[Paper Review] Nonassociative Algebras and Nonperturbative Field Theory for Hierarchical Models

Andreas Pordt, C. Wieczerkowski|ArXiv.org|Jun 7, 1994

Quantum chaos and dynamical systems22 references3 citations

TL;DR

This paper introduces nonassociative algebras as a novel mathematical framework to rigorously analyze global renormalization group (RG) flows and infrared fixed points in hierarchical quantum field theories. By formulating the RG transformation as a nonassociative algebraic structure, the authors derive convergent expansion methods and prove local Borel summability of the ε-expansion, establishing a continuous two-dimensional manifold of nontrivial fixed points described by theta functions.

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 $ε$- expansion for n-well fixed points is proved by algebraic methods.

Motivation & Objective

• To develop a rigorous algebraic framework for analyzing global renormalization group (RG) flows in hierarchical field theories.
• To address the challenge of identifying nontrivial infrared fixed points in nonperturbative quantum field theories.
• To establish convergent expansion methods using algebra norms for handling infinite-dimensional systems.
• To prove local Borel summability of the ε-expansion via algebraic techniques.
• To construct and characterize a continuous two-dimensional manifold of nontrivial fixed points using theta functions.

Proposed method

• Modeling hierarchical RG transformations using nonassociative algebras to encode the nonperturbative structure of the flow.
• Defining algebra norms to analyze convergence of series expansions in the nonassociative algebra framework.
• Formulating the infrared fixed point condition as a quadratic equation with infinitely many unknowns in the algebra.
• Constructing explicit solutions via Jacobi theta functions to describe a continuous two-dimensional family of fixed points.
• Applying algebraic methods to prove local Borel summability of the ε-expansion for n-well fixed points.
• Using functional analytic techniques within the nonassociative algebra setting to ensure mathematical rigor.

Experimental results

Research questions

• RQ1 Can nonassociative algebras provide a rigorous framework for studying global renormalization group flows in hierarchical models?
• RQ2 How can one systematically identify and characterize nontrivial infrared fixed points in nonperturbative field theories?
• RQ3 What algebraic structure underlies the convergence of perturbative expansions in the ε-expansion for hierarchical models?
• RQ4 Is the ε-expansion for n-well fixed points locally Borel summable, and can this be proven algebraically?
• RQ5 Can a continuous manifold of nontrivial fixed points be explicitly constructed and described using special functions?

Key findings

• The infrared fixed points of the hierarchical model are shown to be solutions of a quadratic equation with an infinite number of unknowns in the nonassociative algebra framework.
• A continuous two-dimensional manifold of nontrivial fixed points is explicitly constructed using Jacobi theta functions.
• Local Borel summability of the ε-expansion for n-well fixed points is rigorously proven using algebraic methods.
• Convergent expansion methods based on algebra norms are developed and analyzed, enabling rigorous treatment of nonperturbative flows.
• The nonassociative algebraic structure provides a new and powerful tool for analyzing global RG dynamics beyond perturbation theory.
• The framework successfully unifies the description of fixed points and convergence properties in a mathematically consistent manner.