Massless particle in 2d spacetime with constant curvature

TL;DR

This paper studies the behavior of massless particles in two-dimensional spacetimes with constant curvature, analyzing symmetries, phase space, and quantization to understand their quantum representations.

Contribution

It constructs the dynamical integrals from spacetime symmetries, characterizes the physical phase space, and performs canonical quantization leading to unitary group representations.

Findings

Dynamical integrals form an $sl(2,\mathbb{R})$ algebra.

Physical phase space is a cone defined by the mass-shell condition.

Quantization yields unitary irreducible representations of $SO_\uparrow(2,1)$.

Abstract

We consider dynamics of massless particle in 2d spacetimes with constant curvature. We analyze different examples of spacetime. Dynamical integrals are constructed from spacetime symmetry related to $sl(2.{\bf R})$ algebra. Mass-shell condition restricts dynamical integrals to a cone (without vertex) which defines physical-phase space. We parametrize the cone by canonical coordinates. Canonical quantization with definite choice of operator ordering leads to unitary irreducible representations of $SO_\uparrow (2.1)$ group.