Stochastic quantization and holographic Wilsonian renormalization group of scalar theory with generic mass, self-interaction and multiple trace deformation

Abstract

We explore the mathematical relationship between holographic Wilsonian renormalization group (HWRG) and stochastic quantization (SQ) of scalar field theory with its generic mass, self-interaction and [Formula: see text]-multiple-trace deformation on the [Formula: see text]-dimensional conformal boundary defined in AdS[Formula: see text] space–time. We understand that once we define our Euclidean action, [Formula: see text] as [Formula: see text], then the stochastic process will reconstruct the HWRG data via solving Langevin equation and computing stochastic correlation functions. The [Formula: see text] is given by [Formula: see text], where [Formula: see text] is the boundary counter term and [Formula: see text] is the boundary deformation which gives a boundary condition. In our study, we choose the boundary condition adding (marginal)[Formula: see text]-multiple-trace deformation to the holographic dual field theory. In this theory, we establish maps between fictitious time, [Formula: see text] evolution of stochastic [Formula: see text]-point, ([Formula: see text])-point correlation functions and the (AdS)radial, [Formula: see text] evolution of [Formula: see text]-multiple-trace and ([Formula: see text])-multiple-trace deformations, respectively, once we take identifications of [Formula: see text] and between some of the constants appearing in both sides.