Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 305 / 1 / c / Solution

The helicity of a massless state is the component of angular momentum along its momentum,When the translation part of the massless little group is trivial, the Pauli-Lubanski pseudovector obeysA rotation through angle $\theta$ about $\mathbf{P}$ acts by the phase $e^{-i\lambda \theta }$. On the double cover, a $4\pi$ rotation is the identity, soHence $\lambda =0,\pm \frac{1}{2},\pm 1,\pm \frac{3}{2},\dots$. A parity-invariant theory pairs the $\lambda$ and $-\lambda$ representations, although one chiral massless representation need not contain both.