Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 305 / 1 / a / Solution

The little group of a massive four-momentum is $SO(3)$, so a massive spin-one one-particle state has $2s+1=3$ polarizations. For $p^{\mu }=(E,0,0,p)$ and metric signature $(+---)$, one convenient basis isAll three obey $p_{\mu }{ϵ}^{\mu }=0$ and ${ϵ}^{\ast }\cdot ϵ=-1$. Their completeness relation isFor a massless momentum, the finite-helicity representations of the $ISO(2)$ little group carry only the two helicity states $\lambda =\pm 1$. Gauge equivalence removes the timelike and longitudinal polarizations; correspondingly, ${ϵ}_L$ has no finite $M_X\to 0$ limit.