= Noether identity for abelian scalar gauge symmetry
{c}
{title2=$-\partial_\mu\mathcal E_A^\mu-ie\phi\mathcal E_\phi+ie\phi^*\mathcal E_{\phi^*}\equiv0$}
For $\delta A_\mu=\partial_\mu\alpha$, $\delta\phi=-ie\alpha\phi$ and $\delta\phi^*=ie\alpha\phi^*$, <Noether second theorem> gives
$$
-\partial_\mu\mathcal E_A^\mu-ie\phi\mathcal E_\phi+ie\phi^*\mathcal E_{\phi^*}\equiv0.
$$
This 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>.
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