Local gauge theory and coarse graining

TL;DR

This paper investigates how to coarse grain discrete gauge theories, like spin foam models, by identifying macroscopic data that captures essential features and relates configurations through local deformations, with implications for quantum gravity models.

Contribution

It characterizes macroscopic variables in discrete gauge theories and shows how they relate configurations, providing insights for improving spin foam models of gravity.

Findings

Macroscopic data includes holonomy evaluations and homotopy classes.

Configurations sharing macroscopic data are related by local deformations.

Including all relevant macroscopic degrees of freedom could improve spin foam models.

Abstract

Within the discrete gauge theory which is the basis of spin foam models, the problem of macroscopically faithful coarse graining is studied. Macroscopic data is identified; it contains the holonomy evaluation along a discrete set of loops and the homotopy classes of certain maps. When two configurations share this data they are related by a local deformation. The interpretation is that such configurations differ by "microscopic details". In many cases the homotopy type of the relevant maps is trivial for every connection; two important cases in which the homotopy data is composed by a set of integer numbers are: (i) a two dimensional base manifold and structure group U(1), (ii) a four dimensional base manifold and structure group SU(2). These cases are relevant for spin foam models of two dimensional gravity and four dimensional gravity respectively. This result suggests that if spin foam models for two-dimensional and four-dimensional gravity are modified to include all the relevant macroscopic degrees of freedom -the complete collection of macroscopic variables necessary to ensure faithful coarse graining-, then they could provide appropriate effective theories at a given scale.