Remarks on the formulation of quantum mechanics on noncommutative phase spaces

Muthukumar, B.

doi:10.1088/1126-6708/2007/01/073

Cited by 7 publications

(14 citation statements)

References 35 publications

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“…In Refs. [35,36,37] the authors consider the two dimensional plane with only spatial noncommutativity. In this case, the extended Heisenberg algebra simplifies considerably:…”

Section: Comparison With Other Proposalsmentioning

confidence: 99%

“…The results of Refs. [35,36,37] are nevertheless useful. We argued that our formula (71) is apparently dependent on the choice of SW map, but not the physical predictions.…”

Section: Lemma 41mentioning

confidence: 99%

“…The most natural representation of the algebra (1) formulae for the quasi-distribution on the plane with only spatial noncommutativity [35,36,37] or with phase space noncommutativity [38].…”

Section: Introductionmentioning

confidence: 99%

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Weyl–Wigner formulation of noncommutative quantum mechanics

Bastos¹,

Bertolami²,

Dias³

et al. 2008

Journal of Mathematical Physics

119

219

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We address the phase space formulation of a noncommutative extension of quantum mechanics in arbitrary dimension, displaying both spatial and momentum noncommutativity. By resorting to a covariant generalization of the Weyl-Wigner transform and to the Seiberg-Witten 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 starproduct, 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 Seiberg-Witten 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 2-dimensional spaces.2

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“…In Refs. [35,36,37] the authors consider the two dimensional plane with only spatial noncommutativity. In this case, the extended Heisenberg algebra simplifies considerably:…”

Section: Comparison With Other Proposalsmentioning

confidence: 99%

“…The results of Refs. [35,36,37] are nevertheless useful. We argued that our formula (71) is apparently dependent on the choice of SW map, but not the physical predictions.…”

Section: Lemma 41mentioning

confidence: 99%

See 1 more Smart Citation

Weyl–Wigner formulation of noncommutative quantum mechanics

Bastos¹,

Bertolami²,

Dias³

et al. 2008

Journal of Mathematical Physics

119

219

View full text Add to dashboard Cite

show abstract

“…However, in consideration that the momentum operators are defined as the derivatives with respect to coordinates, it is natural to consider a further extension with noncommutative momentum operators defined by the algebra (we will use the notation P μ to denote the momentum vectors in this space) [35,41,[45][46][47][48][49][50][51],…”

Section: Noncommutative Hamiltonianmentioning

confidence: 99%

“…Physical effects of the above extension have been studied in various aspects [35,41,[45][46][47][48][49][50][51]. And recently, a more general extension on the generators of the whole Poincaré group was conducted in Ref.…”

Section: Noncommutative Hamiltonianmentioning

confidence: 99%

Probing noncommutativities of phase space by using persistent charged current and its asymmetry

Ren

Wang

2018

Phys. Rev. D

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Nontrivial algebra structures of the coordinate and momentum operators are potentially important for describing possible new physics. The persistent charged current in a metal ring is expected to be sensitive to the nontrivial dynamics due to noncommutativities of phase space. In this paper, we propose a new asymmetric observable for probing the noncommutativity of momentum operators. We also analyzed the temperature dependence of this observable, and we find that the asymmetry holds at a finite temperature. The critical temperature, above which the correction due to coordinate noncommutativity is negligible, is also derived.

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Deformation of noncommutative quantum mechanics

Jiang

Chowdhury

2016

Journal of Mathematical Physics

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In this paper, the Lie group G α,β,γ NC , of which the kinematical symmetry group G NC of noncommutative quantum mechanics (NCQM) is a special case due to fixed nonzero α, β and γ, is three-parameter deformation quantized using the method suggested by Ballesteros and Musso in [1]. A certain family of QUE algebras, corresponding to G α,β,γ NC with two of the deformation parameters approaching zero, is found to be in agreement with the existing results of the literature on quantum Heisenberg group. Finally, we dualize the underlying QUE algebra to obtain an expression for the underlying * -product between smooth functions on G α,β,γ NC . I IntroductionNoncommutative quantum mechanics (NCQM) is a vibrant field of research these days. In addition to the canonical position-momentum noncommutativity, it also demands for the noncommutativity between the two position coordinates and the two momenta coordinates respectively for a system of two degrees of freedom. The noncommutativity of the position coordinates was first proposed by H. S. Snyder [13] in his quest for the quantized nature of space-time. Such a model of space-time in scales as small as Planck length is also proposed, among others by Doplicher et al. [11] in order to avoid creation of microscopic black holes to the effect of losing the operational meaning of localization in space-time. The noncommutativity of the momenta coordinates, on the other hand, emerges if one introduces a constant background magnetic field to the underlying system of two degrees of freedom.The defining group of a two-dimensional quantum mechanical system is the wellknown Heisenberg group (denoted by G H in the sequel). What runs parallel to G H in two dimensions, is the triply extended group of translations of R 4 (denoted as G NC in the

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Remarks on the formulation of quantum mechanics on noncommutative phase spaces

Cited by 7 publications

References 35 publications

Weyl–Wigner formulation of noncommutative quantum mechanics

Weyl–Wigner formulation of noncommutative quantum mechanics

Probing noncommutativities of phase space by using persistent charged current and its asymmetry

Deformation of noncommutative quantum mechanics

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