Handbook of Quantum Logic and Quantum Structures: Quantum Structures

Kurt Engesser, Dov M. Gabbay, Daniel Lehmann0444528709, 9780444528704

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

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