Noether second theorem (${\sum }_a[{\mathcal{E}}_a{R}_a-{\partial }_{\mu }({\mathcal{E}}_a{R}_a^{\mu })]\equiv 0$)

Suppose an action is invariant under a local transformation $\delta {\Phi }_a=R_a\alpha +R_a^{\mu }{\partial }_{\mu }\alpha$ for every compactly supported function $\alpha$. Variation and integration by parts give the off-shell identityThus arbitrary local symmetry parameters imply dependencies among the field equations. In gauge theory, these identities accompany constraints rather than independent physical charges for each function. This complements the conserved-current statement of Noether theorem.

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.