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  1.  49
    Quantum Mechanics on Hilbert Manifolds: The Principle of Functional Relativity. [REVIEW]Alexey A. Kryukov - 2006 - Foundations of Physics 36 (2):175-226.
    Quantum mechanics is formulated as a geometric theory on a Hilbert manifold. Images of charts on the manifold are allowed to belong to arbitrary Hilbert spaces of functions including spaces of generalized functions. Tensor equations in this setting, also called functional tensor equations, describe families of functional equations on various Hilbert spaces of functions. The principle of functional relativity is introduced which states that quantum theory (QT) is indeed a functional tensor theory, i.e., it can be described by functional tensor (...)
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  2. On the Problem of Emergence of Classical Space—Time: The Quantum-Mechanical Approach.Alexey A. Kryukov - 2003 - Foundations of Physics 34 (8):1225-1248.
    The Riemannian manifold structure of the classical (i.e., Einsteinian) space-time is derived from the structure of an abstract infinite-dimensional separable Hilbert space S. For this S is first realized as a Hilbert space H of functions of abstract parameters. The space H is associated with the space of states of a macroscopic test-particle in the universe. The spatial localization of state of the particle through its interaction with the environment is associated with the selection of a submanifold M of realization (...)
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  3.  80
    On the Measurement Problem for a Two-Level Quantum System.Alexey A. Kryukov - 2007 - Foundations of Physics 37 (1):3-39.
    A geometric approach to quantum mechanics with unitary evolution and non-unitary collapse processes is developed. In this approach the Schrödinger evolution of a quantum system is a geodesic motion on the space of states of the system furnished with an appropriate Riemannian metric. The measuring device is modeled by a perturbation of the metric. The process of measurement is identified with a geodesic motion of state of the system in the perturbed metric. Under the assumption of random fluctuations of the (...)
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  4.  58
    Coordinate Formalism on Abstract Hilbert Space: Kinematics of a Quantum Measurement. [REVIEW]Alexey A. Kryukov - 2002 - Foundations of Physics 33 (3):407-443.
    Coordinate form of tensor algebra on an abstract (infinite-dimensional) Hilbert space is presented. The developed formalism permits one to naturally include the improper states in the apparatus of quantum theory. In the formalism the observables are represented by the self-adjoint extensions of Hermitian operators. The unitary operators become linear isometries. The unitary evolution and the non-unitary collapse processes are interpreted as isometric functional transformations. Several experiments are analyzed in the new context.
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  5.  5
    On the Motion of Macroscopic Bodies in Quantum Theory.Alexey A. Kryukov - unknown
    Quantum observables can be identified with vector fields on the sphere of normalized states. The resulting {\it vector representation} is used in the paper to undertake a simultaneous treatment of macroscopic and microscopic bodies in quantum mechanics. Components of the velocity and acceleration of state under Schr{\"o}dinger evolution are given a clear physical interpretation. Solutions to Schr{\"o}dinger and Newton equations are shown to be related beyond the Ehrenfest results on the motion of averages. A formula relating the normal probability distribution (...)
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