Scale Relativity, Fractal Space-Time, and Quantum Mechanics

This paper describes the present state of an attempt at understanding the quantum behaviour of microphysics in terms of a nondifferentiable space-time continuum having fractal (i.e., scale-dependent) properties. The fundamental principle upon which we rely is that of scale relativity, which generalizes to scale transformations Einstein’s principle of relativity. After having related the fractal and renormalization group approaches, we develop a new version of stochastic quantum mechanics, in which the correspondence principle and the Schrödinger equation are demonstrated by replacing the classical time derivative by a ”quantumcovariant” derivative. Then we recall that the principle of scale relativity leads one to generalize the standard “Galilean” laws of scale transformation into a Lorentzian form, in which the Planck length-scale becomes invariant under dilations, and so plays for scale laws the same role as played by the velocity of light for motion laws. We conclude by an application of our new framework to the problem of the mass spectrum of elementary particles.