Addition theorems for spin spherical harmonics. II Results

TL;DR

This paper extends addition theorems for spin spherical harmonics, providing explicit formulas for various spin combinations and a general theorem for scalar and tensor harmonics, advancing mathematical tools in quantum physics.

Contribution

It derives the general form of addition theorems for spin spherical harmonics and provides explicit results for specific spin cases and tensor harmonics, building on previous work.

Findings

Explicit addition theorems for spin-1/2, 1, 3/2 harmonics

General addition theorem for scalar and tensor spherical harmonics

Explicit bilocal sums and spherical harmonic expressions

Abstract

Based on the results of part I, we obtain the general form of the addition theorem for spin spherical harmonics and give explicit results in the cases involving one spin-$s'$ and one spin-$s$ spherical harmonics with $s',s=1/2$, 1, 3/2, and $|s'-s|=0$, 1. We obtain also a fully general addition theorem for one scalar and one tensor spherical harmonic of arbitrary rank. A variety of bilocal sums of ordinary and spin spherical harmonics are given in explicit form, including a general explicit expression for bilocal spherical harmonics.