The Mathematical Principles Of Scale Relativity Physics

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: Develops the concept of scale relativity interpreted according to its initial definition enticed by the birth of wave and quantum mechanics Provides the fundamental equations necessary for interpretation of matter, describing the ensembles of free particles according to the concepts of confinement and asymptotic freedom 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.

Provides a comprehensive survey of the state-of-the-art in the development of the relativity theory of scales Transcends and integrates the classical and the quantum regimes Enables quantum mechanics to be based on the principle of relativity provided this principle is extended to scale transformations of the reference system Collects and organizes developments and applications from diverse fields for easy reference

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: Develops the concept of scale relativity interpreted according to its initial definition enticed by the birth of wave and quantum mechanics Provides the fundamental equations necessary for interpretation of matter, describing the ensembles of free particles according to the concepts of confinement and asymptotic freedom 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.

This book provides a comprehensive survey of the development of the theory of scale relativity and fractal space-time. It suggests an original solution to the disunified nature of the classical-quantum transition in physical systems, enabling the basis of quantum mechanics on the principle of relativity, provided this principle is extended to scale transformations of the reference system. In the framework of such a newly generalized relativity theory (including position, orientation, motion and now scale transformations), the fundamental laws of physics may be given a general form that unifies and thus goes beyond the classical and quantum regimes taken separately. A related concern of this book is the geometry of space-time, which is described as being fractal and nondifferentiable. It collects and organizes theoretical developments and applications in many fields, including physics, mathematics, astrophysics, cosmology and life sciences. Contents:General IntroductionTheory:IntroductionStructure of the TheoryNondifferentiable Geometry and Fractal Space-TimeLaws of Scale TransformationsFrom Fractal Space to Non-Relativistic Quantum MechanicsFrom Fractal Space-Time to Relativistic Quantum MechanicsGeometric Theory of Gauge Fields in Scale RelativityQuantum-type Mechanics in Scale SpaceApplications:IntroductionApplications to Laboratory and Earth Scale PhysicsApplications to Elementary Particle and High Energy PhysicsApplications to CosmologyApplications to Gravitational Structuring in AstrophysicsApplications to Life Sciences and Other Sciences Readership: Academics, post-graduates and professionals interested in astrophysics and cosmology. Keywords:Scale Relativity;Fractals;Quantum Mechanics;Space-Time GeometryKey Features:No competing title: it describes the state-of-the art of the theory of scale relativity and of its applications to various sciences by its founderApplications of the theory to various branches of physical science, including elementary particle physics, cosmology, and life sciences, are presented in detail in individual chaptersReviews:“The sweep of this book is enormous. It has implications for every aspect of physics. But it also has important implications for biological science. Systems Biology is multi-scale since integration of function can occur at many different levels from genes to whole organisms and populations in what some of us have called a principle of biological relativity. Could scale relativity provide the framework for systematizing that principle? Models at different levels could be seen to be working with different scale resolutions, or even scale dimensions. The prospect is tantalising and forms a major challenge to mathematical systems biologists. Those who would welcome that challenge will need to read this book.”Denis Noble, FRS Oxford University author of The Music of Life “Scale Relativity seeks to refine our understanding of the nature of space itself. Extended to its natural limits, it has the potential to unify the description of phenomena one associates with Quantum Mechanics, Biology and even Astrophysics and Cosmology.”Jean-Patrick Connerade Emeritus Professor and Distinguished Research Fellow Imperial College London President of the European Academy of Science, Arts and Literature “This is a rich and lovely book. The elegant approach to unifying many concepts in physics provides new ways of seeing things — some things we know in new ways and some very new ideas. It is also exciting to see some conceptual tools for thinking about problems as complex as those we face in understanding biological systems.”David J Galas Professor and Senior Vice President Institute for Systems Biology “Nottale's new book, from the first paragraph to the last section, is a bold, courageous and thoughtful work that addresses this problem directly through the paradigm of scale relativity. The book contains a wealth of insight to accompany an astonishing collection of numerical data from many scales of observation. Nottale has succeeded in going beyond the limitations inherent in conventional differential approaches to produce a convincing picture of scale-dependent physics. The book is a must-read for any scientist contemplating bridging the gap between the large and small-scale approximations of modern physics.”Garnet Ord Ryerson University

This is the first detailed account of a new approach to microphysics based on two leading ideas: (i) the explicit dependence of physical laws on scale encountered in quantum physics, is the manifestation of a fundamental principle of nature, scale relativity. This generalizes Einstein's principle of (motion) relativity to scale transformations; (ii) the mathematical achievement of this principle needs the introduction of a nondifferentiable space-time varying with resolution, i.e. characterized by its fractal properties. The author discusses in detail reactualization of the principle of relativity and its application to scale transformations, physical laws which are explicitly scale dependent, and fractals as a new geometric description of space-time.

Ebenezer Cunningham was a British mathematician with an intense interest in the theory of special relativity, a subject that was just beginning to be recognised as he wrote. This thoroughly and engagingly written 1914 book was one of the first treatises in the English language to focus on special relativity.

The revised and updated 2nd edition of this established textbook provides a self-contained introduction to the general theory of relativity, describing not only the physical principles and applications of the theory, but also the mathematics needed, in particular the calculus of differential forms. Updated throughout, the book contains more detailed explanations and extended discussions of several conceptual points, and strengthened mathematical deductions where required. It includes examples of work conducted in the ten years since the first edition of the book was published, for example the pedagogically helpful concept of a "river of space" and a more detailed discussion of how far the principle of relativity is contained in the general theory of relativity. Also presented is a discussion of the concept of the 'gravitational field' in Einstein's theory, and some new material concerning the 'twin paradox' in the theory of relativity. Finally, the book contains a new section about gravitational waves, exploring the dramatic progress in this field following the LIGO observations. Based on a long-established masters course, the book serves advanced undergraduate and graduate level students, and also provides a useful reference for researchers.

This monograph explains and analyzes the principles of a quantum-geometric framework for the unification of general relativity and quantum theory. By taking advantage of recent advances in areas like fibre and superfibre bundle theory, Krein spaces, gauge fields and groups, coherent states, etc., these principles can be consistently incorporated into a framework that can justifiably be said to provide the foundations for a quantum extrapolation of general relativity. This volume aims to present this approach in a way which places as much emphasis on fundamental physical ideas as on their precise mathematical implementation. References are also made to the ideas of Einstein, Bohr, Born, Dirac, Heisenberg and others, in order to set the work presented here in an appropriate historical context.