Probabilistic cellular automaton for quantum particle in a potential

TL;DR

This paper introduces a probabilistic cellular automaton model that reproduces quantum particle behavior in a potential, deriving Dirac and Schrödinger equations from simple automaton rules.

Contribution

It presents a novel cellular automaton framework that models quantum particles and derives fundamental quantum equations from classical probabilistic rules.

Findings

Automaton reproduces Dirac and Schrödinger equations in continuum limit

Quantum observables follow classical statistical laws

Model captures quantum behavior with simple deterministic rules

Abstract

We propose that a quantum particle in a potential in one space dimension can be described by a probabilistic cellular automaton. While the simple updating rule of the automaton is deterministic, the probabilistic description is introduced by a probability distribution over initial conditions. The proposed automaton involves right- and left-movers, jumping from one cell to a neighboring one. They change their direction of motion at each randomly distributed disorder or scattering point. The continuum limit of an infinite number of cells yields a Dirac equation, and in the non-relativistic limit the familiar Schr\"odinger equation, with potential determined by the spacetime-distribution of scattering points. These equations describe the time evolution of the probabilistic information for the position of the particle. All quantum rules for observables, both for discrete possible measurement values and continuous expectation values, follow from the classical statistical laws.