# Spherical harmonics and integration in superspace - High Energy Physics - Theory

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Abstract: In this paper the classical theory of spherical harmonics in R^m is extendedto superspace using techniques from Clifford analysis. After defining asuper-Laplace operator and studying some basic properties of polynomialnull-solutions of this operator, a new type of integration over the supersphereis introduced by exploiting the formal equivalence with an old result ofPizzetti. This integral is then used to prove orthogonality of sphericalharmonics of different degree, Green-like theorems and also an extension of theimportant Funk-Hecke theorem to superspace. Finally, this integration over thesupersphere is used to define an integral over the whole superspace and it isproven that this is equivalent with the Berezin integral, thus providing a moresound definition of the Berezin integral.

Author: ** Hendrik De Bie, Frank Sommen**

Source: https://arxiv.org/