Angular decomposition of tensor products of a vector

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 $(p_{i_1}...p_{i_L}{)}_{\ell }^L$, 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 $(p_{i_1}...p_{i_L}{)}_{\ell }^L$ 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.

9 pages

References in corpus (1)

• Three-dimensional Fourier transforms, integrals of spherical Bessel functions, and novel delta function identities