Wolfram Science Group
Registered: Oct 2003
Paper on Spacetime Crystallography
Here is a recent paper from the NKS bibliography:
Toffoli, T. A pedestrian's introduction to spacetime crystallography
IBM Journal of Research & Development
Date: January 1, 2004 Volume: 48, Issue: 1, Pages: 31-45
Ordinary crystallography deals with regular, discrete, static arrangements in space. Of course, dynamic considerations— and thus the additional dimension of time—must be introduced when one studies the origin of crystals (since they are emergent structures) and their physical properties such as conductivity and compressibility. The space and time of the dynamics in which the crystal is embedded are assumed to be those of ordinary continuous mechanics. In this paper, we take as the starting point a spacetime crystal, that is, the spacetime structure underlying a discrete and regular dynamics. A dynamics of this kind can be viewed as a “crystalline computer.” After considering transformations that leave this structure invariant, we turn to the possible states of this crystal, that is, the discrete spacetime histories that can take place in it and how they transform under different crystal transformations. This introduction to spacetime crystallography provides the rationale for making certain definitions and addressing specific issues; presents the novel features of this approach to crystallography by analogy and by contrast with conventional crystallography; and raises issues that have no counterpart there.
They develop the idea of a "spacetime crystal" - which is a nice idea, and discuss ordinary crystallography as a starting point for this. They find that in terms of isometry not much is new.
They think in terms of "causal networks" of computational events (as does the NKS book) - partially ordered sets of events.
Along the way, there is a nice discussion of "lattice gases", the Critters rule, invertibility, and emergent properties of lattice gases.
A main part of the paper is showing that lorentz invariance naturally emerges in this framework.
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