Noether identity for abelian scalar gauge symmetry ($-{\partial }_{\mu }{\mathcal{E}}_A^{\mu }-ie\varphi {\mathcal{E}}_{\varphi }+ie{\varphi }^{\ast }{\mathcal{E}}_{{\varphi }^{\ast }}\equiv 0$)

For $\delta A_{\mu }={\partial }_{\mu }\alpha$, $\delta \varphi =-ie\alpha \varphi$ and $\delta {\varphi }^{\ast }=ie\alpha {\varphi }^{\ast }$, Noether second theorem givesThis holds without imposing the field equations. On the matter equations it becomes a divergence identity for the electromagnetic equation, expressing compatibility with charge conservation. The Hamiltonian description carries the corresponding Gauss law constraint in gauge theory.