Hilbert space for quantum mechanics on superspace

Journal Article · Wed Jun 15 04:00:00 EDT 2011 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.3592602· OSTI ID:21501351

Coulembier, K; De Bie, H ^[1]

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1. Department of Mathematical Analysis, Faculty of Engineering, Ghent University, Krijgslaan 281, 9000 Gent (Belgium)

In superspace a realization of sl{sub 2} 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, Proc. London Math. Soc. (accepted) 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 sl{sub 2}-representation is proven. Finally, the Heisenberg uncertainty principle for the super Fourier transform is constructed.

• OSTI ID: 21501351
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 6 Vol. 52; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

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INTEGRAL CALCULUS
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