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Physics Letters A

Particular solutions of a non-linear Schrödinger equation carrying particle-like singularities represent possible models of de Broglie's double solution theory

Jean Pierre Vigier

Published Feb 1, 1989• 34 citations
Publisher
DOI
10.1016/0375-9601(89)90653-1

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The theory of the double preparation: discerned and indiscerned particles

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In this paper we propose a deterministic and realistic quantum mechanics interpretation which may correspond to Louis de Broglie's "double solution theory". Louis de Broglie considers two solutions to the Schrödinger equation, a singular and physical wave u representing the particle (soliton wave) and a regular wave representing probability (statistical wave). We return to the idea of two solutions, but in the form of an interpretation of the wave function based on two different preparations of the quantum system. We demonstrate the necessity of this double interpretation when the particles are subjected to a semi-classical field by studying the convergence of the Schrödinger equation when the Planck constant tends to 0. For this convergence, we reexamine not only the foundations of quantum mechanics but also those of classical mechanics, and in particular two important paradox of classical mechanics: the interpretation of the principle of least action and the the Gibbs paradox. We find two very different convergences which depend on the preparation of the quantum particles: particles called indiscerned (prepared in the same way and whose initial density is regular, such as atomic beams) and particles called discerned (whose density is singular, such as coherent states). These results are based on the Minplus analysis, a new branch of mathematics that we have developed following Maslov, and on the Minplus path integral which is the analog in classical mechanics of the Feynman path integral in quantum mechanics. The indiscerned (or discerned) quantum particles converge to indiscerned (or discerned) classical particles and we deduce that the de Broglie-Bohm pilot wave is the correct interpretation for the indiscerned quantum particles (wave statistics) and the Schrödinger interpretation is the correct interpretation for discerned quantum particles (wave soliton). Finally, we show that this double interpretation can be extended to the non semi-classical case.

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Louis de Broglie Double Solution Theory Confirms the Wave-Particle Duality Principle

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Louis de Broglie in the beginning 20th century voices his theory of a double solution, according to which a pilot wave accompanies a particle, simulated as a point singularity, along the most optimal path from its creation on a source up to the detection. The pilot wave is a real hidden wave, which is similar to the wave function resulting from the solution of the Schrödinger equation. This theory is in agreement with de Broglie’s postulate about the matter waves. In this article we are based on the Helmholtz decomposition theorem according to which any velocity may be represented as a sum of two velocities – irrotational and solenoidal ones. The first velocity stems from the gradient of the scalar field. The second occurs from a pseudo-vector field. We proclaim that the gradients of the scalar field define guiding paths of the pilot wave. While the pseudo-vector field defines a particle solenoidal filling. We give mathematical models of the irrotational and solenoidal flows simulating the position of a particle in a guiding wave. Modified Navier-Stokes equation in pair with the continuity equation resulting in the Schrödinger equation gives such solutions consisting of superposition of the irrotational and solenoidal flow. It is declared that the guiding wave forms from the irrotational flows. In turn, the solenoidal flows underlie forming particles that travel along optimal paths slave by the guiding wave. There are described mathematical solutions of the spherical particle washed by the superfluid representing the guiding wave along which it travels.

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Quantum-Phase-Field: From de Broglie–Bohm Double-Solution Program to Doublon Networks

J. Kundin, Ingo Steinbach

Abstract Different forms of linear and non-linear field equations, so-called ‘phase-field’ equations, are studied in relation to the de Broglie–Bohm double-solution program. This defines a framework in which elementary particles are described by localised non-linear wave solutions moving by the guidance of a pilot wave, defined by the solution of a Schrödinger-type equation. First, we consider the phase-field order parameter as the phase for the linear pilot wave, second as the pilot wave itself, and third as a moving soliton interpreted as a massive particle. In the last case, we introduce the equation for a superwave, the amplitude of which can be considered as a particle moving in accordance to the de Broglie–Bohm theory. Lax pairs for the coupled problems are constructed in order to discover possible non-linear equations that can describe the moving particle and to propose a framework for investigating coupled solutions. Finally, doublons in 1 + 1 dimensions are constructed as self-similar solutions of a non-linear phase-field equation forming a finite space object. Vacuum quantum oscillations within the doublon determine the evolution of the coupled system. Applying a conservation constraint and using general symmetry considerations, the doublons are arranged as a network in 1 + 1 + 2 dimensions, where nodes are interpreted as elementary particles. A canonical procedure is proposed to treat charge and electromagnetic exchange.

arXiv:1911.11571•Zeitschrift für Naturforschung A•Jan 24, 2020•6 citations
In this paper we propose a deterministic and realistic quantum mechanics interpretation which may correspond to Louis de Broglie's "double solution theory". Louis de Broglie considers two solutions to the Schrödinger equation, a singular and physical wave u representing the particle (soliton wave) and a regular wave representing probability (statistical wave). We return to the idea of two solutions, but in the form of an interpretation of the wave function based on two different preparations of the quantum system. We demonstrate the necessity of this double interpretation when the particles are subjected to a semi-classical field by studying the convergence of the Schrödinger equation when the Planck constant tends to 0. For this convergence, we reexamine not only the foundations of quantum mechanics but also those of classical mechanics, and in particular two important paradox of classical mechanics: the interpretation of the principle of least action and the the Gibbs paradox. We find two very different convergences which depend on the preparation of the quantum particles: particles called indiscerned (prepared in the same way and whose initial density is regular, such as atomic beams) and particles called discerned (whose density is singular, such as coherent states). These results are based on the Minplus analysis, a new branch of mathematics that we have developed following Maslov, and on the Minplus path integral which is the analog in classical mechanics of the Feynman path integral in quantum mechanics. The indiscerned (or discerned) quantum particles converge to indiscerned (or discerned) classical particles and we deduce that the de Broglie-Bohm pilot wave is the correct interpretation for the indiscerned quantum particles (wave statistics) and the Schrödinger interpretation is the correct interpretation for discerned quantum particles (wave soliton). Finally, we show that this double interpretation can be extended to the non semi-classical case.
Louis de Broglie in the beginning 20th century voices his theory of a double solution, according to which a pilot wave accompanies a particle, simulated as a point singularity, along the most optimal path from its creation on a source up to the detection. The pilot wave is a real hidden wave, which is similar to the wave function resulting from the solution of the Schrödinger equation. This theory is in agreement with de Broglie’s postulate about the matter waves. In this article we are based on the Helmholtz decomposition theorem according to which any velocity may be represented as a sum of two velocities – irrotational and solenoidal ones. The first velocity stems from the gradient of the scalar field. The second occurs from a pseudo-vector field. We proclaim that the gradients of the scalar field define guiding paths of the pilot wave. While the pseudo-vector field defines a particle solenoidal filling. We give mathematical models of the irrotational and solenoidal flows simulating the position of a particle in a guiding wave. Modified Navier-Stokes equation in pair with the continuity equation resulting in the Schrödinger equation gives such solutions consisting of superposition of the irrotational and solenoidal flow. It is declared that the guiding wave forms from the irrotational flows. In turn, the solenoidal flows underlie forming particles that travel along optimal paths slave by the guiding wave. There are described mathematical solutions of the spherical particle washed by the superfluid representing the guiding wave along which it travels.
Abstract Different forms of linear and non-linear field equations, so-called ‘phase-field’ equations, are studied in relation to the de Broglie–Bohm double-solution program. This defines a framework in which elementary particles are described by localised non-linear wave solutions moving by the guidance of a pilot wave, defined by the solution of a Schrödinger-type equation. First, we consider the phase-field order parameter as the phase for the linear pilot wave, second as the pilot wave itself, and third as a moving soliton interpreted as a massive particle. In the last case, we introduce the equation for a superwave, the amplitude of which can be considered as a particle moving in accordance to the de Broglie–Bohm theory. Lax pairs for the coupled problems are constructed in order to discover possible non-linear equations that can describe the moving particle and to propose a framework for investigating coupled solutions. Finally, doublons in 1 + 1 dimensions are constructed as self-similar solutions of a non-linear phase-field equation forming a finite space object. Vacuum quantum oscillations within the doublon determine the evolution of the coupled system. Applying a conservation constraint and using general symmetry considerations, the doublons are arranged as a network in 1 + 1 + 2 dimensions, where nodes are interpreted as elementary particles. A canonical procedure is proposed to treat charge and electromagnetic exchange.