Spin-weighted spherical harmonics as massless angular momentum eigenstates and their role in obstructing spin-orbital decompositions

TL;DR

This paper demonstrates that massless particles with helicity have angular momentum eigenstates described by spin-weighted spherical harmonics, revealing a topological obstruction to spin-orbital decomposition due to their unique multiplet structure.

Contribution

It establishes the role of spin-weighted spherical harmonics as eigenstates for massless helicity particles and highlights the topological reasons preventing spin-orbital angular momentum splitting.

Findings

Massless helicity particles' angular momentum states are given by spin-weighted spherical harmonics.

The multiplet structure for massless particles differs from massive ones, with at most one multiplet per angular momentum j.

The structure obstructs the usual spin-orbital decomposition of angular momentum in massless particles.

Abstract

We show that for massless helicity $h$ particles, the angular momentum eigenstates are given in an appropriate coordinate system by the spin-weighted spherical harmonics ${_{-h}Y_{jm}}$ of spin-weight $-h$. In particular, these are simultaneous eigenstates of the Hamiltonian, helicity, $J^2$, and $J_z$. The appearance of the spin-weighted spherical harmonics as opposed to the ordinary spherical harmonics reflects the nontrivial topological structure of massless particles with nonzero helicity. The resultant angular momentum multiplet structure is quite different than that of massive particles, with at most one multiplet for each angular momentum $j$ and with $|h|$ acting as a lower bound on $j$. This illustrates the obstruction to a spin-orbital decomposition of the angular momentum for massless particles, as such a sparse multiplet structure is not consistent with any reasonable spin-orbital splitting.