Canonical Structure of Noncommutative Quantum Mechanics as Constraint System

TL;DR

This paper investigates the canonical structure of noncommutative quantum mechanics with both coordinate and momentum noncommutativity, using the projection operator method and star-product to derive exact commutator algebras.

Contribution

It introduces a systematic approach to derive the exact canonical structure in noncommutative quantum mechanics, including all orders of noncommutativity parameters, and relates it to existing methods like Seiberg-Witten map.

Findings

Derived the commutator algebra including all orders of noncommutativity.

Constructed an exact canonical conjugate set equivalent to known formulations.

Discussed alternative Lagrangians for realizing noncommutativities.

Abstract

Starting with the first-order singular Lagrangian, the canonical structure in the noncommutative quantum mechanics with the noncommutativities both of coordinates and momenta is investgated. Using the projection operator method (POM) for the constraint systems and the constraint star-product, the noncommutative quantum system is constructed and the commutator algebra of {\it projected} canonically conjugate set(CCS) of the system is derived in the form including all orders of the noncommutativity parameters. We discuss the alternative CCS, which obeys the ordinary noncommutative commutator algebra. The {\it exact} CCS is constructed in the framework of the POM, and which is shown to be equivalent to the CCS constructed through the Seiberg-Witten map and the Bopp shift. We further discess the alternative Lagrangian to realize the noncommutativities both of coordinates and momenta.