Scale relativity in Cantorian E-(infinity) space and average dimensions ofour world

A Cantorian-fractal space-time, a family member of von Neumann's noncommuta tive geometry, is introduced as a geometry underlying a new relativity theo ry which is similar to the relation between general relativity and Riemanni an geometry. Based on this model and the new relativity theory, an ensemble distribution of all the dimensions of quantum space-time is derived with t he 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 b y 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 n onexistent, since the universe self-organizes and self-tunes according to t he renormalization group (RG) flow with respect to a local scaling microsco pic arrow of time. This implies that the world emerged as a result of a non equilibrium 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 trans ition 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.