Justifying Born's rule $P_\alpha=|\Psi_\alpha|^2$ using deterministic chaos, decoherence, and the de Broglie-Bohm quantum theory

TL;DR

This paper derives Born's rule within de Broglie-Bohm theory by showing that entanglement and chaos induce rapid relaxation to quantum probabilities, supported by a toy model and kinetic theory analogy.

Contribution

It presents a novel derivation of Born's rule using deterministic chaos and decoherence in a Bohmian framework with a simplified qubit environment.

Findings

Entanglement and chaos cause fast relaxation to Born's rule.

A toy model demonstrates the relaxation process.

A kinetic theory analogy supports the quantum equilibrium concept.

Abstract

In this work we derive Born's rule from the pilot-wave theory of de Broglie and Bohm. Based on a toy model involving a particle coupled to a environement made of "qubits" (i.e., Bohmian pointers) we show that entanglement together with deterministic chaos lead to a fast relaxation from any statistitical distribution $\rho(x)$ (of finding a particle at point $x$) to the Born probability law $|\Psi(x)|^2$. Our model is discussed in the context of Boltzmann's kinetic theory and we demonstrate a kind of H theorem for the relaxation to the quantum equilibrium regime.