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Quantum observables algebras and abstract differential geometry: the topos-theoretic dynamics of diagrams of commutative algebraic localizations. (English) Zbl 1113.81071
Summary: 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), à 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.
MSC:
81R15Operator algebra methods (quantum theory)
18F20Categorical methods in sheaf theory
18D30Fibered categories
14F05Sheaves, derived categories of sheaves, etc.
53B50Applications of local differential geometry to physics
58A03Topos-theoretic approach to differentiable manifolds
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