[Paper Review] Noncommutative Quantum Mechanics on a Curved Space

Masayoshi Nakamura|arXiv (Cornell University)|Mar 23, 2015

Noncommutative and Quantum Gravity Theories6 references3 citations

TL;DR

This paper develops a noncommutative quantum mechanics framework on a curved submanifold embedded in higher-dimensional Euclidean space using the projection operator method (POM) and Dirac-bracket quantization. It constructs the canonical structure of a noncommutative system with derivative-type constraints, deriving a commutator algebra that includes quantum corrections up to first order in ℏ and noncommutativity parameters, revealing nontrivial noncommutativity induced by the curved geometry.

ABSTRACT

Starting with the first-order singular Lagrangian, the canonical structures of the noncommutative quantum system on a submanifold embedded in the higher-dimensional Euclidean space are investigated with the projection operator method (POM) and the Dirac-bracket formulation in the case of the derivative-type constraint. Using the successive projection procedure and the iterativity of the Dirac bracket, the noncommutative quantum system is constructed in the form including all orders of the noncommutativity-parameters. When the noncommutative quantum system is constrained to a curved space, the commutator algebra of the system is presented within the 1st-order approximation with respect to Dirac-const. and the noncommutativity-parameters.

Motivation & Objective

• To extend noncommutative quantum mechanics to systems constrained on curved submanifolds embedded in higher-dimensional Euclidean space.
• To resolve the canonical structure of a noncommutative system with derivative-type constraints using constrained Hamiltonian formalism.
• To derive the commutator algebra and Hamiltonian of the system within first-order approximations in ℏ and noncommutativity parameters.
• To investigate how curvature and noncommutativity jointly modify the quantum algebra and dynamics.

Proposed method

• Formulates a first-order singular Lagrangian with dynamical constraints involving the constraint function G(x) and its time derivative.
• Applies the projection operator method (POM) to systematically project the unconstrained system onto the constraint surface, preserving noncommutative structure.
• Uses the Dirac-bracket formalism to derive the consistent quantum algebra, incorporating constraints via the inverse of the constraint Poisson bracket matrix W.
• Introduces antisymmetric matrices Θ and Ξ to model coordinate and momentum noncommutativity, with G = ΘΞ defining the noncommutativity structure.
• Employs iterative Dirac bracket procedures to construct the quantum algebra to all orders in noncommutativity parameters.
• Derives the final commutator algebra and Hamiltonian in the 1st-order approximation with respect to ℏ and noncommutativity parameters.

Experimental results

Research questions

• RQ1 How does noncommutativity arise in quantum systems constrained to a curved submanifold via derivative-type constraints?
• RQ2 What is the structure of the commutator algebra for noncommutative quantum mechanics on a curved space, including quantum corrections?
• RQ3 How do the projection operator method (POM) and Dirac-bracket formalism jointly preserve consistency in the presence of noncommutativity and curvature?
• RQ4 What role do the matrices Θ, Ξ, and G = ΘΞ play in shaping the noncommutative dynamics on the curved manifold?
• RQ5 How does the Hamiltonian of the system transform under the Dirac-bracket quantization procedure in the presence of noncommutativity?

Key findings

• The commutator algebra of the noncommutative quantum system on a curved space includes first-order corrections in both ℏ and noncommutativity parameters, with nontrivial modifications due to the constraint function G(x).
• The Dirac-bracket algebra reveals that [x^i, x^j]_DB = Θ^*_{ij}, indicating coordinate noncommutativity induced by the constraint structure.
• The momentum-momentum commutator [v_i, v_j]_DB = Ξ^*_{ij} shows that momentum noncommutativity is preserved and modified by the geometric structure.
• The algebra includes nontrivial terms like [x^i, p^x_j]_DB = (I - ½G^*)_{ij} - λΘ^*_{ik}G_{kj}(x), showing coupling between noncommutativity and the Lagrange multiplier λ.
• The Hamiltonian contains quantum corrections due to noncommutativity, with terms involving G_{ij}^*, (GΘ)^*, and {G_{ik}(x), G_{jl}(x)} in the p^x-p^x commutator.
• The system's dynamics are consistently quantized via POM and Dirac brackets, yielding a closed algebra that includes all first-order quantum and noncommutative corrections.