A model of the two-dimensional quantum harmonic oscillator in an $AdS_3$ background

TL;DR

This paper models a 2D quantum harmonic oscillator within an $AdS_3$ background using a generalized Schr"odinger picture, revealing a connection between quantum operators and spacetime symmetries.

Contribution

It introduces a novel approach to quantum oscillators in curved spacetime by linking Schr"odinger operators to the $AdS_3$ Killing vectors, with implications for quantum uncertainty relations.

Findings

Operators form the Lie algebra of $AdS_3$ Killing vectors

Spacetime independence of operators in this model

Modified Heisenberg uncertainty relations

Abstract

In this paper we study a model of the two-dimensional quantum harmonic oscillator in a 3-dimensional anti-de Sitter background. We use a generalized Schr\"odinger picture in which the analogs of the Schr\"odinger operators of the particle are independent of both the time and the space coordinates in different representations. The spacetime independent operators of the particle induce the Lie algebra of Killing vector fields of the $AdS_3$ spacetime. In this picture, we have a metamorphosis of the Heisenberg's uncertainty relations.