Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 304 / 3 / a / Solution

Under the global U(1) gauge symmetry, the phases in $\bar{\psi }\psi$ cancel and $\alpha$ is constant, soBoth the fermion term and $F_{\mu \nu }{F}^{\mu \nu }$ are therefore invariant. To extract the Noether current, temporarily promote $\alpha$ to a function. The variation of the action isThe Euler-Lagrange equations then imply the current conservation law ${\partial }_{\mu }{j}^{\mu }=0$.