Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 301 / 1 / a / Solution

Under $A_{\mu }\mapsto A_{\mu }+{\partial }_{\mu }\alpha$, the field strength is unchanged, but the mass term in the Proca action changes bywhich is not generally a total derivative. The mass therefore breaks gauge invariance.

The Euler-Lagrange field equation isTaking its divergence and using the antisymmetry of $F^{\mu \nu }$ gives $m^2{\partial }_{\nu }{A}^{\nu }=0$. Since $m>0$, the Lorenz constraint in Proca theory follows. Substitution back into the field equation then gives