Invariant Integration over the Unitary Group

TL;DR

This paper introduces a new method for integrating products of unitary matrix elements over the U(n) group that avoids complex summations and relies solely on unitarity and Haar measure invariance.

Contribution

The paper develops an alternative approach to group-theoretical integrals over U(n), simplifying calculations by eliminating the need for multiple sums and group theory.

Findings

Provides a closed-form expression for integrals over the unitary group

Simplifies calculations by avoiding tedious summations

Applicable to integrals over hyperspheres as well

Abstract

Integrals for the product of unitary-matrix elements over the U(n) group will be discussed. A group-theoretical formula is available to convert them into a multiple sum, but unfortunately the sums are often tedious to compute. In this paper, we develop an alternative method in which these sums are avoided, and group theory is rendered unnecessary. Only unitarity and the invariance of the Haar measure are required for the computation. The method can also be used to get a closed expression for the simpler integral of monomials over a hypersphere.