(Quantum) Space-Time as a Statistical Geometry of Fuzzy Lumps and the Connection with Random Metric Spaces

TL;DR

This paper proposes a pregeometric framework combining fuzzy lumps, random graphs, and probabilistic metric spaces to better model microphysical phenomena within a dynamic, stochastic geometric environment.

Contribution

It introduces a novel approach integrating fuzzy set theory with random metric spaces and evolving networks to describe microphysics.

Findings

Development of a web of fuzzy lumps as a pregeometry

Representation of lumps as entangled subgraphs in a dynamic network

Integration of fuzzy set concepts into probabilistic metric space models

Abstract

We develop a kind of pregeometry consisting of a web of overlapping fuzzy lumps which interact with each other. The individual lumps are understood as certain closely entangled subgraphs (cliques) in a dynamically evolving network which, in a certain approximation, can be visualized as a time-dependent random graph. This strand of ideas is merged with another one, deriving from ideas, developed some time ago by Menger et al, that is, the concept of probabilistic- or random metric spaces, representing a natural extension of the metrical continuum into a more microscopic regime. It is our general goal to find a better adapted geometric environment for the description of microphysics. In this sense one may it also view as a dynamical randomisation of the causal-set framework developed by e.g. Sorkin et al. In doing this we incorporate, as a perhaps new aspect, various concepts from fuzzy set theory.