Unified Description of Quantum Mechanics on a Curved Space

TL;DR

This paper develops a unified operatorial quantization framework for constrained quantum systems on curved spaces, revealing relationships between different constraint types and analyzing quantum corrections.

Contribution

It introduces a unified approach to quantize systems with various constraints on curved manifolds using the projection operator method.

Findings

Constraint quantum systems with different constraints are derived from a single Lagrangian.

The usual constraint system is shown to be a subsystem of the derivative-type constraint system.

Quantum corrections to Hamiltonians are explicitly discussed.

Abstract

Starting with the first-order singular Lagrangian, the problem of the quantization of a dynamical system constrained to a submanifold embedded in the higher-dimensional Euclidean space is investigated within the framework of operatorial quantization formalism. Through the projection operator method (POM) with the constraint star-products, it is shown that both of the constraint quantum system with the usual constraint and that with the derivative-type constraint are naturally constructed from one Lagarangian. It is proved that the system with the usual constraint is the sub-system of that with the derivative-type one. Furthermore, the quantization of the dynamical system subject to both of the usual constraint and the derivative-type one is investigated by the POM, and the quantum corrections in the resultant Hamiltonians are discussed.