### Abstract

Original language | English |
---|---|

Title of host publication | Deep beauty : understanding the quantum world through mathematical innovation |

Editors | H. Halvorson |

Place of Publication | Cambridge |

Publisher | Cambridge University Press |

Pages | 271-313 |

Number of pages | 486 |

ISBN (Print) | 978-1-107-00570-9 |

Publication status | Published - 2011 |

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### Cite this

*Deep beauty : understanding the quantum world through mathematical innovation*(pp. 271-313). Cambridge: Cambridge University Press.

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*Deep beauty : understanding the quantum world through mathematical innovation.*Cambridge University Press, Cambridge, pp. 271-313.

**Bohrification.** / Heunen, C.; Landsman, N.P.; Spitters, B.A.W.

Research output: Chapter in Book/Report/Conference proceeding › Chapter › Academic

TY - CHAP

T1 - Bohrification

AU - Heunen, C.

AU - Landsman, N.P.

AU - Spitters, B.A.W.

PY - 2011

Y1 - 2011

N2 - 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].

AB - 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].

M3 - Chapter

SN - 978-1-107-00570-9

SP - 271

EP - 313

BT - Deep beauty : understanding the quantum world through mathematical innovation

A2 - Halvorson, H.

PB - Cambridge University Press

CY - Cambridge

ER -