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Interpreting observables in a quantum world from the categorial standpoint. (English) Zbl 1065.81011

The paper starts by discussing observables in quantum and classical theories, pointing out that in either theory an observable is assigned real values and can therefore be represented as a homomorphic mapping from the Borel sets of the real line to the event structure of the theory. In this sense the real line and it’s Borel structure serve as a “modelling object” for the event algebra of the observed system, which in the quantum case is the lattice of closed subspaces of Hilbert Space (p. 266). The Kochen-Specker theorem is understood as showing that a quantum mechanical system cannot be “comprehended” with a single “Boolean device” and so “Boolean observables play the role of coordinatizng objects in an attempt to probe the quantum world.” (p. 266) It is in this sense of dependence on the Boolean observable, that the author calls his approach “relativistic”. The stated purpose of this paper is to provide a “mathematical scheme” for this approach based on category-theoretical methods. (Ibid) “The Boolean event algebras modelling objects, being formed by observational procedures, give rise to Boolean localization systems, which, in turn, provide structure preserving maps from the domain of variable Boolean probing objects to quantum algebras of events”. (p. 267)

As promised the paper is a work in category theory, developing a framework for these ideas. After the introduction in section 1, section 2 introduces categories associated with observable structures. Section 3 constructs Boolean shaping and Boolean presheaf observable functors, with fibrations over Boolean observables. Section 4 concerns the existence of an adjunction between the topos of presheaves of Boolean observables and the category of quantum observables. Section 5 shows the adjunctive correspondence is based on a tensor product construction. Section 6 introduces systems of localization for measurement of observables over quantum event algebras, and section 7 establishes the representation of these algebras as manifolds of Boolean measurement systems. Section 8 very briefly considers semantics, and section 9 even more briefly touches on truth-values, promising future work that may provide “the basis for a natural interpretation of the logic of quantum propositions in terms of fuzzy Boolean truth values.” (p. 293) Section 10 argues that Abstract Differential Geometry, an extension of Differential Geometry provides a suitable framework for analysing the sheaves of algebras that correspond to the quantum observables. Lastly section 11 provides less than a page summing up the paper, indicating future research and expressing the hope it could substantiate the dictum that ”Algebraic Quantum Geometry = Geometric Quantum Logic”. (p. 297)

MSC:
81P10Logical foundations of quantum mechanics; quantum logic
03G12Quantum logic
03G30Categorical logic, topoi
18C10Theories, structure, and semantics
18F20Categorical methods in sheaf theory
58A03Topos-theoretic approach to differentiable manifolds
03B52Fuzzy logic; logic of vagueness