# Fractal Space-Time from Spin-Foams

## TL;DR

This paper calculates the spectral dimension of 4D quantum spacetime using the Barrett-Crane spin-foam model, revealing a scale-dependent transition from approximately 2 to 4 as energy decreases.

## Contribution

It introduces a method to compute the spectral dimension in spin-foam quantum gravity, showing its running behavior across different energy scales and dependence on area spectra.

## Key findings

- Spectral dimension runs from ~2 at high energy to 4 at low energy.
- Dimension at Planck scale varies with the chosen area spectrum.
- Transition occurs over a phase of approximately 1.5 in scale.

## Abstract

In this paper we perform the calculation of the spectral dimension of spacetime in 4d quantum gravity using the Barrett-Crane (BC) spinfoam model. We realize this considering a very simple decomposition of the 4d spacetime already used in the graviton propagator calculation and we introduce a boundary state which selects a classical geometry on the boundary. We obtain that the spectral dimension of the spacetime runs from $\approx 2$ to 4, across a $\approx 1.5$ phase, when the energy of a probe scalar field decreases from high $E \lesssim E_P/25$ to low energy. The spectral dimension at the Planck scale $E \approx E_P$ depends on the areas spectrum used in the calculation. For three different spectra $l_P^2 \sqrt{j(j+1)}$, $l_P^2 (2 j+1)$ and $l_P^2 j$ we find respectively dimension $\approx 2.31$, 2.45 and 2.08.