The aim of this paper is to compare the two topos-theoretic approaches to quantum mechanics that may be found in the literature to date. The first approach, which we will call the contravariant approach, was originally proposed by Isham and Butterfield, and was later extended by Doering and Isham. The second approach, which we will call the covariant approach, was developed by Heunen, Landsman and Spitters. Motivated by coarse-graining and the Kochen-Specker theorem, the contravariant approach uses the topos of presheaves on a specific context category, defined as the poset of commutative von Neumann subalgebras of some given von Neumann algebra. In particular, the approach uses the spectral presheaf. The intuitionistic logic of this approach is given by the (complete) Heyting algebra of closed open subob jects of the spectral presheaf. We show that this Heyting algebra is, in a natural way, a locale in the ambient topos, and compare this locale with the internal Gelfand spectrum of the covariant approach. In the covariant approach, a non-commutative C*-algebra (in the topos Set) defines a commutative C*-algebra internal to the topos of covariant functors from the context category to the category of sets. We give an explicit description of the internal Gelfand spectrum of this commutative C*-algebra, from which it follows that the external spectrum is spatial. Using the daseinisation of self-adjoint operators from the contravariant approach, we give a new definition of the daseinisation arrow in the covariant approach and compare it with the original version. States and state-proposition pairing in both approaches are compared. We also investigate the physical interpretation of the covariant approach.
The aim of this paper is to compare the two topos-theoretic approaches to quantum mechanics that may be found in the literature to date. The first approach, which we will call the contravariant approach, was originally proposed by Isham and Butterfield, and was later extended by Doering and Isham. The second approach, which we will call the covariant approach, was developed by Heunen, Landsman and Spitters. Motivated by coarse-graining and the Kochen-Specker theorem, the contravariant approach uses the topos of presheaves on a specific context category, defined as the poset of commutative von Neumann subalgebras of some given von Neumann algebra. In particular, the approach uses the spectral presheaf. The intuitionistic logic of this approach is given by the (complete) Heyting algebra of closed open subob jects of the spectral presheaf. We show that this Heyting algebra is, in a natural way, a locale in the ambient topos, and compare this locale with the internal Gelfand spectrum of the covariant approach. In the covariant approach, a non-commutative C*-algebra (in the topos Set) defines a commutative C*-algebra internal to the topos of covariant functors from the context category to the category of sets. We give an explicit description of the internal Gelfand spectrum of this commutative C*-algebra, from which it follows that the external spectrum is spatial. Using the daseinisation of self-adjoint operators from the contravariant approach, we give a new definition of the daseinisation arrow in the covariant approach and compare it with the original version. States and state-proposition pairing in both approaches are compared. We also investigate the physical interpretation of the covariant approach.
T he aim of this paper is to com pare th e two topos-theoretic approaches to quantum m echanics th a t may be found in th e literatu re to date.T he first approach, which we will call th e contravariant approach, was pro posed by Isham and B utterfield, and was later extended by Doering and Isham .T he second approach, which we will call th e covariant approach, was developed by H eunen, L andsm an and S pitters.M otivated by coarse-graining and th e Kochen-Specker theorem , the contravariant approach uses th e topos of presheaves on a specific context category, defined as th e poset of com m utative von N eum ann subalgebras of some given von N eum ann algebra.In p articu lar th e approach uses the spectral presheaf.T he intuitionistic logic of this approach is presented by th e (com plete) H eyting algebra of closed open subobjects of th e spectral presheaf.We dem onstrate th a t in a n atu ra l way, this H eyting algebra defines a locale, internal to th e given presheaf topos.T his locale is not regular, which is connected to undesirable properties of th e H eyting nega tion.T his in tu rn is closely connected to th e incom patibility betw een the coarse-graining m otivation and th e internal language of th e presheaf topos seen as a K ripke model.In th e covariant approach approach a non-com m utative C*-algebra (in th e topos S e t) defines a com m utative C*-algebra internal to th e topos of covariant functors from th e context category to th e category of sets.We give an explicit description of th e internal G elfand spectrum of th is com m utative C ^-algebra,1 from which it follows th a t th e external spectrum is spatial.1In th e earlier p a p e r [46], th e G elfand sp e c tru m is com puted in te rm s of form al topology.T h is a pproach yields a n explicit id en tity for th e p o in ts of th e e x te rn al d escription of th e sp ectru m .U sing different m eth o d s we will find th ese sam e p o in ts in th is paper. 1 4T h is is not am biguous, as it does not m a tte r if we ta k e p € C a n d ap p ro x im ate it in C", or if we ta k e S° (p)c' € C ' a n d ap p ro x im ate it in C " .T h e outcom e is th e sam e.5T h is is a collection tc C V(A) such th a t if C € tc , th e n C' C C a n d if C" C C' € t c , th e n C" € tc .N ote t h a t a sieve in V(A) is th e sam e as an ideal of th is poset.
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 Σ(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 Σ , and self-adjoint elements of A define continuous functions (more precisely, locale maps) from Σ to Scott’s interval domain. Noting that open subsets of Σ(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.
What is the role of topos theory in the topos models for quantum theory as used by Isham, Butterfield, Doring, Heunen, Landsman, Spitters and others? In other words, what is the interplay between physical motivation for the models and the mathematical framework used in these models? Concretely, we show that the presheaf topos model of Butterfield, Isham and Doring resembles classical physics when viewed from the internal language of the presheaf topos, similar to the copresheaf topos model of Heunen, Landsman and Spitters. Both the presheaf and copresheaf models provide a `quantum logic' in the form of a complete Heyting algebra. Although these algebras are natural from a topos theoretic stance, we seek a physical interpretation for the logical operations. Finally, we investigate dynamics. In particular we describe how an automorphism on the operator algebra induces a homeomorphism (or isomorphism of locales) on the associated state spaces of the topos models, and how elementary propositions and truth values transform under the action of this homeomorphism. Also with dynamics the focus is on the internal perspective of the topos.
A decade ago, Isham and Butterfield proposed a topos-theoretic approach to quantum mechanics, which meanwhile has been extended by Döring and Isham so as to provide a new mathematical foundation for all of physics. Last year, three of the present authors redeveloped and refined these ideas by combining the C*-algebraic approach to quantum theory with the so-called internal language of topos theory (Heunen et al. in arXiv:0709.4364 ). The goal of the present paper is to illustrate our abstract setup through the concrete example of the C*-algebra M n (ℂ) of complex n×n matrices. This leads to an explicit expression for the pointfree quantum phase space Σ n and the associated logical structure and Gelfand transform of an n-level system. We also determine the pertinent non-probabilisitic state-proposition pairing (or valuation) and give a very natural topos-theoretic reformulation of the Kochen–Specker Theorem. In our approach, the nondistributive lattice ℘(M n (ℂ)) of projections in M n (ℂ) (which forms the basis of the traditional quantum logic of Birkhoff and von Neumann) is replaced by a specific distributive lattice O(Σn) of functions from the poset C(Mn(C)) of all unital commutative C*-subalgebras C of M n (ℂ) to ℘(M n (ℂ)). The lattice O(Σn) is essentially the (pointfree) topology of the quantum phase space Σ n , and as such defines a Heyting algebra. Each element of O(Σn) corresponds to a “Bohrified” proposition, in the sense that to each classical context C∈C(Mn(C)) it associates a yes-no question (i.e. an element of the Boolean lattice ℘(C) of projections in C), rather than being a single projection as in standard quantum logic. Distributivity is recovered at the expense of the law of the excluded middle (Tertium Non Datur), whose demise is in our opinion to be welcomed, not just in intuitionistic logic in the spirit of Brouwer, but also in quantum logic in the spirit of von Neumann.
What is the role of topos theory in the topos models for quantum theory as used by Isham, Butterfield, Döring, Heunen, Landsman, Spitters, and others? In other words, what is the interplay between physical motivation for the models and the mathematical framework used in these models? Concretely, we show that the presheaf topos model of Butterfield, Isham, and Döring resembles classical physics when viewed from the internal language of the presheaf topos, similar to the copresheaf topos model of Heunen, Landsman, and Spitters. Both the presheaf and copresheaf models provide a “quantum logic” in the form of a complete Heyting algebra. Although these algebras are natural from a topos theoretic stance, we seek a physical interpretation for the logical operations. Finally, we investigate dynamics. In particular, we describe how an automorphism on the operator algebra induces a homeomorphism (or isomorphism of locales) on the associated state spaces of the topos models, and how elementary propositions and truth values transform under the action of this homeomorphism. Also with dynamics the focus is on the internal perspective of the topos.
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 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 Σ(A) in T (A), which in our approach plays the role of a 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