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arXiv:0706.3650 · Journal of Physics A Mathematical and Theoretical

The Weyl–Heisenberg group on the noncommutative two-torus: a zoo of representations

Jan Govaerts A, Frederik G. Scholtz

Published Sep 25, 2007• 4 citations
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Abstract

In order to assess possible observable effects of noncommutativity in deformations of quantum mechanics, all irreducible representations of the noncommutative Heisenberg algebra and Weyl–Heisenberg group on the two-torus are constructed. This analysis extends the well-known situation for the noncommutative torus based on the algebra of the noncommuting position operators only. When considering the dynamics of a free particle for any of the identified representations, no observable effect of noncommutativity is implied.

DOI
10.1088/1751-8113/40/41/011

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In order to assess possible observable effects of noncommutativity in deformations of quantum mechanics, all irreducible representations of the noncommutative Heisenberg algebra and Weyl–Heisenberg group on the two-torus are constructed. This analysis extends the well-known situation for the noncommutative torus based on the algebra of the noncommuting position operators only. When considering the dynamics of a free particle for any of the identified representations, no observable effect of noncommutativity is implied.
This paper is devoted to the construction and analysis of the Wigner functions for noncommutative quantum mechanics, their marginal distributions, and star-products, following a technique developed earlier, viz, using the unitary irreducible representations of the group GNC, which is the three fold central extension of the Abelian group of ℝ4. These representations have been exhaustively studied in earlier papers. The group GNC is identified with the kinematical symmetry group of noncommutative quantum mechanics of a system with two degrees of freedom. The Wigner functions studied here reflect different levels of non-commutativity—both the operators of position and those of momentum not commuting, the position operators not commuting and finally, the case of standard quantum mechanics, obeying the canonical commutation relations only.
In two-dimensional space a subtle point that for the case of both space–space and momentum–momentum non-commuting, different from the case of only space–space non-commuting, the deformed Heisenberg–Weyl algebra in non-commutative space is not completely equivalent to the undeformed Heisenberg–Weyl algebra in commutative space is clarified. It follows that there is no well-defined procedure to construct the deformed position–position coherent state or the deformed momentum–momentum coherent state from the undeformed position–momentum coherent state. Identifications of the deformed position–position and deformed momentum–momentum coherent states with the lowest energy states of a cold Rydberg atom in special conditions and a free particle, respectively, are demonstrated.
We show that a possible violation of the Robertson-Schr\"odinger uncertainty principle may signal the existence of a deformation of the Heisenberg-Weyl algebra. More precisely, we prove that any Gaussian in phase space (even if it violates the Robertson-Schr\"odinger uncertainty principle) will always be a quantum state of an appropriate noncommutative extension of quantum mechanics. Conversely, all canonical noncommutative extensions of quantum mechanics display states that violate the Robertson-Schr\"odinger uncertainty principle.
"The possibility of describing noncommuting operators in quantum mechanics by classical type functions, and the associated expression of operator multiplication, is of considerable interest. The well known Wigner-Weyl-Moyal theory is an important example. Another is the diagonal representation of operators using standard coherent states. We develop general necessary and sufficient conditions for the existence of the diagonal representation in the context of a family of generalised coherent states associated with any unitary irreducible Lie group representation. Several examples illustrating these conditions, and interesting results in the Heisenberg-Weyl case, are presented".