Emergence of Space-Time on the Planck Scale described as an Unfolding Phase Transition within the Scheme of Dynamical Cellular Networks and Random Graphs

Abstract

As in an earlier paper we start from the hypothesis that physics on the Planck scale should be described by means of concepts taken from ``discrete mathematics''. This goal is realized by developing a scheme being based on the dynamical evolution of a particular class of ``cellular networks'' being capable of performing an ``unfolding phase transition'' from a (presumed) chaotic initial phase towards a new phase which acts as an ``attractor'' in total phase space and which carries a fine or super structure which is identified as the discrete substratum underlying ordinary continuous space-time (or rather, the physical vacuum). Among other things we analyze the internal structure of certain particular subclusters of nodes/bonds (maximal connected subsimplices, $mss$) which are the fundamental building blocks of this new phase and which are conjectured to correspond to the ``physical points'' of ordinary space-time. Their mutual entanglement generates a certain near- and far-order, viz. a causal structure within the network which is again set into relation with the topological/metrical and causal/geometrical structure of continuous space-time. The mathematical techniques to be employed consist mainly of a blend of a fair amount of ``stochastic mathematics'' with several relatively advanced topics of discrete mathematics like the ``theory of random graphs'' or ``combinatorial graph theory''. Our working philosophy is it to create a scenario in which it becomes possible to identify both gravity and quantum theory as the two dominant but derived(!) aspects of an underlying discrete and more primordial theory (dynamical cellular network) on a much coarser level of resolution, viz. continuous space-time.