Quantum Fractals : From Heisenberg's Uncertainty to Barnsley's Fractality

358 Pages·2014·11.601 MB·English

by  Arkadiusz Jadczyk

Quantum
Fractals
From Heisenberg's Uncertainty
to Barnsley's Fractality
8992_9789814569866_tp.indd 1 17/6/14 9:08 am
May2,2013 14:6 BC:8831-ProbabilityandStatisticalTheory PST˙ws
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Quantum
Fractals
From Heisenberg's Uncertainty
to Barnsley's Fractality
Arkadiusz Jadczyk
Quantum Future Group Inc., USA
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
8992_9789814569866_tp.indd 2 17/6/14 9:08 am
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Jadczyk, Arkadiusz.
Quantum fractals : from Heisenberg’s uncertainty to Barnsley’s fractality / Arkadiusz Jadczyk,
Quantum Future Group Inc, USA.
pages cm
Includes bibliographical references and index.
ISBN 978-9814569866 (hardcover : alk. paper)
1. Fractals. 2. Mathematical physics. 3. Quantum theory. I. Title.
QC20.7.F73J33 2014
530.1201'514742--dc23
2014013078
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2014 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance
Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy
is not required from the publisher.
In-house Editor: Ng Kah Fee
Printed in Singapore
KahFee - Quantum Fractals.indd 1 17/6/2014 12:13:33 PM
May29,2014 10:6 WorldScientificBook-9inx6in QuantumFractals3 pg.v/0
To Laura
for her support of the creative principle
v
May2,2013 14:6 BC:8831-ProbabilityandStatisticalTheory PST˙ws
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
May29,2014 10:6 WorldScientificBook-9inx6in QuantumFractals3 pg.vii/0
Preface
This book provides a handy introduction to quantum fractals — a new
kind of fractals arising in quantum-like jump random processes involving
non-commuting operations. It describes the basic theoretical concepts, al-
gorithms and also touches upon philosophical questions of the foundations
of quantum theory.
Anoverview Thescienceoffractalsisyoungandgrowingfast. Quantum
fractals are even younger and are still crawling on all fours. But the time
seems to be ripe for them to get up and look around. As we hope it will
become clear from this book — various possible applications abound.
Roughly: quantum fractals are patterns generated by iterated function
systems, with place dependent probabilities, of M¨obius transformations on
spheresoronmoregeneralprojectivespaces. Inquantumphysicsquantum
fractalscanbeinterpretedastracesofquantumjumpsduringsimultaneous
monitoring of several non-commuting observables. These quantum jumps
accompanyeventswithinformationexchangesbetweenthequantumsystem
and the classical information processing devices.
Whilemathematicallycompletelyclearsuchaconceptbringsanalmost
revolutionary novelty into quantum physics. Until now it has usually been
assumedthatsimultaneous“measurements”ofnon-commutingobservables
makes no sense, and that it cannot lead to any useful predictions. In this
book we challenge the standard position by proposing that such experi-
ments may lead to organized chaotic behavior that can be experimentally
verified. Thephenomenonisgeneralenoughtobepresentinapplicationsof
the quantum formalism beyond physics and beyond quantum computing,
for instance in quantum games, quantum psychology etc. Here possible
deviations from linearity are also touched upon.
vii
May29,2014 10:6 WorldScientificBook-9inx6in QuantumFractals3 pg.viii/0
viii Quantum Fractals: From Heisenberg’sUncertainty toBarnsley’s Fractality
Another area where quantum fractals may appear is relativistic light
aberration, though there the particular form of place dependent probabili-
ties that are derived from linear event enhanced quantum theory (EEQT)
is not, at the present moment, justified.
About the Book This book combines a number of different topics such
as: fractalsgenerationandanalysis,elementsofgeometry,linearandmulti-
linear algebra and group theory, special relativity, quantum measurements
and, in particular, Heisenberg’s uncertainty relations and their interpreta-
tion, as well as some elements of random processes.
Since it is rather unusual to find a single person that would be inter-
ested in all ofthese areas, this book has been organizedin sucha waythat
the reader should be able to extract from it the information that is of a
particular interest in her/his research. Nevertheless the primary idea of
the book is to bring together a diversity of ideas and, in this way, encour-
agecooperationandstimulate mutual interestbetween variousbranchesof
quantum physics and fractal researchfor the benefit of all.
For this reason the book has not been organized in strict linear or-
der. Tofacilitatethe processofextractingthe informationofinterestthere
are repetitions: the same concept may appear in the book several times,
thoughinasomewhatdifferentcontextandstressingdifferentaspects. The
readerthatwouldliketoknowmoreaboutagivenconceptcanalwaysfind
additional information by perusing the index.
For those who wish to start with looking first at examples: they can
start with “the impossible quantum fractal” — Sec. 2.5 — and then check
examples of hyperbolic quantum fractals in Sec. 3.1.
Those who are simply looking for algorithms and examples of the code
thatwereusedforgeneratingtheseexamplesmayliketostartdirectlywith
Sec. 3.4.
On the other hand readers interested in the foundations of quantum
theory can start with Chap. 4 or one of its sections.
May29,2014 10:6 WorldScientificBook-9inx6in QuantumFractals3 pg.ix/0
Contents
Preface vii
1. Introduction 1
2. What are Quantum Fractals? 11
2.1 Cantor set . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 Cantor set through “Chaos Game” . . . . . . . . 13
2.2 Iterated function systems . . . . . . . . . . . . . . . . . . 15
2.2.1 Definition of IFS . . . . . . . . . . . . . . . . . . . 18
2.2.2 Frobenius-Perronoperator . . . . . . . . . . . . . 22
2.3 Cantor set through matrix eigenvector . . . . . . . . . . . 23
2.4 Quantum iterated function systems . . . . . . . . . . . . . 25
2.5 Example: The “impossible” quantum fractal. . . . . . . . 29
2.5.1 24 symmetries — the octahedral group . . . . . . 30
2.5.2 Construction of the 24-elements SQIFS . . . . . . 31
2.5.3 Open problems. . . . . . . . . . . . . . . . . . . . 37
2.6 Action on the plane . . . . . . . . . . . . . . . . . . . . . 41
2.7 Lorentz group, SL(2,C), and relativistic aberration. . . . 43
2.7.1 The Lorentz group. . . . . . . . . . . . . . . . . . 43
2.7.2 Action of the Lorentz group on the sphere . . . . 45
2.7.3 The group SL(2,C) . . . . . . . . . . . . . . . . . 48
2.7.4 Action of SL(2,C) on the two-sphere S2 . . . . . 50
2.7.5 Projection operators representations of the Bloch
sphere S2 . . . . . . . . . . . . . . . . . . . . . . . 55
2.7.6 Visualization of quantum spin states and state
vectors . . . . . . . . . . . . . . . . . . . . . . . . 60
ix