Exact master equation for a noncommutative Brownian particle

TL;DR

This paper derives an exact master equation for a noncommutative Brownian particle, revealing how noncommutativity influences quantum dynamics and the transition to classical behavior.

Contribution

It provides the first exact derivation of the Hu-Paz-Zhang master equation in a noncommutative setting, including new results on equilibrium states and decoherence times.

Findings

Exact master equation derived for noncommutative Brownian motion

Equilibrium Wigner distribution for noncommutative oscillators obtained

Transition scale from noncommutative to ordinary quantum mechanics estimated

Abstract

We derive the Hu-Paz-Zhang master equation for a Brownian particle linearly coupled to a bath of harmonic oscillators on the plane with spatial noncommutativity. The results obtained are exact to all orders in the noncommutative parameter. As a by-product we derive some miscellaneous results such as the equilibrium Wigner distribution for the reservoir of noncommutative oscillators, the weak coupling limit of the master equation and a set of sufficient conditions for strict purity decrease of the Brownian particle. Finally, we consider a high-temperature Ohmic model and obtain an estimate for the time scale of the transition from noncommutative to ordinary quantum mechanics. This scale is considerably smaller than the decoherence scale.