## Handbook of Quantum Logic and Quantum Structures: Quantum StructuresSince 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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### Contents

xi | |

xv | |

1 | |

55 | |

75 | |

Chapter 4 Quantum Probability | 121 |

Chapter 5 Quantum Logics as Underlying Structures of Generalized Probability Theory | 147 |

Chapter 6 Quantum Logic and Partially Ordered Abelian Groups | 215 |

Chapter 9 DPosets | 367 |

Chapter 10 Wigners Theorem and its Generalizations | 429 |

Chapter 11 Propositional Systems Hilbert Lattices and Generalized Hilbert Spaces | 477 |

Chapter 12 Equations and Hilbert Lattices | 525 |

Chapter 13 The Source of the Orthomodular Law | 555 |

Chapter 14 Starting from the Convex Set of States | 587 |

Chapter 15 Quantum Logic and Automata Theory | 619 |

Chapter 16 Quantum Logic and Quantum Computation | 755 |

### Other editions - View all

Handbook of Quantum Logic and Quantum Structures: Quantum Logic Kurt Engesser,Dov M. Gabbay,Daniel Lehmann No preview available - 2008 |

Handbook of Quantum Logic and Quantum Structures: Quantum Logic Kurt Engesser,Dov M. Gabbay,Daniel Lehmann No preview available - 2008 |

### Common terms and phrases

a s b a,b e atoms automata automaton automorphism Beltrametti bijection binary Boolean algebra bounded C*-algebra called Chetcuti classical closed subspaces commutative compatible convex COROLLARY countable decomposition defined DEFINITION denote Dvurečenskij elements equations equivalent example exists Foulis function fuzzy sets Gleason's theorem Greechie diagram Gudder Hamhalter hence Hilbert lattice Hilbert space homomorphism implies inner product space isomorphic Kalmbach Lemma linear mapping Math mathematical Mayet morphism MV-algebra Navara Neumann algebra notion observables operator orthoalgebra orthogonal orthomodular lattice orthomodular lattice-valued orthomodular poset pairwise partially ordered Phys poset probability measure projection lattice projective geometry Proof prove pseudo effect algebra pseudo MV-algebra Pták Pulmannová quantum logic quantum mechanics quantum structures recş result satisfies self-adjoint sequence subalgebra sublogic subset theory unigroup unique unital group valued von Neumann algebra Wigner's theorem

### Popular passages

Page xiii - Department of Computer Science and Technology Tsinghua University Beijing 100084, PR China...

Page 50 - Free sums of groups and their generalizations. An analysis of the associative law', Amer.

Page 790 - This volume contains the proceedings of the Workshop on Quantum Logic held in Erice (Sicily), December 2-9, 1979, at the Ettore Majorana Center for Scientific Culture.