Canonical Structure of Noncommutative Quantum Mechanics as Constraint System

Understand

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.