Wigner Measures in Noncommutative Quantum Mechanics

Abstract

We study the properties of quasi-distributions or Wigner measures in the context of noncommutative quantum mechanics. In particular, we obtain necessary and sufficient conditions for a phase-space function to be a noncommutative Wigner measure, for a Gaussian to be a noncommutative Wigner measure, and derive certain properties of the marginal distributions which are not shared by ordinary Wigner measures. Moreover, we derive the Robertson-Schrödinger uncertainty principle. Finally, we show explicitly how the set of noncommutative Wigner measures relates to the sets of Liouville and (commutative) Wigner measures.

31 pages, Latex file

References in corpus (4)

• Weyl-Wigner Formulation of Noncommutative Quantum Mechanics
• Berry Phase in the Gravitational Quantum Well and the Seiberg-Witten map
• Quantum States and Hardy's Formulation of the Uncertainty Principle : a Symplectic Approach
• Exact master equation for a noncommutative Brownian particle

Cited by in corpus (24)

• A comparative review of four formulations of noncommutative quantum mechanics
• Aspects of Phase-Space Noncommutative Quantum Mechanics
• Composite system in noncommutative space and the equivalence principle
• Entropic Gravity, Phase-Space Noncommutativity and the Equivalence Principle
• 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
• A Deformation Quantization Theory for Non-Commutative Quantum Mechanics
• Natural Cutoffs via Compact Symplectic Manifolds
• Phase-space noncommutative extension of the Robertson-Schroedinger formulation of Ozawa's uncertainty principle
• Bell operator and Gaussian squeezed states in noncommutative quantum mechanics
• A metaplectic perspective of uncertainty principles in the Linear Canonical Transform domain
• Heat flow and noncommutative quantum mechanics in phase-space
• Quantum mechanics in phase space: The Schrödinger and the Moyal representations
• Wigner Functions for Noncommutative Quantum Mechanics: a group representation based construction
• Noncommutative phase-space effects in thermal diffusion of Gaussian states
• A symplectic extension map and a new Shubin class of pseudo-differential operators
• States in the Hilbert space formulation and the phase space formulation of quantum mechanics
• Uncertainty relations for a non-canonical phase-space noncommutative algebra
• Tuning the separability in noncommutative space
• Deformation of Noncommutative Quantum Mechanics
• The orthogonal Lie algebra of operators: ideals and derivations