Stochastic quantization and holographic Wilsonian renormalization group

TL;DR

This paper explores the connection between stochastic quantization and holographic Wilsonian renormalization group flow, showing that radial flows of boundary deformations in AdS space are captured by stochastic time evolution, with explicit examples provided.

Contribution

It establishes a precise relation between radial flow of double trace deformations and stochastic dynamics, linking holographic RG flow with stochastic quantization methods.

Findings

Radial flow of double trace couplings matches stochastic time evolution.

Radial evolution of boundary actions equals stochastic Fokker-Planck evolution.

Validated relations with examples in AdS2 and AdS4 contexts.

Abstract

We study relation between stochastic quantization and holographic Wilsonian renormalization group flow. Considering stochastic quantization of the boundary on-shell actions with the Dirichlet boundary condition for certain $AdS$ bulk gravity theories, we find that the radial flows of double trace deformations in the boundary effective actions are completely captured by stochastic time evolution with identification of the $AdS$ radial coordinate `$r$' with the stochastic time '$t$' as $r=t$. More precisely, we investigate Langevin dynamics and find an exact relation between radial flow of the double trace couplings and 2-point correlation functions in stochastic quantization. We also show that the radial evolution of double trace deformations in the boundary effective action and the stochastic time evolution of the Fokker-Planck action are the same. We demonstrate this relation with a couple of examples: (minimally coupled)massless scalar fields in $AdS_2$ and U(1) vector fields in $AdS_4$.