Thermal Noise and Stochastic Strings in AdS/CFT

TL;DR

This paper investigates thermal noise in AdS/CFT by modeling a heavy quark string with stochastic boundary conditions, revealing how fluctuations relate to dissipation and two-point functions in the dual field theory.

Contribution

It introduces a stochastic boundary condition framework for analyzing thermal noise in AdS/CFT, connecting string fluctuations to dissipation and correlators.

Findings

Derived the frequency-dependent random force on the boundary quark.

Showed the stochastic motion matches bulk two-point functions.

Provided a bulk picture of the boundary force as a flip-flopping string.

Abstract

We clarify the structure of thermal noise in AdS/CFT by studying the dynamics of an equilibrated heavy quark string. Using the Kruskal extension of the correspondence to generate the dynamics of the field theory on the Keldysh contour, we show that the motion of the string is described by the classical equations of motion with a stochastic boundary condition on the stretched horizon. The form of the stochastic boundary condition is consistent with the dissipation on this surface and is found by integrating out the fluctuations inside of the stretched horizon. Solving the equations of motion for the fluctuating string we determine the full frequency dependence of the random force on the boundary quark and show that it is consistent with the frequency dependent dissipation. We show further that the stochastic motion reproduces the bulk to bulk two point functions of the Kruskal formalism. These turn out to be related to the usual retarded bulk to bulk propagator by KMS relations. Finally we analyze the stochastic equations and give a bulk picture of the random boundary force as a flip-flopping trailing string solution. The basic formalism can be applied to the fluctuations of gravitons, dilatons, and other fields.