Diffusion in quantum geometry

TL;DR

This paper models the scale-dependent effective dimension of quantum spacetime using a multiscale fractional diffusion equation, revealing a stochastic process that explains the spectral dimension flow from 2 to 4.

Contribution

It introduces a novel multiscale fractional diffusion framework to describe dimensional flow in quantum gravity models, including the first construction of the spectral dimension profile for multifractional spaces.

Findings

Spectral dimension decreases to 2 at small scales and increases to 4 at large scales.

The model uses a fractional telegraph process to describe quantum spacetime.

First explicit profile of spectral dimension for multifractional spaces.

Abstract

The change of the effective dimension of spacetime with the probed scale is a universal phenomenon shared by independent models of quantum gravity. Using tools of probability theory and multifractal geometry, we show how dimensional flow is controlled by a multiscale fractional diffusion equation, and physically interpreted as a composite stochastic process. The simplest example is a fractional telegraph process, describing quantum spacetimes with a spectral dimension equal to 2 in the ultraviolet and monotonically rising to 4 towards the infrared. The general profile of the spectral dimension of the recently introduced multifractional spaces is constructed for the first time.