Momentum-space Langevin dynamics of holographic Wilsonian RG flow: self-interacting massive scalar field theory

TL;DR

This paper establishes a mathematical connection between holographic Wilsonian RG flow and stochastic quantization for a self-interacting massive scalar field in AdS space, using momentum-space Langevin dynamics to describe boundary deformations.

Contribution

It introduces a novel mapping between stochastic Langevin dynamics and holographic RG flow in momentum space for scalar fields with self-interactions.

Findings

Map between stochastic time and AdS radial coordinate.

Successful description of marginal multi-trace deformation via Langevin dynamics.

Clarification of relaxation processes in holographic RG through stochastic quantization.

Abstract

We explore mathematical relationship between holographic Wilsonian renormalization group(HWRG) and stochastic quantization(SQ) motivated by the similarity of the monotonicity in RG flow with Langevin dynamics of non-equilibrium thermodynamics. We look at scalar field theory in AdS space with its generic mass, self-interaction, and boundary deformation in the momentum space. Identifying the stochastic time $t$ with radial coordinate $r$ in AdS, we establish maps between the fictitious time evolution of stochastic multi point correlation function and the radial evolution of multi-trace deformation, which respectively, express the relaxation process of Langevin dynamics and holographic RG flow. We especially consider marginal multi-trace deformation on the AdS boundary which is successfully captured by a Langevin dynamics of SQ.