Fractal Dimension in 3d Spin-Foams

TL;DR

This paper calculates the spectral dimension of 3d quantum gravity space-time using Ponzano-Regge and Turaev-Viro models, revealing a scale-dependent transition from 2 to 3 dimensions influenced by energy and cosmological constant.

Contribution

It introduces a simple boundary state approach to compute spectral dimension in 3d quantum gravity models, highlighting the effects of energy scale and cosmological constant.

Findings

Spectral dimension transitions from 2 to 3 across a specific energy range.

At high energy, spectral dimension increases with the cosmological constant in the Turaev-Viro model.

At low energy, the cosmological constant does not affect the spectral dimension.

Abstract

In this paper we perform the calculation of the spectral dimension of the space-time in 3d quantum gravity using the dynamics of the Ponzano-Regge vertex (PR) and its quantum group generalization (Turaev-Viro model (TV)). We realize this considering a very simple decomposition of the 3d space-time and introducing a boundary state which selects a classical geometry on the boundary. We obtain that the spectral dimension of the space-time runs from 2 to 3, across a 1.5 phase, when the energy of a probe scalar field decreases from high to low energy. For the TV model the spectral dimension at hight energy increase with the value of the cosmological constant. At low energy the presence of the cosmological constant does not change the spectral dimension.