Noether charge of a complex scalar field ($Q=i\int ({\psi }^{\ast }\dot{\psi }-\psi {\dot{\psi }}^{\ast })d^{3}x$)

For the global phase symmetry of a complex scalar field oriented as $\delta \psi =-i\epsilon \psi$, the Noether current is $j^{\mu }=i({\psi }^{\ast }{\partial }^{\mu }\psi -\psi {\partial }^{\mu }{\psi }^{\ast })$. The Euler-Lagrange field equations $\square \psi +V^{\prime }(|\psi {|}^2)\psi =0$ and its complex conjugate give ${\partial }_{\mu }{j}^{\mu }=i({\psi }^{\ast }\square \psi -\psi \square {\psi }^{\ast })=0$ 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.