Dynamical dimensional reduction in toy models of 4D causal quantum gravity

TL;DR

This paper introduces a simplified toy model derived from causal dynamical triangulations that analytically reproduces the scale-dependent spectral dimension of space-time, confirming previous numerical findings in quantum gravity.

Contribution

It provides an analytical understanding of the scale-dependent spectral dimension in quantum gravity using a reduced toy model derived from CDT.

Findings

Spectral dimension varies from four to two with scale

Analytical continuum limit reproduces numerical spectral dimension behavior

Model confirms conjectured functional form of spectral dimension transition

Abstract

In recent years several approaches to quantum gravity have found evidence for a scale dependent spectral dimension of space-time varying from four at large scales to two at small scales of order of the Planck length. The first evidence came from numerical results on four-dimensional causal dynamical triangulations (CDT) [Ambjorn et al., Phys. Rev. Lett. 95 (2005) 171]. Since then little progress has been made in analytically understanding the numerical results coming from the CDT approach and showing that they remain valid when taking the continuum limit. Here we argue that the spectral dimension can be determined from a model with fewer degrees of freedom obtained from the CDTs by "radial reduction". In the resulting "toy" model we can take the continuum limit analytically and obtain a scale dependent spectral dimension varying from four to two with scale and having functional behaviour exactly of the form which was conjectured on the basis of the numerical results.