Reasoning in Quantum Theory: Sharp and Unsharp Quantum Logics

300 Pages·2004·6.413 MB·English

by  M. Dalla Chiara, R. Giuntini, R. Greechie (auth.)

REASONING IN QUANTUM THEORY
TRENDS IN LOGIC
Studia Logica Library
VOLUME 22
Managing Editor
Ryszard Wójcicki, Institute of Philosophy and Sociology,
Polish Academy of Sciences, Warsaw, Poland
Editors
Vincent F. Hendricks, Department of Philosophy and Science Studies,
Roskilde University,
Denmark
Daniele Mundici, Department of Mathematics “Ulisse Dini”,
University of Florence, Italy
Ewa Orłowska, National Institute of Telecommunications,
Warsaw, Poland
Krister Segerberg, Department of Philosophy, Uppsala University,
Sweden
Heinrich Wansing, Institute of Philosophy, Dresden University of Technology,
Germany
SCOPE OF THE SERIES
Trends in Logic is a bookseries covering essentially the same area as the journal
Studia Logica – that is, contemporary formal logic and its applications and
relations to other disciplines. These include artificial intelligence, informatics,
cognitive science, philosophy of science, and the philosophy of language.
However, this list is not exhaustive, moreover, the range of applications, com-
parisons and sources of inspiration is open and evolves over time.
Volume Editor
Ryszard Wójcicki
The titles published in this series are listed at the end of this volume.
REASONING IN
QUANTUM THEORY
Sharp and Unsharp Quantum Logics
by
M. DALLA CHIARA
University of Florence,
Italy
R. GIUNTINI
University of Cagliari,
Italy
and
R. GREECHIE
Louisiana Tech University,
Ruston, U.S.A.
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-6562-9 ISBN 978-94-017-0526-4 (eBook)
DOI 10.1007/978-94-017-0526-4
Printed on acid-free paper
All Rights Reserved
©2004 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2004
No part of this work may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, electronic, mechanical, photocopying, microfilming, recording
or otherwise, without written permission from the Publisher, with the exception
of any material supplied specifically for the purpose of being entered
and executed on a computer system, for exclusive use by the purchaser of the work.
Contents
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi
PART I Mathematical and Physical Background . . . . . . . 1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Chapter 1. The mathematical scenario of quantum theory and von
Neumann’s axiomatization . . . . . . . . . . . . . . . . . 9
1.1. Algebraic structures . . . . . . . . . . . . . . . . . . . . . . . 9
1.2. The geometry of quantum theory . . . . . . . . . . . . . . . . 24
1.3. The axiomatization of orthodox QT. . . . . . . . . . . . . . . 31
1.4. The “logic” of the quantum events . . . . . . . . . . . . . . . 34
1.5. The logico-algebraic approach to QT . . . . . . . . . . . . . . 38
Chapter 2. Abstract axiomatic foundations of sharp QT . . . . . . . 41
2.1. Mackey’s minimal axiomatization of QT . . . . . . . . . . . . 42
2.2. Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.3. Event-state systems . . . . . . . . . . . . . . . . . . . . . . . . 52
2.4. Event-state systems and preclusivity spaces . . . . . . . . . . 55
Chapter 3. Back to Hilbert space . . . . . . . . . . . . . . . . . . . . 65
3.1. Events as closed subspaces . . . . . . . . . . . . . . . . . . . . 65
3.2. Events as projections . . . . . . . . . . . . . . . . . . . . . . . 67
3.3. Hilbert event-state systems . . . . . . . . . . . . . . . . . . . 68
3.4. From abstract orthoposets of events to Hilbert lattices . . . . 70
Chapter 4. The emergence of fuzzy events in Hilbert space quantum
theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.1. The notion of effect . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2. Effect-Brouwer Zadeh posets . . . . . . . . . . . . . . . . . . . 77
4.3. Mac Neille completions . . . . . . . . . . . . . . . . . . . . . . 81
4.4. Unsharp preclusivity spaces . . . . . . . . . . . . . . . . . . . 82
Chapter 5. Effect algebras and quantum MV algebras . . . . . . . . 87
v
vi CONTENTS
5.1. Effect algebras and Brouwer Zadeh effect algebras . . . . . . . 87
5.2. The L(cid:3) ukasiewicz operations . . . . . . . . . . . . . . . . . . . 94
5.3. MV algebras and QMV algebras . . . . . . . . . . . . . . . . . 96
5.4. Quasi-linear QMV algebras and effect algebras. . . . . . . . . 107
Chapter6. Abstractaxiomaticfoundationsofunsharpquantumtheory115
6.1. A minimal axiomatization of unsharp QT . . . . . . . . . . . 115
6.2. The algebraic structure of abstract effects . . . . . . . . . . . 119
6.3. The sharply dominating principle . . . . . . . . . . . . . . . . 124
6.4. Abstract unsharp preclusivity spaces . . . . . . . . . . . . . . 127
6.5. Sharp and unsharp abstract quantum theory . . . . . . . . . . 131
Chapter 7. To what extent is quantum ambiguity ambiguous? . . . . 137
7.1. Algebraic notions of “sharp” . . . . . . . . . . . . . . . . . . . 137
7.2. Probabilistic definitions of “sharpness” . . . . . . . . . . . . . 142
PART II Quantum Logics as Logic . . . . . . . . . . . . . . . . 147
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Chapter 8. Sharp quantum logics . . . . . . . . . . . . . . . . . . . . 155
8.1. Algebraic and Kripkean semantics for sharp quantum logics . 155
8.2. Algebraic and Kripkean realizations of Hilbert event-state
systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
8.3. The implication problem in quantum logic . . . . . . . . . . . 164
8.4. Five polynomial conditionals . . . . . . . . . . . . . . . . . . . 165
8.5. Thequantumlogicalconditionalasacounterfactualconditional167
8.6. Implication-connectives . . . . . . . . . . . . . . . . . . . . . . 168
Chapter 9. Metalogical properties and anomalies of quantum logic . 171
9.1. The failure of the Lindenbaum property . . . . . . . . . . . . 171
9.2. A modal interpretation of sharp quantum logics . . . . . . . . 174
Chapter 10. An axiomatization of OL and OQL . . . . . . . . . . . 179
10.1. The calculi for OL and OQL. . . . . . . . . . . . . . . . . . 179
10.2. The soundness and completeness theorems . . . . . . . . . . 181
Chapter 11. The metalogical intractability of orthomodularity . . . . 185
11.1. Orthomodularity is not elementary . . . . . . . . . . . . . . 186
11.2. The embeddability problem . . . . . . . . . . . . . . . . . . . 188
11.3. Hilbert quantum logic and the orthomodular law . . . . . . . 189
Chapter 12. First-order quantum logics and quantum set theories . . 193
12.1. First-order semantics . . . . . . . . . . . . . . . . . . . . . . 193
12.2. Quantum set theories . . . . . . . . . . . . . . . . . . . . . . 198
Chapter 13. Partial classical logic, the Lindenbaum property and the
hidden variable problem . . . . . . . . . . . . . . . . . . 201
13.1. Partial classical logic . . . . . . . . . . . . . . . . . . . . . . 201
CONTENTS vii
13.2. Partial classical logic and the Lindenbaum property . . . . . 208
13.3. States on partial Boolean algebras . . . . . . . . . . . . . . . 210
13.4. The Lindenbaum property and the hidden variable problem. 214
Chapter 14. Unsharp quantum logics . . . . . . . . . . . . . . . . . . 217
14.1. Paraconsistent quantum logic . . . . . . . . . . . . . . . . . . 217
14.2. ε-Preclusivity spaces . . . . . . . . . . . . . . . . . . . . . . . 220
14.3. An aside: similarities of PQL and historiography . . . . . . 222
Chapter 15. The Brouwer Zadeh logics . . . . . . . . . . . . . . . . . 225
15.1. The weak Brouwer Zadeh logic . . . . . . . . . . . . . . . . . 225
15.2. The pair semantics and the strong Brouwer Zadeh logic . . . 228
15.3. BZL3-effect realizations . . . . . . . . . . . . . . . . . . . . 234
Chapter 16. Partial quantum logics and L(cid:3) ukasiewicz’ quantum logic 237
16.1. Partial quantum logics . . . . . . . . . . . . . . . . . . . . . 237
16.2. L(cid:3) ukasiewicz quantum logic . . . . . . . . . . . . . . . . . . . 241
16.3. The intuitive meaning of the L(cid:3) ukasiewicz’ quantum logical
connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
Chapter 17. Quantum computational logic . . . . . . . . . . . . . . . 249
17.1. Quantum logical gates. . . . . . . . . . . . . . . . . . . . . . 252
17.2. The probabilistic content of the quantum logical gates . . . . 259
17.3. Quantum computational semantics. . . . . . . . . . . . . . . 262
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
Synoptic tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
Index of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
Index of Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
List of Figures
1.1.1 The Benzene ring . . . . . . . . . . . . . . . . . . . . . 14
1.1.2 MO2: the smallest OML that is not a BA . . . . . . . . 15
1.1.3 The Greechie diagram of G12 . . . . . . . . . . . . . . . 17
1.1.4 The Hasse diagram of G12 . . . . . . . . . . . . . . . . . 18
1.4.1 Failure of bivalence in QT . . . . . . . . . . . . . . . . . 37
2.2.1 J18: the smallest OMP that is not an OML . . . . . . . 51
2.4.1 The Greechie diagram of GGM410 . . . . . . . . . . . . 59
2.4.2 The state s0 on GGM410 . . . . . . . . . . . . . . . . . . 60
5.1.1 WT: the smallest OA that is not an OMP . . . . . . . 91
5.3.1 M4: the smallest QMV that is not an MV. . . . . . . . 106
5.4.1 The operation ⊕ of M . . . . . . . . . . . . . . . . . . 108
wl
5.4.2 The Hasse diagram of M . . . . . . . . . . . . . . . . 109
wl
6.5.1 The Greechie diagram of G52 . . . . . . . . . . . . . . . 134
6.5.2 The Greechie diagram of G58 . . . . . . . . . . . . . . . 135
9.1.1 Quasi-model for γ in R2 . . . . . . . . . . . . . . . . . . 173
11.3.1 The Greechie diagram of G30 . . . . . . . . . . . . . . . 190
17.1.1 A noncontinuous fuzzy square root of the negation . . . 260
ix
List of Tables
1 The quantum structures . . . . . . . . . . . . . . . . . . 271
2 The labyrinth of quantum logics . . . . . . . . . . . . . 273
xi
Preface
The term “quantum logic” has entered our languages as a synonym
for something that doesn’t make sense to our everyday rationality. Or,
somewhat more technically but still in the common literature, it signifies
some generic sort of mystification of classical logic understood only by the
illuminati. Inthetechnicalliterature,itismostfrequentlyusedtodesignate
the set of projections Π(H) on a Hilbert space H or the set of positive
operators E(H) which fall between the smallest and the largest projections
onHinasuitableordering-orsomealgebraicgeneralizationofoneofthese.
Thus,wehavetwoconcreteorstandardquantumlogics. Thesearestruc-
turescloselyrelatedtotheusualmathematicalformalismthatunderliesthe
foundationofquantumtheory(QT).ThesetΠ(H)isthebasisforthesharp
theory and E(H) for the unsharp theory, in much the same way that classi-
cal logic is based, in its sharp and unsharp manifestations, on (subalgebras
of products of) the two-element set {0,1} and the real unit interval [0,1],
respectively.
There is a still unfolding panorama of structures that generalize these
two standard models. The theory of the foundations of quantum mechanics
called “quantum logic” studies the standard models and their abstractions.
Confusion has persisted as to just what quantum logic is and how it
should be construed as a veritable logic. The purpose of this book is to
delineate (what we know of) the quantum logics, to explain of what the
panorama of quantum logics consists and to present actual logics whose al-
gebraic or Kripkean semantics are based on the algebraic models that are
historically referred to as quantum logics. Our position is that there is not
onebutthattherearemanyquantumlogics. Theselogicshavevariousmod-
els,usuallyoneoftheorthomodularstructures,whichincludeorthomodular
lattices, orthomodular posets, orthoalgebras, and effect algebras.
We present sufficient historical background to give the reader an idea of
how the theory developed. However, far more is presented than is needed
to simply develop the logic, so that the novice may pick up the motivat-
ing aspects of the subject. Readers not interested in the technical logical
details of quantum logics may gain an accounting of the mathematical as-
pects of “quantum logic,” the models of the theory and how they relate
to one another, by reading only the initial chapters. Readers wanting to
learnmoreaboutthesealgebraicstructuresarereferredto(Kalmbach,1983;
Dvureˇcenskij and Pulmannova´, 2000).
xiii