= Global phase current of a linearly coupled complex scalar
{title2=$J^\mu=i(\phi^*\partial^\mu\phi-\phi\partial^\mu\phi^*)-2eA^\mu|\phi|^2$}
For $\delta\phi=-i\epsilon\phi$, the conserved current is $J^\mu=i(\phi^*\partial^\mu\phi-\phi\partial^\mu\phi^*)-2eA^\mu|\phi|^2$. The conjugate field equations imply
$$
\phi^*\Box\phi-\phi\Box\phi^*
=-2ie\,\partial_\mu(A^\mu|\phi|^2),
$$
which directly cancels the <divergence> of the explicit background term. Conservation requires no <Lorenz gauge condition>; constancy of total charge additionally requires vanishing boundary flux.
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