Tensor and Spinor Spherical Harmonics and the Spin-s Harmonics sYlm(θ, φ)

Abstract

A set of Cartesian tensor spherical harmonics is constructed from the spin weighted harmonics of Newman and Penrose, sYlm(θ, φ). It is shown that these tensor harmonics are eigenfunctions of total angular momentum, z component of total angular momentum, total spin and radial component of spin. In particular, − s may be thought of as a helicity for outgoing radiation. Tensor operators are introduced which lower and raise this helicity. They are shown to correspond to the operators ð and ð̄ introduced by Newman and Penrose. Because the sYlm(θ, φ) can be defined for half-integer values of l, m, and s, a set of spinor spherical harmonics is also constructed which has properties paralleling those of the tensor harmonics.