The $\star$-value Equation and Wigner Distributions in Noncommutative Heisenberg algebras

Abstract

We consider the quantum mechanical equivalence of the Seiberg-Witten map in the context of the Weyl-Wigner-Groenewold-Moyal phase-space formalism in order to construct a quantum mechanics over noncommutative Heisenberg algebras. The formalism is then applied to the exactly soluble Landau and harmonic oscillator problems in the 2-dimensional noncommutative phase-space plane, in order to derive their correct energy spectra and corresponding Wigner distributions. We compare our results with others that have previously appeared in the literature.

arXiv comment19 pages

References in corpus (11)

• String Theory and Noncommutative Geometry
• Hydrogen Atom Spectrum and the Lamb Shift in Noncommutative QED
• Quantum mechanics on the noncommutative plane and sphere
• Noncommutative Field Theories and Gravity
• Isotropic representation of noncommutative 2D harmonic oscillator
• Gauge Theory on a Quantum Phase Space
• The noncommutative harmonic oscillator in more than one dimensions
• Noncommutative 3D harmonic oscillator
• Phases in noncommutative quantum mechanics on (pseudo)sphere
• Wigner Functions for the Landau Problem in Noncommutative Spaces
• A Note on Noncommutative and False Noncommutative spaces

Cited by in corpus (25)

• Weyl-Wigner Formulation of Noncommutative Quantum Mechanics
• Magnetic fields in noncommutative quantum mechanics
• Landau Analog Levels for Dipoles in the Noncommutative Space and Phase Space
• Entanglement due to noncommutativity in the phase-space
• Phase-space noncommutative formulation of Ozawa's uncertainty principle
• Probing phase-space noncommutativity through quantum mechanics and thermodynamics of free particles and quantum rotors
• Exact solution of two dimensional Dunkl harmonic oscillator in Non-Commutative phase-space
• Violation of the Robertson-Schrödinger uncertainty principle and non-commutative quantum mechanics
• A Deformation Quantization Theory for Non-Commutative Quantum Mechanics
• Phase-space noncommutative extension of the Robertson-Schroedinger formulation of Ozawa's uncertainty principle
• Bell operator and Gaussian squeezed states in noncommutative quantum mechanics
• Exact master equation for a noncommutative Brownian particle
• Noncommutative Field Theory from Quantum Mechanical Space-Space Noncommutativity
• Deformed squeezed states in noncommutative phase space
• Non-Hermitian noncommutative quantum mechanics
• A new kind of representations on noncommutative phase space
• Heat flow and noncommutative quantum mechanics in phase-space
• Noncommutative phase-space effects in thermal diffusion of Gaussian states
• Coherent quantum squeezing due to the phase space noncommutativity
• Quantum cloning and teleportation fidelity in the noncommutative phase-space
• Negativity-Mutual Information conversion and coherence in two-coupled harmonic oscillators
• Entanglement and separability in the noncommutative phase-space scenario
• Quantum information aspects of noncommutative quantum mechanics
• A Phase-Space Noncommutative Picture of Nuclear Matter
• Emergent time crystals from phase-space noncommutative quantum mechanics