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arXiv:1302.1190

Wave Mechanics and the Fifth Dimension

Paul S. Wesson, JAMES M. OVERDUIN

Published Jan 28, 2013• 0 citations
arXivHTMLPDF

Abstract

Replacing 4D Minkowski space by 5D canonical space leads to a clearer derivation of the main features of wave mechanics, including the wave function and the velocity of de Broglie waves. Recent tests of wave-particle duality could be adapted to investigate whether de Broglie waves are basically 4D or 5D in nature.
DOI
10.48550/arxiv.1302.1190

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It is argued that the de Broglie wave is not the wave usually supposed, but the relativistically induced modulation of an underlying carrier wave that moves with the velocity of the particle. In the rest frame of the particle this underlying structure has the form of a standing wave. De Broglie also assumed the existence of this antecedent standing wave, but it would appear that he failed to notice its survival as a carrier wave in the Lorentz transformed wave structure. Identified as a modulation, the de Broglie wave acquires a physically reasonable ontology, evidencing a more natural unity between matter and radiation than might otherwise be contemplated, and avoiding the necessity of recovering the particle velocity from a superposition of such waves. Because the Schrödinger and other wave equations for massive particles were conceived as equations for the de Broglie wave, this interpretation of the wave is also relevant to such issues in quantum mechanics as the meaning of the wave function, the nature of wave-particle duality, and the possibility of well-defined particle trajectories.
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In his famous doctoral dissertation, de Broglie assumed that a massive particle is surrounded in its rest frame by a standing wave. He argued that as observed from another inertial frame this wave becomes the superluminal wave now known as the de Broglie wave. It is shown here that under a Lorentz transformation, such a standing wave becomes, not the de Broglie wave, but a modulated wave moving at the velocity of the particle. It is the modulation that has the superluminal velocity of the de Broglie wave and should be recognized as the true de Broglie wave. De Broglie's demonstrations relied, variously, on his "theorem of the harmony of phases", on a mechanical model, and on a spacetime diagram. It is shown that in each case the underlying wave was inadvertently suppressed. Identified as a modulation, the de Broglie wave acquires a physically reasonable ontology, avoiding the awkward device of recovering the particle velocity from a superposition of such waves. The deeper wave structure implied by this de Broglie wave must also impinge on such issues in quantum mechanics as the meaning of the wave function and the nature of wave-particle duality.
Louis de Broglie put forward the idea of wave-particle duality in 1923-24, less than two years before the advent of Quantum Mechanics. This last theory assumes that duality, but in a more abstract and counter-intuitive manner than de Broglie did. In this article the wave behavior of matter is presented in a didactic way following de Broglie´s ideas. This kind of presentation may (hopefully) be helpful to better understand the quantum features in a more intuitive way. It is shown that this wave mechanics, in the framework of the theory of relativity, is compatible with Planck´s quantum and explains as well the particle´s dynamics, provided the uncertainty relations are properly incorporated. The main difficulty with this view lies in its interpreting the wave as a non-local entity