An algebraic approach to coarse graining

TL;DR

This paper introduces an algebraic framework using Hopf algebras of rooted trees to analyze coarse graining in complex statistical and quantum systems, extending methods from quantum field theory.

Contribution

It applies Kreimer's renormalization approach to block spin renormalization in inhomogeneous systems, including quantum gravity models and irregular lattice gauge theories.

Findings

Successfully analyzed Z_2 lattice gauge theory on irregular lattices

Extended algebraic methods to Ising/Potts models with variable bonds

Applied to (1+1)-dimensional spin foam models

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.