A pedestrian's introduction to spacetime crystallography

IBM Journal of Research and Development, Jan 2004 by Toffoli, T

Ordinary crystallography deals with regular, discrete, static arrangements in space. Of course, dynamic considerationsand 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.

1. Introduction

Traditional crystallography enumerates and classifies the regular arrangements of motifs in Euclidean space, where by regular one means invariant with respect to a discrete group of spatial isometrics that contains as a subgroup the Abelian group freely generated by three independent translations, and by motif one means an arbitrary geometrical figure in that space.

We recall that, starting with the 14 Bravais lattices and keeping one point of the lattice fixed, one obtains the 32 point groups [1]. If the latter are combined with translations, one obtains the 230 space groups (ascertained in 1891). The corresponding regular arrangements arc exhibited in all their technical glory in crystallography manuals such as [1-3], and justified in a more abstract mathematical fashion in tracts such as [4].

Besides purely geometrical data such as atom positions, the motif may be augmentedby features that are not strictly geometrical, such as charge, polarization, and magnetic susceptibility, insofar as they arc imagined to be acted upon in some definite way by those isometry transformations. (What is the mirror image of the south pole of a magnet-north or south?-and why?) Here the boundary that divides an a priori deductive exercise from an inductive, experimental discipline becomes uncertain and full of surprises, as shown by Altman's admirable essay [5]. In this sense, crystal classification is still a somewhat open-ended enterprise.1

Though started as a specialized, stamp-collecting-like discipline useful for organizing a collection of minerals, crystallography eventually matured into a versatile tool of scientific analysis, useful for inferring the size of atoms, the shape of molecules, and the nature of chemical bonds. The deep emphasis on symmetry and covariant transformations that permeates modern physics [6] owes much to the successes of crystallography as a conceptual discipline.

Finally, the aspect that ultimately matters in crystallography is not a particular geometric lattice per ig, but certain functions (such as charge distributions) defined on this lattice. Statistical mechanics and quantum mechanics take this approach even further insofar as the constructs with which they deal are linear combinations (with real coefficients for statistical distributions and complex coefficients for quantum states) of lattice functions. While the functions themselves (or these higher-order constructs) may not have any special properties vis-a-vis the lattice, nevertheless they may be expressible in terms of functions (e.g., eigenvectors of lattice transformations) that each have particular properties with respect to the lattice itself [7, 8]. In this way, the host lattice becomes a key to describing the guest functions in terms of coordinates that arc most natural to the given context. In general, this is really all that one can hope for-and it is, as a matter of fact, quite a lot.

In this paper we ask four questions:

1. How is the concept of crystal naturally to be extended if one adds a time dimension to the dimensions of space? And what does it mean to add a "time" dimension?

2. What kinds of regularity (strict, approximate, or emergent) are relevant when the repeated motifs are no longer static tiles but dynamic function-composition constructs made out of signals and events, and thus the whole regular arrangement represents a kind of computer?

3. What functions-on this lattice computer explicitly "unfurled" in spacetime-are legitimate computational histories? How do these histories transform when the lattice coordinates are transformed? What properties of these histories are left invariant?

4. Does the uniformity group of the computational histories coincide with that of the underlying computational lattice? Or can the latter be augmented by new, emergent symmetries? In this context, certain tradeoffs between periodicity of structure and of function surprisingly follow from mere computability arguments.