A Categorial Semantic Representation of Quantum Event Structures

TL;DR

This paper introduces a category-theoretic, topos-based framework for representing quantum event structures, enabling a logical and realist interpretation of quantum propositions through Boolean localization and truth-value objects.

Contribution

It develops a novel topos-theoretic approach to quantum logic, replacing set-theoretic methods with sheaves of Boolean frames for better semantic and truth-value modeling.

Findings

Categorical representation of quantum algebras via sheaves of Boolean frames.

Incorporation of a truth-value object for quantum propositions.

Comparison with other categorial approaches to quantum logic.

Abstract

The overwhelming majority of the attempts in exploring the problems related to quantum logical structures and their interpretation have been based on an underlying set-theoretic syntactic language. We propose a transition in the involved syntactic language to tackle these problems from the set-theoretic to the category-theoretic mode, together with a study of the consequent semantic transition in the logical interpretation of quantum event structures. In the present work, this is realized by representing categorically the global structure of a quantum algebra of events (or propositions) in terms of sheaves of local Boolean frames forming Boolean localization functors. The category of sheaves is a topos providing the possibility of applying the powerful logical classification methodology of topos theory with reference to the quantum world. In particular, we show that the topos-theoretic representation scheme of quantum event algebras by means of Boolean localization functors incorporates an object of truth values, which constitutes the appropriate tool for the definition of quantum truth-value assignments to propositions describing the behavior of quantum systems. Effectively, this scheme induces a revised realist account of truth in the quantum domain of discourse. We also include an appendix, where we compare our topos-theoretic representation scheme of quantum event algebras with other categorial and topos-theoretic approaches.