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

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.

On the Structure of Abstract H*-Algebras

“…A principal result of the topos approach is that the Kochen-Specker theorem [19] -which asserts the contextual nature of quantum theory -is equivalent to the statement that the Gelfand spectrum has no global sections [12, Corollary 9.1]. Hence studying the global sections of Spec G allows us to address a more general notion of contextuality which we develop in [11].…”

“…The Gelfand spectrum of a C * -algebra is not just a set, but a compact Hausdorff topological space. In [11] we showed that for A in A -Alg(X ) the Gelfand spectrum Spec G (A) comes naturally equipped with the structure of a compact topological space.…”

“…The present work is part of an ongoing project [10,11] to bridge the monoidal approach to quantum theory of Abramsky and Coecke [1], and the topos approach to quantum theory of Butterfield, Doering and Isham [17,9]. Both the monoidal and topos approaches to quantum theory are algebraic, in that they seek to represent some aspect of physical reality with algebraic structures.…”