= Noether current of scalar electrodynamics
{c}
{title2=$J^\mu=iq[\phi^*D^\mu\phi-\phi(D^\mu\phi)^*]$}
For $D_\mu=\partial_\mu+iqA_\mu$, the constant phase transformation $\delta\phi=-iq\epsilon\phi$ has the displayed <Noether current>. Varying the <gauge field> gives $\partial_\mu F^{\mu\nu}=J^\nu$. In this convention $J^\mu=iq(\phi^*\partial^\mu\phi-\phi\partial^\mu\phi^*)-2q^2A^\mu|\phi|^2$. The matter equations imply $\partial_\mu J^\mu=0$; the integrated <electric charge> additionally requires zero boundary flux. Local phase invariance gives a <Noether second theorem> identity, rather than independent physical charges for arbitrary functions.<Quantum Field Theory by David Tong>[§1.3.4, phase current; §§6.1.1 and 6.3.2, gauge symmetry and scalar coupling; signs here use the explicitly stated $D_\mu$ convention]{ref}
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