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

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