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Alexey Kryukov [12]Alexey A. Kryukov [1]
  1. Alexey Kryukov, Geometric Derivation of Quantum Uncertainty.
    Quantum observables can be identified with vector fields on the sphere of normalized states. Consequently, the uncertainty relations for quantum observables become geometric statements. In the Letter the familiar uncertainty relation follows from the following stronger statement: Of all parallelograms with given sides the rectangle has the largest area.
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  2. Alexey Kryukov (2007). On the Measurement Problem for a Two-Level Quantum System. 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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  3. Alexey Kryukov, The Double-Slit Experiment: A Paradox-Free Kinematic Description.
    The paradoxes of the double-slit experiment with an electron are shown to originate in the implicit assumption that the electron is always located in the classical space. It is demonstrated that there exists a natural substitute for this assumption that provides a method of resolving the paradoxes.
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  4. Alexey Kryukov, The EPR Experiment: A Paradox-Free Definition of Reality.
    The paradoxes of the EPR experiment with two particles are shown to originate in the implicit assumption that the particles are always located in the classical space. There exists a substitute for this assumption that yields a new definition of reality and offers a resolution of the paradoxes.
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  5. Alexey A. Kryukov (2006). Quantum Mechanics on Hilbert Manifolds: The Principle of Functional Relativity. [REVIEW] 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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  6. Alexey Kryukov (2004). On the Problem of Emergence of Classical Space—Time: The Quantum-Mechanical Approach. 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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  7. Malcolm R. Forster & Alexey Kryukov (2003). The Emergence of the Macroworld: A Study of Intertheory Relations in Classical and Quantum Mechanics. Philosophy of Science 70 (5):1039-1051.
    Classical mechanics is empirically successful because the probabilistic mean values of quantum mechanical observables follow the classical equations of motion to a good approximation (Messiah 1970, 215). We examine this claim for the one‐dimensional motion of a particle in a box, and extend the idea by deriving a special case of the ideal gas law in terms of the mean value of a generalized force used to define “pressure.” The examples illustrate the importance of probabilistic averaging as a method of (...)
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  8. Alexey Kryukov (2003). Coordinate Formalism on Abstract Hilbert Space: Kinematics of a Quantum Measurement. [REVIEW] 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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  9. Alexey Kryukov, Coordinate Formalism on Hilbert Manifolds: String Bases of Eigenvectors.
    Coordinate formalism on Hilbert manifolds developed in \cite{Kryukov}, \cite{Kryukov1} is further analyzed. The main subject here is a comparison of the ordinary and the string bases of eigenvectors of a linear operator as introduced in \cite{Kryukov}. It is shown that the string basis of eigenvectors is a natural generalization of its classical counterpart. It is also shown that the developed formalism forces us to consider any Hermitian operator with continuous spectrum as a restriction to a space of square integrable functions (...)
     
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  10. Alexey Kryukov, Coordinate Formalism on Hilbert Manifolds.
    Infinite-dimensional manifolds modelled on arbitrary Hilbert spaces of functions are considered. It is shown that changes in model rather than changes of charts within the same model make coordinate formalisms on finite and infinite-dimensional manifolds deeply similar. In this context the infinite-dimensional counterparts of simple notions such as basis, dual basis, orthogonal basis, etc. are shown to be closely related to the choice of a model. It is also shown that in this formalism a single tensor equation on an infinite-dimensional (...)
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  11. Alexey Kryukov, Coordinate Formalism on Abstract Hilbert Space.
    Coordinate formalism on Hilbert manifolds developed in \cite{Kryukov} is reviewed. The results of \cite{Kryukov} are applied to the simpliest case of a Hilbert manifold: the abstract Hilbert space. In particular, functional transformations preserving properties of various linear operators on Hilbert spaces are found. Any generalized solution of an arbitrary linear differential equation with constant coefficients is shown to be related to a regular solution by a (functional) coordinate transformation. The results also suggest a way of using generalized functions to solve (...)
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  12. Alexey Kryukov, Conformal Transformations of Space-Time as Vector Bundle Automorphisms.
    Conformal group of Minkowski space-time M is considered as a group of bundle automorphisms of a vector bundle U over M. 4-component spin-vectors (4-spinors) are sections of a subbundle of the tangent bundle over U. Isotropic 4-vectors are images of 4-spinors under projection. This leads to a particularly clear interpretation of the spin properties of Nature.
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  13. Alexey Kryukov, Nine Theorems on the Unification of Quantum Mechanics and Relativity.
    A mathematical framework that unifies the standard formalisms of special relativity and quantum mechanics is proposed. For this a Hilbert space H of functions of four variables x,t furnished with an additional indefinite inner product invariant under Poincare transformations is introduced. For a class of functions in H that are well localized in the time variable the usual formalism of non-relativistic quantum mechanics is derived. In particular, the interference in time for these functions is suppressed; a motion in H becomes (...)
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