Resolution-scale relativistic formulation of non-differentiable mechanics

TL;DR

This paper develops a scale relativistic framework for non-differentiable mechanics, linking it to quantum mechanics, and demonstrates how classical dynamics generalizes to quantum behavior through non-differentiable paths and complex velocities.

Contribution

It introduces a complex scale-covariant operator and reformulates Newton's laws for non-differentiable paths, connecting classical mechanics to quantum mechanics within a unified scale relativity approach.

Findings

Langevin equation describes non-differentiable dynamics

Numerical results match Schrödinger equation solutions

Quantum mechanics interpreted as non-differentiable classical dynamics

Abstract

This article motivates and presents the scale relativistic approach to non-differentiability in mechanics and its relation to quantum mechanics. It stems from the scale relativity proposal to extend the principle of relativity to resolution-scale transformations, which leads to considering non-differentiable dynamical paths. We first define a complex scale-covariant time-differential operator and show that mechanics of non-differentiable paths is implemented in the same way as classical mechanics but with the replacement of the time derivative and velocity with the time-differential operator and associated complex velocity. With this, the generalized form of Newton's fundamental relation of dynamics is shown to take the form of a Langevin equation in the case of stationary motion characterized by a null average classical velocity. The numerical integration of the Langevin equation in the case of a harmonic oscillator taken as an example reveals the same statistics as the stationary solutions of the Schrodinger equation for the same problem. This motivates the rest of the paper, which shows Schrodinger's equation to be a reformulation of Newton's fundamental relation of dynamics as generalized to non-differentiable geometries and leads to an alternative interpretation of the other axioms of standard quantum mechanics in a coherent picture. This exercise validates the scale relativistic approach and, at the same time, it allows to envision macroscopic chaotic systems observed at resolution time-scales exceeding their horizon of predictability as candidates in which to search for quantum-like dynamics and structures.