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Advances in Mathematical Physics

Extended Minimal Atomicity through Nondifferentiability: A Mathematical-Physical Approach

Gabriel Gavriluţ, Alina Cristiana Gavriluţ, Maricel Agop

Published Jan 10, 2019• 4 citations
Publisher

Abstract

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.

DOI
10.1155/2019/8298691

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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.
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;
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.
One of the main results of scale relativity as regards the foundation of quantum mechanics is its explanation of the origin of the complex nature of the wave function. The scale relativity theory introduces an explicit dependence of physical quantities on scale variables, founding itself on the theorem according to which a continuous and non-differentiable space-time is fractal (i.e., scale-divergent). In the present paper, the nature of the scale variables and their relations to resolutions and differential elements are specified in the non-relativistic case (fractal space). We show that, owing to the scale-dependence which it induces, non-differentiability involves a fundamental two-valuedness of the mean derivatives. Since, in the scale relativity framework, the wave function is a manifestation of the velocity field of fractal space-time geodesics, the two-valuedness of velocities leads to write them in terms of complex numbers, and yields therefore the complex nature of the wave function, from which the usual expression of the Schrödinger equation can be derived.
First remark: Feynman's discovery in accordance of which quantum trajectories are of fractal nature (continuous everywhere but nowhere differentiable) suggests describing the dynamics of such systems by explicitly introducing the Brownian motion of fractional order in their equations. The second remark is that, apparently, it is only in the complex plane that the Brownian motion of fractional order with independent increments can be generated, by using random walks defined with the complex roots of the unity; in such a manner that, as a result, the use of complex variables would be compulsory to describe quantum systems. Here one proposes a very simple set of axioms in order to expand the consequences of these remarks. Loosely speaking, a one-dimensional system with real-valued coordinate is in fact the average observation of a one-dimensional system with complex-valued coordinate: It is a strip modeling. Assuming that the system is governed by a stochastic differential equation driven by a complex valued fractional Brownian of order n, one can then obtain the explicit expression of the corresponding covariant stochastic derivative with respect to time, whereby we switch to the extension of Lagrangian mechanics. One can then derive a Schrödinger equation of order n in quite a direct way. The extension to relativistic quantum mechanics is outlined, and a generalized Klein–Gordon equation of order n is obtained. As a by-product, one so obtains a new proof of the Schrödinger equation.
A new application of a fractal concept to quantum physics has been developed. The fractional path integrals over the paths of the Levy flights are defined. It is shown that if fractality of the Brownian trajectories leads to standard quantum mechanics, then the fractality of the Levy paths leads to fractional quantum mechanics. The fractional quantum mechanics has been developed via the new fractional path integrals approach. A fractional generalization of the Schrodinger equation has been discovered. The new relationship between the energy and the momentum of the nonrelativistic fractional quantum-mechanical particle has been established, and the Levy wave packet has been introduced into quantum mechanics. The equation for the fractional plane wave function has been found. We have derived a free particle quantum-mechanical kernel using Fox's H-function. A fractional generalization of the Heisenberg uncertainty relation has been found. As physical applications of the fractional quantum mechanics we have studied a free particle in a square infinite potential well, the fractional "Bohr atom" and have developed a new fractional approach to the QCD problem of quarkonium. We also discuss the relationships between fractional and the well-known Feynman path integral approaches to quantum mechanics. (c) 2000 American Institute of Physics.
In this paper the fractal model of the atom, using the hydrodynamic approach of the scale relativity theory, is obtained. Thus, assuming that the electron motion around the nucleus takes place on fractal curves of fractal dimension DF (continuous but non-differentiable curves), it is shown that its dynamics, in the second order approximation of the equation of motion, is described in complex speed field by a generalized Navier-Stokes type equation with imaginary viscosity coefficient. Applying this model to study the atom, it resulted that the real part of the complex velocity field describes the electron averaged movement. The electron moves on stationary orbits according to a quantification condition and the imaginary part of the complex velocity describes the fractality through a fractal potential. In the DF=2 fractal dimension and for the D=h / 2m viscosity coefficient, the classical results of quantum mechanics for the hydrogen atom are obtained.