Handbook of Quantum Logic and Quantum Structures: Quantum Structures
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Table of contents
- Cover
- Table of Contentsxv
- Forewordv
- Editorial Prefacevii
- List of Authorsxi
- Chapter 1 New Quantum Structures1
- 1 INTRODUCTION1
- 2 QUANTUM LOGICS, EFFECT ALGEBRAS AND D-POSETS4
- 3 PSEUDO MV-ALGEBRAS19
- 4 PSEUDO EFFECT ALGEBRAS32
- 5 CONCLUSION49
- ACKNOWLEDGEMENTS50
- BIBLIOGRAPHY50
- Chapter 2 Quantum Structures and Fuzzy Set Theory55
- 1 INTRODUCTION55
- 2 BASIC NOTIONS AND DEFINITIONS56
- 3 MACZYŃSKI'S FUNCTIONAL MODEL OF B-VN QUANTUM LOGIC59
- 4 THE GENERAL FUZZY SET MODEL OF B-VN QUANTUM LOGIC60
- 5 TWO PAIRS OF BINARY OPERATIONS62
- 6 GENERAL QUANTUM STRUCTURES AND FUZZY SETS64
- 7 EFFECT ALGEBRAS OF FUZZY SETS GENERATED BY NILPOTENT TRIANGULAR NORMS67
- 8 FUZZY SET MODELS OF QUANTUM PROBABILITY70
- 9 SUMMARY72
- ACKNOWLEDGMENTS73
- BIBLIOGRAPHY73
- Chapter 3 Algebraic and Measure-theoretic Properties of Classes of Subspaces of an Inner Product Spa75
- 1 INTRODUCTION75
- 2 FAMILIES OF CLOSED SUBSPACES OF AN INNER PRODUCT SPACE S76
- 3 ALGEBRAIC STRUCTURE OF P(S), C(S), E(S), Eq(S), F(S) AND W(S)80
- 4 ALGEBRAIC COMPLETENESS CRITERIA OF INNER PRODUCT SPACES85
- 5 MEASURES ON P(S), C(S), E(S), Eq(S), F(S) AND W(S)97
- 6 GLEASON AND DOROFEEV-SHERSTNEV THEOREMS98
- 7 IS EVERY REGULAR CHARGE ON L(H) COMPLETELY-ADDITIVE?101
- 8 MEASURE-THEORETIC COMPLETENESS CRITERIA OF INNER PRODUCT SPACES104
- 9 CONVERGENCE OF CHARGES109
- 10 INFINITE-VALUED MEASURES114
- ACKNOWLEDGEMENTS117
- BIBLIOGRAPHY117
- Chapter 4 Quantum Probability121
- 1 INTRODUCTION121
- 2 MOTIVATIONAL CALCULATIONS122
- 3 NOTATION AND DEFINITIONS125
- 4 PROBABILITY AND CONDITIONAL PROBABILITY126
- 5 INDEPENDENCE129
- 6 PROPERTIES OF THE SEQUENTIAL PRODUCT131
- 7 ALMOST SHARP EFFECTS132
- 8 NON-DISTURBANCE FOR FUZZY QUANTUM MEASUREMENTS135
- 9 SEQUENTIAL EFFECT ALGEBRAS140
- BIBLIOGRAPHY146
- Chapter 5 Quantum Logics as Underlying Structures of Generalized Probability Theory147
- 1 INTRODUCTION147
- 2 BASIC DEFINITIONS AND FACTS148
- 3 COMPATIBLE SUBSETS OF A LOGIC153
- 4 STATES ON A LOGIC157
- 5 OBSERVABLES ON A LOGIC165
- 6 PARTIAL COMPATIBILITY AND JOINT DISTRIBUTIONS OF OBSERVABLES168
- 7 THE LOGIC OF CLOSED SUBSPACES OF A HILBERT SPACE184
- 8 JOINT DISTRIBUTIONS ON THE HILBERT SPACE LOGIC189
- 9 APPENDIX 1: UNCERTAINTY RELATIONS194
- 10 APPENDIX 2: BELL INEQUALITIES ON QUANTUM LOGICS195
- 11 QUANTUM LOGICS AND PHYSICAL SYSTEMS196
- 12 BELL INEQUALITIES201
- BIBLIOGRAPHY208
- Chapter 6 Quantum Logic and Partially Ordered Abelian Groups215
- 1 INTRODUCTION215
- 2 ORTHODOX QUANTUM MECHANICS216
- 3 PROJECTIONS AND COMPRESSIONS220
- 4 SYMMETRIES221
- 5 MIXED STATES AND DENSITY OPERATORS223
- 6 EFFECT OPERATORS AND POV-MEASURES225
- 7 ABSTRACTION FROM P(h) AND E(h)„EFFECT ALGEBRAS228
- 8 CLASSIFICATION OF EFFECT ALGEBRAS231
- 9 PARTIALLY ORDERED ABELIAN GROUPS236
- 10 UNIVERSAL GROUPS238
- 11 ABSTRACTION FROM G(h)„UNITAL GROUPS241
- 12 SEMISIMPLICIAL UNITAL GROUPS245
- 13 INTERPOLATION UNIGROUPS AND THEIR UNIT INTERVALS249
- 14 UNITAL GROUPS OF REAL-VALUED FUNCTIONS253
- 15 EFFECT-ORDERED RINGS257
- 16 ABSTRACTION FROM P(h)„CB-GROUPS260
- 17 THE RICKART PROJECTION PROPERTY AND RC-GROUPS266
- 18 SPECTRAL THEORY IN AN ARC-GROUP269
- 19 RETROSPECTIVE276
- BIBLIOGRAPHY280
- Chapter 7 Quantum Structures and Operator Algebras285
- 1 INTRODUCTION AND PRELIMINARIES285
- 2 QUANTUM INTEGRATION291
- 3 BASIC PRINCIPLES OF NONCOMMUTATIVE MEASURE THEORY308
- 4 NONCOMMUTATIVE PROPERTIES OF MEASURES323
- ACKNOWLEDGMENT327
- BIBLIOGRAPHY327
- Chapter 8 Constructions of Quantum Structures335
- 1 INTRODUCTION335
- 2 QUANTUM STRUCTURES335
- 3 STANDARD CONSTRUCTIONS342
- 4 PASTING OF BOOLEAN ALGEBRAS349
- 5 ADDITIONAL CONSTRUCTIONS BASED ON PASTING359
- 6 MISCELLANEOUS TOPICS363
- ACKNOWLEDGEMENT363
- BIBLIOGRAPHY364
- Chapter 9 D-Posets367
- 1 INTRODUCTION367
- 2 DIFFERENCE POSETS368
- 3 COMPATIBILITY IN D-POSETS385
- 4 D-HOMOMORPHISMS OF D-POSETS404
- 5 IDEALS AND FILTERS IN D-POSETS420
- ACKNOWLEDGEMENTS425
- BIBLIOGRAPHY425
- Chapter 10 Wigner's Theorem and its Generalizations429
- 1 INTRODUCTION429
- 2 THE CLASSICAL HILBERT SPACE FORMULATION OF QUANTUM MECHANICS430
- 3 THE ORIGIN OF WIGNER'S THEOREM AND A SHORT HISTORY OF ITS PROOFS433
- 4 AN ELEMENTARY PROOF OF WIGNER'S THEOREM435
- 5 UHLHORN'S VERSION OF WIGNER'S THEOREM445
- 6 WIGNER'S THEOREM VIEWED BY THE GENEVA SCHOOL449
- 7 GENERALIZATIONS TO INDEFINITE INNER PRODUCT SPACES451
- 8 SOME OTHER GENERALIZATIONS454
- 9 QUATERNIONIC HILBERT SPACES460
- 10 A TOPOLOGICAL AND LATTICE APPROACH461
- 11 SOME OTHER SYMMETRY GROUPS469
- 12 FROM AUTOMORPHISMS TO THE HAMILTONIAN472
- BIBLIOGRAPHY473
- Chapter 11 Propositional Systems, Hilbert Lattices and Generalized Hilbert Spaces477
- 1 INTRODUCTION477
- 2 PROJECTIVE GEOMETRIES, PROJECTIVE LATTICES480
- 3 IRREDUCIBLE COMPONENTS488
- 4 THE FUNDAMENTAL THEOREMS OF PROJECTIVE GEOMETRY493
- 5 HILBERT GEOMETRIES, HILBERT LATTICES, PROPOSITIONAL SYSTEMS499
- 6 IRREDUCIBLE COMPONENTS AGAIN508
- 7 THE REPRESENTATION THEOREM FOR PROPOSITIONAL SYSTEMS512
- 8 FROM HERE ON514
- 9 APPENDIX: NOTIONS FROM LATTICE THEORY519
- ACKNOWLEDGEMENT521
- BIBLIOGRAPHY521
- Chapter 12 Equations and Hilbert Lattices525
- 1 INTRODUCTION525
- 2 DEFINITIONS AND BASIC FACTS526
- 3 ORTHOARGUESIAN EQUATIONS AND SOME OTHER ONES528
- 4 EQUATIONS CONNECTED WITH REAL-VALUED STATES531
- 5 OTHER EQUATIONS HOLDING IN MOST GHLS540
- 6 EQUATIONS CONNECTED WITH H-STATES546
- 7 ORTHOSYMMETRIC ORTHOLATTICES550
- 8 CONCLUDING REMARKS552
- BIBLIOGRAPHY553
- Chapter 13 The Source of the Orthomodular Law555
- 1 INTRODUCTION555
- 2 SURJECTIVE MAPS AND QUOTIENTS558
- 3 DECOMPOSITIONS560
- 4 SURJECTIONS AND DECOMPOSITIONS FOR FINITE SETS564
- 5 DECOMPOSITIONS OF SETS WITH STRUCTURE567
- 6 COMPATIBILITY OF DECOMPOSITIONS569
- 7 DECOMPOSITIONS AND QUANTUM LOGIC572
- 8 FURTHER RESULTS AND OPEN PROBLEMS581
- 9 CONCLUSIONS584
- BIBLIOGRAPHY584
- Chapter 14 Starting from the Convex Set of States587
- 1 INTRODUCTION587
- 2 OBSERVABLES AND FACES OF THE SET OF STATES589
- 3 CONVEXITY MODELS592
- 4 EFFECT ALGEBRAS AND CONVEXITY MODELS595
- 5 THE OPERATIONAL FRAMEWORK597
- 6 THE BELL EFFECT600
- 7 CLASSICAL AND NONCLASSICAL CORRELATIONS606
- 8 OPERATIONAL EXTENSION OF THE QUANTUM MODEL611
- BIBLIOGRAPHY615
- Chapter 15 Quantum Logic and Automata Theory619
- 1 INTRODUCTION619
- 2 PRELIMINARIES627
- 3 ORTHOMODULAR LATTICE-VALUED (NONDETERMINISTIC) FINITE AUTOMATA643
- 4 ORTHOMODULAR LATTICE-VALUED PUSHDOWN AUTOMATA709
- 5 CONCLUSION744
- 6 BIBLIOGRAPHICAL NOTES749
- ACKNOWLEDGEMENTS751
- BIBLIOGRAPHY751
- Chapter 16 Quantum Logic and Quantum Computation755
- 1 INTRODUCTION755
- 2 HILBERT LATTICE759
- 3 GREECHIE DIAGRAMS760
- 4 GEOMETRY: GENERALIZED ORTHOARGUESIAN EQUATIONS763
- 5 STATES: GODOWSKI EQUATIONS769
- 6 STATES: MAYET-GODOWSKI EQUATIONS773
- 7 STATE VECTORS: MAYET'S E-EQUATIONS781
- 8 CONCLUSION786
- BIBLIOGRAPHY790
- Index793
Since its inception in the famous 1936 paper by Birkhoff and von Neumann entitled “The logic of quantum mechanics quantum logic, i.e. the logical investigation of quantum mechanics, has undergone an enormous development. Various schools of thought and approaches have emerged and there are a variety of technical results.
Quantum logic is a heterogeneous field of research ranging from investigations which may be termed logical in the traditional sense to studies focusing on structures which are on the border between algebra and logic. For the latter structures the term quantum structures is appropriate.
The chapters of this Handbook, which are authored by the most eminent scholars in the field, constitute a comprehensive presentation of the main schools, approaches and results in the field of quantum logic and quantum structures. Much of the material presented is of recent origin representing the frontier of the subject.
The present volume focuses on quantum structures. Among the structures studied extensively in this volume are, just to name a few, Hilbert lattices, D-posets, effect algebras MV algebras, partially ordered Abelian groups and those structures underlying quantum probability.
- Written by eminent scholars in the field of logic
- A comprehensive presentation of the theory, approaches and results in the field of quantum logic
- Volume focuses on quantum structures
Quantum logic is a heterogeneous field of research ranging from investigations which may be termed logical in the traditional sense to studies focusing on structures which are on the border between algebra and logic. For the latter structures the term quantum structures is appropriate.
The chapters of this Handbook, which are authored by the most eminent scholars in the field, constitute a comprehensive presentation of the main schools, approaches and results in the field of quantum logic and quantum structures. Much of the material presented is of recent origin representing the frontier of the subject.
The present volume focuses on quantum structures. Among the structures studied extensively in this volume are, just to name a few, Hilbert lattices, D-posets, effect algebras MV algebras, partially ordered Abelian groups and those structures underlying quantum probability.
- Written by eminent scholars in the field of logic
- A comprehensive presentation of the theory, approaches and results in the field of quantum logic
- Volume focuses on quantum structures