Position and momentum operators for a moving particle in bulk

TL;DR

This paper develops a framework to describe a moving particle in the bulk of a gravitational system via dual conformal field theories, constructing corresponding states and operators that reflect the particle's geodesic motion.

Contribution

It introduces a method to construct dual CFT states and operators for a moving particle, linking geodesic dynamics to quantum operators and classical Poisson brackets.

Findings

Operators' expectation values match geodesic solutions

Quantum commutators reduce to classical Poisson brackets

Framework enhances understanding of gravitational attraction in dual CFTs

Abstract

In this paper we explore how to describe a bulk moving particle in the dual conformal field theories (CFTs). One aspect of this problem is to construct the dual state of the moving particle. On the other hand one should find the corresponding operators associated with the particle. The dynamics of the particle, i.e., the geodesic equation, can be formulated as a Hamiltonian system with canonical variables. The achievements of our paper are to construct the dual CFT states and the operators corresponding to the canonical variables. The expectation values of the operators give the expected solutions of the geodesic line, and the quantum commutators reduce to the classical Poisson brackets to leading order in the bulk gravitational coupling. Our work provides a framework to understand the geodesic equation, that is gravitational attraction, in the dual CFTs.