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

摘要

As in an earlier paper we start from the hypothesis that physics on thePlanck scale should be described by means of concepts taken from ``discretemathematics''. This goal is realized by developing a scheme being based on thedynamical evolution of a particular class of ``cellular networks'' beingcapable of performing an ``unfolding phase transition'' from a (presumed)chaotic initial phase towards a new phase which acts as an ``attractor'' intotal phase space and which carries a fine or super structure which isidentified as the discrete substratum underlying ordinary continuous space-time(or rather, the physical vacuum). Among other things we analyze the internalstructure of certain particular subclusters of nodes/bonds (maximal connectedsubsimplices, $mss$) which are the fundamental building blocks of this newphase and which are conjectured to correspond to the ``physical points'' ofordinary space-time. Their mutual entanglement generates a certain near- andfar-order, viz. a causal structure within the network which is again set intorelation with the topological/metrical and causal/geometrical structure ofcontinuous space-time. The mathematical techniques to be employed consistmainly of a blend of a fair amount of ``stochastic mathematics'' with severalrelatively advanced topics of discrete mathematics like the ``theory of randomgraphs'' or ``combinatorial graph theory''. Our working philosophy is it tocreate a scenario in which it becomes possible to identify both gravity andquantum theory as the two dominant but derived(!) aspects of an underlyingdiscrete and more primordial theory (dynamical cellular network) on a muchcoarser level of resolution, viz. continuous space-time.