Boolean Localization Of Quantum Events

Elias Zafiris

Shmuel Fishman

Elias Zafiris

Homologous operational localization processes are effectuated in terms of generalized topological covering systems on structures of physical events. We study localization systems of quantum events' structures by means of Gtothendieck topologies on the base category of Boolean events' algebras. We show that a quantum events algebra is represented by means of a Grothendieck sheaf-theoretic fibred structure, with respect to the global partial order of quantum events' fibres over the base category of local Boolean frames.

Elias Zafiris

We construct a sheaf-theoretic representation of quantum events structures, in terms of Boolean localization systems. These covering systems are constructed as ideals of structure-preserving morphisms of quantum event algebras from varying Boolean domains, identified with physical contexts of measurement. The modeling sheaf-theoretic 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 also 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 events structures. We construct the object of quantum truth values explicitly, and furthermore, demonstrate its functioning for the valuation of propositions in a typical quantum measurement situation.

Elias Zafiris

An observable effects a schematization of the Quantum event structure by correlating Boolean algebras picked by measurements with the Borel algebra of the real line. In a well-defined sense Boolean observables play the role of coordinatizing objects in the Quantum world, by picking Boolean figures and subsequently opening Boolean windows for the perception of the latter, interpreted as local measurement charts. A mathematical scheme for the implementation of this thesis is being proposed based on Category theoretical methods. The scheme leads to a manifold representation of Quantum structure in terms of topos-theoretical Boolean reference frames. The coordinatizing objects give rise to structure preserving maps with the modeling objects as their domains, effecting finally an isomorphism between quantum event algebra objects and Boolean localization systems for the masurement of observables.

Dick J. Hoekzema