Invariant integration over the orthogonal group

TL;DR

This paper extends a recent integration method from the unitary to the orthogonal group, providing explicit formulas and recursion relations for integrals involving vectors, enhancing tools for mathematical physics and group theory applications.

Contribution

It introduces a novel adaptation of an existing unitary group integration method to the orthogonal group, including explicit formulas and recursive techniques.

Findings

Derived explicit formulas for one, two, and three-vector integrals over the orthogonal group.

Developed recursion formulas for more complex integrals.

Extended the applicability of integration techniques to the orthogonal group.

Abstract

I adapt a recently introduced method for integrating over the unitary group (S. Aubert and C.S. Lam, J.Math.Phys. 44, 6112-6131 (2003)) to the orthogonal group. I derive explicit formulas for a number of one, two and three-vector integrals, as well as recursion formulas for more complicated cases.