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Deformation quantization of noncommutative quantum mechanics and dissipation

Research paper by C. Bastos, O. Bertolami, N. C. Dias, J. N. Prata

Indexed on: 21 Dec '06Published on: 21 Dec '06Published in: High Energy Physics - Theory



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.