# Fractal Spacetimes in Stochastic Gravity? -- Views from Anomalous Diffusion and the Correlation Hierarchy

## TL;DR

This paper investigates whether non-Gaussian noise from quantum matter fields' correlations can indicate fractal structures in spacetime, using stochastic gravity and anomalous diffusion as tools to connect quantum gravity theories with observable phenomena.

## Contribution

It introduces a correlation hierarchy kinetic theory framework in stochastic gravity to link quantum matter correlations with potential fractal spacetime structures.

## Key findings

- Hierarchy of correlations in quantum fields influences geometric correlations via Einstein-Langevin equations.
- Anomalous diffusion models test particle behavior in fractal spacetimes.
- The approach bridges low-energy effective theories with quantum gravity predictions.

## Abstract

We explore in stochastic gravity theory whether non-Gaussian noises from the higher order correlation functions of the stress tensor for quantum matter fields when back-reacting on the spacetime may reveal hints of multi-scale structures. Anomalous diffusion may depict how a test particle experiences in a fractal spacetime. The hierarchy of correlations in quantum matter field induces the hierarchy of correlations in geometric objects via the set of Einstein-Langevin equations for each correlation order. This correlation hierarchy kinetic theory conceptual framework, aided by the characteristics of stochastic processes, may serve as a conduit for connecting the low energy `Bottom-Up' approach with the `Top-Down' theories of quantum gravity which predict the appearance of fractal spacetimes at the Planck scale.