Nonassociative Algebras and Nonperturbative Field Theory for Hierarchical Models

TL;DR

This paper explores the use of nonassociative algebras in hierarchical renormalization group analysis, providing new tools for understanding fixed points and nonperturbative phenomena in field theory.

Contribution

It introduces a novel algebraic framework for RG flows, including convergence analysis, fixed point solutions, and Borel summability proofs for epsilon expansions.

Findings

Infrared fixed points characterized by quadratic equations with infinitely many variables

Existence of a continuous manifold of 2D periodic fixed points via theta functions

Proof of local Borel summability for epsilon expansions using algebraic methods

Abstract

Hierarchical renormalization group (RG) transformations are related to nonassociative algebras. These algebras serve as a new basic tool for a rigorous treatment of global RG flows and the search of nontrivial infrared fixed points. Convergent expansion methods are presented and analyzed in terms of algebra norms. It is shown that the infrared fixed points are solutions of a quadratic equation with an infinite number of unknowns. A continuous manifold of two dimensional periodic nontrivial fixed points is presented in terms of theta functions. Local Borel summability of the $\epsilon$- expansion for n-well fixed points is proved by algebraic methods.