The Mathematical Principles of Scale Relativity Physics

The Concept of Interpretation

Nicolae Mazilu, Maricel Agop, Ioan Merches

• Publisher: CRC Press
• Year: 2019
• ISBN:
  9781000751260
• Edition: 1
• Topic: Mathematics
• Subtopic: Arithmetic
• Index: Mathematics

The Mathematical Principles of Scale Relativity Physics: The Concept of Interpretation explores and builds upon the principles of Laurent Nottale's scale relativity. The authors address a variety of problems encountered by researchers studying the dynamics of physical systems. It explores Madelung fluid from a wave mechanics point of view, showing that confinement and asymptotic freedom are the fundamental laws of modern natural philosophy. It then probes Nottale's scale transition description, offering a sound mathematical principle based on continuous group theory. The book provides a comprehensive overview of the matter to the reader via a generalization of relativity, a theory of colors, and classical electrodynamics.

Key Features:

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• Develops the concept of scale relativity interpreted according to its initial definition enticed by the birth of wave and quantum mechanics

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• Provides the fundamental equations necessary for interpretation of matter, describing the ensembles of free particles according to the concepts of confinement and asymptotic freedom

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• Establishes a natural connection between the Newtonian forces and the Planck's law from the point of view of space and time scale transition: both are expressions of invariance to scale transition

The work will be of great interest to graduate students, doctoral candidates, and academic researchers working in mathematics and physics.

Chapter 1. Introduction

Among the newest theories of physics, the Laurent Nottale’s scale theory of relativity deserves, in our opinion, a special attention. The scale relativity theory (SRT in what follows) really means business, and big business at that, and we are set here on demonstrating this fact: SRT targets in fact the very foundations of our positive knowledge. The proof will be effectively done by showing that SRT follows a line of essential achievements of the physical knowledge of the world, and follows it properly. As a matter of fact the bottom line of our conclusion here is that, once the principle of scale invariance is adopted, there is no other way to follow but the right way, which is the line of thought marked by those essential achievements of knowledge. All of the works to date of Laurent Nottale, regarding the problems raised by scale relativity testify of a well guided thinking, and such a guidance cannot come but from an inherent fundamental principle of knowledge. If there is an ambition from our part here, that would therefore be none other than to make this principle as obvious as possible, maybe even by giving it an explicit verbalization. In doing this, we make use both of common and own results upon the fractal theory of space, expounded though along a special line indicated by Laurent Nottale himself, in an evaluation of thirty years of development of the theory. We quote the final words from a relatively recent book of Nottale:

Giving up the differentiability hypothesis, i.e. generalizing the geometric description to general continuous manifolds, differentiable or not, involves an extremely large number of new possible structures to be investigated and described. In view of the immensity of the task, we have chosen to proceed by steps, using presently-known physics as a guide. Such an approach is rendered possible by the result according to which the small scale structures, which manifest the nondifferentiability, are smoothed out beyond some relative transitions toward the large scales. One therefore recovers the standard classical differentiable theory as a large scale approximation of this generalized approach. But one also obtains a new geometric theory, which allows one to understand quantum mechanics as a manifestation of an underlying nondifferentiable and fractal geometry and finally to suggest generalizations of it and new domains of application for these generalizations.

Now the difficulty that also makes their interest with theories of relativity is that they are meta-theories rather than theories of some particular systems. Hence, after the construction of special relativity of motion at the beginning of the 20 th century, the whole of physics needed to be rendered relativistic (from the viewpoint of motion), a task that is not yet fully achieved.

The same is true regarding the program of constructing a fully scale-relativistic science. Whatever the already-obtained successes, the task remains huge, in particular when one realizes that it is no longer only physics that is concerned, but also many other sciences. Its ability to go beyond the frontiers between sciences may be one of the main interests of the scale relativity theory, opening the hope of a refoundation on mathematical principles and on predictive differential equations of a philosophy of nature in which physics would no longer be separated from other sciences. [(Nottale, 2011), p. 712; our Italics]

The Italics in this excerpt roughly mark our points of intervention with the present work, ‘using presently-known… SRT as a guide’. In broad strokes, we aim here to clarify the idea of “general continuous manifolds”...

Table of contents

• Cover
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Chapter 1. Introduction
• Chapter 2. Madelung Fluid Dynamics
• Chapter 3. De Broglie’s Interpretation of Wave Function
• Chapter 4. The Planetary Model as a Dynamical Kepler Problem
• Chapter 5. The Light in a Schrödinger Apprenticeship
• Chapter 6. The Wave Theory of Geometric Phase
• Chapter 7. The Physical Point of View in the Theory of Surfaces
• Chapter 8. Nonconstant Curvature
• Chapter 9. The Nonstationary Description of Matter
• Chapter 10. The Idea of Continuity in Fluid Dynamics
• Chapter 11. A Hertz-type Labelling in a Madelung Fluid
• Chapter 12. Theory of Nikolai Alexandrovich Chernikov
• Conclusions: Concept of Interpretation and Necessary Further Elaborations
• References
• Subject index