 Solution

The local gauge symmetry contains the constant-phase subgroup. For constant $\alpha$, $\delta A_{\mu }=0$, and Noether's theorem applied to the matter fields givesThus ${\partial }_{\mu }{J}^{\mu }=0$ on the matter field equations. More explicitly, allowing $\alpha (x)$ in the matter variation while holding $A$ fixed gives $\delta \mathcal{L}=J^{\mu }{\partial }_{\mu }\alpha$. On shell it also equals ${\partial }_{\mu }(\alpha J^{\mu })$, so equality for arbitrary $\alpha$ implies conservation. In the full gauge transformation, varying $A$ cancels the $J^{\mu }{\partial }_{\mu }\alpha$ term; this is the related Noether second-theorem identity, not a new independent charge for every gauge function.