Nonlocality and aperiodic tilings

Date: Fri, 14 Mar 1997 00:58:28 -0800 (PST)
From: Mitchell Porter <mitch@thehub.com.au>
To: quantum-d@teleport.com
Subject: Nonlocality and aperiodic tilings
           
In _The Emperor's New Mind_ Roger Penrose discusses aperiodic
tilings of the plane - coverings of the plane by numerous
copies of a few basic shapes, Escher-fashion, which do not
repeat. It seems that these tilings are determined by the
specification of the space to be tesselated (e.g. the plane),
the shapes which are allowed, and a few rules constraining
the ways in which they may join.

The laws of these tilings are "local" in the sense that they
are of the form, "Such-and-such tile can only have such-and-such
neighbors". Yet attempting to satisfy such local rules apparently
leads to nonlocal constraints relating distant parts of the
tiling: Penrose says (p436, in my copy) that "the assembly [of 
the tilings] is necessarily non-local", in that one must examine 
the state of the pattern many tiles away from the point of 
assembly, in order to figure out what tile to place next.

He then describes some materials called quasicrystals, which
seem to exhibit properties (10-fold rotational symmetry)
resembling some of the aperiodic tilling patterns, and
speculates that some nonlocal quantum process is behind
the assembly of quasicrystals.

I wonder if we could turn this around, and seek to explain
quantum nonlocality as the nonlocal product of local
constraints, as in the aperiodic tilings.

Think of the spin networks which are the subject of much
research in quantum gravity at the moment. Suppose one
had a set of rules constraining a network, perhaps of
the form: the spin labels of the edges meeting at a 
point must sum to X; or, the valencies of the neighbors
of a vertex with valency X must sum to f(X); etc.
What if one could deduce quantum transition probabilities
from such a set of rules?

There has apparently already been extensive mathematical
investigation of one's ability to infer distant features of 
a tiling, given knowledge of a particular neighborhood, 
in which probabilistic notions were introduced: see
http://www.ams.org/publications/notices/199604/radin.html.
("We measure the degree of order of the tiling by the extent 
to which we can infer, from the features near us, features of 
the tiling far away from us. We use a probabilistic approach 
to justify such inferences...")

-mitch
http://www.thehub.com.au/~mitch