On the Explanation of Born-Rule Statistics in the de Broglie-Bohm Pilot-Wave Theory

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.

De Broglie–Bohm Pilot‐Wave Theory: Many Worlds in Denial?

Many Worlds? ◽

10.1093/acprof:oso/9780199560561.003.0019 ◽

2010 ◽

pp. 476-509 ◽

Cited By ~ 7

Author(s):

Antony Valentini

Keyword(s):

Wave Theory ◽

Many Worlds ◽

Pilot Wave ◽

Pilot Wave Theory ◽

De Broglie Bohm

Relaxation to quantum equilibrium for Dirac fermions in the de Broglie–Bohm pilot-wave theory

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2011.0549 ◽

2011 ◽

Vol 468 (2140) ◽

pp. 1116-1135 ◽

Cited By ~ 9

Author(s):

Samuel Colin

Keyword(s):

Wave Function ◽

Wave Theory ◽

Dirac Fermions ◽

Two Dimensional ◽

Step Potential ◽

Pilot Wave ◽

Equilibrium Distributions ◽

Chaotic Nature ◽

Pilot Wave Theory ◽

De Broglie Bohm

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).

Bouncing oil droplets, de Broglie's quantum thermostat and convergence to equilibrium

10.20944/preprints201808.0475.v1 ◽

2018 ◽

Author(s):

Mohamed Hatifi ◽

Ralph Willox ◽

Samuel Colin ◽

Thomas Durt

Keyword(s):

Wave Theory ◽

Convergence To Equilibrium ◽

Potential Well ◽

Oil Droplets ◽

Stochastic Version ◽

Pilot Wave ◽

Wave Particle Duality ◽

H Theorem ◽

Pilot Wave Theory ◽

De Broglie Bohm

Recently, the properties of bouncing oil droplets, also known as `walkers', have attracted much attention because they are thought to offer a gateway to a better understanding of quantum behaviour. They indeed constitute a macroscopic realization of wave-particle duality, in the sense that their trajectories are guided by a self-generated surrounding wave. The aim of this paper is to try to describe walker phenomenology in terms of de Broglie-Bohm dynamics and of a stochastic version thereof. In particular, we first study how a stochastic modification of the de Broglie pilot-wave theory, à la Nelson, affects the process of relaxation to quantum equilibrium, and we prove an H-theorem for the relaxation to quantum equilibrium under Nelson-type dynamics. We then compare the onset of equilibrium in the stochastic and the de Broglie-Bohm approaches and we propose some simple experiments by which one can test the applicability of our theory to the context of bouncing oil droplets. Finally, we compare our theory to actual observations of walker behavior in a 2D harmonic potential well.

Bouncing Oil Droplets, de Broglie’s Quantum Thermostat, and Convergence to Equilibrium

Entropy ◽

10.3390/e20100780 ◽

2018 ◽

Vol 20 (10) ◽

pp. 780 ◽

Cited By ~ 2

Author(s):

Mohamed Hatifi ◽

Ralph Willox ◽

Samuel Colin ◽

Thomas Durt

Keyword(s):

Wave Theory ◽

Convergence To Equilibrium ◽

Oil Droplets ◽

Stochastic Version ◽

Pilot Wave ◽

Wave Particle Duality ◽

Quantum Behavior ◽

H Theorem ◽

Pilot Wave Theory ◽

De Broglie Bohm

Recently, the properties of bouncing oil droplets, also known as “walkers,” have attracted much attention because they are thought to offer a gateway to a better understanding of quantum behavior. They indeed constitute a macroscopic realization of wave-particle duality, in the sense that their trajectories are guided by a self-generated surrounding wave. The aim of this paper is to try to describe walker phenomenology in terms of de Broglie–Bohm dynamics and of a stochastic version thereof. In particular, we first study how a stochastic modification of the de Broglie pilot-wave theory, à la Nelson, affects the process of relaxation to quantum equilibrium, and we prove an H-theorem for the relaxation to quantum equilibrium under Nelson-type dynamics. We then compare the onset of equilibrium in the stochastic and the de Broglie–Bohm approaches and we propose some simple experiments by which one can test the applicability of our theory to the context of bouncing oil droplets. Finally, we compare our theory to actual observations of walker behavior in a 2D harmonic potential well.

The Pilot-Wave Theory

Undergraduate Lecture Notes in Physics - Foundations of Quantum Mechanics ◽

10.1007/978-3-319-65867-4_7 ◽

2017 ◽

pp. 177-213

Author(s):

Travis Norsen

Keyword(s):

Wave Theory ◽

Pilot Wave ◽

Pilot Wave Theory

The ‘beables’ of relativistic pilot-wave theory

From Physics to Philosophy ◽

10.1017/cbo9780511597947.006 ◽

1999 ◽

pp. 71-89 ◽

Cited By ~ 2

Author(s):

Simon Saunders

Keyword(s):

Wave Theory ◽

Pilot Wave ◽

Pilot Wave Theory

Geometrical interpretation of the pilot wave theory and manifestation of spinor fields

Progress of Theoretical and Experimental Physics ◽

10.1093/ptep/ptaa106 ◽

2020 ◽

Vol 2020 (9) ◽

Author(s):

Mariya Iv. Trukhanova ◽

Gennady Shipov

Keyword(s):

Wave Theory ◽

Geometrical Interpretation ◽

Real Field ◽

Spin Vector ◽

Spinor Wave Function ◽

Spinning Particle ◽

Pilot Wave ◽

Intrinsic Angular Momentum ◽

Spinor Wave ◽

Pilot Wave Theory

Abstract Using the hydrodynamical formalism of quantum mechanics for a Schrödinger spinning particle developed by Takabayashi, Vigier, and followers, which involves vortical flows, we propose a new geometrical interpretation of the pilot wave theory. The spinor wave in this interpretation represents an objectively real field, and the evolution of a material particle controlled by the wave is a manifestation of the geometry of space. We assume this field to have a geometrical nature, basing on the idea that the intrinsic angular momentum, the spin, modifies the geometry of the space, which becomes a manifold, represented as a vector bundle with a base formed by the translational coordinates and time, and the fiber of the bundle, specified at each point by the field of a tetrad $e^a_{\mu}$, forms from bilinear combinations of the spinor wave function. It has been shown that the spin vector rotates following the geodesic of the space with torsion, and the particle moves according to the geometrized guidance equation. This fact explains the self-action of the spinning particle. We show that the curvature and torsion of the spin vector line is determined by the space torsion of the absolute parallelism geometry.