= Global phase symmetry of a complex scalar field
{title2=$\psi\mapsto e^{-i\alpha}\psi$}
For $\mathcal L=\partial_\mu\psi^*\partial^\mu\psi-V(|\psi|^2)$, constant $\alpha$ preserves both terms. This is an <internal symmetry of a classical field theory> with <circle group> $U(1)$. A spacetime-dependent phase would introduce derivative terms, so the ordinary-derivative density has only the global symmetry. The associated current is the <Noether charge of a complex scalar field> construction; the orientation $e^{-i\alpha}$ fixes its overall sign.
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