[Paper Review] How to justify Born's rule using the pilot wave theory of de Broglie?

Aurélien Drezet|arXiv (Cornell University)|Sep 28, 2016

Quantum Mechanics and Applications8 references5 citations

TL;DR

This paper presents multiple dynamical derivations of Born's rule within the de Broglie-Bohm pilot wave theory, showing that quantum equilibrium—where particle positions follow the Born probability distribution—emerges naturally from deterministic dynamics under mild initial conditions. Using chaotic maps like the Bernoulli map and entanglement-induced relaxation, the authors demonstrate that inhomogeneities in initial probability densities decay exponentially, leading to uniform quantum equilibrium in the long-time limit.

ABSTRACT

In this article we discuss few new derivations of the so called Born's rule for quantum probability in the context of the pilot wave theory proposed by de Broglie in 1927.

Motivation & Objective

• To provide new dynamical derivations of Born's rule within the de Broglie-Bohm pilot wave framework.
• To investigate the conditions under which quantum equilibrium emerges from deterministic initial particle distributions.
• To explore the role of chaos and entanglement in the relaxation to quantum equilibrium.
• To assess the robustness of quantum equilibrium in pilot wave theory, especially in non-equilibrium regimes.
• To lay the groundwork for a future double solution theory free of contradictions that fully reproduces standard quantum mechanics and pilot wave trajectories.

Proposed method

• Uses the non-relativistic Schrödinger equation as the foundational dynamics for N spinless particles.
• Applies the de Broglie-Bohm guidance equation to define particle trajectories guided by the wavefunction.
• Models relaxation to quantum equilibrium using the Bernoulli map: $ y_{n+1} = 2y_n \mod 1 $, which induces exponential mixing.
• Employs a Perron-Frobenius iterative relation: $ \rho_{n+1}(y) = \frac{1}{2}(\rho_n(y/2) + \rho_n((y+1)/2)) $, describing density evolution.
• Expands the probability density in terms of Bernoulli polynomials: $ \rho_n(y) = \sum_{m=0}^\infty C_m e^{-n m \ln 2} B_m(y) $, showing exponential decay of non-constant modes.
• Derives a continuous-time relaxation equation: $ \frac{d}{dt}\rho_t(y) = -\frac{\rho_t(y) - 1}{\tau} $, analogous to kinetic relaxation in statistical mechanics.

Experimental results

Research questions

• RQ1 Under what conditions does the pilot wave theory naturally lead to the Born probability distribution?
• RQ2 How does deterministic chaos in the guidance dynamics contribute to the relaxation of initial probability distributions to quantum equilibrium?
• RQ3 What role does entanglement play in enabling trajectory crossings and enabling the emergence of chaotic dynamics in the pilot wave framework?
• RQ4 Can the relaxation to quantum equilibrium be modeled as a kinetic process akin to the Boltzmann equation or Prigogine’s dissipative systems?
• RQ5 Is quantum equilibrium a robust outcome of the pilot wave dynamics, and can non-equilibrium states be used to test deviations from standard quantum mechanics?

Key findings

• The probability density $ \rho_n(y) $ converges to a uniform distribution $ \rho_n(y) \to 1 $ as $ n \to \infty $, regardless of initial inhomogeneities.
• The relaxation process is exponentially fast, with the decay rate of non-uniform components governed by $ e^{-n \ln 2} $, corresponding to a Lyapunov exponent of $ \ln 2 $.
• The asymptotic relaxation to equilibrium is described by a continuous-time Fokker-Planck-like equation: $ \frac{d}{dt}\rho_t(y) = -\frac{\rho_t(y) - 1}{\tau} $, with $ \tau \propto \tau_0 (1 - e^{-\ln 2})^{-1} $.
• The convergence to quantum equilibrium is robust under mild initial conditions, as higher-order Bernoulli polynomial modes decay exponentially.
• The system exhibits mixing behavior typical of K-systems or Bernoulli shifts, suggesting a deep connection between quantum equilibrium and deterministic chaos.
• The results support the idea that quantum equilibrium is not an arbitrary postulate but a natural outcome of deterministic dynamics with chaotic evolution.