Deformation quantization of noncommutative quantum mechanics

Abstract

The Wigner phase-space distribution function provides the basis for Moyal's deformation quantization alternative to the more conventional Hilbert space and path integral quantization. In this paper, we investigate the basic aspects of deformation quantization for noncommutative quantum mechanics (NCQM). We first prove some general relations of the Weyl correspondence in non-commuting phase-space. Then we derive explicit form of the Wigner Function (WF) for NCQM starting from fundamental principle of the Weyl correspondence, and show that it satisfies a generalized -genvalue equation. We also demonstrate that the new WFs possess orthonormality and completeness, so they can be used as a basis to expand all phase-space functions. Some example is discussed to support our results.