Holographic Brownian Motion in 1+1 Dimensions

TL;DR

This paper uses holographic techniques to analyze Brownian motion of a heavy particle in a 1+1 dimensional CFT at finite temperature, revealing exact solutions and dissipation effects even at zero temperature.

Contribution

It provides an exact solution for the string equations in BTZ background and derives a generalized Langevin equation for the boundary particle, including dissipation at zero temperature.

Findings

Exact Green function for the boundary force

Dissipation observed at zero temperature

Drag force remains zero at constant velocity

Abstract

We study the motion of a stochastic string in the background of a BTZ black hole. In the 1+1 dimensional boundary theory this corresponds to a very heavy external particle (e.g, a quark), interacting with the fields of a CFT at finite temperature, and describing Brownian motion. The equations of motion for a string in the BTZ background can be solved exactly. Thus we can use holographic techniques to obtain the Schwinger-Keldysh Green function for the boundary theory for the force acting on the quark. We write down the generalized Langevin equation describing the motion of the external particle and calculate the drag and the thermal mass shift. Interestingly we obtain dissipation even at zero temperature for this 1+1 system. Even so, this does not violate boost (Lorentz) invariance because the drag force on a constant velocity quark continues to be zero. Furthermore since the Green function is exact, it is possible to write down an effective membrane action, and thus a Langevin equation, located at a "stretched horizon" at an arbitrary finite distance from the horizon.