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Boolean coverings of quantum observable structure: a setting for an abstract differential geometric mechanism. (English) Zbl 1068.18013

The main guiding idea of the author is based on the employment of objects belonging to the Boolean species of observable structure, as covers, for the understanding of the objects belonging to the quantum species of observable structure. The language of category theory proves to be suitable for implementation of this idea in a universal way. The conceptual essence of this scheme is the development of a sheaf theoretical perspective on quantum observable structures. The physical interpretation of the categorical framework make use of the analogy with geometric manifold theory. Namely, it is associated with the development of a Boolean manifold picture that takes place trough the identification of Boolean charts in systems of localization for quantum event algebras with reference frames, relative to which the results of measurements can be coordinatized. In this sense, any Boolean chart in a localization system covering a quantum algebra of events, corresponds to a set of Boolean events which become realizable in the experimental context of a measurement situation. This identification amounts to the introduction of a relativity principle in quantum theory, suggesting a contextual interpretation of its descriptive apparatus.

In quantum logical approaches the notion of event, associated with the measurement of an observable is taken to be equivalent to a proposition describing the behavior of a physical system. This formulation of quantum theory is based on the identification of propositions with projection opertors on a complex Hilbert space. In this sense, the Hilbert space formalism of quantum theory associates events with closed subspaces of a separable, complex Hilbert space corresponding to a quantum system. Then, the quantum event algebra is identified with the lattice of closed subspaces of the Hilbert space, ordered by inclusion and carrying an orthocomplementation opertion which is given by the orthogonal complements of the closed subspaces. Equivalently, it is isomorphic to the partial Boolean algebra of closed subspaces of the Hilbert space of the system, or alternatively the partial Boolean algebra of projection operators of the system. The author argues that the set theoretical axiomatizations of quantum observable structures hides the intrinsic significance of Boolean localizing systems in the formation of these structures. The main thesis of the paper is that the objectification of a quantum observable structure takes place through Boolean reference frames that can be pasted together using category theoretical means.

Contextual topos theoretical approaches to quantum structures have been considered, from a different viewpoint by C. J. Isham and J. Butterfield [“Topos perspective on the Kochen-Specker theorem. I: Quantum states as generalized valuations”, Int. J. Theor. Phys. 37, 2669–2733 (1998; Zbl 0979.81018); “Some possible roles for topos theory in quantum theory and quantum gravity”, Found. Phys. 30, 1707–1835 (2000)], and discussed by J. P. Rawling and S. A. Selesnick [“Orthologic and quantum logic: models and computational elements”, J. ACM 47, 721–751 (2000)] and I. Raptis [“Presheaves, sheaves and their topoi in quantum gravity and quantum logic”, arXiv.org/abs/gr-qc/0110064].

The author defines event and observable structures in a category theoretical language, introduces the functorial concepts of Boolean coordinatizations and Boolean presheaves, develops the idea of fibrations over Boolean observables. He proves the existence of an adjunction between the topos of presheaves of Boolean observables and the category of quantum observables. Then (section 5) the author defines systems of localization for measurement of observables over a quantum event algebra and talks about isomorphic representations of quantum algebras in terms of Boolean localization systems using the adjunction established. In section 7 he examines the consequences of the scheme related to the interpretation of the logic of quantum propositions and in section 8 he discusses the implications of covering systems in relation to the possibility of development of a differential geometric machinery suitable for the quantum regime. Finally, there exists a section with conclusions.

Reviewer: Ioan Pop (Iaşi)
MSC:
18F05Local categories and functors
18F20Categorical methods in sheaf theory
18D30Fibered categories
14F05Sheaves, derived categories of sheaves, etc.
53B50Applications of local differential geometry to physics
81P10Logical foundations of quantum mechanics; quantum logic