We construct a sheaf-theoretic representation of quantum observables algebras over a base category equipped with a Grothendieck topology, consisting of epimorphic families of commutative observables algebras, playing the role of local arithmetics in measurement situations. This construction makes possible the adaptation of the methodology of Abstract Differential Geometry (ADG), a la Mallios, in a topos-theoretic environment, and hence, the extension of the "mechanism of differentials" in the quantum regime. The process of gluing information, within diagrams of commutative algebraic localizations, generates dynamics, involving the transition from the classical to the quantum regime, formulated cohomologically in terms of a functorial quantum connection, and subsequently, detected via the associated curvature of that connection.We construct a sheaf-theoretic representation of quantum observables algebras over a base category equipped with a Grothendieck topology, consisting of epimorphic families of commutative observables algebras, playing the role of local arithmetics in measurement situations. This construction makes possible the adaptation of the methodology of Abstract Differential Geometry (ADG), a la Mallios, in a topos-theoretic environment, and hence, the extension of the "mechanism of differentials" in the quantum regime. The process of gluing information, within diagrams of commutative algebraic localizations, generates dynamics, involving the transition from the classical to the quantum regime, formulated cohomologically in terms of a functorial quantum connection, and subsequently, detected via the associated curvature of that connection.

arXiv:gr-qc/0405009May 2, 200418 citations

A Topos for Algebraic Quantum Theory

Chris Heunen, Nicolaas P. Landsman, Spitters, BAW Bas

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Differential Geometry of Quantum States, Observables and Evolution

Florio M. Ciaglia, Alberto Ibort, Giuseppe Marmo

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Chapter 19 Dynamical Deformation of Algebras of Observables

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A Topos for Algebraic Quantum Theory

Chris Heunen, Nicolaas P. Landsman, Bas Spitters

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 ${\mathcal{T}(A)}$ in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra ${\underline{A}}$ . According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum ${\underline{\Sigma}(\underline{A})}$ in ${\mathcal{T}(A)}$ , 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 ${\underline{\Sigma}}$ , and self-adjoint elements of A define continuous functions (more precisely, locale maps) from ${\underline{\Sigma}}$ to Scott’s interval domain. Noting that open subsets of ${\underline{\Sigma}(\underline{A})}$ 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 ${\mathcal{T}(A)}$ .These results were inspired by the topos-theoretic approach to quantum physics proposed by Butterfield and Isham, as recently generalized by Döring and Isham.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 $${\mathcal{T}(A)}$$ in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra $${\underline{A}}$$ . According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum $${\underline{\Sigma}(\underline{A})}$$ in $${\mathcal{T}(A)}$$ , 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 $${\underline{\Sigma}}$$ , and self-adjoint elements of A define continuous functions (more precisely, locale maps) from $${\underline{\Sigma}}$$ to Scott’s interval domain. Noting that open subsets of $${\underline{\Sigma}(\underline{A})}$$ 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 $${\mathcal{T}(A)}$$ .These results were inspired by the topos-theoretic approach to quantum physics proposed by Butterfield and Isham, as recently generalized by Döring and Isham.

arXiv:0709.4364Communications in Mathematical PhysicsJul 20, 200982 citations

{\mathcal{T}(A)}

in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra

{\underline{A}}

. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum

{\underline{\Sigma}(\underline{A})}

{\mathcal{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

{\underline{\Sigma}}

, and self-adjoint elements of A define continuous functions (more precisely, locale maps) from

{\underline{\Sigma}}

to Scott’s interval domain. Noting that open subsets of

{\underline{\Sigma}(\underline{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

{\mathcal{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.