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).

34 pages, Latex, no figures, probably being published in Journ.Phys.A MathGen

References in corpus (6)

• Dual formulation of spin network evolution
• Dimension Theory of Graphs and Networks
• Causal evolution of spin networks
• The future of spin networks
• Pregeometric Concepts on Graphs and Cellular Networks as Possible Models of Space-Time at the Planck-Scale
• A New Approach to Functional Analysis on Graphs, the Connes-Spectral Triple and its Distance Function

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• Emergent Space-Time via a Geometric Renormalization Method
• Space-Time as an Orderparameter Manifold in Random Networks and the Emergence of Physical Points
• Semiclassical Quantum Gravity: Obtaining Manifolds from Graphs
• Dirac Operators and the Calculation of the Connes Metric on arbitrary (Infinite) Graphs
• Let's call it Nonlocal Quantum Physics
• Scale Free Small World Networks and the Structure of Quantum Space-Time
• Projective Systems of Noncommutative Lattices as a Pregeometric Substratum
• Emergent Properties in Structurally Dynamic Disordered Cellular Networks
• Classical and Quantum Causality in Quantum Field Theory. Or, "The Quantum Universe"
• Wormhole Spaces, Connes' "Points, Speaking to Each Other", and the Translocal Structure of Quantum Theory
• Graph-Laplacians and Dirac Operators on (Infinite) Graphs and the Calculation of the Connes-Distance-Functional
• Proper time and Minkowski structure on causal graphs
• On a correspondence principle between discrete differential forms, graph structure and multi-vector calculus on symmetric lattices
• Proceedings to the 'Euroconference on Symmetries Beyond the Standard Model', 12. - 17. July 2003, Portoroz, Slovenia (Part 1 of 2)
• Fuzzy Continuity in Scalable Topology
• Planck Scale Physics, Pregeometry and the Notion of Time