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arXiv:0904.0790

Three Wave Hypothesis, Gear Model and the Rest Mass

Mohammed I. Sanduk

Published Apr 5, 2009• 0 citations
arXivHTMLPDF

Abstract

Three Wave Hypothesis (TWH) is of a relativistic quantum foundation. The formulation of TWH has perfect similarities with bevel gear model. The rest mass is considered within TWH and as a consequence of similarity, between TWH and the gear model, it is found that rest mass frequency is related the frequency of touch point of the gear in relative to the large wheel. In regarding this frequency instead of the small wheel frequency, the gear model leads to four-vector representation of a particle.

DOI
10.48550/arxiv.0904.0790

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Three Wave Hypothesis (TWH) is of a relativistic quantum foundation. The formulation of TWH has perfect similarities with bevel gear model. The rest mass is considered within TWH and as a consequence of similarity, between TWH and the gear model, it is found that rest mass frequency is related the frequency of touch point of the gear in relative to the large wheel. In regarding this frequency instead of the small wheel frequency, the gear model leads to four-vector representation of a particle.
The three wave hypothesis (TWH) is an attempt to relate the particle to an associated wave phenomenon. This hypothesis regards that the particle is associated with three waves: one transformed and two dispersive waves. Since the two dispersive waves are associated with a single particle, in this work we try to get a single representation for these two dispersions. The single representation exhibits similarities with those of a classical gear consisting of two perpendicular wheels. According to this similarity, the parameters of dispersive waves correspond to those of the wheels, and the transformed Compton wave corresponds to the system parameters (the combination). This similarity between the two models may possibly point to a hidden structure.
Based on de Broglie’s wave hypothesis and the covariant ether, the Three Wave Hypothesis (TWH) has been proposed and developed in the last century. In 2007, the author found that the TWH may be attributed to a kinematical classical system of two perpendicular rolling circles. In 2012, the author showed that the position vector of a point in a model of two rolling circles in plane can be transformed to a complex vector under a proposed effect of partial observation. In the present project, this concept of transformation is developed to be a lab observation concept. Under this transformation of the lab observer, it is found that velocity equation of the motion of the point is transformed to an equation analogising the relativistic quantum mechanics equation (Dirac equation). Many other analogies has been found, and are listed in a comparison table. The analogy tries to explain the entanglement within the scope of the transformation. These analogies may suggest that both quantum mechanics and special relativity are emergent, both of them are unified, and of the same origin. The similarities suggest analogies and propose questions of interpretation for the standard quantum theory, without any possible causal claims.
The Three Wave Hypothesis (TWH) has been proposed by Horodecki in 1981. Sanduk attributed TWH to a classical kinematical model of two rolling circles in 2007. In a previous project in 2012, it is shown that the position vector of a point in a system of two rolling circles can be transformed to a complex vector under the effect of partial observation. The present work tries to develop this concept of transformation. Under this transformation, it is found that the kinematical equations of the motion of point can be transformed to equations analogise the relativistic quantum mechanics equations. Many analogies have been found and are listed in a comparison table. These analogies may sagest that both of the quantum mechanics and the special relativity are emergent, and are of the same origin.
In the wave-particle duality, a free particle can be considered as a wave packet. Is there a wave property that corresponds to the rest mass of the particle? This is an interesting question that has not been extensively explored before. We suggest that this problem may be approached by treating the rest mass on the same footing as energy and momentum. Here we demonstrate that, by assuming that the matter wave of a particle behaves like a photon, one could derive the mass-energy relation from the solution of a simplified wave equation. This solution suggests that the rest mass of a particle is associated with a "transverse wave number", which characterizes the radial variation of the wave function in the transverse plane. This finding has several appealing features. For example, it shows a consistent geometrical relationship between mass, energy and momentum in both the wave and particle perspectives. Also, it predicts that a massless particle must travel at the constant speed of light. Furthermore, from a philosophical point of view, this model suggests that the mass-energy relation attributed to the special theory of relativity may have a root in the wave-particle duality. Indeed, it is shown that the Klein-Gordon equation can be derived naturally based on this model.
It is well known that the mass of a particle has properties different from Newtonian mechanics. First, it is speed-dependent. Second, it is convertible to energy. These properties were generally thought to be derived from the principle of relativity (PR). We have conducted a careful examination of the historical records and found that the non-Newtonian properties of mass were derived not so much based on PR, but more based on Einstein’s intuitive thinking that radiation and matters behave similarly. This gives us a hint: Since both photon and electron can behave as a particle as well as a wave, can such a wave nature account for the deviations from Newtonian mechanics? Thus, we have developed a wave model to describe the motion of a free particle with or without rest mass. We found that both the speed-dependence of mass and the mass-energy equivalence can indeed be derived based on the wave properties of a particle. This wave hypothesis has several advantages; it can naturally explain why particles can be created in the vacuum and why a particle cannot travel faster than the speed of light.
We present a phase-based formulation of special relativity in which the kinematical structure of the theory is reconstructed from the requirement of phase coherence of localized wave states. Starting from the assumption that physical propagation is associated with surfaces of constant phase and that matter admits an intrinsic rest-frame oscillation, we show that the relativistic relations for time dilation, energy, and momentum follow from the invariant accumulation of phase along particle trajectories. In this framework, proper time is identified operationally as the phase count of an internal clock, and the Minkowski interval arises as the quadratic form consistent with phase invariance across inertial observers. The relation between mass and rest-frame frequency emerges naturally, providing a unified interpretation of relativistic kinematics in wave-mechanical terms. This formulation does not introduce new dynamics but offers a coherent structural perspective that bridges relativistic kinematics and wave propagation.