Quantization of a relativistic particle on the SL(2,R) manifold based on Hamiltonian reduction

TL;DR

This paper develops a quantum theory for a relativistic particle on the SL(2,R) manifold using Hamiltonian reduction, revealing a Hilbert space composed of discrete series representations with a positive spectrum.

Contribution

It introduces a novel Hamiltonian reduction approach to quantize a relativistic particle on SL(2,R), connecting geometric and representation-theoretic methods.

Findings

Hilbert space consists of discrete series of SL(2,R) representations

Spectrum of the system is positive and discrete

Method effectively splits the system into coadjoint orbits

Abstract

A quantum theory is constructed for the system of a relativistic particle with mass m moving freely on the SL(2,R) group manifold. Applied to the cotangent bundle of SL(2,R), the method of Hamiltonian reduction allows us to split the reduced system into two coadjoint orbits of the group. We find that the Hilbert space consists of states given by the discrete series of the unitary irreducible representations of SL(2,R), and with a positive-definite, discrete spectrum.