Alternative Approach to Noncommutative Quantum Mechanics on a Curved Space

TL;DR

This paper develops an exact method for constructing noncommutative quantum mechanics on curved spaces using constraint star-product quantization, revealing quantum corrections in the Hamiltonian and exploring noncommutativity of coordinates and momenta.

Contribution

It introduces a novel approach to noncommutative quantum mechanics on curved spaces by applying the constraint star-product quantization to eliminate redundant variables and derive the Hamiltonian.

Findings

Exact construction of noncommutative quantum systems on curved spaces.

Quantum corrections explicitly included in the Hamiltonian.

Discussion of constraints for noncommutativity of both coordinates and momenta.

Abstract

Starting with the first-order singular Lagrangian containing the redundant variables, the noncommutative quantum mechanics on a curved space is investigated by the constraint star-product quantization formalism of the projection operator method. Imposing the additional constraints to eliminate the reduntant degrees of freedom, the noncommutative quantum system with noncommutativity among the coordinates on the curved space is exactly constructed. Then, it is shown that the resultant Hamiltonian contains the quantum corrections in the exact form. We further discuss the additional constraints to realize the noncommutativities both of coordinates and momenta on the curved space.