A topos for algebraic quantum theory

The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr's idea that the empirical content of quantum physics is accessible only through classical physics, we show how a C*-algebra of observables A induces a topos T(A) in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum S(A) in T(A), which in our approach plays the role of a quantum phase space of the system. Thus we associate a locale (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which is the noncommutative notion of a space). In this setting, states on A become probability measures (more precisely, valuations) on S(A), and self-adjoint elements of A define continuous functions (more precisely, locale maps) from S(A) to Scott's interval domain. Noting that open subsets of S(A) correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by A is essentially turned into a classical theory, internal to the topos T(A).Comment: 52 pages, final version, to appear in Communications in Mathematical Physic

Working Document on Gloss Ontology

This document describes the Gloss Ontology. The ontology and associated class model are organised into several packages. Section 2 describes each package in detail, while Section 3 contains a summary of the whole ontology

Integrals and Valuations

We construct a homeomorphism between the compact regular locale of integrals on a Riesz space and the locale of (valuations) on its spectrum. In fact, we construct two geometric theories and show that they are biinterpretable. The constructions are elementary and tightly connected to the Riesz space structure.Comment: Submitted for publication 15/05/0

Bohrification

New foundations for quantum logic and quantum spaces are constructed by merging algebraic quantum theory and topos theory. Interpreting Bohr's "doctrine of classical concepts" mathematically, given a quantum theory described by a noncommutative C*-algebra A, we construct a topos T(A), which contains the "Bohrification" B of A as an internal commutative C*-algebra. Then B has a spectrum, a locale internal to T(A), the external description S(A) of which we interpret as the "Bohrified" phase space of the physical system. As in classical physics, the open subsets of S(A) correspond to (atomic) propositions, so that the "Bohrified" quantum logic of A is given by the Heyting algebra structure of S(A). The key difference between this logic and its classical counterpart is that the former does not satisfy the law of the excluded middle, and hence is intuitionistic. When A contains sufficiently many projections (e.g. when A is a von Neumann algebra, or, more generally, a Rickart C*-algebra), the intuitionistic quantum logic S(A) of A may also be compared with the traditional quantum logic, i.e. the orthomodular lattice of projections in A. This time, the main difference is that the former is distributive (even when A is noncommutative), while the latter is not. This chapter is a streamlined synthesis of 0709.4364, 0902.3201, 0905.2275.Comment: 44 pages; a chapter of the first author's PhD thesis, to appear in "Deep Beauty" (ed. H. Halvorson

On the representation theory of Galois and Atomic Topoi

We elaborate on the representation theorems of topoi as topoi of discrete actions of various kinds of localic groups and groupoids. We introduce the concept of "proessential point" and use it to give a new characterization of pointed Galois topoi. We establish a hierarchy of connected topoi: [1. essentially pointed Atomic = locally simply connected], [2. proessentially pointed Atomic = pointed Galois], [3. pointed Atomic]. These topoi are the classifying topos of, respectively: 1. discrete groups, 2. prodiscrete localic groups, and 3. general localic groups. We analyze also the unpoited version, and show that for a Galois topos, may be pointless, the corresponding groupoid can also be considered, in a sense, the groupoid of "points". In the unpointed theories, these topoi classify, respectively: 1. connected discrete groupoids, 2. connected (may be pointless) prodiscrete localic groupoids, and 3. connected groupoids with discrete space of objects and general localic spaces of hom-sets, when the topos has points (we do not know the class of localic groupoids that correspond to pointless connected atomic topoi). We comment and develop on Grothendieck's galois theory and its generalization by Joyal-Tierney, and work by other authors on these theories.Comment: This is a revised version of arXiv.org/math.CT/02008222 to appear in JPA