Weyl-Wigner formulation of noncommutative quantum mechanics

Journal Article · Tue Jul 15 04:00:00 EDT 2008 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.2944996· OSTI ID:21100334

Bastos, Catarina; Bertolami, Orfeu ^[1]; Dias, Nuno Costa; Prata, Joao Nuno ^[2]

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1. Departamento de Fisica, Instituto Superior Tecnico, Avenida Rovisco Pais 1, 1049-001 Lisboa (Portugal)
2. Departamento de Matematica, Universidade Lusofona de Humanidades e Tecnologias, Av. Campo Grande, 376, 1749-024 Lisboa (Portugal)

We address the phase-space formulation of a noncommutative extension of quantum mechanics in arbitrary dimension, displaying both spatial and momentum noncommutativities. By resorting to a covariant generalization of the Weyl-Wigner transform and to the Darboux map, we construct an isomorphism between the operator and the phase-space representations of the extended Heisenberg algebra. This map provides a systematic approach to derive the entire structure of noncommutative quantum mechanics in phase space. We construct the extended star product and Moyal bracket and propose a general definition of noncommutative states. We study the dynamical and eigenvalue equations of the theory and prove that the entire formalism is independent of the particular choice of the Darboux map. Our approach unifies and generalizes all the previous proposals for the phase-space formulation of noncommutative quantum mechanics. For concreteness we rederive these proposals by restricting our formalism to some two-dimensional spaces.

• OSTI ID: 21100334
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 7 Vol. 49; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ALGEBRA
COMMUTATION RELATIONS
EIGENFUNCTIONS
EIGENVALUES
EQUATIONS
MAPS
PHASE SPACE
QUANTUM MECHANICS
TWO-DIMENSIONAL CALCULATIONS
WEYL UNIFIED THEORY
WIGNER THEORY