Bohrification §

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 ‘Bohrification’ 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 Σ(A), which is a locale internal to the topos [C(A),Set]. We interpret its external description ΣA (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 ΣA correspond to (atomic) propositions, so that the ‘Bohrified’ quantum logic of A is given by the Heyting algebra structure of ΣA. 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 ΣA 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 difference is that ΣA 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]. Radboud Universiteit Nijmegen, Institute for Mathematics, Astrophysics, and Particle Physics, Heyendaalseweg 135, 6525 AJ NIJMEGEN, THE NETHERLANDS. Radboud Universiteit Nijmegen, Institute for Computing and Information Sciences, Heyendaalseweg 135, 6525 AJ NIJMEGEN, THE NETHERLANDS. Current address: Wolfson Building, Parks Road, OXFORD OX1 3QD, UNITED KINGDOM. Eindhoven University of Technology, Department of Mathematics and Computer Science, P.O. Box 513, 5600 MB EINDHOVEN, THE NETHERLANDS. To appear in Deep Beauty, ed. H. Halvorson (Cambridge University Press, 2010).