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REASONING in QUANTUM THEORY Sharp and Unsharp Quantum Logics Contents Maria Luisa Dalla Chiara Roberto Giuntini Richard Greechie REASONING IN QUANTUM THEORY Sharp and Unsharp Quantum Logics 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. TheLukasiewicz operations . 94 5.3. MV algebras and QMV algebras . 96 5.4. Quasi-linear QMV algebras and effect algebras . 107 Chapter 6. Abstract axiomatic foundations of unsharp quantum theory115 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. The quantum logical conditional as a counterfactual conditional167 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 andLukasiewicz’ quantum logic 237 16.1. Partial quantum logics . 237 16.2.Lukasiewicz quantum logic . 241 16.3. The intuitive meaning of theLukasiewicz’ 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 Mwl . 108 5.4.2 The Hasse diagram of Mwl . 109 6.5.1 The Greechie diagram of G52 . 134 6.5.2 The Greechie diagram of G58 . 135 2 9.1.1 Quasi-model for γ in R . 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. In the technical literature, it is most frequently used to designate 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 on H in a suitable ordering - or some algebraic generalization of one of these. Thus, we have two concrete or standard quantum logics. These are struc- tures closely related to the usual mathematical formalism that underlies the foundation of quantum theory (QT). The set Π(H) is the basis for the sharp 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 one but that there are many quantum logics. These logics have various mod- els, usually one of the orthomodular structures, which include orthomodular 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 learn more about these algebraic structures are referred to (Kalmbach, 1983; Dvureˇcenskij and Pulmannov´a,2000). xiii xiv PREFACE Even the purist (non-quantum) logician may have some difficulty read- ing the technical logic in the second half of the book without the earlier preparation; such readers could, however, begin in the logical sections and refer to the earlier sections as needed. Quantum logicians, however, may proceed directly to the latter chapters. We are writing for a multidisciplinary audience, and we warn the reader that we at times are too verbose for mathematicians, too pedantic for physi- cists, too glib for logicians, and too technical for philosophers of science. We assure the reader that we have had the whole readership in mind as we made our compromises and we beg her indulgence. Here is an outline of the organization of the book. There are two parts. Part I, which consists of the first seven chapters, presents the historical background and the algebraic developments that motivate the syntax and underlie the semantics of the sequel. Part II studies a variety of quantum logics; in this Part, the term “logic” is used in the traditional sense, as a theory for a consequence relation that may hold between well-formed formu- las of a given language. Semantical characterizations are introduced, both algebraic semantics and Kripkean semantics. We set the stage in Chapter 1 by presenting some abstract notions needed later and by sketching the historical underpinnings of the subject. We present some of the basic notions of ordered sets followed by a quick review of Hilbert space and operators thereon. We recall von Neumann’s axioms for quantum theory and Birkhoff and von Neumann’s inseminal idea that propositions about quantum systems can be viewed as forming a kind of logic more appropriately modeled by projections on a Hilbert space than by a Boolean algebra.
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