Stochastic quantization and holographic Wilsonian renormalization group of scalar theories with arbitrary mass

Abstract

We explore the mathematical relation between stochastic quantization (SQ) and the holographic Wilsonian renormalization group (HWRG) of a massive scalar field defined in asymptotically anti–de Sitter space. We compute the stochastic two-point correlation function by quantizing the boundary on-shell action (it is identified with the Euclidean action in our stochastic frame) of the scalar field, requiring the initial value of the stochastic field Dirichlet boundary condition, and study its relationship with the double-trace deformation in HWRG computation. It turns out that the stochastic two-point function precisely corresponds to the double-trace deformation through the relation proposed in [J. High Energy Phys. 11 (2012) 144] even in the case that the scalar field mass is arbitrary. In our stochastic framework, the Euclidean action constituting the Langevin equation is not the same as that in the original stochastic theory; in fact, it contains the stochastic time “t-dependent” kernel in it. A justification for the exotic Euclidean action is provided by proving that it transforms to the usual form of the Euclidean action in a new stochastic frame by an appropriate rescaling of both the stochastic fields and time. We also apply the Neumann boundary condition to the stochastic fields to study the relation between SQ and the HWRG when alternative quantization is allowed. It turns out that the application of the Neumann boundary condition to the stochastic fields generates the radial evolution of the single-trace operator as well as the double-trace term.