The Mathematical Principles of Scale Relativity Physics: The Concept of Interpretation

Description

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. Table of Contents Chapter 1. Introduction Chapter 2. Madelung Fluid Dynamics The Madelung Fluid Classical and Quantum Conservation Laws Hydrodynamics of Free Point Particles: Universality of the Schr¨odinger Equation A Definition of the Interpretation Chapter 3. De Broglie’s Interpretation of Wave Function The Appropriate Geometry of de Broglie’s Idea Lessons and Mandatory Developments Chapter 4. The Planetary Model as a Dynamical Kepler Problem A Newtonian Brief on Density The Concept of Confinement A Clasic Example of Affine Reference Frame: Maxwell Stress Tensor Chapter 5. The Light in a Schr¨odinger Apprenticeship A Special Contribution of Helmholtz Determination of similar color pairs. Enters Erwin Schr¨odinger Chapter 6. The Wave Theory of Geometric Phase Enters Sir Michael Berry A Kepler Motion Analysis: the Geometrical Condition of YangMills Fields The Berry Moment of Human Knowledge A Classical Implementation of the Idea of Interpretation A Characterization of the Hertz’s Material Point The General Meaning of Berry’s Curvature Chapter 7. The Physical Point of View in the Theory of Surfaces A Few Mathematical Prerequisites The Differential Theory of Surfaces Rainich’s Description of Surrounding Space A Physical Parametrization of Surface The Three-Dimensional Space of Accelerations Force at an Outward Distance Chapter 8. Nonconstant Curvature The Infinitesimal Deformation Summing up the Differential Geometry of Curvature Parameters A Definition of Surface Tension The Statistics of Fluxes on a Material Point The Stress by a Statistic The Tensions: Conclusions and Outlook Chapter 9. The Nonstationary Description of Matter The Louis de Broglie Moment Airy Moment of Berry and Balazs Cosmological Moment of Berry and Klein Chapter 10. The Idea of Continuity in Fluid Dynamics The Mass Transport in a Volume Element Transport Theorem in Finite Volume Some Classic Physical Examples The Hamiltonian Transport in Finite Volume Transcendence between Volume Element and a Control Volume Chapter 11. A Hertz-type Labelling in a Madelung Fluid Torsion Induced by Space Variations of Density The Reference Frame and the Torsion The Torsion and the Waves Chapter 12. Theory of Nikolai Alexandrovich Chernikov Enters Chernikov Chernikov’s Theory in the Three-Dimensional Case Conclusions: Concept of Interpretation and Necessary Further Elaborations Subject Index A B C D E F G H I K M N P R S T U V W