Quantum fractals : from Heisenberg's uncertainty to Barnsley's fractality / Arkadiusz Jadczyk.

Format:

Book

Author/Creator:

Jadczyk, Arkadiusz, author.

Language:

English

Subjects (All):

Fractals.

Mathematical physics.

Quantum theory.

Physical Description:

1 online resource (358 p.)

Place of Publication:

Singapore : World Scientific Publishing Co. Pte. Ltd., 2014.

Language Note:

English

Summary:

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 sta

Contents:

Preface; Contents; 1. Introduction; 2. What are Quantum Fractals?; 2.1 Cantor set; 2.1.1 Cantor set through "Chaos Game"; 2.2 Iterated function systems; 2.2.1 Definition of IFS; 2.2.2 Frobenius-Perron operator; 2.3 Cantor set throughmatrix eigenvector; 2.4 Quantum iterated function systems; 2.5 Example: The "impossible" quantum fractal; 2.5.1 24 symmetries - the octahedral group; 2.5.2 Construction of the 24-elements SQIFS; 2.5.3 Open problems; 2.6 Action on the plane; 2.7 Lorentz group, SL(2,C), and relativistic aberration; 2.7.1 The Lorentz group

3.1 Hyperbolic quantum fractals3.1.1 The circle; 3.1.1.1 Contracting and expanding regions; 3.1.1.2 Place dependent probabilities; 3.1.1.3 Images; 3.1.1.4 Frobenius-Perron operator; 3.1.1.5 Details of the calculations; 3.1.2 Platonic quantum fractals for a qubit; 3.1.2.1 Platonic solids; 3.1.2.2 Approximations to the invariant measure; 3.1.2.3 Geometric interpretation of transformations and probabilities; 3.1.2.4 Quantum Tetrahedron; 3.1.2.5 Quantum Octahedron; 3.1.2.6 Quantum Cube; 3.1.2.7 Quantum Icosahedron; 3.1.2.8 Quantum Dodecahedron

3.2 Controlling chaotic behavior and fractal dimension3.3 Quantum fractals on n-spheres; 3.3.1 Clifford algebras; 3.3.1.1 Notation; 3.3.1.2 Examples of Clifford algebras; 3.3.1.3 Even and odd parts, principal automorphism and anti-automorphism, conjugation; 3.3.1.4 Examples; 3.3.1.5 The norm function; 3.3.1.6 Spin group; 3.3.1.7 Vector space isomorphism between the Clifford and the Grassmann algebra; 3.3.1.8 Paravectors; 3.3.1.9 Example; 3.3.1.10 The trace; 3.3.1.11 Algebra isomorphism between C+(V 1,Q1) and R(2, C(V,Q)); 3.3.1.12 Mobius transformations

3.3.1.13 Conformal spin geometry of n-spheres

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on print version record.

ISBN:

981-4569-87-9