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A Topos for Algebraic Quantum Theory
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2009
"... 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 ..."
Abstract

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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 topostheoretical 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 selfadjoint 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 topostheoretic approach to quantum physics proposed by Butterfield and Isham, as recently generalized by Döring and Isham.
On the ordered Dedekind real numbers in toposes
"... In this paper it is shown that the ordered structure of the Dedekind real numbers is effectively homogeneous in any topos with natural numbers object. This result is obtained within the framework of local set theory. ..."
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In this paper it is shown that the ordered structure of the Dedekind real numbers is effectively homogeneous in any topos with natural numbers object. This result is obtained within the framework of local set theory.
Constructions for Modeling Product Structure
"... Abstract. This paper identifies constructions needed for modeling product structure, shows which ones can be represented in OWL2 and suggests extensions for those that do not have OWL2 representations. A simplified mobile robot specification is formalized as a Knowledge Base (KB) in an extended logi ..."
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Abstract. This paper identifies constructions needed for modeling product structure, shows which ones can be represented in OWL2 and suggests extensions for those that do not have OWL2 representations. A simplified mobile robot specification is formalized as a Knowledge Base (KB) in an extended logic. A KB is constructed from a signature of types (classes), typed properties, and typed variables and operators. Modeling product structure requires part decompositions, connections between parts, data valued properties, typed operations with variables, and constraints between property values. Data valued properties represent observable properties assumed or measured regarding a product. Operations with variables are used to define constraint properties such as fact that the total product weight is the some of weights of product components. Constructors, which take arguments from the signature and have properties as values, are used to specify families of properties such as part properties. These constructions are illustrated for the mobile robot. Operators and variables are represented as type, property, and operations within type theory.