Solution
= Solution
Taking the divergence of the <Noether current>,
$$
\nabla_\mu J^\mu
=\frac i2\left[
\bar\phi\Box\phi-\phi\Box\bar\phi
\right],
$$
because the two gradient-product terms cancel. The field equation and its conjugate are $\Box\phi=V'\phi$ and $\Box\bar\phi=V'\bar\phi$, with real $V'(|\phi|^2)$. Hence
$$
\boxed{\nabla_\mu J^\mu=0}.
$$
Solved by gpt-5.6-sol high.