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Wigner Measures in Noncommutative Quantum Mechanics

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Abstract

We study the properties of quasi-distributions or Wigner measures in the context of noncommutative quantum mechanics. In particular, we obtain necessary and sufficient conditions for a phase-space function to be a noncommutative Wigner measure, for a Gaussian to be a noncommutative Wigner measure, and derive certain properties of the marginal distributions which are not shared by ordinary Wigner measures. Moreover, we derive the Robertson-Schrödinger uncertainty principle. Finally, we show explicitly how the set of noncommutative Wigner measures relates to the sets of Liouville and (commutative) Wigner measures.

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Authors and Affiliations

  1. Departamento de Física and Instituto de Plasmas e Fusão Nuclear, Instituto Superior Técnico, Avenida Rovisco Pais 1, 1049-001, Lisboa, Portugal

    C. Bastos

  2. Departamento de Matemática, Universidade Lusófona de Humanidades e Tecnologias, Av. Campo Grande 376, 1749-024, Lisboa, Portugal

    N. C. Dias & J. N. Prata

  3. Grupo de Física Matemática, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003, Lisboa, Portugal

    N. C. Dias & J. N. Prata

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  1. C. Bastos
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  2. N. C. Dias
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  3. J. N. Prata
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Correspondence to N. C. Dias.

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Communicated by H.-T. Yau

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Bastos, C., Dias, N.C. & Prata, J.N. Wigner Measures in Noncommutative Quantum Mechanics. Commun. Math. Phys. 299, 709–740 (2010). https://doi.org/10.1007/s00220-010-1109-5

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