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

For the positive-energy massless orbit choose $k=(E,E,0)$. Its connected little group is the one-parameter group of null rotations, isomorphic to $(\mathbb{R},+)$. Its unitary irreducible representations are the characters $t\mapsto e^{i\rho t}$, $\rho \in \mathbb{R}$, and induction gives the massless one-particle representations. The physically usual finite-component representation has trivial little-group action, $\rho =0$; there is no $SO(2)$ helicity subgroup for a null momentum in $(2+1)$ dimensions.