Solitons, chaos, and quantum phenomena: a deterministic approach to the Schr\"odinger equation

TL;DR

This paper presents a deterministic framework where solitons in a field theory exhibit quantum phenomena through background fluctuations, deriving the Schrödinger equation from classical soliton dynamics and simulating quantum tunneling.

Contribution

It introduces a novel deterministic approach linking soliton dynamics and quantum mechanics, deriving the Schrödinger equation from classical soliton behavior influenced by background fluctuations.

Findings

Solitons follow classical trajectories on zero background.

Quantum phenomena emerge from fluctuations in a chaotic background.

Simulation results match predictions of the Schrödinger equation for tunneling.

Abstract

We show that the Schr\"odinger equation describes the ensemble mean dynamics of solitons in a Galilean invariant field theory where we interpret solitons as particles. On a zero background, solitons move classically, following Newton`s second law, however, on a non-zero amplitude chaotic background, their momentum and position fluctuate fulfilling an exact uncertainty relation, which give rise to the emergence of quantum phenomena. The Schrodinger equation for the ensemble of solitons is obtained from this exact uncertainty relation, and the amplitude of the background fluctuations is what corresponds to the value of $\hbar$. We confirm our analytical results running simulations of solitons moving against a potential barrier and comparing the ensemble probabilities with the predictions of the time dependent Schr\"odinger equation, providing a deterministic version of the quantum tunneling effect. We conclude with a discussion of how our theory does not present statistical independence between measurement and experiment outcome.