Scale relativity and Schrödingerʹs equation

The theory of scale relativity generalizes the application domain of Einsteinʹs principle of relativity to scale transformations of space-time resolutions. In this theory, we no longer assume that the space-time continuum is differentiable, this implying its fractal character. Both classical and quantum laws may emerge from a unique, more profound, scale law. The effects of nondifferentiability (complex nature of wave function, new terms in differential equations of mean motion) are accounted for by a scale-covariant derivative that transforms the equations of classical mechanics into the Schrödinger equation. Using an intermediate description in terms of a ‘fractal potential’, we finally establish the m−1 dependence of the Compton—de Broglie wavelength.