Requardt M.
Emergence of Space-Time on the Planck Scale described as an Unfolding
Phase Transition within the Scheme of Dynamical Cellular Networks and
Random Graphs
(96K, Latex)
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 {\it discrete mathematics}. This goal is realized by
developing a scheme being based on the dynamical evolution of a
particular class of {\it cellular networks} being capable of
performing an {\it unfolding phase transition} from a (presumed)
chaotic initial phase towards a new phase which acts as an {\it
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 {\it 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 {\it stochastic mathematics} with several
relatively advanced topics of discrete mathematics like the {\it
theory of random graphs} or {\it 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.