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.
Highlights
The functional integral formalism underlies many modern theories from various fields of science [1,2,3,4,5,6,7,8,9,10,11,12,13,14]
We study in detail a translation-invariant pair of hierarchies: equations for the vacuum holographic mean, two-particle and four-particle holographic Green functions (GFs), as well as for the vacuum mean and two-particle GF, corresponding to the continuity functional equation
Restricting ourselves to the zero and second terms in the expansion (4), we arrive at the following equation in the condensed notation: δS [Λ, φ]
Summary
The functional (path) integral formalism underlies many modern theories from various fields of science [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. The main results obtained by the authors and presented in this paper are: rigorous derivation of the functional HJ equation hierarchy, solution of the first equations of the hierarchy for an arbitrary field Λ, rigorous derivation of the functional Schrödinger equation hierarchy (quantum HJ and continuity functional equations hierarchies, the semiclassical solution of the first equations of the hierarchies (with optical potential), rigorous derivation of the WP functional equation from the functional master equation (the equation containing the mode coarse graining growth functional), semiclassical solution of the first equations of the WP functional equation hierarchy (with Litim regulator), rigorous derivation of conditions when the mode coarse graining growth functional is geometric
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