Wigner Functions for the Landau Problem in Noncommutative Spaces

Dayi, Omer Faruk; Kelleyane, Lara T.

doi:10.1142/s0217732302008356

Cited by 66 publications

(60 citation statements)

References 5 publications

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“…(We thank Dr. J. Prata for bringing to our attention this reference.) Dayi and Kelleyane [29] derived the Wigner functions for the Landau problem when the plane is noncommutative. They introduced a generalized *-genvalue equation for this problem and found solutions for it.…”

Section: A Superstar Wigner-moyal Equationmentioning

confidence: 99%

The classical and commutative limits of noncommutative quantum mechanics: a superstar * Wigner-Moyal equation

Eftekharzadeh

2005

Braz. J. Phys.

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We are interested in the similarities and differences between the quantum-classical (Q-C) and the noncommutative-commutative (NC-Com) correspondences. As one useful platform to address this issue we derive the superstar Wigner-Moyal equation for noncommutative quantum mechanics (NCQM). A superstar -product combines the usual phase space * star and the noncommutative star-product. Having dealt with subtleties of ordering present in this problem we show that the Weyl correspondence of the NC Hamiltonian has the same form as the original Hamiltonian, but with a non-commutativity parameter θ-dependent, momentumdependent shift in the coordinates. Using it to examine the classical and the commutative limits, we find that there exist qualitative differences between these two limits. Specifically, if θ = 0 there is no classical limit. Classical limit exists only if θ → 0 at least as fast as → 0, but this limit does not yield Newtonian mechanics, unless the limit of θ/ vanishes as θ → 0. For another angle towards this issue we formulate the NC version of the continuity equation both from an explicit expansion in orders of θ and from a Noether's theorem conserved current argument. We also examine the Ehrenfest theorem in the NCQM context. AimIn this program of investigation we ask the question whether there is any structural similarity or conceptual connection between the quantum-classical (Q-C) and the noncommutative-commutative (NC-Com) correspondences. We want to see if our understanding of the quantum-classical correspondence acquired in the last decade can aid us in any way to understand the physical attributes and meanings of a noncommutative space from the vantage point of the ordinary commutative space. We find that the case of quantum to classical transition in the context of noncommutative geometry is quite different from that in the ordinary (commutative) space. Specifically, if θ = 0 there is no classical limit. Classical limit exists only if θ → 0 at least as fast as → 0, but this limit does not yield Newtonian mechanics, unless the limit of θ/ vanishes as θ → 0. We make explicit this relationship by deriving a superstar Wigner-Moyal equation for noncommutative quantum mechanics (NCQM) and identifying the difference between the classical and the commutative limits. A superstar -product combines the usual phase space * star and the noncommutative -product [1].In this paper we focus on the nature of the commutative and classical limits of noncommutative quantum physics. We point out some subtleties which arise due to the ordering problem. When these issues are properly addressed we show that the classical correspondent to the NC Hamiltonian is indeed one with a θ-dependent, momentum-dependent shift in the coordinates. For another angle towards this issue we formulate the NC version of the continuity equation both from an explicit expansion in orders of θ and from a Noether's theorem conserved current argument. We also examine the Ehrenfest theorem in the NCQM context. I. CRITERIA FOR CLASSICALITYWe open this di...

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Section: A Superstar Wigner-moyal Equationmentioning

confidence: 99%

The classical and commutative limits of noncommutative quantum mechanics: a superstar * Wigner-Moyal equation

Eftekharzadeh

2005

Braz. J. Phys.

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“…This was initially motivated by studies of the low energy effective theory of D-brane with a nonzero Neveu-Schwarz B field background. Many efforts have been devoted to the various aspects of noncommutative quantum mechanics, such as Quantum Hall effect [10,14], Landau problem on noncommutative plane [1,11,16], the two-dimensional quantum system with arbitrary central potential [2,17], and the DKP oscillator in a noncommutative space [15], etc. The noncommutative phase space is characterized by the fact that their coordinate operators satisfy the equation [4],…”

Section: Introductionmentioning

confidence: 99%

DKP Oscillator with Spin-0 in Three-dimensional Noncommutative Phase Space

et al. 2010

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“…The recent interest in non-commutative quantum mechanics was motivated by studies of the low-energy effective theory of D-branes in the background of a Neveu-Schwarz B-field in a non-commutative space [17][18][19][20]. Among recent applications, let us mention the quantum Hall effect on noncommutative spaces [21][22][23][24], the Landau problem on the non-commutative plane [25][26][27][28], planar quantum systems with central potentials [29,30], and studies of the relativistic DKP oscillator in a non-commutative space [31][32][33][34][35]. Papers investigating Galilei-invariant systems with non-commutative geometry are in Refs.…”

Section: Introductionmentioning

confidence: 99%

Spinless Duffin-Kemmer-Petiau Oscillator in a Galilean Non-commutative Phase Space

2012

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We examine Galilei-invariant linear wave equations in a non-commutative phase space. Specifically, we establish and solve the Galilean covariant Duffin-Kemmer-Petiau equation for spin-0 fields in a harmonic oscillator potential. We obtain these wave equations with a Galilean covariant approach, based on a (4 + 1)-dimensional manifold with light-cone coordinates followed by a reduction to a (3 + 1)-dimensional spacetime. We find the exact wave functions and their energy levels, and we examine the effects of non-commutativity.

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Wigner Functions for the Landau Problem in Noncommutative Spaces

Cited by 66 publications

References 5 publications

The classical and commutative limits of noncommutative quantum mechanics: a superstar * Wigner-Moyal equation

The classical and commutative limits of noncommutative quantum mechanics: a superstar * Wigner-Moyal equation

DKP Oscillator with Spin-0 in Three-dimensional Noncommutative Phase Space

Spinless Duffin-Kemmer-Petiau Oscillator in a Galilean Non-commutative Phase Space

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