COBE SATELLITE MEASUREMENT, HYPERSPHERES, SUPERSTRINGS AND THE DIMENSION OF SPACETIME

The first part of the paper attempts to establish connections between hypersphere backing in infinite dimensions, the expectation value of d im E-(infinity) spacetime and the COBE measurement of the microwave ba ckground radiation. One of the main results reported here is that the mean sphere in S-(infinity) spans a four dimensional manifold and is t hus equal to the expectation value of the topological dimension of E-( infinity). In the second part we introduce within a general theory, a probabolistic justification for a compactification which reduces an in finite dimensional spacetime E-(infinity) (n=infinity) to a four dimen sional one (D-T = n = 4). The effective Hausdorff dimension of this sp ace is given by [dim(H)E((infinity))]=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 m anifolds, noncommutative geometry, quasi periodic tiling and Fredholm operators. In addition some relevant analogies between E-(infinity), s tatistical mechanics and Jones polynomials are drawn. This allows a be tter insight into the nature of the proposed compactification, the ass ociated E-(infinity) space and the Pisot-Vijayvaraghavan number 1/phi( 3)=4.236067977 representing it's dimension. This dimension is in turn shown to be capable of a natural interpretation in terms of Jones' kno t invariant and the signature of four manifolds. This brings the work near to the context of Witten and Donaldson topological quantum field theory. (C) 1998 Elsevier Science Ltd. All rights reserved.