Topos Quantum Theory Reduced by Context-Selection Functors

TL;DR

This paper develops a method to simplify topos-based quantum theories by selecting specific contexts and integrating quantum probabilities as truth-values, resulting in more manageable and interpretable quantum frameworks.

Contribution

It introduces a functor-based context selection and constructs Grothendieck topologies to reduce and interpret topos quantum theories with probabilities as truth-values.

Findings

Reduction of presheaf-based topos quantum theory via sheafification.

Quantum probabilities represented as intuitionistic truth-values.

Construction of a quantum theory with integrated probabilities on a sheaf topos.

Abstract

In this paper, we deal with quantum theories on presheaves and sheaves on context categories consisting of commutative von Neumann algebras of bounded operators on a Hilbert space, from two viewpoints. One is to reduce presheaf-based topos quantum theory via sheafification, and the other is to import quantum probabilities to the reduced sheaf quantum theory. The first is done by means of a functor that selects some expedient contexts. It defines a Grothendieck topology on the category consisting of all contexts, hence, induces a sheaf topos on which we construct a downsized quantum theory. Also, we show that the sheaf quantum theory can be replaced by an equivalent, more manageable presheaf quantum theory. Quantum probabilities are imported by means of a Grothendieck topology that is defined on a category consisting of probabilities and enables to regard them as intuitionistic truth-values. From these topologies, we construct another Grothendieck topology that is defined on the product of the context category and the probability category and reflects the selection of contexts and the identification of probabilities with truth-values. We construct a quantum theory equipped with quantum probabilities as truth-values on the sheaf topos induced by the Grothendieck topology.