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Published Online: 02 July 2008
Accepted: May 2008

Weyl–Wigner formulation of noncommutative quantum mechanics

Journal of Mathematical Physics 49, 072101 (2008); https://doi.org/10.1063/1.2944996
Catarina Bastos1, a), Orfeu Bertolami1, b), Nuno Costa Dias2, c), and João Nuno Prata2, d)
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ABSTRACT
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.

ACKNOWLEDGMENTS

The work of C.B. is supported by Fundação para a Ciência e a Tecnologia (FCT) under Fellowship No. SFRH/BD/24058/2005. The work of N.C.D. and J.N.P. was partially supported by Grant Nos. POCTI/MAT/45306/2002 and POCTI/0208/2003 of the FCT.

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