Quantum mechanics and geodesic deviation in the brane world

TL;DR

This paper explores how geodesic deviation equations behave in brane world models, revealing a connection between quantum quantization rules and gravitational dynamics in the Randall-Sundrum scenario.

Contribution

It demonstrates that the Bohr-Sommerfeld quantization rule emerges from the consistency of geodesic and geodesic deviation equations in the first Randall-Sundrum model.

Findings

The Newtonian limit of the geodesic deviation equation is derived.

The Bohr-Sommerfeld quantization rule is obtained from gravitational considerations.

Test particle paths are quantized in units related to Planck's constant.

Abstract

We investigate the induced geodesic deviation equations in the brane world models, in which all the matter forces except gravity are confined on the 3-brane. Also, the Newtonian limit of induced geodesic deviation equation is studied. We show that in the first Randall-Sundrum model the Bohr-Sommerfeld quantization rule is as a result of consistency between the geodesic and geodesic deviation equations. This indicates that the path of test particle is made up of integral multiples of a fundamental Compton-type unit of length $h/mc$.