Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 301 / 2 / a / v / Solution

Positivity selectsfor which the kinetic terms combine, up to normalization and a total derivative, into the Proca field form $-F_{\mu \nu }{F}^{\mu \nu }/4$. Taking the divergence of the equation of motion givesFor the selected relation and $m\ne 0$, this implies the transverse vector field condition ${\partial }_{\mu }{V}^{\mu }=0$. The remaining three massive polarizations have positive on-shell energy when $\tilde{a}>0$.