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Operator Spin Foams: holonomy formulation and coarse graining

Benjamin Bahr

TL;DR

The paper addresses how to compare operator spin foam models across different discretizations and to formulate a renormalization group flow for background-independent quantum gravity. It introduces a dual holonomy formulation, expressing $Z[\nabla]$ as both a sum over face representations and an integral over group holonomies, connected by face amplitudes $\hat{S}_f(\rho_f)$ and edge functions $C_e$. A coarse graining scheme is developed via a partial order on $2$-complexes, yielding a cylindrical consistency condition that defines a renormalization trajectory on the poset and points toward a continuum limit via a projective limit, akin to generalized connections in LQG. The framework encompasses BF, BC, and EPRL-FK-type models and provides a principled route to study renormalizability and potential restoration of diffeomorphism symmetry at the discrete level through RG flow.

Abstract

A dual holonomy version of operator spin foam models is presented, which is particularly adapted to the notion of coarse graining. We discuss how this leads to a natural way of comparing models on different discretization scales, and a notion of renormalization group flow on the partially ordered set of 2-complexes.

Operator Spin Foams: holonomy formulation and coarse graining

TL;DR

The paper addresses how to compare operator spin foam models across different discretizations and to formulate a renormalization group flow for background-independent quantum gravity. It introduces a dual holonomy formulation, expressing as both a sum over face representations and an integral over group holonomies, connected by face amplitudes and edge functions . A coarse graining scheme is developed via a partial order on -complexes, yielding a cylindrical consistency condition that defines a renormalization trajectory on the poset and points toward a continuum limit via a projective limit, akin to generalized connections in LQG. The framework encompasses BF, BC, and EPRL-FK-type models and provides a principled route to study renormalizability and potential restoration of diffeomorphism symmetry at the discrete level through RG flow.

Abstract

A dual holonomy version of operator spin foam models is presented, which is particularly adapted to the notion of coarse graining. We discuss how this leads to a natural way of comparing models on different discretization scales, and a notion of renormalization group flow on the partially ordered set of 2-complexes.

Paper Structure

This paper contains 5 sections, 12 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Operator spin foams
  3. Dual formulation
  4. Coarse graining
  5. Summary

Figures (3)

  • Figure 1: An edge $e$ and four incident faces, with agreeing or opposite orientations.
  • Figure 2: The holonomy $g_f$ around a face $f$ is given by an ordered product of $h_{(e,f)}$, $e\subset\partial f$. In this case $g_f=h_{(e_1,f)}h_{(e_2,f)}^{-1}h_{(e_3,f)}^{-1}h_{(e_4,f)}h_{(e_5,f)}^{-1}$
  • Figure 3: Microscopic holonomies $h'_{(e'_k,f'_k)}$ are composed to macroscopic holonomy $h_{(e,f)}$.