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

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.