Extended Minimal Atomicity through Nondifferentiability: A Mathematical-Physical Approach
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...
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Wiley
2019-01-01
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| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2019/8298691 |
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| author | Gabriel Gavriluţ Alina Gavriluţ Maricel Agop |
| author_facet | Gabriel Gavriluţ Alina Gavriluţ Maricel Agop |
| author_sort | Gabriel Gavriluţ |
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| description | 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. |
| format | Article |
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| institution | OA Journals |
| issn | 1687-9120 1687-9139 |
| language | English |
| publishDate | 2019-01-01 |
| publisher | Wiley |
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| series | Advances in Mathematical Physics |
| spelling | doaj-art-145c15a19acf4f74ba58fd1c8ce304472025-08-20T02:04:52ZengWileyAdvances in Mathematical Physics1687-91201687-91392019-01-01201910.1155/2019/82986918298691Extended Minimal Atomicity through Nondifferentiability: A Mathematical-Physical ApproachGabriel Gavriluţ0Alina Gavriluţ1Maricel Agop2Faculty of Physics, Alexandru Ioan Cuza University, Carol I Bd. 11, Iaşi, 700506, RomaniaFaculty of Mathematics, Alexandru Ioan Cuza University, Carol I Bd. 11, Iaşi, 700506, RomaniaDepartment of Physics, Gheorghe Asachi Technical University of Iaşi, RomaniaThe 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.http://dx.doi.org/10.1155/2019/8298691 |
| spellingShingle | Gabriel Gavriluţ Alina Gavriluţ Maricel Agop Extended Minimal Atomicity through Nondifferentiability: A Mathematical-Physical Approach Advances in Mathematical Physics |
| title | Extended Minimal Atomicity through Nondifferentiability: A Mathematical-Physical Approach |
| title_full | Extended Minimal Atomicity through Nondifferentiability: A Mathematical-Physical Approach |
| title_fullStr | Extended Minimal Atomicity through Nondifferentiability: A Mathematical-Physical Approach |
| title_full_unstemmed | Extended Minimal Atomicity through Nondifferentiability: A Mathematical-Physical Approach |
| title_short | Extended Minimal Atomicity through Nondifferentiability: A Mathematical-Physical Approach |
| title_sort | extended minimal atomicity through nondifferentiability a mathematical physical approach |
| url | http://dx.doi.org/10.1155/2019/8298691 |
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