The Landau Problem and Noncommutative Quantum Mechanics

Gamboa, J.; Méndez, F.; Loewe, M.; Rojas, J. C.

doi:10.1142/s0217732301005345

Cited by 164 publications

(145 citation statements)

References 5 publications

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“…We conclude that the known similarity between an oscillator in a noncommutative space and a particle in a constant magnetic field [10,11,13,14,20,21] can be extended to a relativistic motion. The problem is exactly solvable in the spinless cases.…”

Section: Discussionmentioning

confidence: 80%

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The Klein-Gordon and the Dirac Oscillators in a Noncommutative Space

Mirza

Mohadesi

2004

Commun. Theor. Phys.

151

127

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We study the Dirac and the klein-Gordon oscillators in a noncommutative space. It is shown that the Klein-Gordon oscillator in a noncommutative space has a similar behaviour to the dynamics of a particle in a commutative space and in a constant magnetic field. The Dirac oscillator in a noncommutative space has a similar equation to the equation of motion for a relativistic fermion in a commutative space and in a magnetic field, however a new exotic term appears, which implies that a charged fermion in a noncommutative space has an electric dipole moment.

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Section: Discussionmentioning

confidence: 80%

“…[6]. A simple insight on the role of noncommutativity in field theory can be obtained by studying the one particle sector, which prompted an interest in the study of noncommutative quantum mechanics (NCQM) [7,8,9,10,11,12,13,14]. In these studies some attention was paid to two-dimensional NCQM and its relation to the Landau problem.…”

Section: Introductionmentioning

confidence: 99%

The Klein-Gordon and the Dirac Oscillators in a Noncommutative Space

Mirza

Mohadesi

2004

Commun. Theor. Phys.

151

127

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“…This can be regarded as a "true" quantum field theory limit, in which both ultraviolet and infrared cutoffs have been removed. It is defined by 18) and it thereby corresponds to keeping all length scales in between the infrared and ultraviolet regimes (no "zooming" in or out). We believe that such intermediate scaling limits will only lead to an interacting quantum field theory if some non-planar limit of the matrix model is used.…”

Section: Ultraviolet Limitmentioning

confidence: 99%

Exact Solution of Quantum Field Theory on Noncommutative Phase Spaces

Langmann

Szabo

Zarembo

2004

J. High Energy Phys.

107

198

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We present the exact solution of a scalar field theory defined with noncommuting position and momentum variables. The model describes charged particles in a uniform magnetic field and with an interaction defined by the Groenewold-Moyal star-product. Explicit results are presented for all Green's functions in arbitrary even spacetime dimensionality. Various scaling limits of the field theory are analysed non-perturbatively and the renormalizability of each limit examined. A supersymmetric extension of the field theory is also constructed in which the supersymmetry transformations are parametrized by differential operators in an infinite-dimensional noncommutative algebra.

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“…Applications of noncommutative theories were also found in condensed matter physics, for instance in the Quantum Hall effect [13? ] and the non-commutative Landau problem [15,16,17] i.e., a quantum particle in the noncommutative plane, coupled to a constant magnetic field with a constant selected θ parameter as usual.In this letter, we generalize the quantum mechanics in non-commutative geometry by promoting the θ parameter with a new field obeying its own field equations. Note that some authors, (for example [18]) introduced a position dependent θ field using a Kontsevich product [19] in the study of gauge theory.…”

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confidence: 99%

Monopole and Berry phase in momentum space in noncommutative quantum mechanics

Bérard

Mohrbach

2004

Phys. Rev. D

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To build genuine generators of the rotations group in noncommutative quantum mechanics, we show that it is necessary to extend the noncommutative parameter θ to a field operator, which one proves to be only momentum dependent. We find consequently that this field must be obligatorily a dual Dirac monopole in momentum space. Recent experiments in the context of the anomalous Hall effect provide evidence for a monopole in the crystal momentum space. We suggest a connection between the noncommutative field and the Berry curvature in momentum space which is at the origine of the anomalous Hall effect. . Noncommutative gauge theories were also found as being naturally related to string and M-theory [4].In this framework an antisymmetric θ µν parameter usually taken to be constant [5,11] is introduced in the commutation relation of the coordinates in the space time manifold [x µ , x υ ] = iθ µν . This relation leads to the violation of the Lorentz symmetry, a possibility which is intensively studied theoretically and experimentally [12]. Applications of noncommutative theories were also found in condensed matter physics, for instance in the Quantum Hall effect [13? ] and the non-commutative Landau problem [15,16,17] i.e., a quantum particle in the noncommutative plane, coupled to a constant magnetic field with a constant selected θ parameter as usual.In this letter, we generalize the quantum mechanics in non-commutative geometry by promoting the θ parameter with a new field obeying its own field equations. Note that some authors, (for example [18]) introduced a position dependent θ field using a Kontsevich product [19] in the study of gauge theory. Contrary to these approaches we find that the θ field must be momentum dependent.The physical motivations of our work are twofold: (i) For a constant θ field, we show that a quantum particle in a harmonic potential has a behavior similar to a particle in a constant magnetic field θ in standard quantum mechanics, since a paramagnetic term appears in the Hamiltonian. Moreover the particle in the presence of the θ field acquires an effective dual mass in the same way that an electron moving in a periodic potential in solid state physics. Thus it is legitimate to interpret this field like a field having properties of the vacuum.In this context it is natural to extend the theory to a non-constant field. This proposal is strongly enforced by the lack of rotation generators in noncommutative space with a constant θ parameter, i.e. the angular momentum does not satisfy the usual angular momentum algebra. We then show that this θ field is only momentum dependent and that the requirement of the angular momentum algebra, that is the existence of an angular momentum, necessarily imposes a dual Dirac monopole in momentum space field configuration. Thereafter we will intensely use the concept of duality between the quantities defined in momentum space compared with those defined in the position space.(ii) The second motivation comes for recent theoretical works [20] concerning the anomal...

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The Landau Problem and Noncommutative Quantum Mechanics

Cited by 164 publications

References 5 publications

The Klein-Gordon and the Dirac Oscillators in a Noncommutative Space

The Klein-Gordon and the Dirac Oscillators in a Noncommutative Space

Exact Solution of Quantum Field Theory on Noncommutative Phase Spaces

Monopole and Berry phase in momentum space in noncommutative quantum mechanics

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