A Deformation Quantization Theory for Non-Commutative Quantum Mechanics

Abstract

We show that the deformation quantization of non-commutative quantum mechanics previously considered by Dias and Prata can be expressed as a Weyl calculus on a double phase space. We study the properties of the star-product thus defined, and prove a spectral theorem for the star-genvalue equation using an extension of the methods recently initiated by de Gosson and Luef.

Submitted for publication

References in corpus (12)

• Weyl-Wigner Formulation of Noncommutative Quantum Mechanics
• Berry Phase in the Gravitational Quantum Well and the Seiberg-Witten map
• Magnetic fields in noncommutative quantum mechanics
• Black Holes and Phase Space Noncommutativity
• Wigner Measures in Noncommutative Quantum Mechanics
• The singularity problem and phase-space noncanonical noncommutativity
• Newton equation for canonical, Lie-algebraic and quadratic deformation of classical space
• Exact master equation for a noncommutative Brownian particle
• Quantum theta functions and Gabor frames for modulation spaces
• What kind of noncommutative geometry for quantum gravity ?
• On the Usefulness of Modulation Spaces in Deformation Quantization
• Minimalisation of uncertainty relations in noncommutative quantum mechanics

Cited by in corpus (16)

• Noncommutative Graphene
• Entanglement due to noncommutativity in the phase-space
• Phase-space noncommutative formulation of Ozawa's uncertainty principle
• Non-Canonical Phase-Space Noncommutativity and the Kantowski-Sachs singularity for Black Holes
• Violation of the Robertson-Schrödinger uncertainty principle and 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
• Noncommutative Mapping from the symplectic formalism
• A metaplectic perspective of uncertainty principles in the Linear Canonical Transform domain
• Quantum mechanics in phase space: The Schrödinger and the Moyal representations
• Triply Extended Group of Translations of ${\mathbb{R}}^4$ as Defining Group of NCQM: relation to various gauges
• Metaplectic formulation of the Wigner transform and applications
• A symplectic extension map and a new Shubin class of pseudo-differential operators
• Entanglement and separability in the noncommutative phase-space scenario
• New parameters of Non-commutativity in Quantum Mechanics
• Deformation of Noncommutative Quantum Mechanics