On the transition from the classical to the quantum regime in fractal space–time theory

In the scale-relativity theory, space–time is described as a nondifferentiable continuum and the trajectories as its geodesics. In such a space–time, the coordinates are defined as the sum of a ‘classical part’ that remains differentiable, and a fluctuating, ‘fractal part’, that is divergent and nondifferentiable. The nondifferentiable geometry has three minimal consequences, namely infinite number, fractality and irreversibility of geodesics. These three effects are accounted for by the introduction of three new terms in the total derivative acting on the ‘classical part’ of the coordinates. When it is written using this total derivative, Newton’s equation is integrated in terms of a Schrödinger equation. Such an equation is therefore both classical and quantum. In the present paper, we use this property to analyze the specific roles played by each of the individual contributions, in order to shed some light on the complicated transition from the classical to the quantum regime.