Statistical Mechanics Applied to a
Class of Graphs Yielding a Spatial Dimension

The idea and model presented here stems from papers by Freeman Dyson [Dyson 1969a] [Dyson 1969b] where the one dimensional Ising model is reconsidered using a long range interaction rather than the usual local interaction.

It is well known that the Ising Model of Ferromagnetism in one dimension does not exhibit spontaneous magnetization, and that for higher dimensions it does. The Ising model in any dimension is usually defined as a physical lattice in n-space where the vertices of the lattice are assumed to have up-down spin states. Each vertex is assumed to interact (spin-spin interaction) only with its nearest neighbors. Clearly, as the dimension increases, so does the number of nearest neighbors. It is presumably this increase in the number of nearest neighbors that provides the vertex correlation that is ultimately the spontaneous magnetization.

Dyson increases the effective number of nearest neighbors in the one dimensional model by extending the range of the interaction, and, in fact, remarks on the relation between range of interaction and dimension.

Consider space to be a set of "pointlike" objects that are connected by some sort of interaction between them. The points are in motion and their interactions are in flux. Suppose that there is a single fundamental interaction between these structureless points, so that thinking graph theoretically, the graph lines must all be of length equal to some fundamental length. I've seen no name for graphs of this type, so I'll call them isometric graphs. The energy of a graph, one could reasonably expect to be the product of some energetic constant and the number of lines.

A truly fundamental length that is a genuine limit to the distiguishability of points is the Planck length which has quantum theoretical associations. There is no real quantum theory here, essentially because I don't know how to do it, but also because the solution to classical problems are relatively easier than quantum problems. There is however a suggestion of a transition to a quantized picture of the Big Bang (BB). Considering only an interaction type of energy, the BB should look like a complete n-vertex graph, with very large n, alternatively an n-simplex. If one makes some handwaving argument about this having to be a pregeometrical condition where the very notions of space and time are not applicable and where semiclassically things have been pushed back to a set of connected verticies or points, then the natural transition to a quantum picture is to map the points to vectors that span a Hilbert space Hilb(n). The simplex then becomes an associated complex in the language of elementary homology. This idea happens to fit neatly with a developing quantum set theory on other pages here, as well as with the notion of finite canonical commutation relations.

This is in the nature of an exercise and of a simple construction of a model where the dimension of a space is statistically and dynamically determined.

The model is essentially then a generalization of the statistical mechanical model of a classical hard sphere gas. The generalization does not specify at the outset, the dimension of the space in which the "gas" exists.

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