## Categorical Foundations of Quantum Logics and Their Truth Values Structures (2004)

Citations: | 1 - 1 self |

### BibTeX

@MISC{Zafiris04categoricalfoundations,

author = {Elias Zafiris},

title = {Categorical Foundations of Quantum Logics and Their Truth Values Structures},

year = {2004}

}

### OpenURL

### Abstract

We introduce a foundational sheaf theoretical scheme for the comprehension of quantum event structures, in terms of localization systems consisting of Boolean coordinatization coverings induced by measurement. The scheme is based on the existence of a categorical adjunction between presheaves of Boolean event algebras and Quantum event algebras. On the basis of this adjoint correspondence we prove the existence of an object of truth values in the category of quantum logics, characterized as subobject classifier. This classifying object plays the equivalent role that the two-valued Boolean truth values object plays in classical event structures. We construct the object of quantum truth values explicitly and argue that it constitutes the appropriate choice for the valuation of propositions describing the behavior of quantum systems.

### Citations

474 |
Categories for the working mathematician
- MacLane
- 1998
(Show Context)
Citation Context ...omC(X, Y ) × HomC(Y, Z) → HomC(X, Z) For morphisms g ∈ HomC(X, Y ), h ∈ HomC(Y, Z) the image of the pair (g, h) is called the composition; it is denoted h◦g. The composition operation is associative. =-=[4]-=-. For any f ∈ HomC(X, Y ) we have idY ◦ f = f ◦ idX = f. For an arbitrary category C the opposite category C op is defined in the following way: the objects are the same, but HomC op(X, Y ) = HomC(Y, ... |

448 |
Théorie des topos et cohomologie étale des schémas
- Artin, Grothendieck, et al.
- 1977
(Show Context)
Citation Context ...simply the set P(B), although its elements are written as pairs so as to form a disjoint union. The construction of the fibration induced by P, is an instance of the general Grothendieck construction =-=[7]-=-. G(P, B) GP ������������������ ������������������ B P Sets 4 The Fundamental Adjunction The adjunctive correspondence, which will be proved in what follows, provides the conceptual ground, concerning... |

265 |
Neumann The Logic of Quantum Mechanics
- Birkhoff, Von
- 1936
(Show Context)
Citation Context ...stem in the domain of the corresponding theory. Naturally, the typical mathematical structure associated with logic is an ordered structure. The original quantum logical formulation of Quantum theory =-=[1, 2]-=- depends in an essential way on the identification of propositions with projection operators on a complex Hilbert space. A non-classical, nonBoolean logical structure is effectively induced which has ... |

220 |
The problem of hidden variables in quantum mechanics
- Kochen, Specker
- 1967
(Show Context)
Citation Context ...of quantum event algebras, as it is encoded in Boolean localization systems. The fact that a quantum event algebra is actually a non-trivial global object is fully justified by Kochen-Specker theorem =-=[11]-=-. According to this there are no two-valued homomorphisms on the algebra of quantum propositions. Consequently a quantum logical algebra cannot be embedded into a Boolean one. We note that a two-value... |

71 | A topos perspective on the KochenSpecker theorem: III. Von Neumann algebras as the base category
- Hamilton, Isham, et al.
- 1999
(Show Context)
Citation Context ...ns of propositions describing the behavior of quantum systems. 4Contextual topos theoretical approaches to quantum structures of truth values have been also considered, from a different viewpoint in =-=[12, 13]-=-, and discussed in [14-16]. Of particular relevance to the present work, regarding the specification of a quantum truth values object, although not based on category theory methods, seems to be the ap... |

46 | Toposes and Local Set Theories - Bell - 1988 |

19 | Some Possible Roles for Topos Theory - Isham - 1999 |

11 | From absolute to local mathematics, Synthese - Bell - 1986 |

6 |
Geometry of Quantum Mechanics
- Varadarajan
- 1968
(Show Context)
Citation Context ...stem in the domain of the corresponding theory. Naturally, the typical mathematical structure associated with logic is an ordered structure. The original quantum logical formulation of Quantum theory =-=[1, 2]-=- depends in an essential way on the identification of propositions with projection operators on a complex Hilbert space. A non-classical, nonBoolean logical structure is effectively induced which has ... |

5 | Orthologic and Quantum Logic - Rawling, Selesnick - 2000 |

5 |
Two Applications of Logic to
- Takeuti
- 1978
(Show Context)
Citation Context ...nt work, regarding the specification of a quantum truth values object, although not based on category theory methods, seems to be the approach to the foundations of quantum logic by Takeuti and Davis =-=[17, 18]-=-, according to whom, quantization of a proposition of classical physics is equivalent to interpreting it in a Boolean extension of a set theoretical universe, where B is a complete Boolean algebra of ... |

5 |
A Relativity Principle in Quantum Mechanics
- Davis
- 1977
(Show Context)
Citation Context ...nt work, regarding the specification of a quantum truth values object, although not based on category theory methods, seems to be the approach to the foundations of quantum logic by Takeuti and Davis =-=[17, 18]-=-, according to whom, quantization of a proposition of classical physics is equivalent to interpreting it in a Boolean extension of a set theoretical universe, where B is a complete Boolean algebra of ... |

3 |
Quantum Event Structures from the perspective of Grothendieck Topoi
- Zafiris
- 2004
(Show Context)
Citation Context ... this paper we will avoid to mention Grothendieck topologies explicitly in order to avoid unnecessary technical complications in the exposition of the arguments, but the interested reader may consult =-=[10]-=- for details. The category of sheaves is a topos, and consequently, comes naturally equipped with an object of generalized truth values, called subobject classifier. This object of truth values, being... |