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An algebraic approach to coarse graining

Fotini Markopoulou

TL;DR

The paper develops a Hopf-algebra-based framework to coarse-grain inhomogeneous spin systems and spin foams, extending Kreimer's diagram renormalization to block-spin transformations on irregular lattices. By representing partitioned lattices and spin foams as algebra elements and introducing a shrinking antipode $S_R$, it provides a non-perturbative, iterative method to derive effective couplings and weights under renormalization, with exact and approximate schemes. The authors demonstrate the construction explicitly for a $Z_2$ lattice gauge theory and a 1D Ising model, and extend the approach to a 1+1 spin foam, showing how RG flow can be encoded as algebraic relations $O_R$ that vanish under suitable equivalences. The significance lies in offering a structured, potentially scalable tool for RG in nonuniform geometries and quantum gravity contexts, with prospects for numerical implementation and application to Lorentzian spin foams and higher dimensions.

Abstract

We propose that Kreimer's method of Feynman diagram renormalization via a Hopf algebra of rooted trees can be fruitfully employed in the analysis of block spin renormalization or coarse graining of inhomogeneous statistical systems. Examples of such systems include spin foam formulations of non-perturbative quantum gravity as well as lattice gauge and spin systems on irregular lattices and/or with spatially varying couplings. We study three examples which are Z_2 lattice gauge theory on irregular 2-dimensional lattices, Ising/Potts models with varying bond strengths and (1+1)-dimensional spin foam models.

An algebraic approach to coarse graining

TL;DR

The paper develops a Hopf-algebra-based framework to coarse-grain inhomogeneous spin systems and spin foams, extending Kreimer's diagram renormalization to block-spin transformations on irregular lattices. By representing partitioned lattices and spin foams as algebra elements and introducing a shrinking antipode , it provides a non-perturbative, iterative method to derive effective couplings and weights under renormalization, with exact and approximate schemes. The authors demonstrate the construction explicitly for a lattice gauge theory and a 1D Ising model, and extend the approach to a 1+1 spin foam, showing how RG flow can be encoded as algebraic relations that vanish under suitable equivalences. The significance lies in offering a structured, potentially scalable tool for RG in nonuniform geometries and quantum gravity contexts, with prospects for numerical implementation and application to Lorentzian spin foams and higher dimensions.

Abstract

We propose that Kreimer's method of Feynman diagram renormalization via a Hopf algebra of rooted trees can be fruitfully employed in the analysis of block spin renormalization or coarse graining of inhomogeneous statistical systems. Examples of such systems include spin foam formulations of non-perturbative quantum gravity as well as lattice gauge and spin systems on irregular lattices and/or with spatially varying couplings. We study three examples which are Z_2 lattice gauge theory on irregular 2-dimensional lattices, Ising/Potts models with varying bond strengths and (1+1)-dimensional spin foam models.

Paper Structure

This paper contains 10 sections, 60 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Coarse-graining the $Z_2$ lattice gauge theory on a 2-dimensional lattice
  3. The Hopf algebra of partitioned $Z_2$ lattices and their weights
  4. Example
  5. The shrinking "antipode"
  6. Shrinking example
  7. An alternative definition of $S_R$ on the $Z_2$ lattice
  8. The Hopf algebra renormalization of the 1-d Ising model
  9. Basics of the Hopf algebra renormalization of a 1+1 spin foam
  10. Discussion

Figures (1)

  • Figure 1: A 2-dimensional lattice with its plaquettes marked.