The Continuum Limit of Discrete Geometries

TL;DR

This paper develops a mathematical framework for deriving continuum limits from discrete geometries, crucial for understanding space-time in quantum gravity, by combining metric space concepts and geometric renormalisation techniques.

Contribution

It introduces a systematic approach to construct continuum limits from discrete spaces using intrinsic scaling dimensions and a geometric renormalisation group.

Findings

Characterization of the continuum limit process

Conditions for the limit space to resemble classical space-time

Insights into the dimension and smoothness of the limit space

Abstract

In various areas of modern physics and in particular in quantum gravity or foundational space-time physics it is of great importance to be in the possession of a systematic procedure by which a macroscopic or continuum limit can be constructed from a more primordial and basically discrete underlying substratum, which may behave in a quite erratic and irregular way. We develop such a framework within the category of general metric spaces by combining recent work of our own and ingeneous ideas of Gromov et al, developed in pure mathematics. A central role is played by two core concepts. For one, the notion of intrinsic scaling dimension of a (discrete) space or, in mathematical terms, the growth degree of a metric space at infinity, on the other hand, the concept of a metrical distance between general metric spaces and an appropriate scaling limit (called by us a geometric renormalisation group) performed in this metric space of spaces. In doing this we prove a variety of physically interesting results about the nature of this limit process, properties of the limit space as e.g. what preconditions qualify it as a smooth classical space-time and, in particular, its dimension.