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)
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.