Past exam of the theoretical physics major option course of the University of Oxford / 2017 / 5 / a / Solution

The Proca action is invariant under spacetime translations and proper continuous Lorentz transformations, together forming the connected Poincaré group. The fixed nonzero mass excludes scale invariance. In four dimensions, a dimensionless action and the canonical kinetic term implyThese are mass dimensions. The field-strength term is invariant under $A_{\mu }\to A_{\mu }+{\partial }_{\mu }\chi$, but the mass term is not: its change contains $m_A^2{A}^{\mu }{\partial }_{\mu }\chi +\frac{1}{2}{m}_A^2(\partial \chi {)}^2$. Thus the displayed massive-vector theory has no Maxwell-type local gauge invariance. The massless limit has a different constraint structure.