Addition theorems for spin spherical harmonics. I Preliminaries

TL;DR

This paper develops a systematic approach to deriving addition theorems for spin spherical harmonics, focusing on technical results related to tensor products, Clebsch-Gordan coefficients, and matrix elements, laying groundwork for future theorems.

Contribution

It introduces a systematic method for deriving addition theorems for spin spherical harmonics and related bilocal sums, with explicit results for matrix elements of tensor products.

Findings

Factorization of orbital and spin degrees of freedom.

Explicit matrix elements for tensor products of position and angular-momentum operators.

Foundation for addition theorems in subsequent work.

Abstract

We develop a systematic approach to deriving addition theorems for, and some other bilocal sums of, spin spherical harmonics. In this first part we establish some necessary technical results. We discuss the factorization of orbital and spin degrees of freedom in certain products of Clebsch-Gordan coefficients, and obtain general explicit results for the matrix elements in configuration space of tensor products of arbitrary rank of the position and angular-momentum operators. These results are the basis of the addition theorems for spin spherical harmonics obtained in part II.