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

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

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.

Pilot-Wave Theory for a Relativistic Spin Zero Particle

In the usual approach to the pilot-wave theory for a spin zero particle one starts with the Klein-Gordon equation, which is the relativistic generalization of the Schrodinger equation. This approach encounters several difficulties including superluminal motion and particle trajectories that move backwards in time. In this paper I start with the relativistic classical Hamilton-Jacobi equation and introduce the quantum potential in a way that avoids the above mentioned difficulties. Particle trajectories are timelike or null and are future pointing. The wave equation satisfied by the field is a nonlinear generalization of the Klein-Gordon equation.

Hamiltonian formulation of the pilot-wave theory

In the pilot-wave theory of quantum mechanics particles have definite positions and velocities and the system evolves deterministically. The velocity of a particle is determined by the wave function of the system (the guidance equation) and the wave function evolves according to Schrödinger’s equation. In this paper I first construct a Hamiltonian that gives Schrödinger’s equation and the guidance equation for the particle. I then find the Hamiltonian for a relativistic particle in Dirac’s theory and for a quantum scalar field.

Energy velocity and reactive fields

Conventional definitions of ‘near fields’ set bounds that describe where near fields may be found. These definitions tell us nothing about what near fields are, why they exist or how they work. In 1893, Heaviside derived the electromagnetic energy velocity for plane waves. Subsequent work demonstrated that although energy moves in synchronicity with radiated electromagnetic fields at the speed of light, in reactive fields the energy velocity slows down, converging to zero in the case of static fields. Combining Heaviside's energy velocity relation with the field Lagrangian yields a simple parametrization for the reactivity of electromagnetic fields that provides profound insights to the behaviour of electromagnetic systems. Fields guide energy. As waves interfere, they guide energy along paths that may be substantially different from the trajectories of the waves themselves. The results of this paper not only resolve the long-standing paradox of runaway acceleration from radiation reaction, but also make clear that pilot wave theory is the natural and logical consequence of the need for quantum mechanics correspond to the macroscopic results of the classical electromagnetic theory. This article is part of the theme issue ‘Celebrating 125 years of Oliver Heaviside's ‘Electromagnetic Theory’’.

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

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.

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

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.