 Solution

For $P^2=m^2>0$, choose the future-pointing rest momentum $p^{\mu }=(m,0,0,0)$. Its little group is the rotation group $SO(3)$, or $SU(2)$ on the double cover. Its finite-dimensional unitary irreducible representations are labelled byand have dimension $2j+1$. These are the spin states of a massive particle.

For $P^2=0$, choose $p^{\mu }=(E,0,0,E)$. Its little group is the two-dimensional Euclidean groupNontrivial unitary action of the translation subgroup produces infinite-dimensional continuous-spin representations. For the finite-helicity particles observed in ordinary relativistic field theory, that subgroup acts trivially, leaving a one-dimensional representation of rotations about $\mathbf{p}$. Such representations are labelled by helicity.