Emergence of complex and spinor wave functions in scale relativity. I. Nature of scale variables

TL;DR

This paper explains how the complex nature of the wave function in quantum mechanics arises from the fractal, non-differentiable structure of space-time in scale relativity, linking scale variables to wave function properties.

Contribution

It clarifies the role of scale variables and non-differentiability in deriving the complex wave function within the scale relativity framework.

Findings

Complex wave functions originate from two-valued mean derivatives.

The fractal structure of space-time leads to complex velocity fields.

Schrödinger equation emerges from scale-dependent non-differentiability.

Abstract

One of the main results of Scale Relativity as regards the foundation of quantum mechanics is its explanation of the origin of the complex nature of the wave function. The Scale Relativity theory introduces an explicit dependence of physical quantities on scale variables, founding itself on the theorem according to which a continuous and non-differentiable space-time is fractal (i.e., scale-divergent). In the present paper, the nature of the scale variables and their relations to resolutions and differential elements are specified in the non-relativistic case (fractal space). We show that, owing to the scale-dependence which it induces, non-differentiability involves a fundamental two-valuedness of the mean derivatives. Since, in the scale relativity framework, the wave function is a manifestation of the velocity field of fractal space-time geodesics, the two-valuedness of velocities leads to write them in terms of complex numbers, and yields therefore the complex nature of the wave function, from which the usual expression of the Schr\"odinger equation can be derived.