Cellular networks as models for Planck-scale physics

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).

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

“…[25], cf. also section 3.2 of [6]). If we represent this simplicial complex by a new (clique-) graph with only the maximal simplices occurring as meta-nodes we loose, on the other side, some information, as we do not keep track of, to give an example, situations where, say, three lumps or cliques have a common overlap.…”

Emergent space-time via a geometric renormalization method

“…As in the spin network case we can define Hilbert spaces over the vertices and edges and then graph operators such as discrete Laplacians and Dirac operators. This procedure also establishes a connection to Connes' noncommutative geometry (for details see for example [23], [30] or [31]). In section 4 of [32] we showed that the evolving dynamics belongs to the same general class of graph transformations or dynamics, as is the case for spin network dynamics or causal set dynamics.…”

“…We developed such concepts in e.g. [23] as well as a certain discrete calculus (which has relations to non-commutative geometry). We showed for example that one can develop a kind of (co)homology theory (see section 3.2) by associating simplices to subsets of DoF with elementary interactions existing between all the respective pairs of DoF in the subset.…”

Local and Nonlocal Discrete Regularization on Weighted Graphs for Image and Mesh Processing

“…Analog definitions and properties have also been used in the context of functional analysis on graphs (Requardt 1997(Requardt , 1998Bensoussan and Menaldi 2005), semisupervised learning (Zhou and Schölkopf 2005;Hein et al 2007) and image processing (Bougleux and Elmoataz 2005;Bougleux et al 2007a). …”