Handbook of Quantum Logic and Quantum Structures: Quantum Structures

✍ Scribed by Kurt Engesser (editor), Dov M. Gabbay (editor), Daniel Lehmann (editor)

• Publisher: Elsevier Science
• Year: 2007
• Tongue: English
• Leaves: 821
• Edition: 1
• Category: Library

✦ Synopsis

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

✦ Table of Contents

Handbook of Quantum Logic and Quantum Structures
Copyright Page
Foreword
Editorial Preface
Contributors
Contents
Chapter 1 New Quantum Structures
1 INTRODUCTION
2 QUANTUM LOGICS, EFFECT ALGEBRAS AND D-POSETS
3 PSEUDO MV-ALGEBRAS
4 PSEUDO EFFECT ALGEBRAS
5 CONCLUSION
ACKNOWLEDGEMENTS
BIBLIOGRAPHY
Chapter 2 Quantum Structures and Fuzzy Set Theory
1 INTRODUCTION
2 BASIC NOTIONS AND DEFINITIONS
3 MACZYNSKI'S FUNCTIONAL MODEL OF B-VN QUANTUM LOGIC
4 THE GENERAL FUZZY SET MODEL OF B-VN QUANTUM LOGIC
5 TWO PAIRS OF BINARY OPERATIONS
6 GENERAL QUANTUM STRUCTURES AND FUZZY SETS
7 EFFECT ALGEBRAS OF FUZZY SETS GENERATED BY NILPOTENT TRIANGULAR NORMS
8 FUZZY SET MODELS OF QUANTUM PROBABILITY
9 SUMMARY
ACKNOWLEDGMENTS
BIBLIOGRAPHY
Chapter 3 Algebraic and Measure-theoretic Properties of Classes of Subspaces of an Inner Product Space
1 INTRODUCTION
2 FAMILIES OF CLOSED SUBSPACES OF AN INNER PRODUCT SPACE S
3 ALGEBRAIC STRUCTURE OF P(S), C(S), E(S), Eq(S), F(S) AND W(S)
4 ALGEBRAIC COMPLETENESS CRITERIA OF INNER PRODUCT SPACES
5 MEASURES ON P(S), C(S), E(S), Eq(S), F(S) AND W(S)
6 GLEASON AND DOROFEEV-SHERSTNEV THEOREMS
7 IS EVERY REGULAR CHARGE ON L(H) COMPLETELY-ADDITIVE?
8 MEASURE-THEORETIC COMPLETENESS CRITERIA OF INNER PRODUCT SPACES
9 CONVERGENCE OF CHARGES
10 INFINITE-VALUED MEASURES
ACKNOWLEDGEMENTS
BIBLIOGRAPHY
Chapter 4 Quantum Probability
1 INTRODUCTION
2 MOTIVATIONAL CALCULATIONS
3 NOTATION AND DEFINITIONS
4 PROBABILITY AND CONDITIONAL PROBABILITY
5 INDEPENDENCE
6 PROPERTIES OF THE SEQUENTIAL PRODUCT
7 ALMOST SHARP EFFECTS
8 NON-DISTURBANCE FOR FUZZY QUANTUM MEASUREMENTS
9 SEQUENTIAL EFFECT ALGEBRAS
BIBLIOGRAPHY
Chapter 5 Quantum Logics as Underlying Structures of Generalized Probability Theory
1 INTRODUCTION
2 BASIC DEFINITIONS AND FACTS
3 COMPATIBLE SUBSETS OF A LOGIC
4 STATES ON A LOGIC
5 OBSERVABLES ON A LOGIC
6 PARTIAL COMPATIBILITY AND JOINT DISTRIBUTIONS OF OBSERVABLES
7 THE LOGIC OF CLOSED SUBSPACES OF A HILBERT SPACE
8 JOINT DISTRIBUTIONS ON THE HILBERT SPACE LOGIC
9 APPENDIX 1: UNCERTAINTY RELATIONS
10 APPENDIX 2: BELL INEQUALITIES ON QUANTUM LOGICS
11 QUANTUM LOGICS AND PHYSICAL SYSTEMS
12 BELL INEQUALITIES
BIBLIOGRAPHY
Chapter 6 Quantum Logic and Partially Ordered Abelian Groups
1 INTRODUCTION
2 ORTHODOX QUANTUM MECHANICS
3 PROJECTIONS AND COMPRESSIONS
4 SYMMETRIES
5 MIXED STATES AND DENSITY OPERATORS
6 EFFECT OPERATORS AND POV-MEASURES
7 ABSTRACTION FROM P(h) AND E(h)—EFFECT ALGEBRAS
8 CLASSIFICATION OF EFFECT ALGEBRAS
9 PARTIALLY ORDERED ABELIAN GROUPS
10 UNIVERSAL GROUPS
11 ABSTRACTION FROM G(h)—UNITAL GROUPS
12 SEMISIMPLICIAL UNITAL GROUPS
13 INTERPOLATION UNIGROUPS AND THEIR UNIT INTERVALS
14 UNITAL GROUPS OF REAL-VALUED FUNCTIONS
15 EFFECT-ORDERED RINGS
16 ABSTRACTION FROM P(h)—CB-GROUPS
17 THE RICKART PROJECTION PROPERTY AND RC-GROUPS
18 SPECTRAL THEORY IN AN ARC-GROUP
19 RETROSPECTIVE
BIBLIOGRAPHY
Chapter 7 Quantum Structures and Operator Algebras
1 INTRODUCTION AND PRELIMINARIES
2 QUANTUM INTEGRATION
3 BASIC PRINCIPLES OF NONCOMMUTATIVE MEASURE THEORY
4 NONCOMMUTATIVE PROPERTIES OF MEASURES
ACKNOWLEDGMENT
BIBLIOGRAPHY
Chapter 8 Constructions of Quantum Structures
1 INTRODUCTION
2 QUANTUM STRUCTURES
3 STANDARD CONSTRUCTIONS
4 PASTING OF BOOLEAN ALGEBRAS
5 ADDITIONAL CONSTRUCTIONS BASED ON PASTING
6 MISCELLANEOUS TOPICS
ACKNOWLEDGEMENT
BIBLIOGRAPHY
Chapter 9 D-Posets
1 INTRODUCTION
2 DIFFERENCE POSETS
3 COMPATIBILITY IN D-POSETS
4 D-HOMOMORPHISMS OF D-POSETS
5 IDEALS AND FILTERS IN D-POSETS
ACKNOWLEDGEMENTS
BIBLIOGRAPHY
Chapter 10 Wigner's Theorem and its Generalizations
1 INTRODUCTION
2 THE CLASSICAL HILBERT SPACE FORMULATION OF QUANTUM MECHANICS
3 THE ORIGIN OF WIGNER'S THEOREM AND A SHORT HISTORY OF ITS PROOFS
4 AN ELEMENTARY PROOF OF WIGNER'S THEOREM
5 UHLHORN'S VERSION OF WIGNER'S THEOREM
6 WIGNER'S THEOREM VIEWED BY THE GENEVA SCHOOL
7 GENERALIZATIONS TO INDEFINITE INNER PRODUCT SPACES
8 SOME OTHER GENERALIZATIONS
9 QUATERNIONIC HILBERT SPACES
10 A TOPOLOGICAL AND LATTICE APPROACH
11 SOME OTHER SYMMETRY GROUPS
12 FROM AUTOMORPHISMS TO THE HAMILTONIAN
BIBLIOGRAPHY
Chapter 11 Propositional Systems, Hilbert Lattices and Generalized Hilbert Spaces
1 INTRODUCTION
2 PROJECTIVE GEOMETRIES, PROJECTIVE LATTICES
3 IRREDUCIBLE COMPONENTS
4 THE FUNDAMENTAL THEOREMS OF PROJECTIVE GEOMETRY
5 HILBERT GEOMETRIES, HILBERT LATTICES, PROPOSITIONAL SYSTEMS
6 IRREDUCIBLE COMPONENTS AGAIN
7 THE REPRESENTATION THEOREM FOR PROPOSITIONAL SYSTEMS
8 FROM HERE ON
9 APPENDIX: NOTIONS FROM LATTICE THEORY
ACKNOWLEDGEMENT
BIBLIOGRAPHY
Chapter 12 Equations and Hilbert Lattices
1 INTRODUCTION
2 DEFINITIONS AND BASIC FACTS
3 ORTHOARGUESIAN EQUATIONS AND SOME OTHER ONES
4 EQUATIONS CONNECTED WITH REAL-VALUED STATES
5 OTHER EQUATIONS HOLDING IN MOST GHLS
6 EQUATIONS CONNECTED WITH H-STATES
7 ORTHOSYMMETRIC ORTHOLATTICES
8 CONCLUDING REMARKS
BIBLIOGRAPHY
Chapter 13 The Source of the Orthomodular Law
1 INTRODUCTION
2 SURJECTIVE MAPS AND QUOTIENTS
3 DECOMPOSITIONS
4 SURJECTIONS AND DECOMPOSITIONS FOR FINITE SETS
5 DECOMPOSITIONS OF SETS WITH STRUCTURE
6 COMPATIBILITY OF DECOMPOSITIONS
7 DECOMPOSITIONS AND QUANTUM LOGIC
8 FURTHER RESULTS AND OPEN PROBLEMS
9 CONCLUSIONS
BIBLIOGRAPHY
Chapter 14 Starting from the Convex Set of States
1 INTRODUCTION
2 OBSERVABLES AND FACES OF THE SET OF STATES
3 CONVEXITY MODELS
4 EFFECT ALGEBRAS AND CONVEXITY MODELS
5 THE OPERATIONAL FRAMEWORK
6 THE BELL EFFECT
7 CLASSICAL AND NONCLASSICAL CORRELATIONS
8 OPERATIONAL EXTENSION OF THE QUANTUM MODEL
BIBLIOGRAPHY
Chapter 15 Quantum Logic and Automata Theory
1 INTRODUCTION
2 PRELIMINARIES
3 ORTHOMODULAR LATTICE-VALUED (NONDETERMINISTIC) FINITE AUTOMATA
4 ORTHOMODULAR LATTICE-VALUED PUSHDOWN AUTOMATA
5 CONCLUSION
6 BIBLIOGRAPHICAL NOTES
ACKNOWLEDGEMENTS
BIBLIOGRAPHY
Chapter 16 Quantum Logic and Quantum Computation
1 INTRODUCTION
2 HILBERT LATTICE
3 GREECHIE DIAGRAMS
4 GEOMETRY: GENERALIZED ORTHOARGUESIAN EQUATIONS
5 STATES: GODOWSKI EQUATIONS
6 STATES: MAYET-GODOWSKI EQUATIONS
7 STATE VECTORS: MAYET'S E-EQUATIONS
8 CONCLUSION
BIBLIOGRAPHY
Index