## A Topos for Algebraic Quantum Theory (2009)

Venue: | COMMUNICATIONS IN MATHEMATICAL PHYSICS |

Citations: | 11 - 3 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.

### Citations

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Citation Context ...resenting 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 between quantum theory and relativity theory in [23] more precise; s... |

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Citation Context ...space’, will be met by a combination of two profound notions of generalized space that have been around for some time.1 INTRODUCTION 6 1. First, let us recall the strategy of noncommutative geometry =-=[24, 25]-=-. One starts with the replacement of a compact topological space X by the associated algebra of complex-valued continuous functions C(X, C). If X fails to be Hausdorff, this step loses information, bu... |

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Citation Context ... ∧Dq−a = ⊥, and hence Da−q = Da−q ∧ ⊤ = Da−q ∧ (Db ∨Dq−a) = Da−q ∧Db � Db = ∨ U0. 31 Alternatively, writing Da ⊳0 U iff U ⊇ Da, the covering relation ⊳ is inductively generated by ⊳0, as explained in =-=[27, 79]-=-. The triple (LA, �,⊳0) is a flat site as defined in [79]. ։ □A GENERATING LATTICES FOR FRAMES 43 Thus we have two alternative expressions for the spectrum: O(Σ) ∼ = {U ∈ Idl(LA) | ∀q>0Da−q ∈ U ⇒ Da ... |

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(Show Context)
Citation Context ... π1 ◦ pt = id are just (unconstrained) locale maps X → [T]. 1.6 Observation and approximation Our construction of the locale map δ(a) : Σ → IR in Section 1.4 involves the so-called interval domain IR =-=[63]-=-. To motivate its definition, consider the approximation of real numbers by nested intervals with endpoints in Q. For example, the real number π can be described by specifying the sequence [3, 4], [3.... |

30 |
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(Show Context)
Citation Context ...hr’s doctrine mathematically precise. We will naturally arrive at the notion of a quasi-state that causes the Mackey-Gleason problem [8], and contribute to the programme of Isham and co-workers (like =-=[35, 11, 12, 13, 14, 26, 27, 28, 29]-=-) to address the related interpretational problems posed by the Kochen-Specker theorem. Toposes The technical tool enabling the construction in this article is a topos. Originally invented by Grothend... |

27 |
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(Show Context)
Citation Context ... (28) (r, ∞) ↦→ Da−r. (29) As â −1 is a frame map, for bounded open intervals (r,s) we therefore obtain 5 â −1 : (r,s) ↦→ Ds−a ∧Da−r. (30) We now recall an explicit construction of the Dedekind reals =-=[39]-=-, [51, D4.7.4 & D4.7.5]. Define the propositional geometric theory TR generated by formal symbols (p,q) ∈ Q × Q with p < q, ordered as (p,q) � (p ′ ,q ′ ) iff p ′ � p and q � q ′ , subject to the foll... |

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Citation Context ...V (where U∧V = {u∧v | u ∈ U, v ∈ V }). For example, if (L, ≤) = (O(X), ⊆) one may take x ⊳ U iff x = ∨ U, i.e. if U covers x. Alternatively, one may define x ⊳f U iff U is a finite cover of x. 70 See =-=[61]-=- for an overview, and [58] for the connection with domain theory. Ref. [3] shows that formal topology is valid in constructive set theory, and hence in particular in topos theory. 71 A coverage on a m... |

24 | Topoi: the Categorical Analysis of Logic - Goldblatt - 2006 |

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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... |

23 | About Stone’s notion of spectrum
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(Show Context)
Citation Context ...ation theorem, which characterises them completely as algebras of complex-valued continuous functions on a compact Hausdorff space. It can be shown that the essence of this theorem holds in any topos =-=[4, 19]-=-, a fact we recollect in Theorem 4 below. The paraphrase leading up to it is somewhat verbose, because the specifics are of importance in Section 4. It follows a presentation that avoids the Banach-Ma... |

23 |
The Logical Analysis of Quantum Mechanics
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Citation Context ...mmutative C*-algebra (assumed unital), and hinges on three ideas: 1. Algebraic quantum theory [34, 39, 50]; 2. Constructive Gelfand duality [7, 8, 9, 23, 24]; 3. Bohr’s doctrine of classical concepts =-=[13, 62, 51]-=-. 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). As to the second, it turns out that the notion of a C... |

21 |
Toposes & Local Set Theories: an Introduction
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(Show Context)
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... |

21 |
Riesz spaces
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(Show Context)
Citation Context ... the integral can be extended to this new space. This ‘embedding’ identifies x, y when |x − y| ≤ 1 n for all n ∈ N. It is thus an embedding when the ordered vector space is an Archimedean Riesz space =-=[42, 56]-=-. This, however, can not be defined geometrically. The situation is thus similar to that of seminorms on pre-C*algebras. In a sense, C(X) with a positive linear functional is also the only example. Al... |

20 |
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(Show Context)
Citation Context ...ze in this subsection. This construction associates 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 noncommuta... |

20 | Open locales and exponentiation - Johnstone - 1984 |

18 |
Aspects of general topology in constructive set theory
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(Show Context)
Citation Context ...tivate its definition, consider the approximation of real1 INTRODUCTION 16 numbers by nested intervals with endpoints in Q. For example, the real number π can be described by specifying the sequence =-=[3,4]-=-,[3.1,3.2],[3.14,3.15], [3.141,3.142],... Each individual interval may be interpreted as finitary information about the real number under scrutiny, involving the single observation that the real numbe... |

18 |
C.J.: A globalisation of the Gelfand duality theorem
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(Show Context)
Citation Context ...ation theorem, which characterises them completely as algebras of complex-valued continuous functions on a compact Hausdorff space. It can be shown that the essence of this theorem holds in any topos =-=[4, 19]-=-, a fact we recollect in Theorem 4 below. The paraphrase leading up to it is somewhat verbose, because the specifics are of importance in Section 4. It follows a presentation that avoids the Banach-Ma... |

18 | Entailment relations and distributive lattices
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(Show Context)
Citation Context ...al spaces ‘without points’. A compact regular locale can be conveniently represented by a normal distributive lattice [18], and in turn such lattices can be represented by normal entailment relations =-=[15]-=-. In topological (or better localic) terms, a lattice presents a basis while an entailment presents a subbasis. In logical terms, these entailment relations provide a propositional geometric theory an... |

18 |
Metric spaces, generalized logic, and closed
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(Show Context)
Citation Context ...nto a norm, and then complete this normed space. Both of these constructions are, however, not geometric and thus better avoided. Fortunately, there is a satisfactory solution: the localic completion =-=[41, 54]-=-. The localic completion assigns to every generalised metric space X a locale, the points of which are those of the completion of X. A C*-algebra is the localic completion of a pre-C*-algebra. Every C... |

18 |
Generalized real numbers in constructive mathematics
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(Show Context)
Citation Context ...A). To obtain a correspondence as in the previous lemma, we then consider cuts of Q instead of Dedekind cuts. This so-called interval domain [50] is often seen as a generalisation of the real numbers =-=[48]-=-. It is generated by rational intervals, ordered by reverse inclusion: a smaller interval means that we have more information (about the number that the ever smaller intervals converge to). However, b... |

17 |
The generally covariant locality principle—a new paradigm for local quantum field theory
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(Show Context)
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 ... |

17 | Kochen–Specker theorem for Von Neumann algebras
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(Show Context)
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... |

17 |
the Categorial Analysis of Logic
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(Show Context)
Citation Context ...spect of topos theory is that it unifies two seemingly wholly distinct mathematical subjects: on the one hand, topology and algebraic geometry and on the other hand, logic and set theory. We refer to =-=[41, 63, 50, 51]-=- for accounts of topos theory; see also [9, 65, 57] for historical details. Briefly, a topos is a category in which one can essentially reason as in the category Sets of all sets (with functions as ar... |

16 |
Handbook of Categorical Algebra 3: Categories of Sheaves. Encyclopedia of Mathematics and its Applications 52
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(Show Context)
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... |

14 |
2000): The spectral theory of commutative C*-algebras: the constructive spectrum
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(Show Context)
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)... |

14 | Formal topology and constructive mathematics: the Gelfand and StoneYosida representation theorems - Coquand, Spitters - 1932 |

14 | Between classical and quantum - Landsman - 2007 |

14 | The probabilistic powerdomain for stably compact spaces
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(Show Context)
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... |

13 |
Continuous domains as formal spaces
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(Show Context)
Citation Context ...U, v ∈ V }). For example, if (L, ≤) = (O(X), ⊆) one may take x ⊳ U iff x = ∨ U, i.e. if U covers x. Alternatively, one may define x ⊳f U iff U is a finite cover of x. 70 See [61] for an overview, and =-=[58]-=- for the connection with domain theory. Ref. [3] shows that formal topology is valid in constructive set theory, and hence in particular in topos theory. 71 A coverage on a meet semilattice L is a fun... |

12 | C.J.: A topos foundation for theories of physics: II. Daseinisation and the liberation of quantum theory
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(Show Context)
Citation Context ...hr’s doctrine mathematically precise. We will naturally arrive at the notion of a quasi-state that causes the Mackey-Gleason problem [8], and contribute to the programme of Isham and co-workers (like =-=[35, 11, 12, 13, 14, 26, 27, 28, 29]-=-) to address the related interpretational problems posed by the Kochen-Specker theorem. Toposes The technical tool enabling the construction in this article is a topos. Originally invented by Grothend... |

12 |
An Extension of the Galois Theory of Grothendieck, volume 309 of Memoirs
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(Show Context)
Citation Context ...ales in more general topoi. Our quantum phase spaces, then, will be examples of locales in topoi; in this respect, it is reassuring that topos theorists simply refer to ‘internal’ locales as ‘spaces’ =-=[44, 45, 55]-=-. Returning to the opening words from Mac Lane and Moerdijk just quoted, one might say that the unification in question is exemplified by the idea of an internal locale with its associated Heyting alg... |