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 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 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.

Alternative Approach to Noncommutative Quantum Mechanics on a Curved Space

“…Following our previous works [2,7,8], we here present the brief review of the constraint star-product quantization formalism of the POM together with the explanation of the notations appearing hereafter.…”

“…We shall propose the first-order singular model Lagrangian describing the noncommutative quantum system constrained to the curved space specified by G(x) = 0 in R N . Following the canonical quantization formulation for constraint systems [2,9,10], we shall construct the unconstraint quantum system S.…”

“…Through the consistency conditions for the time evolusions of constraint operators, the consistent set of constraints, K, is given as follows: (1) i , φ (2) i , φ (3) , φ (4) i , ψ (1) },…”

“…[φ (1) i , φ (2) j ] = ihδ ij , [φ (1) i , φ (3) ] = −ihG i (x), [φ (1) i , ψ (1) 2) i , ψ (1) ] = −ihG i (x), [φ (4) i , φ (4) j ] = ihΘ ij , (the others) = 0.…”

“…The problem of the quantization of a dynamical system constrained to a submanifold embedded in the higher-dimensional Euclidean space has been extensively investigated as one of the quantum theories on a curved space until now [1], and the noncommutative extensions of quantum mechanics and quantum field theories have been also done widely [2]. Then, the noncommutative extensions of the quantum theories on the curved spaces have been investigated with much interest [3,4].…”