(Quantum) spacetime as a statistical geometry of fuzzy lumps and the connection with random metric spaces

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.

Geometry, pregeometry and beyond

“…Space is associated, at a deeper level, with a graph whose nodes n i are to be found in a certain state s i (only differences of "charge" s i −s k are meaningful), and its bonds b j in a state J ik that can be equal to 0 or ±1 (vanishing or directed bonds, respectively). A law for the evolution in what he called the "clock-time" of such a graph is simply introduced (Requardt & Roy, 2001, pp. 3042-3043) on the grounds that it provides, by trial and error, some desired consequences.…”

“…More specifically, they are cliques, "the maximal complete subgraphs or maximal subsimplices of a given graph" (Requardt & Roy, 2001, p. 3040).…”

A geometric renormalization group in discrete quantum space–time

“…[23][24][25][26][27][28][29]. It generalizes the concept of cellular automata to so-called cellular networks which live on, in general, very large irregular and dynamical graphs.…”

Constancy of Velocity of Light and Stochastic Background

“…In Ref. 76, and in related works we argued that the lumps emerge from a more primordial discrete dynamical substratum, where the lumps have a specific internal structure. In the approach of Menger et al (at least as far as we can see) they figure as the not-further-resolvable building blocks of spacetime if one approaches the so-called Planck regime discussed above, namely, from the continuum side.…”

“…We argued in Ref. 76 that what we consider to be the elementary building blocks of continuous spacetime-the so-called physical points-are, on a finer scale, actually densely entangled subclusters of 616 sisir roy nodes and bonds of the underlying network or graph. We also called them cliques (which denote in graph theory the maximal complete subgraphs or maximal subsimplices of a given graph).…”