Handbook of Quantum Logic and Quantum Structures (eBook)

Cover 1
Table of Contents 16
Foreword 6
Editorial Preface 8
List of Authors 12
Chapter 1 New Quantum Structures 18
1 INTRODUCTION 18
2 QUANTUM LOGICS, EFFECT ALGEBRAS AND D-POSETS 21
3 PSEUDO MV-ALGEBRAS 36
4 PSEUDO EFFECT ALGEBRAS 49
5 CONCLUSION 66
ACKNOWLEDGEMENTS 67
BIBLIOGRAPHY 67
Chapter 2 Quantum Structures and Fuzzy Set Theory 72
1 INTRODUCTION 72
2 BASIC NOTIONS AND DEFINITIONS 73
3 MACZYNSKI'S FUNCTIONAL MODEL OF B-VN QUANTUM LOGIC 76
4 THE GENERAL FUZZY SET MODEL OF B-VN QUANTUM LOGIC 77
5 TWO PAIRS OF BINARY OPERATIONS 79
6 GENERAL QUANTUM STRUCTURES AND FUZZY SETS 81
7 EFFECT ALGEBRAS OF FUZZY SETS GENERATED BY NILPOTENT TRIANGULAR NORMS 84
8 FUZZY SET MODELS OF QUANTUM PROBABILITY 87
9 SUMMARY 89
ACKNOWLEDGMENTS 90
BIBLIOGRAPHY 90
Chapter 3 Algebraic and Measure-theoretic Properties of Classes of Subspaces of an Inner Product Space 92
1 INTRODUCTION 92
2 FAMILIES OF CLOSED SUBSPACES OF AN INNER PRODUCT SPACE S 93
3 ALGEBRAIC STRUCTURE OF P(S), C(S), E(S), Eq(S), F(S) AND W(S) 97
4 ALGEBRAIC COMPLETENESS CRITERIA OF INNER PRODUCT SPACES 102
5 MEASURES ON P(S), C(S), E(S), Eq(S), F(S) AND W(S) 114
6 GLEASON AND DOROFEEV-SHERSTNEV THEOREMS 115
7 IS EVERY REGULAR CHARGE ON L(H) COMPLETELY-ADDITIVE? 118
8 MEASURE-THEORETIC COMPLETENESS CRITERIA OF INNER PRODUCT SPACES 121
9 CONVERGENCE OF CHARGES 126
10 INFINITE-VALUED MEASURES 131
ACKNOWLEDGEMENTS 134
BIBLIOGRAPHY 134
Chapter 4 Quantum Probability 138
1 INTRODUCTION 138
2 MOTIVATIONAL CALCULATIONS 139
3 NOTATION AND DEFINITIONS 142
4 PROBABILITY AND CONDITIONAL PROBABILITY 143
5 INDEPENDENCE 146
6 PROPERTIES OF THE SEQUENTIAL PRODUCT 148
7 ALMOST SHARP EFFECTS 149
8 NON-DISTURBANCE FOR FUZZY QUANTUM MEASUREMENTS 152
9 SEQUENTIAL EFFECT ALGEBRAS 157
BIBLIOGRAPHY 163
Chapter 5 Quantum Logics as Underlying Structures of Generalized Probability Theory 164
1 INTRODUCTION 164
2 BASIC DEFINITIONS AND FACTS 165
3 COMPATIBLE SUBSETS OF A LOGIC 170
4 STATES ON A LOGIC 174
5 OBSERVABLES ON A LOGIC 182
6 PARTIAL COMPATIBILITY AND JOINT DISTRIBUTIONS OF OBSERVABLES 185
7 THE LOGIC OF CLOSED SUBSPACES OF A HILBERT SPACE 201
8 JOINT DISTRIBUTIONS ON THE HILBERT SPACE LOGIC 206
9 APPENDIX 1: UNCERTAINTY RELATIONS 211
10 APPENDIX 2: BELL INEQUALITIES ON QUANTUM LOGICS 212
11 QUANTUM LOGICS AND PHYSICAL SYSTEMS 213
12 BELL INEQUALITIES 218
BIBLIOGRAPHY 225
Chapter 6 Quantum Logic and Partially Ordered Abelian Groups 232
1 INTRODUCTION 232
2 ORTHODOX QUANTUM MECHANICS 233
3 PROJECTIONS AND COMPRESSIONS 237
4 SYMMETRIES 238
5 MIXED STATES AND DENSITY OPERATORS 240
6 EFFECT OPERATORS AND POV-MEASURES 242
7 ABSTRACTION FROM P(h) AND E(h)„EFFECT ALGEBRAS 245
8 CLASSIFICATION OF EFFECT ALGEBRAS 248
9 PARTIALLY ORDERED ABELIAN GROUPS 253
10 UNIVERSAL GROUPS 255
11 ABSTRACTION FROM G(h)„UNITAL GROUPS 258
12 SEMISIMPLICIAL UNITAL GROUPS 262
13 INTERPOLATION UNIGROUPS AND THEIR UNIT INTERVALS 266
14 UNITAL GROUPS OF REAL-VALUED FUNCTIONS 270
15 EFFECT-ORDERED RINGS 274
16 ABSTRACTION FROM P(h)„CB-GROUPS 277
17 THE RICKART PROJECTION PROPERTY AND RC-GROUPS 283
18 SPECTRAL THEORY IN AN ARC-GROUP 286
19 RETROSPECTIVE 293
BIBLIOGRAPHY 297
Chapter 7 Quantum Structures and Operator Algebras 302
1 INTRODUCTION AND PRELIMINARIES 302
2 QUANTUM INTEGRATION 308
3 BASIC PRINCIPLES OF NONCOMMUTATIVE MEASURE THEORY 325
4 NONCOMMUTATIVE PROPERTIES OF MEASURES 340
ACKNOWLEDGMENT 344
BIBLIOGRAPHY 344
Chapter 8 Constructions of Quantum Structures 352
1 INTRODUCTION 352
2 QUANTUM STRUCTURES 352
3 STANDARD CONSTRUCTIONS 359
4 PASTING OF BOOLEAN ALGEBRAS 366
5 ADDITIONAL CONSTRUCTIONS BASED ON PASTING 376
6 MISCELLANEOUS TOPICS 380
ACKNOWLEDGEMENT 380
BIBLIOGRAPHY 381
Chapter 9 D-Posets 384
1 INTRODUCTION 384
2 DIFFERENCE POSETS 385
3 COMPATIBILITY IN D-POSETS 402
4 D-HOMOMORPHISMS OF D-POSETS 421
5 IDEALS AND FILTERS IN D-POSETS 437
ACKNOWLEDGEMENTS 442
BIBLIOGRAPHY 442
Chapter 10 Wigner's Theorem and its Generalizations 446
1 INTRODUCTION 446
2 THE CLASSICAL HILBERT SPACE FORMULATION OF QUANTUM MECHANICS 447
3 THE ORIGIN OF WIGNER'S THEOREM AND A SHORT HISTORY OF ITS PROOFS 450
4 AN ELEMENTARY PROOF OF WIGNER'S THEOREM 452
5 UHLHORN'S VERSION OF WIGNER'S THEOREM 462
6 WIGNER'S THEOREM VIEWED BY THE GENEVA SCHOOL 466
7 GENERALIZATIONS TO INDEFINITE INNER PRODUCT SPACES 468
8 SOME OTHER GENERALIZATIONS 471
9 QUATERNIONIC HILBERT SPACES 477
10 A TOPOLOGICAL AND LATTICE APPROACH 478
11 SOME OTHER SYMMETRY GROUPS 486
12 FROM AUTOMORPHISMS TO THE HAMILTONIAN 489
BIBLIOGRAPHY 490
Chapter 11 Propositional Systems, Hilbert Lattices and Generalized Hilbert Spaces 494
1 INTRODUCTION 494
2 PROJECTIVE GEOMETRIES, PROJECTIVE LATTICES 497
3 IRREDUCIBLE COMPONENTS 505
4 THE FUNDAMENTAL THEOREMS OF PROJECTIVE GEOMETRY 510
5 HILBERT GEOMETRIES, HILBERT LATTICES, PROPOSITIONAL SYSTEMS 516
6 IRREDUCIBLE COMPONENTS AGAIN 525
7 THE REPRESENTATION THEOREM FOR PROPOSITIONAL SYSTEMS 529
8 FROM HERE ON 531
9 APPENDIX: NOTIONS FROM LATTICE THEORY 536
ACKNOWLEDGEMENT 538
BIBLIOGRAPHY 538
Chapter 12 Equations and Hilbert Lattices 542
1 INTRODUCTION 542
2 DEFINITIONS AND BASIC FACTS 543
3 ORTHOARGUESIAN EQUATIONS AND SOME OTHER ONES 545
4 EQUATIONS CONNECTED WITH REAL-VALUED STATES 548
5 OTHER EQUATIONS HOLDING IN MOST GHLS 557
6 EQUATIONS CONNECTED WITH H-STATES 563
7 ORTHOSYMMETRIC ORTHOLATTICES 567
8 CONCLUDING REMARKS 569
BIBLIOGRAPHY 570
Chapter 13 The Source of the Orthomodular Law 572
1 INTRODUCTION 572
2 SURJECTIVE MAPS AND QUOTIENTS 575
3 DECOMPOSITIONS 577
4 SURJECTIONS AND DECOMPOSITIONS FOR FINITE SETS 581
5 DECOMPOSITIONS OF SETS WITH STRUCTURE 584
6 COMPATIBILITY OF DECOMPOSITIONS 586
7 DECOMPOSITIONS AND QUANTUM LOGIC 589
8 FURTHER RESULTS AND OPEN PROBLEMS 598
9 CONCLUSIONS 601
BIBLIOGRAPHY 601
Chapter 14 Starting from the Convex Set of States 604
1 INTRODUCTION 604
2 OBSERVABLES AND FACES OF THE SET OF STATES 606
3 CONVEXITY MODELS 609
4 EFFECT ALGEBRAS AND CONVEXITY MODELS 612
5 THE OPERATIONAL FRAMEWORK 614
6 THE BELL EFFECT 617
7 CLASSICAL AND NONCLASSICAL CORRELATIONS 623
8 OPERATIONAL EXTENSION OF THE QUANTUM MODEL 628
BIBLIOGRAPHY 632
Chapter 15 Quantum Logic and Automata Theory 636
1 INTRODUCTION 636
2 PRELIMINARIES 644
3 ORTHOMODULAR LATTICE-VALUED (NONDETERMINISTIC) FINITE AUTOMATA 660
4 ORTHOMODULAR LATTICE-VALUED PUSHDOWN AUTOMATA 726
5 CONCLUSION 761
6 BIBLIOGRAPHICAL NOTES 766
ACKNOWLEDGEMENTS 768
BIBLIOGRAPHY 768
Chapter 16 Quantum Logic and Quantum Computation 772
1 INTRODUCTION 772
2 HILBERT LATTICE 776
3 GREECHIE DIAGRAMS 777
4 GEOMETRY: GENERALIZED ORTHOARGUESIAN EQUATIONS 780
5 STATES: GODOWSKI EQUATIONS 786
6 STATES: MAYET-GODOWSKI EQUATIONS 790
7 STATE VECTORS: MAYET'S E-EQUATIONS 798
8 CONCLUSION 803
BIBLIOGRAPHY 807
Index 810