I want to find the conserved current $j^\mu$ associated with Dirac equation coupled with a external electromagnetic field, i.e,
$$\left(i\hbar \gamma^\mu \partial_\mu - \frac{e}{c} \gamma^\mu A_\mu -mc\right)\psi = 0 ,$$
where $\overline{\psi}$ is the conjugate Dirac field.
For that I intended to start with the Lagrangian
$$\mathcal{L} = i \hbar \overline{\psi}\gamma^\mu \partial_\mu \psi - \frac{e}{c} \overline{\psi}\gamma^\mu A_\mu\psi - mc \overline{\psi}\psi$$
and trough Noether's theorem find $j^\mu$. I know that the field has the following symmetry
$$\psi \rightarrow e^{-i\alpha}\psi.$$
My question is: Should I consider the gauge invariance of $A_\mu$ as a symmetry to find $j^\mu$? If not, the current found is exactly the same as the one for a free Dirac field. What is the physical reason for this?
