Requardt M.
Discrete Mathematics and Physics on the Planck-Scale
(52K, Latex)
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 '{\it cellular network}' consisting of cells
interacting with each other via bonds. Both the internal states of
the cells and the "strength" of the bonds are assumed to be dynamical
variables. In section 3 the basis is laid for a version of '{\it
discrete
analysis}' 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, new 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, may change dynamically as a result of a '{\it
phase
transition like}' change in '{\it connectivity}' in the network. A
certain
speculative argument, along the lines of statistical mechanics, is
supplied in favor of the naturalness of dimension 4 of ordinary
(classical) space-time.