# Uncertainty Relations and Quantum Corrections in Noncommutative Quantum Mechanics on a Curved Space

## TL;DR

This paper explores noncommutative quantum mechanics on curved spaces using a constraint star-product formalism, revealing quantum corrections from uncertainty relations among constraints that are absent in traditional Dirac-bracket approaches.

## Contribution

It introduces a novel quantization method for noncommutative quantum systems on curved spaces, highlighting quantum corrections from uncertainty relations among constraints.

## Key findings

- Quantum corrections arise from uncertainty relations among constraint operators.
- Two equivalent constrained quantum systems are identified.
- Standard Dirac-bracket formalism misses certain quantum corrections.

## Abstract

Starting with the first-order singular Lagrangian describing the dynamical system with 2nd-class constraints, 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, it is shown that the resultant noncommutative quantum system on the curved space is represented with two kinds of the constrained quantum systems, which are equivalent with each other. Then, it is shown that the resultant Hamiltonians contain the quantum corrections caused by the uncertainty relations among the constraint-operators in addition to those due to the projections of operators, which are missed in the usual approaches with the Dirac-bracket quantization formalism.