On background-independent renormalization of spin foam models

TL;DR

This work develops a background-independent renormalization framework for spin foam quantum gravity by employing embedded 2-complexes to define a scale-like hierarchy and using cylindrical consistency to relate finite-scale measures across coarse-grainings. The continuum theory emerges as a projective limit $\overline{\mathcal{A}}$ equipped with a family of measures $\{\mu_\Gamma\}$ that satisfy RG-like flow equations without a fixed metric length, while diffeomorphism invariance imposes strong, scale-dependent constraints on these measures. An exactly solvable 2D $U(1)$ example illustrates how the RG flow restricts solutions to diffeomorphism-invariant possibilities such as the Ashtekar-Lewandowski and BF-type measures, and a practical triad of approximation schemes (restricting complexes, observables, or parameter space) enables tractable analysis in more complex settings. The results point toward a promising path to defining finite, background-independent continuum limits in quantum gravity and invite exploration of Lorentzian, noncompact groups and boundary formulations within this RG framework.

Abstract

In this article we discuss an implementation of renormalization group ideas to spin foam models, where there is no a priori length scale with which to define the flow. In the context of the continuum limit of these models, we show how the notion of cylindrical consistency of path integral measures gives a natural analogue of Wilson's RG flow equations for background-independent systems. We discuss the conditions for the continuum measures to be diffeomorphism-invariant, and consider both exact and approximate examples.

Figures (12)

• Figure 1: Left: Embedded $2$-complexes $\Gamma$ consist of vertices $v$, edges $e$ and faces $f$. Right: A configuration is as many group elements $h_{ef}$ per edge $e$ as there are faces $f$ touching it. In this case there are four group elements associated to the edge $e$: $h_{ef_1}$, $h_{ef_2}$, $h_{ef_3}$, and $h_{ef_4}$.
• Figure 2: If there is a background structure, then the RG flow can be organized along a length parameter.
• Figure 3: Without a background structure, the RG flow runs not along a single parameter, but along the partially ordered set of all $2$-complexes embedded in $\mathcal{M}$.
• Figure 4: Diffeomorphisms $\phi$ of $\mathcal{M}$ act naturally on $\overline{\mathcal{A}}$ via the $2$-complexes $\Gamma$.
• Figure 5: Both $\Gamma_1$ and $\Gamma_2=\phi(\Gamma_1)$ can be refined by $\Gamma$ due to cylindrical consistency. Diffeomorphism invariance of expectation values of observables then translates into strong conditions for the measure $\mu_{\Gamma}$.