No, the homogenous pair results that way. To get the inhomogenous ones you need to add in a term for coupling to (charged) matter.I understand that one is able to derive the inhomogenuous pair of Maxwell's equations from varying the field strength tensor Lagrangian [...]
From a geometric perspective, U(1) is embedded in the field.[...] Now implying the U(1) gauge invariance, how is one led to the Maxwell's equations?
Could you give me a few hints on how to do it explicitly or is this already enough to manage with the task of deriving the equations?From a geometric perspective, U(1) is embedded in the field.
It is only justified by the fact that we end up with a theory that agrees with experiment. If gauge theories did not agree with experiment, only mathematicians (and not physicists) would care about them.It seems perfectly reasonable to require that physics stay the same under U(1) symmetry. But adding the vector potential and simply implying an ad hoc transformation rule seems unjustifiable.