Titlebook: Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models; Andrei Khrennikov Book 1997 Kluwer Academic Publishe

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Measurements and Numbers,Since ancient times a mathematical apparatus has played a very important role in the process of the human investigation of physical reality. In our book we try to understand this role in different physical theories, especially in quantum mechanics.

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Non-Kolmogorov Probability and Quantum Physics,We have already seen that the EPR definition of elements of reality is based on two purely mathematical elements: real numbers and real probabilities. As the Kolmogorov probability theory is the main mathematical model for describing real probabilities, we can speak of ..

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Non-Archimedean Analysis,apter 4, where the formalism of .-adic quantum mechanics (which describes a finite precision of measurements) will be presented. We refer to results of the present chapter in only a few places in Chapter 4. The reader can easily return to the present chapter and find the corresponding mathematical facts.

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The Ultrametric Hilbert Space Description of Quantum Measurements with Finite Precision,Chapter 1). The precision of a quantum measurement is used as a new metric on the space of quantum states. This metric is an ultrametric. Thus formally our quantum model can be considered as a representation of canonical commutation relations in ultrametric Hilbert spaces.

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Non-Kolmogorov Probability Theory,lmogorov axiomatics [174] (1933) of the modern theory of probability cannot describe all probabilistic phenomenon which are observed in nature. Hence the theory of probability based on this axiomatics is not universal probabilistic formalism. This is only a particular model, the Kolmogorov probabilistic model.

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Position and Momentum Representations,quantum field theory. Let us realize this formal expression as a differential operator with variables . belonging to the field of .-adic numbers . and study the properties of this operator in a .-adic Hilbert space.

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Mathematics and Its Applicationshttp://image.papertrans.cn/n/image/666845.jpg

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https://doi.org/10.1007/978-94-009-1483-4Hilbert space; calculus; dynamical systems; fractal; quantum mechanics; quantum physics

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on of reality: the role of a mathematical apparatus in a description of reality. We pay our main attention to the role of number systems which are used, or may be used, in this process. We shall show that the picture of reality based on the standard (since the works of Galileo and Newton) methods of