Exact Renormalization Groups As a Form of Entropic Dynamics

Abstract

Abstract:The Renormalization Group (RG) is a set of methods that have been instrumental in tackling problems involving an infinite number of degrees of freedom, such as, for example, in quantum field theory and critical phenomena. What all these methods have in common-which is what explains their success-is that they allow a systematic search for those degrees of freedom that happen to be relevant to the phenomena in question. In the standard approaches the RG transformations are implemented by either coarse graining or through a change of variables. When these transformations are infinitesimal, the formalism can be described as a continuous dynamical flow in a fictitious time parameter. It is generally the case that these exact RG equations are functional diffusion equations. In this paper we show that the exact RG equations can be derived using entropic methods. The RG flow is then described as a form of entropic dynamics of field configurations. Although equivalent to other versions of the RG, in this approach the RG transformations receive a purely inferential interpretation that establishes a clear link to information theory.

Field Theoretical Approach for Signal Detection in Nearly Continuous Positive Spectra I: Matricial Data

“…The distribution ρ for eigenvalues of the inverse matrix can be easily deduced from (40). Figure 6 provides a picture of the numerical flow for a quartic truncation.…”

“…Note that, in regimes where momenta take large values and DE breaks down, other approximation schemes exist, enabling to keep the full dependence of the effective vertices. The most popular being the so-called Blaizot–Mendez–Wschebor (BMW) method [ 21 , 22 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 ], which, combined with exact relations such as Ward identities, gives the possibility to provide exact (i.e., scheme independent) results [ 61 ]. Another current source of disagreement concerns the choice of the regulator.…”

“…The connection between PCA and RG can be traced from information theory [ 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 ] as a consequence of the ability of the RG to extract large scale relevant features of a microscopic system. From an information theory point of view, the RG describes how a theory valid for small distance physics flows toward a simpler theory (i.e., with a reduced set of parameters) at larger distance, as information is progressively lost (due to coarse-graining).…”

“…Working above this wall, we show that only a few number of local couplings are relevant, essentially quartic and sixtic couplings for small perturbations around MP law. We close the section with the Wetterich–Morris non-perturbative formalism [ 21 , 22 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 ], that we discuss in this context. Section 3 is devoted to an analysis of the exact RG equation using standard local potential approximation (LPA) and its improved version (LPA ) which takes into account field renormalization effects and anomalous dimension.…”

Entropic Dynamics on Gibbs Statistical Manifolds

“…ED has been successful in reconstructing dynamical models in physics such as quantum mechanics [28,29], quantum field theory [30], and the renormalization group [31]. Beyond physics, it has been recently applied to the rhw fields of finance [32,33] and neural networks [34].…”