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

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

The symmetric stress-energy tensor obtained by metric variation isIt is conserved on shell. With signature $(+---)$ its energy density isThe canonical stress-energy tensor differs from this symmetric tensor by the divergence of $F^{\mu \rho }{A}^{\nu }$ plus terms proportional to the field equation. Thus its integrated energy agrees after discarding the corresponding total spatial derivative, and positivity holds on shell.

For the plane wave $A^{\mu }(x)={ϵ}^{\mu }(k)e^{-ik\cdot x}$, the equations from part a becomeIn the rest frame $k^{\mu }=(m,0)$, transversality sets ${ϵ}^0=0$ while the three spatial components remain independent. Hence a Proca field has three polarization states, as required for a massive spin-one particle. A covariant orthonormal choice obeys

The quadratic momentum-space operator is inverted by the Proca propagator. With the Feynman propagator boundary prescription,Multiplication by the Fourier-space Proca operator gives $i{\delta }_{\mu }{}^{\nu }$, with the overall sign determined by the convention for the Green function.