Pregeometric Concepts on Graphs and Cellular Networks as Possible Models of Space-Time at the Planck-Scale

TL;DR

This paper explores how discrete structures like graphs and networks can serve as models for space-time at the Planck-scale, aiming to develop protoforms of continuum physics through discrete geometry and analysis.

Contribution

It introduces a framework for constructing discrete geometric and analytical concepts on irregular graph structures as potential models of space-time at the Planck-scale.

Findings

Development of discrete dimensional concepts

Construction of discrete geometric analysis tools

Potential implications for quantum gravity theories

Abstract

Starting from the working hypothesis that both physics and the corresponding mathematics have to be described by means of discrete concepts on the Planck-scale, one of the many problems one has to face is to find the discrete protoforms of the building blocks of continuum physics and mathematics. In the following we embark on developing such concepts for irregular structures like (large) graphs or networks which are intended to emulate (some of) the generic properties of the presumed combinatorial substratum from which continuum physics is assumed to emerge as a coarse grained and secondary model theory. We briefly indicate how various concepts of discrete (functional) analysis and geometry can be naturally constructed within this framework, leaving a larger portion of the paper to the systematic developement of dimensional concepts and their properties, which may have a possible bearing on various branches of modern physics beyond quantum gravity.