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

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.