{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,17]],"date-time":"2026-03-17T14:57:49Z","timestamp":1773759469004,"version":"3.50.1"},"reference-count":53,"publisher":"AIP Publishing","issue":"7","content-domain":{"domain":["pubs.aip.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2008,7,1]]},"abstract":"<jats:p>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\u2013Wigner 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.<\/jats:p>","DOI":"10.1063\/1.2944996","type":"journal-article","created":{"date-parts":[[2008,7,31]],"date-time":"2008-07-31T23:14:53Z","timestamp":1217546093000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":102,"title":["Weyl\u2013Wigner formulation of noncommutative quantum mechanics"],"prefix":"10.1063","volume":"49","author":[{"given":"Catarina","family":"Bastos","sequence":"first","affiliation":[{"name":"Instituto Superior T\u00e9cnico 1 Departamento de F\u00edsica, , Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugal"}]},{"given":"Orfeu","family":"Bertolami","sequence":"additional","affiliation":[{"name":"Instituto Superior T\u00e9cnico 1 Departamento de F\u00edsica, , Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugal"}]},{"given":"Nuno Costa","family":"Dias","sequence":"additional","affiliation":[{"name":"Universidade Lus\u00f3fona de Humanidades e Tecnologias 2 Departamento de Matem\u00e1tica, , Av. Campo Grande, 376, 1749-024 Lisboa, Portugal"}]},{"given":"Jo\u00e3o Nuno","family":"Prata","sequence":"additional","affiliation":[{"name":"Universidade Lus\u00f3fona de Humanidades e Tecnologias 2 Departamento de Matem\u00e1tica, , Av. Campo Grande, 376, 1749-024 Lisboa, Portugal"}]}],"member":"317","published-online":{"date-parts":[[2008,7,2]]},"reference":[{"key":"2023062803572441400_c1","doi-asserted-by":"publisher","first-page":"151602","DOI":"10.1103\/PhysRevLett.88.151602","volume":"88","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"2023062803572441400_c2","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1016\/S0370-2693(01)00339-2","volume":"505","year":"2001","journal-title":"Phys. Lett. B"},{"key":"2023062803572441400_c3","doi-asserted-by":"publisher","first-page":"10097","DOI":"10.1088\/0305-4470\/34\/47\/314","volume":"34","year":"2001","journal-title":"J. Phys. 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