Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models

AvAndrei Y. Khrennikov

Inbunden, Engelska, 1997

Del i serien Mathematics and Its Applications

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Beskrivning

This volume can be recommended as an extensive course on p-adic mathematics, treating subjects such as a p-adic theory of probability and stochastic processes; spectral theory of operators in non-Archimedean Hilbert spaces; dynamic systems; p-adic fractal dimension, infinite-dimensional analysis and Feynman integration based on the Albeverio-Hoegh-Rohn approach; both linear and nonlinear differential and pseudo-differential equations; and complexity of random sequences and a p-adic description of chaos. The book also explores the concept of using fields of p-adic numbers and their corresponding non-Archimedean analysis, a p-adic solution of paradoxes in the foundations of quantum mechanics, and especially the famous Einstein-Podolsky-Rosen paradox to create an epistemological framework for scientific use. This work should be of interest to postgraduate students and researchers with an interest in such diverse disciplines as mathematics, physics, biology and philosophy.

Produktinformation

Utgivningsdatum:1997-09-30
Mått:156 x 234 x 26 mm
Vikt:758 g
Format:Inbunden
Språk:Engelska
Serie:Mathematics and Its Applications
Antal sidor:376
Upplaga:1997
Förlag:Kluwer Academic Publishers
ISBN:9780792348009

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Innehållsförteckning

I. Measurements and Numbers.- 1. Mathematics and Reality.- 2. Measurements and Natural Numbers.- 3. Measurements and Rational Numbers.- 4. Real Numbers: Infinite Exactness of Measurements.- 5. On the Boundary of the Real Continuum.- 6. Finite Exactness and m-adic Numbers.- 7. Rings of m-adic Numbers.- 8. Ultrametric Spaces.- 9. Ultrametric Social Space.- 10. Non-Real Models of Space.- II. Fundamentals.- 1. Einstein-Podolsky-Rosen Paradox.- 2. Foundations of Quantum Mechanics.- 3. Foundations of Probability Theory.- 4. Statistical Interpretation of Quantum mechanics.- 5. Quantum Probabilities; Two Slit Experiment.- 6. Bell’s Inequality and the Death of Reality.- 7. Individual Realists Interpretation and Hidden Variables.- 8. Orthodox Copenhagen Interpretation.- 9. Einstein-Podolsky-Rosen Paradox and Interpretations of Quantum Mechanics.- III. Non-Archimedean Analysis.- 1. Exponential Function.- 2. Normed and Locally Convex Spaces.- 3. Locally Constant Functions.- 4. Kaplansky’s Theorem.- 5. Differentiate Functions.- 6. Analytic Functions.- 7. Complex Non-Archimedean Numbers.- 8. Mahler Basis.- 9. Measures on the Ring of p-adic Integers.- 10. Volkenborn Integral (Uniform Distribution).- 11. The Monna-Springer Integration Theory.- IV. The Ultrametric Hilbert Space Description of Quantum Measurements with a Finite Exactness.- 1. Critique of Interpretations of Quantum Mechanics.- 2. Preparation Procedures and State Spaces.- 3. Ultrametric (m-adic) Hilbert Space.- 4. m-adic (Ultrametric) Axiomatic of Quantum Measurements.- 5. Heisenberg Uncertainty and Inexactness Relations.- 6. Energy Representation for the Harmonic Oscillator.- 7. Einstein-Podolsky-Rosen Paradox and Infinite Exactness of Measurements.- 8. Fuzzy Reality.- 9. Quantum-Classical Heisenberg InexactnessRelation for the Harmonic Oscillator and Free Particle.- V. Non-Kolmogorov Probability Theory.- 1. Frequency Probability Theory.- 2. Measure and Probability.- 3. Densities.- 4. Integration Technique.- 5. Non-Kolmogorov Axiomatics.- 6. Products of Probabilities.- 7. Proportional and Classical Definitions of Probability.- 8. p-adic Asymptotic of Bernoulli Probabilities.- 9. More Complicated p-adic Asymptotics.- 10. p-adic Bernoulli Theorem.- 11. Non-symmetrical Bernoulli Distributions.- 12. The Central Limit Theorem.- VI. Non-Kolmogorov Probability and Quantum Physics.- 1. Dirac, Feynman, Wigner and Negative Probabilities.- 2. p-adic Stochastic Point of View of Bell’s Inequality.- 3. An Example of p-adic Negative Probability Behaviour.- 4. p-adic Stochastic Hidden Variable Model with Violations of Bell’s Inequality.- 5. Quadri Variate Joint Probability Distribution.- 6. Non-Kolmogorov Statistical Theory.- 7. Physical Interpretation of Negative Probabilities in Prugovecki’s Empirical Theory of Measurement.- 8. Experiments to Find p-adic Stochastics in the Two Slit Experiment.- VII. Position and Momentum Representations.- 1. Groups of Unitary Isometric Operators in a p-adic Hilbert Space.- 2. p-adic Valued Gaussian Integration and Spaces of Square Integrable Functions.- 3. A Representation of the Translation Group.- 4. Gaussian Representations for the Position and Momentum Operators.- 5. Unitary Isometric One Parameter Groups Corresponding to the Position and Momentum Operators.- 6. Operator Calculus.- 7. Exactness of a Measurement of Positions and Momenta.- 8. Spectrum of p-adic Position Operator.- 9. L2-space with respect to p-adic Lebesgue distributions.- 10. Fourier Transform of L2-maps and Momentum Representation.- 11. Schrödinger Equation.- VIII. p-adicDynamical Systems with Applications to Biology and Social Sciences.- 1. Roots of Unity.- 2. Dynamical Systems in Non-Archimedean Fields.- 3. Dynamical Systems in the Field of Complex p-adic Numbers.- 4. Dynamical Systems in the Fields of p-adic Numbers.- 5. Computer Calculations for Fuzzy Cycles.- 6. The Human Subconscious as a p-adic Dynamical System.- 7. Ultrametric on the Genealogical Tree.- 8. Social Dynamics.- 9. Human History as a p-adic Dynamical System.- 10. God as p-adic Dynamical System.- 11. Struggle of Civilizations.- 12. Economical and Social Effectiveness.- Open Problems.- 1. Newton’s Method (Hensel Lemma).- 2. Non-Real Reality.- 5. Quantum Mechanics of Vladimirov and Volovich.