96-260 Requardt M.

Discrete Mathematics and Physics on the Planck-Scale exemplified by means of a Class of 'Cellular Network Modells' and their Dynamics (82K, Tex) Jun 13, 96

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Abstract. Starting from the hypothesis that both physics, in particular space-time and the physical vacuum, and the corresponding mathematics are discrete on the Planck scale we develop a certain framework in form of a class of '{\it cellular networks}' consisting of cells (nodes) interacting with each other via bonds according to a certain {\it 'local law'} which governs their evolution. Both the internal states of the cells and the strength/orientation of the bonds are assumed to be dynamical variables. We introduce a couple of candidates of such local laws which, we think, are capable of catalyzing the unfolding of the network towards increasing complexity and pattern formation. In section 3 the basis is laid for a version of '{\it discrete analysis}' and {\it 'discrete topology/geometry'} which, starting from different, perhaps more physically oriented principles, manages to make contact with the much more abstract machinery of Connes et al. and may complement the latter approach. In section 4 a, as far as we can see, promising concept of '{\it topological dimension}' in form of a '{\it degree of connectivity}' for graphs, networks and the like is developed. It is then indicated how this '{\it dimension}', which for continuous structures or lattices being embedded in a continuous background agrees with the ''usual'' notion of dimension (i.e. the respective embedding dimension) , may vary dynamically as a result of a '{\it phase transition like}' change of the '{\it connectivity}' in the network. A certain (highly) speculative argument, along the lines of statistical mechanics, is supplied in favor of the naturalness of dimension 4 of ordinary (classical) space-time.

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