Superstrings, knots, and noncommutative geometry in E(∞) space

Within a general theory, a probabilistic justification for a compactification which reduces an infinite-dimensional spacetime E-(x)(n = infinity) to a four-dimensional one (D-T = n = 4) is proposed. The effective Hausdorff dimension of this space is given by [dim(H) E-(x)] = d(H) = 4 + phi(3), where phi(3) = 1/[4 + phi(3)] is a PV number and phi = (root 5 - 1)/2 is the golden mean. The derivation makes use of various results from knot theory, four-manifolds, noncommutative geometry, quasiperiodic tiling, and Fredholm operators. In addition some relevant analogies between E-(x), statistical mechanics, and Jones polynomials are drawn. This allows a better insight into the nature of the proposed compactification, the associated E-(x) space, and the Pisot-Vijayvaraghavan number 1/phi(3) = 4.236067977 representing its dimension. This dimension is in turn shown to be capable of a natural interpretation in terms of the Jones knot invariant and the signature of four-manifolds. This brings the work near to the context of Witten and Donaldson topological quantum field theory.