Noncommutative quantum mechanics as a constrained system

TL;DR

This paper demonstrates that noncommutative quantum mechanics can be derived from a constrained system framework, allowing quantization directly in noncommutative variables and revealing two phases depending on the potential.

Contribution

It introduces a novel constrained system approach to derive noncommutative quantum mechanics, applicable to different potential types, and connects it with the Moyal product formulation.

Findings

Two formulations for noncommutative quantum mechanics are presented.

Quantization in noncommutative variables is achieved.

The approach explains the emergence of two phases in NQM.

Abstract

It is shown that quantum mechanics on noncommutative spaces (NQM) can be obtained by the canonical quantization of some underlying second class constrained system formulated in extended configuration space. It leads, in particular, to an intriguing possibility of quantization in terms of the initial (noncommutative) variables. Two different formulations are discissed. The first one is appropriate for at most quadratic potential. The noncommutativity parameter and rank of matrix of the constraint brackets depend on the potential. It explains appearance of two phases of the resulting NQM. The second formulation is appropriate for an arbitrary potential. In both cases the corresponding Lagrangian action is presented and quantized, which leads to quantum mechanics with ordinary product replaced by the Moyal product.