Observables IV: The presheaf perspective

TL;DR

This paper introduces a presheaf framework for associating observable structures to von Neumann algebras, highlighting the role of contextual observables and the limitations of global sections.

Contribution

It constructs isomorphic presheaves for von Neumann algebras and explores the relationship between global sections and observables, advancing the presheaf approach in quantum theory.

Findings

Each von Neumann algebra yields a pair of isomorphic presheaves.

Global sections correspond to contextual observables but not all arise from actual observables.

Discussion of states within the presheaf framework.

Abstract

In this fourth of our series of papers on observables we show that one can associate to each von Neumann algebra R a pair of isomorphic presheaves, the upper presheaf O^{+}_{R} and the lower presheaf O^{-}_{R}, on the category of abelian von Neumann subalgebras of R. Each $A \in R_{sa}$ induces a global section of O^{+}_{R} and of O^{-}_{R} respectively. We call them \emph {contextual observables}. But we show that, in general, not every global section of these presheaves arises in this way. Moreover, we discuss states of a von Neumann algebra in the presheaf context.