Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 302 / 2 / d / Solution

For a massive positive-energy orbit, choose $p_{\ast }=(m,0,0)$ with $m>0$. Its little group is $SO(2)$, whose unitary irreducible representations are the characters $e^{is\theta }$. Induction gives one-particle states $|p,s⟩$ on the mass shell $p^2=m^2$, $p^0>0$, labeled by mass and spin. For the Poincare group itself, single-valuedness gives $s\in \mathbb{Z}$; its double cover permits half-integers, and its universal cover permits any real $s$, producing anyonic spin in $(2+1)$ dimensions.