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

Abstract: We show that for massless particles of helicity $h$, 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 $|h|$ acting as a lower bound on $J^2$ and at most one multiplet for each angular momentum $j$. This illustrates the obstruction to a spin-orbital decomposition of the angular momentum for massless particles, as such a sparse multiplet structure does not result from any reasonable spin-orbital splitting.