How one can know the gauge field emerging from the local gauge invariance is actually the EM field?
I understood in a simple scalar field whose Lagrangian is given by
$ \mathcal{L} = \frac{1}{2}(\partial_\mu \phi)^{*}(\partial^\mu \phi) - \frac{1}{2} |\phi|^2 $
needs a vector gauge field in order to satisfy the local gauge invariance, which is simply
the electromagnetic field $A_\mu$.
What I'm trying to understand is that how one can prove that gauge field
is exactly an EM field that satisfy the Maxwell's eqn: $\partial_\mu \partial^{[\space\mu} A^{\nu \space]} = J^\nu$.
For making the Lagrangian be locally gauge invariant, the field $A_\mu$, whatever it is,
I showed it should have a gauge transformation relation
$A_\mu \rightarrow A_\mu ' = A_\mu - \partial_\mu \chi $ as $\phi \rightarrow \phi' = e^{-i\chi}\phi$
I think however, it doesn't guarantee that $A_\mu$ should be the EM field
that satisfies above Maxwell's eqn.
Please let me understand. Thank you.