Quantization of the Particle Motion on the $n$-Dimensional Sphere

TL;DR

This paper introduces a formalism that converts second-class constraints into first-class constraints for a particle on an n-sphere, leading to a Lie algebra structure and identifying the Casimir operator as the observable.

Contribution

It presents a new method to handle constraints in particle motion on spheres, simplifying the quantization process and clarifying the algebraic structure involved.

Findings

Poisson algebra closes for the system

Quantization yields a Lie algebra

Casimir operator corresponds to angular momentum squared

Abstract

We develop here a simple formalism that converts the second-class constraints into first-class ones for a particle moving on the $n$-dimensional sphere. The Poisson algebra generated by the Hamiltonian and the constraints closes and by quantization transforms into a Lie algebra. The observable of the theory is given by the Casimir operator of this algebra and coincides with the square of the angular momentum.