Starting from the working hypothesis that both physics and the corresponding mathematics 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 continuum physics and mathematics. A core concept is the notion of dimension. In the following we develop such a notion for irregular structures like (large) graphs and networks and derive a number of its properties. Among other things we show its stability under a wide class of perturbations which is important if one has 'dimensional phase transitions' in mind. Furthermore we systematically construct graphs with almost arbitrary 'fractal dimension' which may be of some use in the context of 'dimensional renormalization' or statistical mechanics on irregular sets.

In the following we undertake to describe how macroscopic space-time (or rather, a microscopic protoform of it) is supposed to emerge as a superstructure of a web of lumps in a stochastic discrete network structure. As in preceding work (mentioned below), our analysis is based on the working philosophy that both physics and the corresponding mathematics have to be genuinely discrete on the primordial (Planck scale) level. This strategy is concretely implemented in the form of cellular networks and random graphs. One of our main themes is the development of the concept of physical (proto)points or lumps as densely entangled subcomplexes of the network and their respective web, establishing something like (proto)causality. It may perhaps be said that certain parts of our programme are realisations of some early ideas of Menger and more recent ones sketched by Smolin a couple of years ago. We briefly indicate how this two-story-concept of quantum space-time can be used to encode the (at least in our view) existing non-local aspects of quantum theory without violating macroscopic space-time causality!

We study the consequences of the KMS-condition on the properties of quasi-particles, assuming their existence. We establish (i) If the correlation functions decay sufficiently, we can create them by quasi-free field operators.(ii) The outgoing and incoming quasi-free fields coincide, there is no scattering.(iii) There are may age-operators T conjugate to H. For special forms of the dispersion law ε(fc) of the quasi-particles there is a T commuting with the number of quasi-particles and its time-monotonicity describes how the quasiparticles travel to infinity.

A quantum weak energy inequality for spin-one fields in curved space-time

In various areas of modern physics and in particular in quantum gravity or foundational space-time physics, it is of great importance to be in the possession of a systematic procedure by which a macroscopic or continuum limit can be constructed from a more primordial and basically discrete underlying substratum, which may behave in a quite erratic and irregular way. We develop such a framework within the category of general metric spaces by combining recent work of our own and ingeneous ideas of Gromov et al. developed in pure mathematics. A central role is played by two core concepts. For one, the notion of intrinsic scaling dimension of a (discrete) space or, in mathematical terms, the growth degree of a metric space at infinity, on the other hand, the concept of a metrical distance between general metric spaces and an appropriate scaling limit (called by us a geometric renormalization group) performed in this metric space of spaces. In doing this, we prove a variety of physically interesting results about the nature of this limit process, properties of the limit space, e.g., what preconditions qualify it as a smooth classical space-time and, in particular, its dimension.Keywords: Gromov-Hausdorff Distance, discrete metric spaces, continuum limit, geometric growth at infinity, dimension. 285 Int. J. Geom. Methods Mod. Phys. 2006.03:285-313. Downloaded from www.worldscientific.com by YALE UNIVERSITY on 07/07/15. For personal use only.