Past exam of the theoretical physics I course of the University of Cambridge / 2019 / 3 / a

Under $\psi \mapsto e^{i\alpha }\psi$ with constant real $\alpha$, the field and its complex conjugate acquire opposite phases. These cancel in $|\psi {|}^2$, $|\psi {|}^4$, $\nabla {\psi }^{\ast }\cdot \nabla \psi$, and ${\psi }^{\ast }\dot{\psi }-\psi {\dot{\psi }}^{\ast }$. The Lagrangian density therefore has the particle-number phase symmetry, a global circle group $U(1)$.

For a spacetime-dependent real $\alpha (\mathbf{r},t)$,Writing $n=|\psi {|}^2$, the change in the Lagrangian density isThis is not generally zero or a boundary term. Thus the symmetry is global, with no arbitrary local phase symmetry. A compensating gauge field and gauge-covariant derivative of a charged scalar field would be required to promote it to a local redundancy.