Cellular Networks as Models for Planck-Scale Physics

TL;DR

This paper proposes a discrete cellular network model on graphs to explore Planck-scale physics, aiming to bridge quantum gravity concepts with a new mathematical framework based on local interactions and fractal geometry.

Contribution

It introduces a novel cellular network approach on graphs for modeling Planck-scale physics, including discrete analysis, geometry, and intrinsic fractal dimension concepts.

Findings

Development of discrete (functional) analysis on cellular networks

Proposal of a fractal dimension concept for irregular structures

Comparison with spin network approaches in quantum gravity

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 in this enterprise is to find the discrete protoforms of the building blocks of our ordinary continuum physics and mathematics. We base our own approach on what we call `cellular networks', consisting of cells (nodes) interacting with each other via bonds (figuring as elementary interactions) according to a certain `local law'. Geometrically our dynamical networks are living on graphs. Hence a substantial amount of the investigation is devoted to the developement of various versions of discrete (functional) analysis and geometry on such (almost random) webs. Another important topic we address is a suitable concept of intrinsic (fractal) dimension on erratic structures of this kind. In the course of the investigation we make comments concerning both different and related approaches to quantum gravity as, say, the spin network framework. It may perhaps be said that certain parts of our programme seem to be a realisation of ideas sketched by Smolin some time ago (see the introduction).