Maybe we can model the connectivity using techniques from many-body physics, materials science, percolation and the like. Although it is quite a leap to go from piles of sand, fractured rocks, and soap foams to gravitational wormhole foams: a heavy and difficult set of tools and techniques would need to be developed. The going is bound to be slow.

An alternate approach is to just start guessing, guided by a-priori arguments and instincts. Maybe one can get lucky. This is the approach of Cahill & Klinger. There is a popular exposition in newscientist ( mirror). Copies of the technical papers can be found on the R. Cahill home page (mirror). They propose a random matrix model that exhibits a 3D geometry. Their model may be fundamental; alternately, it just might happen to be a good approximation of a spacetime foam (i.e. where matrix elements are the wormhole connections). My gut sense is that they are really onto something, even if the particular model is faulty. They offer a research direction that is unexplored, and has some tantalizing interpretations.

They propose evolving a non-deterministic, chaotic matrix equation over time:

B(t+1) = B(t) - a (B(t)/bHere B(t) is a skew-adjoint (anti-symmetric) n x n real matrix, a and b are real numbers, a > b >> 1, and W(t) is a skew-adjoint matrix whose entries are independent random variables with variance 1. The authors start with B(0) having small entries; then they run the equation and see what happens.^{2}+ B(t)^{-1}) + W(t)

They find that the matrix, after many iterations, can be roughly factored into a direct sum of several pieces. One piece, a 'condensate' doesn't provide anything much interesting. In the other piece, one finds large matrix elements. interpreting these as connections between elements, they find that the connectivity they provide has a fractal dimension of three: i.e. a 3D space forms as a statistical result. Curiously, the 3D space also has topological defects, which they propose as the underlying mechanism for non-local EPR-style quantum correlations. They also see a decay of these topological short-cuts, which they identify with state function collapse during quantum measurement. The net result of the claims is quite exciting, if ultimately shallow. It is the mere possibility of looking things in this other way that is the focus of my excitement. Although the equation is almost certainly wrong, the groping seems in the right direction, and the state of affairs is reminiscent of the turn of the last, 19th century. Read the popular overview, and then their papers.

Cahill & Klinger use a peculiar set of allegorical allusions in to build excitement in their results, referring to Godel, Chaitin, undecidablity and the like. On this page, we'll explore a different mythological foundation for our allegories: quantum wormhole foams. Let us begin.

Imagine matrix element B_ij as being the inverse geodesic length from the center of the throat of wormhole i to the center of the throat of wormhole j. Large values of B_ij means two wormholes are near each other. The anti-symmetric matrices means that the diagonal elements are zero. The sign of the matrix entry determines whether the geodesic is in the future or past light-cone (thus, B_ij = -B_ji). There is no explicit assumption that the wormholes are in a 4D space: they could be in any number of dimensions, ten or 26, although we need a light-cone like idea with which to interpret the sign of the matrix entries and the anti-symmetry of the matrix. There is no explicit assumption of the dynamical equations of the underlying space-time geometry: they could be Einstein's equations, or they could be something else.

If this is our allegory, then what the heck is that inverse matrix
term? Well, maybe we should interpret the matrix elements differently.
Suppose B_ij was the fraction of geodesics that flowed out of wormhole
i and into wormhole j. Then one could argue that I = B B^{-1}
is just a statement that the number of geodesics is conserved (there
are no singularities).

Lets try a different model. Let the matrix D represent the (inverse) geodesic distances. Let the matrix F represent the fraction of geodesics that flow out of i and into j. What are the dynamical equations for D and F? Don't know, but lets look at some constraints. We want a conservation of geodesics/probabilities, so that following a path from i to j, via the throats of any other wormholes, sums to one. The motivation for this is that we want to interpret these paths as the paths in a functional (Feynmann) path integral, and we want to normalize the Jacobian. So we must have something like:

1 = F + For something like that. However, not all paths are of equal length, so maybe we should mix D in there somehow. Use D as a kind of action, maybe: an action that is given by the geodesic length.^{2}+ F^{3}+ ...

We hypothesized in our earlier metaphysical ramblings that the flow of time is caused by the dynamic resolution of topological inconsistencies. So maybe our constraint should apply at each 'instance' of time, where we measure time along geodesics. Thus, with each iteration of F and D, we apply a 'correction' or feedback term that tries to force the total number of geodesics following all possible paths to always add up to one.

The research program is now to start guessing equations that match these words, and then exploring to see if they exhibit any sort Cahill-Klinger type statistical behaviors.

Under construction ...

March 2001

Linas Vepstas linas@linas.org