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arXiv:2109.09353 · Entropy

Justifying Born’s Rule Pα = |Ψα|2 Using Deterministic Chaos, Decoherence, and the de Broglie–Bohm Quantum Theory

Aurélien Drezet

Published Oct 20, 2021• 12 citations
arXivHTMLPDF

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 an environment made of “qubits” (i.e., Bohmian pointers), we show that entanglement together with deterministic chaos leads to a fast relaxation from any statistical distribution ρ(x) of finding a particle at point x to the Born probability law |Ψ(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.

DOI
10.3390/e23111371

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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 an environment made of “qubits” (i.e., Bohmian pointers), we show that entanglement together with deterministic chaos leads to a fast relaxation from any statistical distribution ρ(x) of finding a particle at point x to the Born probability law |Ψ(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.
We study the origin of the Born probability rule ρ = | ψ | 2 in the de Broglie–Bohm pilot–wave formulation of quantum theory. It is argued that quantum probabilities arise dynamically, and have a status similar to thermal probabilities in ordinary statistical mechanics. This is illustrated by numerical simulations for a two–dimensional system. We show that a simple initial ensemble, with a non–standard distribution ρ ≠ | ψ | 2 of particle positions, evolves towards the quantum distribution to high accuracy. The relaxation process ρ →| ψ | 2 is quantified in terms of a coarse–grained H –function (equal to minus the relative entropy of ρ with respect to | ψ | 2 ), which is found to decrease approximately exponentially over time, with a time constant that accords with a simple theoretical estimate.
Numerical simulations indicate that the Born rule does not need to be postulated in the de Broglie–Bohm pilot-wave theory, but tends to arise dynamically (relaxation to quantum equilibrium). These simulations were done for a particle in a two-dimensional box whose wave function obeys the non-relativistic Schrödinger equation and is therefore scalar. The chaotic nature of the de Broglie–Bohm trajectories, thanks to the nodes of the wave function which yield to vortices, is crucial for a fast relaxation to quantum equilibrium. For spinors, we typically do not expect any node. However, in the case of the Dirac equation, the de Broglie–Bohm velocity field has vorticity even in the absence of nodes. This observation raises the question of the origin of relaxation to quantum equilibrium for fermions. In this article, we provide numerical evidence to show that Dirac particles also undergo relaxation, by simulating the evolution of various non-equilibrium distributions for two-dimensional systems (the two-dimensional Dirac oscillator and the Dirac particle in a two-dimensional spherical step potential).
We illustrate through explicit numerical calculations how the Born rule probability densities of non-relativistic quantum mechanics emerge naturally from the particle dynamics of de Broglie–Bohm pilot-wave theory. The time evolution of a particle distribution initially not equal to the absolute square of the wave function is calculated for a particle in a two-dimensional infinite potential square well. Under the de Broglie–Bohm ontology, the box contains an objectively existing ‘pilot wave’ which guides the electron trajectory, and this is represented mathematically by a Schrödinger wave function composed of a finite out-of-phase superposition of M energy eigenstates (with M ranging from 4 to 64). The electron density distributions are found to evolve naturally into the Born rule ones and stay there; in analogy with the classical case this represents a decay to ‘quantum equilibrium’. The proximity to equilibrium is characterized by the coarse-grained subquantum H -function which is found to decrease roughly exponentially towards zero over the course of time. The time scale τ for this relaxation is calculated for various values of M and the coarse-graining length ε . Its dependence on M is found to disagree with an earlier theoretical prediction. A power law, τ ∝ M −1 , is found to be fairly robust for all coarse-graining lengths and, although a weak dependence of τ on ε is observed, it does not appear to follow any straightforward scaling. A theoretical analysis is presented to explain these results. This improvement in our understanding of time scales for relaxation to quantum equilibrium is likely to be of use in the development of models of relaxation in the early Universe, with a view to constraining possible violations of the Born rule in inflationary cosmology.
The predictions of quantum mechanics are probabilistic. Quantum probabilities are extracted using a postulate of the theory called the Born rule, the status of which is central to the problem of quantum mechanics. Efforts to justify the Born rule from other physical principles, and thus elucidate the measurement process, have involved lengthy statistical or information-theoretic arguments. Here we show that Bohm's deterministic formulation of quantum mechanics allows the Born rule for measurements on a single system to be derived, without any statistical assumptions. We solve a simple example where the creation of an ensemble of identical quantum states, together with position measurements on those states, are described by Bohm's quantum dynamics. The calculated measurement outcomes agree with the Born-rule probabilities, which are thus a consequence of deterministic evolution. Our results demonstrate that quantum probabilities can emerge from simple dynamical laws alone, and they support the view that there is no underlying indeterminism in quantum phenomena.
We compare and contrast two distinct approaches to understanding the Born rule in de Broglie-Bohm pilot-wave theory, one based on dynamical relaxation over time (advocated by this author and collaborators) and the other based on typicality of initial conditions (advocated by the 'Bohmian mechanics' school). It is argued that the latter approach is inherently circular and physically misguided. The typicality approach has engendered a deep-seated confusion between contingent and law-like features, leading to misleading claims not only about the Born rule but also about the nature of the wave function. By artificially restricting the theory to equilibrium, the typicality approach has led to further misunderstandings concerning the status of the uncertainty principle, the role of quantum measurement theory, and the kinematics of the theory (including the status of Galilean and Lorentz invariance). The restriction to equilibrium has also made an erroneously-constructed stochastic model of particle creation appear more plausible than it actually is. To avoid needless controversy, we advocate a modest 'empirical approach' to the foundations of statistical mechanics. We argue that the existence or otherwise of quantum nonequilibrium in our world is an empirical question to be settled by experiment.
We extend de Broglie--Bohm (dBB) pilot-wave theory by introducing a complex source field into the Schrödinger equation and examining its effects on quantum equilibrium and nonequilibrium dynamics. In dBB theory, the physical particle distribution P, rather than the Born density ∣ψ∣2|ψ|^2∣ψ∣2, carries the ensemble probability, so a source term may modify the pilot wave without violating conservation of total particle probability. We derive the source-modified Hamilton--Jacobi and continuity equations, the transport equation for the nonequilibrium ratio f=P/∣ψ∣2f=P/|ψ|^2f=P/∣ψ∣2, and an exact entropy-production formula. Three applications follow. First, suitably designed sources can drive exponential relaxation toward quantum equilibrium. Second, the entropy-production rate admits a Prigogine-type bilinear form, providing a basis for a subquantum thermodynamics with entropy-producing and entropy-extracting regimes. Third, a tuned source can exactly cancel the Bohmian quantum potential, yielding classical particle trajectories while the guiding wave retains nontrivial structure and the nonequilibrium ratio remains conserved. The resulting framework provides a unified setting for studying source-driven quantum nonequilibrium, entropy exchange, and classicalization, and motivates further investigation of the physical and ontological status of the source field.