Dimensional flow in discrete quantum geometries

TL;DR

This paper investigates how spectral dimension changes in discrete quantum geometries, showing that superpositions of states can produce a flow from the topological dimension to smaller values, resembling fractal behavior at small scales.

Contribution

It introduces a class of quantum geometries with spectral dimension flow via superpositions of regular complexes, connecting discrete quantum gravity models to fractal-like dimensional behavior.

Findings

Spectral dimension reduces to alpha at small scales for certain superpositions.

Hausdorff dimension varies between 1 and d, walk dimension remains 2.

Reproduces the spacetime spectral dimension of 2 when alpha equals 1.

Abstract

In various theories of quantum gravity, one observes a change in the spectral dimension from the topological spatial dimension $d$ at large length scales to some smaller value at small, Planckian scales. While the origin of such a flow is well understood in continuum approaches, in theories built on discrete structures a firm control of the underlying mechanism is still missing. We shed some light on the issue by presenting a particular class of quantum geometries with a flow in the spectral dimension, given by superpositions of states defined on regular complexes. For particular superposition coefficients parametrized by a real number $0<\alpha<d$, we find that the spatial spectral dimension reduces to $d_s \simeq \alpha$ at small scales. The spatial Hausdorff dimension of such class of states varies between 1 and $d$, while the walk dimension takes the usual value $d_w=2$. Therefore, these quantum geometries may be considered as fractal only when $\alpha=1$, where the "magic number" ${d_s}^{\rm spacetime}\simeq 2$ for the spectral dimension of space\emph{time}, appearing so often in quantum gravity, is reproduced as well. These results apply, in particular, to special superpositions of spin-network states in loop quantum gravity, and they provide more solid indications of dimensional flow in this approach.