The status and programs of scale relativity theory

A review of the most recent results of the scale relativity theory, founded by Nottale and developed further by the present author, is presented. Thes e include an elementary derivation of the black hole entropy-area relation and its logarithmic corrections; the derivation of the string uncertainty r elations and generalizations: the relation between the four-dimensional gra vitational conformal anomaly and the fine structure constant: and the role of noncommutative geometry, negative probabilities and E-(infinity) Cantori an-fractal space-time in the Youngs two-slit experiment. We then generalize the recent construction of the quenched-minisuperspace bosonic p-brane pro pagator in D dimensions [S. Ansoldi. A. Aurilia, C. Castro, E. Spallucci, P hys. Rev. D (to be submitted)] to the lull multidimensional case involving all p-branes: the construction of the multidimensional-particle propagator in Clifford spaces (C-spaces) associated with a nested family of p-loop his tories living in a target D-dim background space-time. We show how the effe ctive C-space geometry is related to extrinsic curvature of ordinary space- time. The motion of rigid particles/branes is studied to explain the natura l emergence of classical spin. The relation among C-space geometry and W , Finsler geometry and (braided) quantum groups is discussed. Some final rema rks about the Riemannian long-distance limit of C-space geometry are made. (C) 2001 Elsevier Science Ltd. All rights reserved.