Abstract :
A review of the most recent results of the scale relativity theory, founded by Nottale and developed further by the present author, is presented. These include an elementary derivation of the black hole entropy–area relation and its logarithmic corrections; the derivation of the string uncertainty relations and generalizations; the relation between the four-dimensional gravitational conformal anomaly and the fine structure constant; and the role of noncommutative geometry, negative probabilities and Cantorian-fractal space-time in the Youngʹs two-slit experiment. We then generalize the recent construction of the quenched-minisuperspace bosonic p-brane propagator in D dimensions [S. Ansoldi, A. Aurilia, C. Castro, E. Spallucci, Phys. Rev. D (to be submitted)] to the full 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 histories living in a target D-dim background space-time. We show how the effective C-space geometry is related to extrinsic curvature of ordinary space-time. The motion of rigid particles/branes is studied to explain the natural emergence of classical spin. The relation among C-space geometry and , Finsler geometry and (braided) quantum groups is discussed. Some final remarks about the Riemannian long-distance limit of C-space geometry are made.
