On Functional Hamilton-Jacobi and Schr\"{o}dinger Equations and Functional Renormalization Group

TL;DR

This paper explores the connections between functional Hamilton-Jacobi, Schr"{o}dinger, and Wilson-Polchinski equations within the framework of the functional renormalization group, introducing a holographic scalar field to derive new integro-differential equations and solutions.

Contribution

It introduces a holographic scalar field to unify and derive functional equations of the RG and quantum mechanics, providing new solution methods and analytical results.

Findings

Derived integro-differential hierarchies for Green functions.

Obtained translation-invariant solutions for two- and four-particle Green functions.

Proposed an approximation scheme for the generalized Wilson-Polchinski equation.

Abstract

Functional Hamilton-Jacobi (HJ) equation, the central equation of the holographic renormalization group (HRG), functional Schr\"{o}dinger equation, and generalized Wilson-Polchinski (WP) equation, the central equation of the functional renormalization group (FRG), are considered in $D$-dimensional coordinate and abstract (formal) spaces. Instead of extra coordinates or an FRG scale, a holographic scalar field $\varLambda$ is introduced. The extra coordinate (or scale) is obtained as the amplitude of delta-field or constant field configurations of $\varLambda$. A rigorous derivation of corresponding integro-differential equation hierarchies for Green functions (GFs) as well as the integration formula for functionals are given. Using the integration formula, the functional (arbitrary configuration of $\varLambda$) solution for the translation-invariant two-particle GF is obtained. For the delta-field and the constant field configurations of $\varLambda$ this solution is studied in detail. Separable solution for two-particle GF is briefly discussed. Then, rigorous derivation of the quantum HJ and the continuity functional equations from the functional Schr\"{o}dinger equation as well as the semiclassical approximation are given. An iterative procedure for solving the functional Schr\"{o}dinger equation is suggested. Translation-invariant solutions for various GFs (both hierarchies) on delta-field configuration of $\varLambda$ are obtained. In context of continuity equation and open quantum field systems an optical potential is briefly discussed. Modes coarse graining growth functional for WP functional is analyzed. An approximation scheme is proposed for the generalized WP equation. With an optimized regulator translation-invariant solutions for two-particle and four-particle amputated GFs from approximated WP hierarchy are found analytically.