Quantum Mechanics on a Poincar\'e Hyperboloid

TL;DR

This paper explores the quantization of a charged particle on a Poincaré hyperboloid under a magnetic field, revealing the algebraic structure and deriving the Hamiltonian.

Contribution

It provides a detailed derivation of the Poisson brackets, quantization process, and algebraic structure for this specific geometric and physical setup.

Findings

Dirac bracket algebra becomes ISO(1,2) after quantization

Explicit representation of the algebra is provided

Hamiltonian of the system is derived

Abstract

We discuss the process to obtain Poisson brackets among the phase-space variables of a system of a charged particle on a Poincar\'e hyperboloid in the presence of a uniform magnetic field. We show that after quantization the Dirac bracket algebra becomes the algebra of ISO(1,2). The representation of this algebra is explicitly analyzed and the Hamiltonian of this system has been derived.