On Functional Hamilton–Jacobi and Schrödinger Equations and Functional Renormalization Group

Abstract

We consider the functional Hamilton–Jacobi (HJ) equation, which is the central equation of the holographic renormalization group (HRG), functional Schrödinger equation, and generalized Wilson–Polchinski (WP) equation, which is the central equation of the functional renormalization group (FRG). These equations are formulated in D-dimensional coordinate and abstract (formal) spaces. Instead of extra coordinates or an FRG scale, a “holographic” scalar field Λ is introduced. The extra coordinate (or scale) is obtained as the amplitude of delta-field or constant-field configurations of Λ. For all the functional equations above a rigorous derivation of corresponding integro-differential equation hierarchies for Green functions (GFs) as well as the integration formula for functionals are given. An advantage of the HJ hierarchy compared to Schrödinger or WP hierarchies is that the HJ hierarchy splits into independent equations. Using the integration formula, the functional (arbitrary configuration of Λ) solution for the translation-invariant two-particle GF is obtained. For the delta-field and the constant-field configurations of Λ, this solution is studied in detail. A separable solution for a two-particle GF is briefly discussed. Then, rigorous derivation of the quantum HJ and the continuity functional equations from the functional Schrödinger equation as well as the semiclassical approximation are given. An iterative procedure for solving the functional Schrödinger equation is suggested. Translation-invariant solutions for various GFs (both hierarchies) on delta-field configuration of Λ are obtained. In context of the continuity equation and open quantum field systems, an optical potential is briefly discussed. The mode coarse-graining growth functional for the WP action (WP functional) is analyzed. Based on this analysis, an approximation scheme is proposed for the generalized WP equation. With an optimized (Litim) regulator translation-invariant solutions for two-particle and four-particle amputated GFs from approximated WP hierarchy are found analytically. For Λ=0 these solutions are monotonic in each of the momentum variables.

Nonlocal Fractional Quantum Field Theory and Converging Perturbation Series

“…After that, in GF we expand the exponent with potential (50) and face the problem: we need to calculate complicated Gaussian expectation value of fractional potential Fourier transform. The solution to this problem arises from the Parseval-Plancherel identity (52), which brings us back to the original potential. Proof of the obtained series convergence for zero sources is presented in Section 4.3.…”

“…Generating functional Z is a regular functional, so it could be expanded in a functional Taylor series at j = 0 [44,48,52]:…”

“…Another complicated but interesting mathematical problem is the rigorous definition of the Dyson-Schwinger, Schwinger-Tomonaga and functional (nonperturbative, exact) renormalization group flow (FRG flow for short) equations in nonlocal QFT [42,44,45,[48][49][50][51][52][53]. These equations are formulated in terms of variational derivatives, therefore, they are differential.…”