Anomalous diffusion and factor ordering in (1+1)-dimensional Lorentzian quantum gravity

TL;DR

This paper investigates the non-manifold-like structure of quantum spacetime in (1+1) dimensions using diffusion processes, revealing anomalous diffusion behavior and the influence of factor ordering on quantum geometry.

Contribution

It introduces a quantum heat kernel approach to analyze diffusion in quantum gravity, highlighting the impact of factor ordering on anomalous diffusion and spectral properties.

Findings

Diffusion is anomalous, similar to porous or fractal structures.

Walk dimension has an infimum of 4, varies with factor ordering.

Return probability expansion depends on factor-ordering parameter.

Abstract

Using properties of diffusion according to a quantum heat kernel constructed as an expectation over classical heat kernels on $S^1$, we probe the non-manifold-like nature of quantized space in a model of (1+1)-dimensional quantum gravity. By computing the mean squared displacement of a diffusing particle, we find that diffusion is anomalous, behaving similarly to that on a porous substrate, network, or fractal over short distances. The walk dimension of the path for a particle diffusing in quantized space is calculated to have an infimum of 4, rising to arbitrarily large values depending on a parameter labeling the choice of factor ordering in the quantum Hamiltonian for our model and figuring in the asymptotic behavior of the wavefunction used to construct the quantum heat kernel. Additionally, we derive an expansion for return probability of a diffusing particle, whose modifications from the classical power-series form depend on the factor-ordering parameter.