Scale relativity in Cantorian E(∞) space and average dimensions of our world

A Cantorian-fractal space-time, a family member of von Neumann's noncommutative geometry, is introduced as a geometry underlying a new relativity theory which is similar to the relation between general relativity and Riemannian geometry. Based on this model and the new relativity theory, an ensemble distribution of all the dimensions of quantum space-time is derived with the help of Fermat's last theorem. The calculated average dimension is very close to the value of 4 + phi (3) (where phi is the golden mean) obtained by El Naschie on the basis of a different approach. It is shown that within the framework of the new relativity, the cosmological constant problem is nonexistent, since the universe self-organizes and self-tunes according to the renormalization group (RG) flow with respect to a local scaling microscopic arrow of time. This implies that the world emerged as a result of a nonequilibrium process of self-organized critical phenomena launched by vacuum fluctuations in Cantorian-fractal space-lime E-infinity. It is shown that we are living in a metastable vacuum and are moving towards a fixed point (D-av = 4 + phi (3)) of the RG. After reaching this point, a new phase transition will drive the universe to a quasi-crystal phase of the lower average dimension of phi (3). (C) 2001 Elsevier Science Ltd. All rights reserved.