Canonical Structure of Noncommutative Quantum Mechanics as Constraint System

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.