A Geometric Renormalisation Group in Discrete Quantum Space-Time

TL;DR

This paper introduces a geometric renormalisation group approach to model quantum space-time as dynamical networks, aiming to explain the emergence of continuous space-time through coarse-graining of discrete, critical geometries.

Contribution

It develops a novel geometric renormalisation group framework for discrete quantum space-time networks, linking microscopic structures to macroscopic continuous space-time.

Findings

The renormalisation process can produce a fixed limit phase resembling continuous space-time.

Analytic and numerical results track geometric changes across scales.

Concepts of dimension and geometric criticality are key to understanding the model.

Abstract

We model quantum space-time on the Planck scale as dynamical networks of elementary relations or time dependent random graphs, the time dependence being an effect of the underlying dynamical network laws. We formulate a kind of geometric renormalisation group on these (random) networks leading to a hierarchy of increasingly coarse-grained networks of overlapping lumps. We provide arguments that this process may generate a fixed limit phase, representing our continuous space-time on a mesoscopic or macroscopic scale, provided that the underlying discrete geometry is critical in a specific sense (geometric long range order). Our point of view is corroborated by a series of analytic and numerical results, which allow to keep track of the geometric changes, taking place on the various scales of the resolution of space-time. Of particular conceptual importance are the notions of dimension of such random systems on the various scales and the notion of geometric criticality.