlin, Heidelberg, 2015.
http://dx.doi.org/10.1007/978-3-642-54618-1