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

Heunen, Chris

Landsman, Nicolaas P.

Spitters, Bas

English

arXiv

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

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Bohrification

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Heunen, C.

Landsman, N.P.

Spitters, B.A.W.

Radboud Repository

Nicolaas P.Landsman Chris Heunen

Bas Spitters

Hans Halvorson

Oxford University Research Archive (ORA)

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 non-commutative (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 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 difference is that SA is distributive (even when A is noncommutative), while Proj(A) is not

Heunen, C

Landsman, NP

Spitters, BAW Bas

Repository TU/e

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]

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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 non-commutative (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 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 difference is that SA is distributive (even when A is noncommutative), while Proj(A) is not

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