Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 301 / 1 / d / Solution

The local U(1) gauge symmetry isThe global phase subgroup has Noether current and chargeMultiplying the fermion equation by $\bar{\psi }$, its adjoint by $\psi$, and subtracting shows ${\partial }_{\mu }{j}^{\mu }=0$ on shell. Therefore $\dot{Q}=-\int d^{3}x\nabla \cdot \mathbf{j}=0$ under vanishing boundary flux.

The source in Maxwell's equation isThe second term is an identically conserved magnetization current, because it is the divergence of an antisymmetric tensor. It changes the local source but contributes only a boundary term to the total charge.