Angular decomposition of tensor products of a vector

TL;DR

This paper derives a compact, explicit expression for the angular decomposition of tensor products of vectors, facilitating Fourier transforms of functions relevant in particle interaction analysis.

Contribution

It provides a general formula for tensor product decomposition into angular momentum components valid for all orders and angular momenta.

Findings

Explicit formula for tensor decomposition derived

Enables efficient Fourier transforms of vector functions

Applicable to analyzing particle interactions

Abstract

The tensor product of $L$ copies of a single vector, such as $p_{i_1} ... p_{i_L}$, can be analyzed in terms of angular momentum. When $p_{i_1} ... p_{i_L}$ is decomposed into a sum of components $( p_{i_1} ... p_{i_L} )^L_\ell$, each characterized by angular momentum $\ell$, the components are in general complicated functions of the $p_i$ vectors, especially so for large $\ell$. We obtain a compact expression for $( p_{i_1} ... p_{i_L} )^L_\ell$ explicitly in terms of the $p_i$ valid for all $L$ and $\ell$. We use this decomposition to perform three-dimensional Fourier transforms of functions like $p^n \hat p_{i_1} ... \hat p_{i_L}$ that are useful in describing particle interactions.