Weyl-Wigner Formulation of Noncommutative Quantum Mechanics

Catarina Bastos, Orfeu Bertolami, Nuno Costa Dias, João Nuno Prata

Published:

We address the phase space formulation of a noncommutative extension of
quantum mechanics in arbitrary dimension, displaying both spatial and momentum
noncommutativity. By resorting to a covariant generalization of the Weyl-Wigner
transform and to the Seiberg-Witten map we construct an isomorphism between the
operator and the phase space representations of the extended Heisenberg
algebra. This map provides a systematic approach to derive the entire structure
of noncommutative quantum mechanics in phase space. We construct the extended
starproduct, Moyal bracket and propose a general definition of noncommutative
states. We study the dynamical and eigenvalue equations of the theory and prove
that the entire formalism is independent of the particular choice of
Seiberg-Witten map. Our approach unifies and generalizes all the previous
proposals for the phase space formulation of noncommutative quantum mechanics.
For concreteness we rederive these proposals by restricting our formalism to
some 2-dimensional spaces.