Spectral Presheaves, Kochen-Specker Contextuality, and Quantale-Valued Relations

TL;DR

This paper extends the topos approach to quantum theory by analyzing the Gelfand spectrum presheaf for categories of quantale-valued relations, providing a non-contextuality result and a topological interpretation relevant to physical theories.

Contribution

It generalizes the Gelfand spectrum in the topos framework to quantale-valued relations and establishes a non-contextuality result for these categories.

Findings

Non-contextuality result for categories of quantale-valued relations

Gelfand spectrum presheaf equipped with a natural topology

Interpretation of the spectrum's topology in physical theories

Abstract

In the topos approach to quantum theory of Doering and Isham the Kochen-Specker Theorem, which asserts the contextual nature of quantum theory, can be reformulated in terms of the global sections of a presheaf characterised by the Gelfand spectrum of a commutative C*-algebra. In previous work we showed how this topos perspective can be generalised to a class of categories typically studied within the monoidal approach to quantum theory of Abramsky and Coecke, and in particular how one can generalise the Gelfand spectrum. Here we study the Gelfand spectrum presheaf for categories of quantale-valued relations, and by considering its global sections we give a non-contextuality result for these categories. We also show that the Gelfand spectrum comes equipped with a topology which has a natural interpretation when thinking of these structures as representing physical theories.