The Fractal Structure of the Quantum Space-Time

Abstract

We report here on the present state of an attempt at understanding micro-physics by basing ourselves on a ‘principle of scale relativity’, according to which the laws of physics should apply to systems of reference whatever their scale. The continuity but non differentiability of quantum mechanical particle paths, the occurrence of infinities in quantum field theories and the universal length and time scale dependence of measurement results implied by Heisenberg’s relations have led us, among other arguments, to suggest the achievement of such a principle by using the mathematical tool of fractals (Nottale 1989). The concept of a continuous and self-avoiding fractal space-time is worked out: arguments are given for generally describing them as families of Riemannian space-times whose curvature tends to infinity when scale approaches zero. We recall some basic results already obtained in this quest: the fractal dimension of quantum trajectories jumps from 1 to 2 for all 4 space-time coordinates, with the fractal/non fractal transition occurring around the de Broglie’s length and time; conversely the Heisenberg relations may be obtained as a consequence of assumed fractal structures; point particles following curves of fractal dimension 2 are naturally endowed with a spin. We develop the interpretation of the wave-particle duality as a property of families of geodesical lines in a fractal space-time and examplify it with the Young’s hole and Einstein-Podolsky-Rosen experiments. Finally some possible consequences concerning gravitation are recalled, with the suggestion that Newton’s law may break down for active gravitational masses smaller than the Planck mass.