Geometry of 2d spacetime and quantization of particle dynamics

TL;DR

This paper explores the classical and quantum behavior of particles in two-dimensional curved spacetimes, revealing how global symmetries influence the phase-space structure and quantum representations.

Contribution

It demonstrates how global spacetime symmetries determine the phase-space and quantum structure of particles in 2D curved spacetimes, using canonical quantization and group representations.

Findings

Global symmetries define phase-space structure.

Canonical quantization yields unitary irreducible representations.

Analysis applies to spacetimes with constant curvature.

Abstract

We analyze classical and quantum dynamics of a particle in 2d spacetimes with constant curvature which are locally isometric but globally different. We show that global symmetries of spacetime specify the symmetries of physical phase-space and the corresponding quantum theory. To quantize the systems we parametrize the physical phase-space by canonical coordinates. Canonical quantization leads to unitary irreducible representations of $SO_\uparrow (2.1)$ group.