Spherical harmonics and integration in superspace

Understand

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.