Probing the quantum nature of spacetime by diffusion

TL;DR

This paper develops new diffusion equations with positive semi-definite solutions to better understand the spectral properties of quantum spacetimes in various quantum gravity theories, providing a refined probe beyond spectral dimension.

Contribution

It introduces a novel set of diffusion equations ensuring positivity, applicable across multiple quantum gravity models, enhancing the analysis of quantum spacetime structures.

Findings

New diffusion equations with positive solutions for quantum gravity

Spectral dimension results are consistent with previous studies

Probability distributions offer a refined probe of quantum spacetime

Abstract

Many approaches to quantum gravity have resorted to diffusion processes to characterize the spectral properties of the resulting quantum spacetimes. We critically discuss these quantum-improved diffusion equations and point out that a crucial property, namely positivity of their solutions, is not preserved automatically. We then construct a novel set of diffusion equations with positive semi-definite probability densities, applicable to Asymptotically Safe gravity, Horava-Lifshitz gravity and Loop Quantum Gravity. These recover all previous results on the spectral dimension and shed further light on the structure of the quantum spacetimes by assessing the underlying stochastic processes. Pointing out that manifestly different diffusion processes lead to the same spectral dimension, we propose the probability distribution of the diffusion process as a refined probe of quantum spacetime.