Deformation quantization of noncommutative quantum mechanics and dissipation

Abstract

We review the main features of the Weyl-Wigner formulation of noncommutative quantum mechanics. In particular, we present a $\star$-product and a Moyal bracket suitable for this theory as well as the concept of noncommutative Wigner function. The properties of these quasi-distributions are discussed as well as their relation to the sets of ordinary Wigner functions and positive Liouville probability densities. Based on these notions we propose criteria for assessing whether a commutative regime has emerged in the realm of noncommutative quantum mechanics. To induce this noncommutative-commutative transition, we couple a particle to an external bath of oscillators. The master equation for the Brownian particle is deduced.

7 pages, Latex file, Based on a talk presented by Joao Prata at D.I.C.E. 2006, Piombino, Italy

References in corpus (8)

• Noncommutative Field Theory
• Noncommutative Gravitational Quantum Well
• Alternative Derivation of the Hu-Paz-Zhang Master Equation for Quantum Brownian Motion
• Weyl-Wigner Formulation of Noncommutative Quantum Mechanics
• Exact positivity of the Wigner and P-functions of a Markovian open system
• Generalized Weyl-Wigner map and Vey quantum mechanics
• Dynamical origin of the ${\star }_{\theta }$-noncommutativity in field theory from quantum mechanics
• Environment-Induced Decoherence in Noncommutative Quantum Mechanics

Cited by in corpus (2)

• Probing phase-space noncommutativity through quantum beating, missing information and the thermodynamic limit
• Probing phase-space noncommutativity through quantum mechanics and thermodynamics of free particles and quantum rotors