Scale Relativity and Fractal Space-Time

The following sections are included:

• PREFACE

• LIST OF FIGURES

• CONTENTS

The following sections are included:

• The principle of relativity

• Premise of scale relativity theory

• Summary of fractal space-time and microphysics

• Content of the book

• Audience

The theory of scale-relativity is an attempt to extend today's theories of relativity, by applying the principle of relativity not only to motion transformations, but also to scale transformations of the reference system. Recall that, in the formulation of Einstein [158], the principle of relativity consists in requiring that "the laws of nature be valid in every system of coordinates, whatever their state". Since Galileo, this principle had been applied to the states of position (origin and orientation of axes) and of motion of the system of coordinates (measured by velocity and acceleration). These states are characterized by their relativity, namely, they are never definable in an absolute way, but only in a relative way. This means that the state of any system (including reference systems) can be defined only with respect to another system…

The following sections are included:

• Development Levels of the Theory

• Evolution of Concepts in Physics

• Founding First Principles in Relativity

• Methods of Scale Relativity Theory

The following sections are included:

• Analysis of the Differentiability Hypothesis

• Beyond Differentiability

• Fractal Geometry

• Fractality of a Nondifferentiable Continuum

• Explicit Scale Dependence on Resolution

• Nature of Resolutions: Covariance Matrix

The following sections are included:

• Introduction

• Scale Invariance and Galilean Scale-relativity

• Generalized Scale Laws

• Special Relativity of Scales

• Metric of a Fractal Space-Time

The following sections are included:

• Introduction

• General Link Between Heisenberg–de Broglie Relations and Fractal Behavior

• Main Consequences of Nondifferentiability

• Covariant Total Derivative Operator

• Complex Covariant Mechanics

• From Newton to Schrödinger Equation

• Born and von Neumann Postulates

• Generalization to Nondifferentiable Wave Functions

• Potential Energy from Fractal Geometry

• Quantum Mechanics of Many Particles

• Schrödinger form of other Fundamental Equations of Physics

• Foundation of Quantum Mechanics on the Principle of Relativity

• Quantum Entanglement in Scale Relativity

• Generalized Schrödinger Equations Founded on Extended Scale Laws

The following sections are included:

• Introduction

• Klein–Gordon Equation

• Dirac Spinors and the Dirac Equation

• Pauli Spinors and Pauli Equation

• Generalization of the Klein–Gordon and Dirac Equations

The following sections are included:

• Introduction

• Statement of the Problem

• Classical Electrodynamics in Scale Relativity

• Quantum Electrodynamics

• Non-Abelian Gauge Theories

• Conclusion and Future Prospect

The following sections are included:

• Motivation

• Schrödinger Equation in Terms of Resolutions

• Schrödinger Equation in Terms of the "djinn"

• Complexergy

Part II of this book has been devoted to the foundation of the theory of scale relativity, under its three aspects of (i) scale laws finding their origin in the continuous and nondifferentiable geometry, (ii) motion laws of a quantum nature being based on these scale laws and (iii) gauge fields emerging as manifestations of the fractal geometry of space-time…

The following sections are included:

• Introduction

• Quantum-Classical Transition

• Macroscopic Quantum-like Theories

• Quantum Laws as Anti-diffusion Laws

• Numerical Simulations of Fractal Geodesics

• Macroscopic Quantum Potentials

The following sections are included:

• Implications of Fractal Space-Time and Special Scale Relativity

• Application of Quantum Mechanics in Scale Space

• Theoretical Expectation of Higgs Boson Mass

• Ultra High Energy Cosmic Rays

The following sections are included:

• Introduction

• Special Scale Relativity and Cosmology

• Theoretical Prediction of the Space Dimensionality

• Cosmological Implications of Scale Relativity

• Consequences for the Early Universe

• The Nature of the Cosmological Constant and of "Dark Energy"

The following sections are included:

• Introduction

• Theory

• Theoretical Predictions

• Solar System

• Extrasolar Planetary Systems

• Galactic Structures

• Planetary Nebulae

• Extragalactic Structures

The following sections are included:

• Fractals in Biology

• Applications of Log-Periodic Laws

• Applications of Scale Relativity to Biology

To conclude, the scale relativity theory amounts to making proposals for the fundamental principles to be adopted in order to set the foundations of the laws that underlie the various physical, biological and other phenomenons with the insistance on the fact that the most fundamental of these principles is the principle of relativity itself. The extraordinary success due to the application of this principle to position, orientation and motion for four centuries is well known…

The following sections are included:

• BIBLIOGRAPHY

• INDEX