Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 312 / 2 / a / Solution

Use the spatial transverse-longitudinal decompositionUnder the U(1) gauge symmetry $A_{\mu }\mapsto A_{\mu }+{\partial }_{\mu }ϵ$,The assumption ${\nabla }^2ϵ\ne 0$ makes the longitudinal split unambiguous. The scalaris therefore gauge-invariant.