The Structurally Dynamic Cellular Network and Quantum Graphity Approaches to Quantum Gravity and Quantum Geometry - A Review and Comparison

Starting from the working hypothesis that both physics and the corresponding mathematics and in particular geometry have to be described by means of discrete concepts on the Planck-scale, one of the many problems one has to face in this enterprise is to find the discrete protoforms of the building blocks of our ordinary continuum physics and mathematics living on a smooth background, and perhaps more importantly find a way how this continuum limit emerges from the mentioned discrete structure. We model this underlying substratum as a structurally dynamic cellular network (basically a generalisation of a cellular automaton). We regard these continuum concepts and continuum spacetime in particular as being emergent, coarse-grained and derived relative to this underlying erratic and disordered microscopic substratum, which we would like to call quantum geometry and which is expected to play by quite different rules, namely generalized cellular automaton rules. A central role in our analysis is played by a geometric renormalization group which creates (among other things) a kind of sparse translocal network of correlations between the points in classical continuous space-time and underlies, in our view, such mysterious phenomena as holography and the black hole entropy-area law. The same point of view holds for quantum theory which we also regard as a low-energy, coarse-grained continuum theory, being emergent from something more fundamental. In this paper we review our approach and compare it to the quantum graphity framework.