scholarly journals Hamiltonian formulation of the pilot-wave theory

2019 ◽  
Vol 97 (4) ◽  
pp. 431-435
Author(s):  
Dan N. Vollick
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Scalar Field ◽  
Relativistic Particle ◽  
Hamiltonian Formulation ◽  
Wave Theory ◽  
Schrödinger’S Equation ◽  
Pilot Wave ◽  
Schrödinger's Equation ◽  
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.

scholarly journals Bohm Quantum Trajectories of Scalar Field in Trans-Planckian Physics

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Jung-Jeng Huang
Keyword(s):  
Scalar Field ◽  
Wave Theory ◽  
Vacuum State ◽  
Quantum Trajectory ◽  
Quantum Trajectories ◽  
De Sitter ◽  
Pilot Wave ◽  
In Trans ◽  
Schrödinger Picture ◽  
Pilot Wave Theory

In lattice Schrödinger picture, we investigate the possible effects of trans-Planckian physics on the quantum trajectories of scalar field in de Sitter space within the framework of the pilot-wave theory of de Broglie and Bohm. For the massless minimally coupled scalar field and the Corley-Jacobson type dispersion relation with sextic correction to the standard-squared linear relation, we obtain the time evolution of vacuum state of the scalar field during slow-roll inflation. We find that there exists a transition in the evolution of the quantum trajectory from well before horizon exit to well after horizon exit, which provides a possible mechanism to solve the riddle of the smallness of the cosmological constant.

scholarly journals The Many-Object "Pilot Wave" Theory

2011 ◽  
Vol 9 ◽  
pp. 69-77
Author(s):  
Yoav Ben-Dov
Keyword(s):  
Quantum Mechanics ◽  
Wave Theory ◽  
Object A ◽  
Pilot Wave ◽  
Supplementary Variables ◽  
The Many ◽  
Pilot Wave Theory

The "pilot wave." supplementary variables version of quantum mechanics diacusaed. It is claimed that in the many-object a semi-class picture of particles "guided" in their motion by waves m 3-spaces is difficult to maintain-Other interpretative schemes are suggested

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

2011 ◽  
Vol 468 (2140) ◽  
pp. 1116-1135 ◽  
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).

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

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.

Pilot-wave theory in retrospect

2013 ◽  
pp. 224-241
Author(s):  
Guido Bacciagaluppi ◽  
Antony Valentini
Keyword(s):  
Wave Theory ◽  
Pilot Wave ◽  
Pilot Wave Theory
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