Linearity of Maxwell's equations as a result of special relativity.

[TrickyDicky]

Thanks Tom.
I guess I interpreted the phrase "a gauge theory is linear if and only if its gauge group is Abelian" in a previous post as meaning "a gauge theory is non-linear if and only if its gauge group is non-Abelian" which are not exactly equivalent, the latter is wrong while the former is right.

 

[Ben Niehoff]

However what I fail to see is how QED contains that non-linear term only with the U(1) abelian gauge symmetry. This might be getting offtopic and not related to relativity, so maybe I should open a new thread.

QED is not a pure gauge theory; it is coupled to fermions. The fermions provide the nonlinearity as Tom described above.

Generally, when we solve classical E&M problems, we consider the sources J to be non-dynamical. They are fixed, or are given by a predetermined forcing function (cf. antennas). In this case, the problem we are left to solve is actually pure E&M.

If you allow the sources to be dynamical--i.e., to allow the fields to move the charges in the way that they actually would in nature--then the theory as a whole is no longer linear. At least I think so. This fact may actually depend on the sources having a nonlinear coupling to the fields, as in the case with fermions.

In any case, we should be careful what we mean by "gauge theory". I've been referring to the pure gauge sector, possibly up to including external, non-dynamical sources. QED, QCD, etc. are "gauge theories coupled to matter".