C. Heunen, N.P. Landsman, B.A.W. Spitters

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The aim of this chapter is to construct new foundations for quantum logic and quantum spaces. This is accomplished by merging algebraic quantum theory and topos theory (encompassing the theory of locales or frames, of which toposes in a sense form the ultimate generalization). In a nutshell, the relation between these fields is as follows. First, our mathematical interpretation of Bohr's `doctrine of classical concepts' is that the empirical content of a quantum theory described by a noncommutative (unital) C*-algebra A is contained in the family of its commutative (unital) C*-algebras, partially ordered by inclusion. Seen as a category, the ensuing poset C(A) canonically defines the topos [C(A); Set] of covariant functors from C(A) to the category Set of sets and functions. This topos contains the `Bohri??cation' A of A, defined as the tautological functor C 7! C, as an internal commutative C*-algebra. Second, according to the topos-valid Gelfand duality theorem of Banaschewski and Mulvey, A has a Gelfand spectrum S(A), which is a locale internal to the topos [C(A); Set]. We interpret its external description SA (in the sense of Joyal and Tierney), as the `Bohrified' phase space of the physical system described by A. As in classical physics, the open subsets of SA correspond to (atomic) propositions, so that the `Bohrified' quantum logic of A is given by the Heyting algebra structure of SA. 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 (as in the case where A is a von Neumann algebra, or, more generally, a Rickart C*-algebra), the intuitionistic quantum logic SA of A may also be compared with the traditional quantum logic Proj(A), i.e. the orthomodular lattice of projections in A. This time, the main di??erence is that SA is distributive (even when A is noncommutative), while Proj(A) is not. This chapter is a streamlined synthesis of our earlier papers in Comm. Math. Phys. (arXiv:0709.4364), Found Phys. (arXiv:0902.3201) and Synthese (arXiv:0905.2275). See also [51].
Original languageEnglish
Title of host publicationDeep beauty : understanding the quantum world through mathematical innovation
EditorsH. Halvorson
Place of PublicationCambridge
PublisherCambridge University Press
Number of pages486
ISBN (Print)978-1-107-00570-9
Publication statusPublished - 2011

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