Discrete Mathematics and Physics on the Planck-Scale exemplified by means of a Class of 'Cellular Network Models' and their Dynamics

TL;DR

This paper explores a discrete framework for space-time and vacuum at the Planck scale using cellular network models, introducing new local laws, discrete analysis on graphs, and topological tools to understand complex network dynamics.

Contribution

It develops a novel class of cellular network models with dynamical cells and bonds, and introduces discrete analysis and topological concepts to study their structure and evolution.

Findings

Networks can exhibit increasing complexity and pattern formation.

A groupoid structure naturally arises in these networks.

A new concept of topological or fractal dimension for networks is proposed.

Abstract

Starting from the hypothesis that both physics, in particular space-time and the physical vacuum, and the corresponding mathematics are discrete on the Planck scale we develop a certain framework in form of a class of ' cellular networks' consisting of cells (nodes) interacting with each other via bonds according to a certain 'local law' which governs their evolution. Both the internal states of the cells and the strength/orientation of the bonds are assumed to be dynamical variables. We introduce a couple of candidates of such local laws which, we think, are capable of catalyzing the unfolding of the network towards increasing complexity and pattern formation. In section 3 the basis is laid for a version of 'discrete analysis' on 'graphs' and 'networks' which, starting from different, perhaps more physically oriented principles, manages to make contact with the much more abstract machinery of Connes et al. and may complement the latter approach. In section 4 several more advanced geometric/topological concepts and tools are introduced which allow to study and classify such irregular structures as (random)graphs and networks. We show in particular that the systems under study carry in a natural way a 'groupoid structure'. In section 5 a, as far as we can see, promising concept of 'topological dimension' (or rather: ' fractal dimension') in form of a 'degree of connectivity' for graphs, networks and the like is developed. The possibility of dimensional phase transitions is discussed.