Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, 1997 Mathematics and Its Applications Series, Vol. 427

N atur non facit saltus? This book is devoted to the fundamental problem which arises contin­ uously in the process of the human investigation 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 real analysis is not the unique possible way of presenting reality in a human brain. There exist other pictures of reality where other num­ ber fields are used as basic elements of a mathematical description. In this book we try to build a p-adic picture of reality based on the fields of p-adic numbers Qp and corresponding analysis (a particular case of so called non-Archimedean analysis). However, this book must not be considered as only a book on p-adic analysis and its applications. We study a much more extended range of problems. Our philosophical and physical ideas can be realized in other mathematical frameworks which are not obliged to be based on p-adic analysis. We shall show that many problems of the description of reality with the aid of real numbers are induced by unlimited applications of the so called Archimedean axiom.

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.