Spectrum Generating Algebras for the free motion in $S^3$

TL;DR

This paper develops the spectrum generating algebra for a free particle on a 3D sphere, providing algebraic solutions for classical motion and ladder operators for quantum states, advancing understanding of quantum systems on curved spaces.

Contribution

It constructs and analyzes the spectrum generating algebra for both classical and quantum free motion on $S^3$, including ladder operators and constants of motion.

Findings

Classical SGA yields algebraic solutions for particle trajectories.

Quantum SGA includes ladder operators for eigenstate construction.

The approach offers a unified framework for classical and quantum analysis on curved spaces.

Abstract

We construct the spectrum generating algebra (SGA) for a free particle in the three dimensional sphere $S^3$ for both, classical and quantum descriptions. In the classical approach, the SGA supplies time-dependent constants of motion that allow to solve algebraically the motion. In the quantum case, the SGA include the ladder operators that give the eigenstates of the free Hamiltonian. We study this quantum case from two equivalent points of view.