Hilbert space for quantum mechanics on superspace

TL;DR

This paper develops a Hilbert space framework for quantum mechanics in superspace, extending an inner product to the super Schwartz space, constructing eigenfunctions, and establishing the super Fourier transform's uncertainty principle.

Contribution

It introduces a new Hilbert space for superspace quantum mechanics, extending previous inner products and analyzing the super Fourier transform.

Findings

Inner product extended to super Schwartz space

Constructed eigenfunctions for orthosymplectically invariant problems

Proved the super Fourier transform satisfies the Heisenberg uncertainty principle

Abstract

In superspace a realization of sl2 is generated by the super Laplace operator and the generalized norm squared. In this paper, an inner product on superspace for which this representation is skew-symmetric is considered. This inner product was already defined for spaces of weighted polynomials (see [K. Coulembier, H. De Bie and F. Sommen, Orthogonality of Hermite polynomials in superspace and Mehler type formulae, arXiv:1002.1118]). In this article, it is proven that this inner product can be extended to the super Schwartz space, but not to the space of square integrable functions. Subsequently, the correct Hilbert space corresponding to this inner product is defined and studied. A complete basis of eigenfunctions for general orthosymplectically invariant quantum problems is constructed for this Hilbert space. Then the integrability of the sl2-representation is proven. Finally the Heisenberg uncertainty principle for the super Fourier transform is constructed.