More on discrete spacetime
Meeting Room 3, CMS
I. AN ONTOLOGICAL INTERPRETATION OF THE STRUCTURE OF SPACETIME
We presuppose three epistemological levels: 1, observations; 2, theoretical models;3, ontological entities.Physical objects of level 2 can be interpreted with philosophical concepts of level 3. We ascribe the concepts of material beings to the fundamental entities of the world. The first property of these beings is to produce effects in other beings. The causal interactions among the fundamental beings can be taken as the ontological background in level 3 for the relational theories of spacetime.
References: E. Coreth, "Methaphysik", Tyrolia, Innsbruck 1961 J. Zubiri, "Espacio.Tiempo.Materia", Alianza, Madrid 2008 M. Lorente, "Some Relational Theories on the Structure of Spacetime", Pensamiento, 64 (2008), n.242, pp. 665-691.
II. A RELATIONAL MODEL FOR THE DISCRETE SPACETIME
>From the properties of material beings (we call them"hylions") infinite number of networks can be constructed, which are considered the arena where the elementary particles are emerging. For this arena we present three particular models described by simple graphs. These graphs are constructed by vertices (hylions) and edges (interactions) which are nowhere and notime.
i) the n-dimensional hypercubic lattice, where each hylion is interacting with 2n different ones, giving rise to n-dimensional spacetime, where straight lines, orthogonal lines, orthogonal coordinates and metric distance can be define intrinsecally. ii) the n-simplicial lattice evolving with discrete time. A set of interacting hylions, forming a layer of (n - 1) simplices which are evolving in time according to Pachner moves. iii) the planar graph with negative discrete curvature. Given a planar graph derived from hyperbolic tessellation by omitting the embedding space, we can define discrete curvature by combinatorial properties of the underlying discrete hyperboloid made out from vertices and edges.
At the end some comparison will be made between our model and some current models on the structure of spacetime: spin networks (Penrose), spin foams (Rovelli et al.), causal sets (Sorkin et al.), quantm causal histories (Markopoulou). Some References: arXiv:hep-lat/0401019 (discrete Lorentz transformation) arXiv:hep-lat/0312045 (discrete Maxwell equations) arXiv:gr-qc/0412094 (discrete curvature) arXiv:math-ph/0401009 (orthogonal polynomials of discrete variable arXiv:quant-ph/0401087 (discrete harmonic oscillator and hydrogen atom