Quantum Fractals

From Heisenberg's Uncertainty to Barnsley's Fractality

Starting with numerical algorithms resulting in new kinds of amazing fractal patterns on the sphere, this book describes the theory underlying these phenomena and indicates possible future applications. The book also explores the following questions:

• What are fractals?
• How do fractal patterns emerge from quantum observations and relativistic light aberration effects?
• What are the open problems with iterated function systems based on Mobius transformations?
• Can quantum fractals be experimentally detected?
• What are quantum jumps?
• Is quantum theory complete and/or universal?
• Is the standard interpretation of Heisenberg's uncertainty relations accurate?
• What is Event Enhanced Quantum Theory and how does it differ from spontaneous localization theories?
• What are the possible applications of quantum fractals?

Contents:

• Introduction

• What are Quantum Fractals?:

  • Cantor Set
  • Iterated Function Systems
  • Cantor Set Through Matrix Eigenvector
  • Quantum Iterated Function Systems
  • Example: The "Impossible" Quantum Fractal
  • Action on the Plane
  • Lorentz Group, SL(2,ℂ), and Relativistic Aberration

• Examples:

  • Hyperbolic Quantum Fractals
  • Controlling Chaotic Behavior and Fractal Dimension
  • Quantum Fractals on n-Spheres
  • Algorithms for Generating Hyperbolic Quantum Fractals

• Foundational Questions:

  • Stochastic Nature of Quantum Measurement Processes
  • Are There Quantum Jumps?
  • Bohmian Mechanics
  • Event Enhanced Quantum Theory
  • Ghirardi–Rimini–Weber Spontaneous Localization
  • Heisenberg's Uncertainty Principle and Quantum Fractals
  • Are Quantum Fractals Real?

• Appendix A. Mathematical Concepts:

  • Metric Spaces
  • Normed Spaces
  • Measure and Integral
  • Markov, Frobenius-Perron and Koopman Operators

• Appendix B. Minkowski Space Generalization of Euler-Rodrigues Formula:

  • Alternative Derivation via SL(2,ℂ)

Readership: Advanced undergraduate students and professionals in quantum chaos, as well as philosophers of science.