## A Topos for Algebraic Quantum Theory (2009)

Venue: | COMMUNICATIONS IN MATHEMATICAL PHYSICS |

Citations: | 14 - 4 self |

### BibTeX

@MISC{Heunen09atopos,

author = {Chris Heunen and Nicolaas P. Landsman and Bas Spitters},

title = { A Topos for Algebraic Quantum Theory},

year = {2009}

}

### OpenURL

### Abstract

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 noncommutative 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 A. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum �(A) in T (A), which in our approach plays the role of the 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 �, and self-adjoint elements of A define continuous functions (more precisely, locale maps) from � to Scott’s interval domain. Noting that open subsets of �(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). These results were inspired by the topos-theoretic approach to quantum physics proposed by Butterfield and Isham, as recently generalized by Döring and Isham.

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Citation Context ... turn, every compact group G yields a C*-algebra C ∗ (G), its 27so-called group C*-algebra. It seems likely that there is a relationship between classical types (i.e. the so-called classical objects =-=[16]-=-) in C, and the finite-dimensional ‘classical snapshots of reality’ in C(C ∗ (G)), the closure of which is the entire C(C ∗ (G)). • Even if p, q ∈ Asa do not commute, Proposition 18 still allows one t... |

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Citation Context ...ing more observables, and hence more information. Let us comment briefly on the internal logic of a topos in the context of our running example. For more information, we must refer the reader to e.g. =-=[43, 38, 5]-=-. The internal logic of a topos assigns a truth value to every proposition of higher-order, many-sorted logic. However, unlike in the topos Set, these truth values can be more general than the Boolean... |

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Citation Context ... in the category of sets (as defined above) to frames in more general topoi. This is indeed possible, as all of the above concepts can be defined in any topos by using its internal language [63]; see =-=[12]-=- for details. In particular, in a topos T one may consider the category FrmT of internal frames and its opposite category LocT of internal locales. The terminal object of the latter is the locale ∗ wh... |

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Citation Context ... topos Set (O(M),⊆) of Setvalued functors on Minkowski space. Presenting an AQFT as a covariant functor of C*-algebras, even in the more complicated case of general relativity, has been emphasised in =-=[7]-=-. (It has also been tried to extend such results to sheaves [32, 49].) One could thus consider another intermediate level between the ambient topos and T (A). This is one way of making the similarity ... |

19 | T.: Entailment relations and distributive lattices
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Citation Context ...ded operators on H. Then the locale Σ in T (A) has no points. The Kochen-Specker theorem holds for a more general C*-algebras then just the collection of all bounded operators on a Hilbert space, see =-=[25]-=- for results on von Neumann algebras. For C*-algebras one has [34]: a simple infinite unital C*-algebra does not admit a dispersion-free quasi-state. The previous theorem also holds for such extension... |

19 |
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Citation Context ...lized Riesz-Markov Theorem, then, is as follows. 55 54 Equivalently, µ satisfies additivity as well as µ(⊥) = 0, and is Scott continuous [67]. Probability valuations extend uniquely to Borel measures =-=[5]-=-. 55 Despite this theorem, from a constructive point of view there is a crucial difference between valuations and integrals. The integral I(f) of a function f ∈ C(X) is a Dedekind real, so that it can... |

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Citation Context ...iates a certain internal locale to a noncommutative C*-algebra (assumed unital), and hinges on three ideas:1 INTRODUCTION 10 1. Algebraic quantum theory [38, 43, 58]; 2. Constructive Gelfand duality =-=[5, 6, 7, 26, 28]-=-; 3. Bohr’s doctrine of classical concepts [11, 73, 60]. From the first, we just adopt the methodology of describing a quantum system by a noncommutative C*-algebra A (defined in the usual topos Sets)... |

15 |
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14 |
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Citation Context ...pure states fail to define truth assignments in the usual binary sense (i.e. true or false) renders the entire notion of truth in quantum mechanics obscure and calls for a complete reanalysis thereof =-=[33, 34, 35, 36, 37]-=-. As also probably first recognized by the same authors, such an analysis can fruitfully be attempted using topos theory, whose internal logic is indeed intuitionistic.� �� 1 INTRODUCTION 5 From our ... |