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Open Journal of Microphysics

Fractal Approximation of Motion and Its Implications in Quantum Mechanics

Maricel Agop, Daniela Magop, Elena Simona Băcăiţă

Published Jan 1, 2012• 1 citation
Publisher

Abstract

Inconsistencies of some standard quantum mechanical models (Madelung’s, de Broglie’s models) are eliminated as- suming the micro particle movements on continuous, but non-differentiable curves (fractal curves). This hypothesis, named by us the fractal approximation of motion, will allow an unitary approach of the phenomena in quantum me-chanics (separation of the physical motion of objects in wave and particle components depending on the scale of resolution, correlated motions of the wave and particle, i.e. wave-particle duality, the mechanisms of duality, by means of both phase wave-particle coherence and wave-particle incoherence, the particle as a clock, particle incorporation into the wave and the implications of such a process). Moreover, correspondences with standard gravitational models (Einstein’s model, string theory) can be also distinguished.

DOI
10.4236/ojm.2012.23005

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Inconsistencies of some standard quantum mechanical models (Madelung’s, de Broglie’s models) are eliminated as- suming the micro particle movements on continuous, but non-differentiable curves (fractal curves). This hypothesis, named by us the fractal approximation of motion, will allow an unitary approach of the phenomena in quantum me-chanics (separation of the physical motion of objects in wave and particle components depending on the scale of resolution, correlated motions of the wave and particle, i.e. wave-particle duality, the mechanisms of duality, by means of both phase wave-particle coherence and wave-particle incoherence, the particle as a clock, particle incorporation into the wave and the implications of such a process). Moreover, correspondences with standard gravitational models (Einstein’s model, string theory) can be also distinguished.
We perform a critical analysis of some quantum mechanical models such as the hydrodynamic model (Madelung's model), de Broglie's theory of double solution etc., specifying both mathematical and physical inconsistencies that occur in their construction.These inconsistencies are eliminated by means of the fractal approximation of motion (physical objects moving on continuous and non-differentiable curves, i.e. fractal curves) developed in the framework of Scale Relativity (SR) (Nottalle, L., 1993;Chaline, J. et al, 2009;
The mathematical concept of minimal atomicity is extended to fractal minimal atomicity, based on the nondifferentiability of the motion curves of physical system entities on a fractal manifold. For this purpose, firstly, different results concerning minimal atomicity from the mathematical procedure of the Quantum Measure Theory and also several physical implications are obtained. Further, an inverse method with respect to the common developments concerning the minimal atomicity concept has been used, showing that Quantum Mechanics is identified as a particular case of Fractal Mechanics at a given scale resolution. More precisely, for fractality through Markov type stochastic processes, i.e., fractalization through stochasticization, the standard Schrödinger equation is identified with the geodesics of a fractal space for motions of the physical system entities on nondifferentiable curves on fractal dimension two at Compton scale resolution. In the one-dimensional stationary case of the fractal Schrödinger type geodesics, a special symmetry induced by the homographic group in Barbilian’s form “makes possible the synchronicity” of all entities of a given physical system. The integral and differential properties of this group under the restriction of defining a parallelism of directions in Levi-Civita’s sense impose correspondences with the “dynamics” of the hyperbolic plane so that harmonic mappings between the ordinary flat space and the hyperbolic one generate (by means of a variational principle) a priori probabilities in Jaynes’ sense. The explicitation of such situation specifies the fact that the hydrodynamical variant of a Fractal Mechanics is more easily approached and, from this, the fact that Quantum Measure Theory can be a particular case of a possible Fractal Measure Theory.
Making sense of the nearly 100-hundred-year-old quantum nature of light and matter, with its — to name a few — wave-particle duality, measurement problem, and entanglement ‘spooky action at the distance’ — remains the greatest questions to physics. Since the 1980s, fractal geometry, a new and exciting field of mathematics — that the fathers of quantum mechanics did not have and has not been tested for quantum properties — has developed. Can the isolated fractal explain the quantum? An experiment was conducted on a simple — but isolated — fractal testing whether the geometry of fractals corresponds to quantum enigmas. It was found the isolated fractal emerges by a duality of propagation — of an oscillating sinusoidal wave — of ‘bits’ — of information — iterating in a superposition of time, scale, and symmetry, with a possible constant speed, demonstrating all the hypothesised. The quantum ‘measurement problem’ was addressed as being a problem of isolated scale-invariant fractals — or fractal landscapes — where position is only ‘known’ when additional (fractal) information is added — which ‘equally’ gives rise to a (quantum-like) ‘uncertainty’ problem. Also, quantum entanglement and other quantum features were explained by the fractal model. Consequences of the model were discussed, notably to its relevance to the nature of light (the speed of it etc.), the behaviour of the atom, time, knowledge and our reality, and the model’s direct inextricable connection with cosmological observations and conjectures. Finally, it was concluded that the model is preliminary but fundamental.
In this study, we try to complement the theory with de-Broglie's idea on wave-pilot and Sakharov's hypothesis on gravitational fluctuations to convert the Method of Indirect Computation into the Comprehensive Quantum Theory. We assume that de-Broglie's wave-pilots are gravitational stochastic ones, and we shall regard micro-objects and test particles as being subject to the influence of such de-Broglie's waves. This role can be played by classical gravitational fields and waves (i.e. Gravity Background) generated by external objects. It has been shown that such stochastic gravitational fields and waves bring about mechanical wave-type fluctuations of a classical test particle. These classical particles in the course of measurements of their coordinates appear smeared in space, and these can described by quantum-mechanical wave functions. And these classical mechanical fluctuations correspond to the quantum mechanical wave function.
Introduction The evolution of microelectronics toward reduced device sizes has proceeded to a degree that renders conventional models, approaches, and theories inapplicable. Indeed, for objects with sizes of 100 nanometers or less it is frequently the case that the length scales associated with fundamental physical processes are comparable to the geometrical size of the device; also, fundamental time scales associated with physical processes are of the order of the time parameters for nanodevice operation. Therefore, on the nanoscale the theories and models underlying modern nanoelectronics become more complicated, and rely more and more on basic science. Generally, in the nanoworld the fundamental laws of physics that govern particles and material fields differ from those that apply to familiar macroscopic phenomena such as the motion of a baseball or a train. Instead of classical mechanics, that works so well for macroscopic phenomena, the motion of particles in the nanoworld is determined by the so-called wave mechanics or quantum mechanics. An underlying principle of central importance for nanophysics is the fundamental concept that all matter , including electrons, nuclei, and electromagnetic fields, behaves as both waves and particles , that is, wave–particle duality is a basic characteristic of all matter. At first glance, wave properties and particle properties for the same physical object are hardly compatible. To understand wave–particle duality, we will briefly review, in the following two subsections, the basic properties of particles and waves known from classical physics.