Boolean representations of physical magnitudes and locality

Roger M. Cooke, Jan Hilgevoord

Elias Zafiris

We construct a sheaf-theoretic representation of quantum probabilistic structures, in terms of covering systems of Boolean measure algebras. These systems coordinatize quantum states by means of Boolean coefficients, interpreted as Boolean localization measures. The representation is based on the existence of a pair of adjoint functors between the category of presheaves of Boolean measure algebras and the category of quantum measure algebras. The sheaf-theoretic semantic transition of quantum structures shifts their physical significance from the orthoposet axiomatization at the level of events, to the sheaf-theoretic gluing conditions at the level of Boolean localization systems.

Elias Zafiris

We propose a sheaf-theoretic framework for the representation of a quantum observable structure in terms of Boolean information sieves. The algebraic representation of a quantum observable structure in the relational local terms of sheaf theory effectuates a semantic transition from the axiomatic set-theoretic context of orthocomplemented partially ordered sets, la Birkhoff and Von Neumann, to the categorical topos-theoretic context of Boolean information sieves, la Grothendieck. The representation schema is based on the existence of a categorical adjunction, which is used as a theoretical platform for the development of a functorial formulation of information transfer, between quantum observables and Boolean localisation devices in typical quantum measurement situations. We also establish precise criteria of integrability and invariance of quantum information transfer by cohomological means.

Charles Paul Enz, Jagdish Mehra

Franco Strocchi

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.