Spherical harmonics and integration in superspace

TL;DR

This paper extends classical spherical harmonics to superspace using Clifford analysis, introduces a new integration method over the supersphere, and connects it with the Berezin integral, enhancing the mathematical framework of superspace analysis.

Contribution

It develops a new integration over the supersphere based on Pizzetti's result, proving orthogonality, Green's theorems, and extending the Funk-Hecke theorem in superspace.

Findings

Established a new integration over the supersphere.

Proved orthogonality of superspace spherical harmonics.

Connected supersphere integration with the Berezin integral.

Abstract

In this paper the classical theory of spherical harmonics in R^m is extended to superspace using techniques from Clifford analysis. After defining a super-Laplace operator and studying some basic properties of polynomial null-solutions of this operator, a new type of integration over the supersphere is introduced by exploiting the formal equivalence with an old result of Pizzetti. This integral is then used to prove orthogonality of spherical harmonics of different degree, Green-like theorems and also an extension of the important Funk-Hecke theorem to superspace. Finally, this integration over the supersphere is used to define an integral over the whole superspace and it is proven that this is equivalent with the Berezin integral, thus providing a more sound definition of the Berezin integral.