= Noether charge of a complex scalar field
{c}
{title2=$Q=i\int(\psi^*\dot\psi-\psi\dot\psi^*)\,d^3x$}
For the <global phase symmetry of a complex scalar field> oriented as $\delta\psi=-i\varepsilon\psi$, the <Noether current> is $j^\mu=i(\psi^*\partial^\mu\psi-\psi\partial^\mu\psi^*)$. The <Euler-Lagrange field equations> $\Box\psi+V'(|\psi|^2)\psi=0$ and its <complex conjugate> give $\partial_\mu j^\mu=i(\psi^*\Box\psi-\psi\Box\psi^*)=0$ for a real differentiable potential. Integrating $j^0$ gives $Q$ when boundary flux vanishes. A charged <scalar> coupled to electromagnetism has <electric charge> $qQ$ with the chosen charge normalization; without that physical identification it is an internal charge.
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