# Riccati equations as a scale-relativistic gateway to quantum mechanics

## TL;DR

This paper develops a scale-relativistic framework linking non-differentiable paths to quantum mechanics, showing that solutions to Riccati equations underpin quantum behavior and extend to multi-dimensional, time-dependent systems.

## Contribution

It introduces a novel scale-relativistic approach that derives quantum mechanics from non-differentiable dynamics using Riccati equations, generalizing to higher dimensions and time dependence.

## Key findings

- Stationary motion corresponds to Ito processes driven by Riccati solutions.
- The probability density matches the squared modulus of Schrödinger solutions.
- Quantum-like states characterize entire systems, not individual particles.

## Abstract

Applying the resolution-scale relativity principle to develop a mechanics of non-differentiable dynamical paths, we find that, in one dimension, stationary motion corresponds to an Ito process driven by the solutions of a Riccati equation. We verify that the corresponding Fokker-Planck equation is solved for a probability density corresponding to the squared modulus of the solution of the Schrodinger equation for the same problem. Inspired by the treatment of the one-dimensional case, we identify a generalization to time dependent problems in any number of dimensions. The Ito process is then driven by a function which is identified as establishing the link between non-differentiable dynamics and standard quantum mechanics. This is the basis for the scale relativistic interpretation of standard quantum mechanics and, in the case of applications to chaotic systems, it leads us to identify quantum-like states as characterizing the entire system rather than the motion of its individual constituents.