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AuthorNagasawa, Masao. author
TitleSchrรถdinger Equations and Diffusion Theory [electronic resource] / by Masao Nagasawa
ImprintBasel : Birkhรคuser Basel : Imprint: Birkhรคuser, 1993
Connect tohttp://dx.doi.org/10.1007/978-3-0348-8568-3
Descript XII, 323 p. online resource

SUMMARY

Schrรถdinger Equations and Diffusion Theory addresses the question "What is the Schrรถdinger equation?" in terms of diffusion processes, and shows that the Schrรถdinger equation and diffusion equations in duality are equivalent. In turn, Schrรถdinger's conjecture of 1931 is solved. The theory of diffusion processes for the Schrรถdinger equation tell us that we must go further into the theory of systems of (infinitely) many interacting quantum (diffusion) particles. The method of relative entropy and the theory of transformations enable us to construct severely singular diffusion processes which appear to be equivalent to Schrรถdinger equations. The theory of large deviations and the propagation of chaos of interacting diffusion particles reveal the statistical mechanical nature of the Schrรถdinger equation, namely, quantum mechanics. The text is practically self-contained and requires only an elementary knowledge of probability theory at the graduate level


CONTENT

I Introduction and Motivation -- 1.1 Quantization -- 1.2 Schrรถdinger Equation -- 1.3 Quantum Mechanics and Diffusion Processes -- 1.4 Equivalence of Schrรถdinger and Diffusion Equations -- 1.5 Time Reversal and Duality -- 1.6 QED and Quantum Field Theory -- 1.7 What is the Schrรถdinger Equation -- 1.8 Mathematical Contents -- II Diffusion Processes and their Transformations -- 2.1 Time Homogeneous Diffusion Processes -- 2.2 Time Inhomogeneous Diffusion Processes -- 2.3 Brownian Motions -- 2.4 Stochastic Differential Equations -- 2.5 Transformation by a Multiplicative Functional -- 2.6 Feynman-Kac Formula -- 2.7 Kacโ{128}{153}s Semi-Group and its Renormalization -- 2.8 Time Change -- 2.9 Dirichlet Problem -- 2.10 Fellerโ{128}{153}s One-Dimensional Diffusion Processes -- 2.11 Fellerโ{128}{153}s Test -- III Duality and Time Reversal of Diffusion Processes -- 3.1 Kolmogoroffโ{128}{153}s Duality -- 3.2 Time Reversal of Diffusion Processes -- 3.3 Duality of Time-Inhomogeneous Diffusion Processes -- 3.4 Schrรถdingerโ{128}{153}s and Kolmogoroff s Representations -- 3.5 Some Remarks -- IV Equivalence of Diffusion and Schrรถdinger Equations -- 4.1 Change of Variable Formulae -- 4.2 Equivalence Theorem -- 4.3 Discussion of the Non-Linear Dependence -- 4.4 A Solution to Schrรถdingerโ{128}{153}s Conjecture -- 4.5 A Unified Theory -- 4.6 On Quantization -- 4.7 As a Diffusion Theory -- 4.8 Principle of Superposition -- 4.9 Remarks -- V Variational Principle -- 5.1 Problem Setting in p-Representation -- 5.2 Csiszarโ{128}{153}s Projection Theorem -- 5.3 Reference Processes -- 5.4 Diffusion Processes in Schrรถdingerโ{128}{153}s Representation -- 5.5 Weak Fundamental Solutions -- 5.6 An Entropy Characterization of the Markov Property -- 5.6 Remarks -- VI Diffusion Processes in q-Representation -- 6.1 A Multiplicative Functional -- 6.2 Flows of Distribution Densities -- 6.3 Discussions on the q-Representation -- 6.4 What is the Feynman Integral -- 6.5 A Remark on Kacโ{128}{153}s Semi-Group -- 6.6 A Typical Case -- 6.7 Hydrogen Atom -- 6.8 A Remark on {?a,?b} -- VII Segregation of a Population -- 7.1 Introduction -- 7.2 Harmonic Oscillator -- 7.3 Segregation of a Finite-System of Particles -- 7.4 A Formulation of the Propagation of Chaos -- 7.5 The Propagation of Chaos -- 7.6 Skorokhod Problem with Singular Drift -- 7.7 A Limit Theorem -- 7.8 A Proof of Theorem 7.1 -- 7.9 Schrรถdinger Equations with Singular Potentials -- VIII The Schrรถdinger Equation can be a Boltzmann Equation -- 8.1 Large Deviations -- 8.2 The Propagation of Chaos in Terms of Large Deviations -- 8.3 Statistical Mechanics for Schrรถdinger Equations -- 8.4 Some Comments -- IX Applications of the Statistical Model for Schrรถdinger Equation -- 9.1 Segregation of a Monkey Population -- 9.2 An Eigenvalue Problem -- 9.3 Septation of Escherichia Coli -- 9.4 The Mass Spectrum of Mesons -- 9.5 Titius-Bode Law -- X Relative Entropy and Csiszarโ{128}{153}s Projection -- 10.1 Relative Entropy -- 10.2 Csiszarโ{128}{153}s Projection -- 10.3 Exponential Families and Marginal Distributions -- XI Large Deviations -- 11.1 Lemmas -- 11.2 Large Deviations of Empirical Distributions -- XII Non-Linearity Induced by the Branching Property -- 12.1 Branching Property -- 12.2 Non-Linear Equations of Branching Processes -- 12.3 Quasi-Linear Parabolic Equations -- 12.4 Branching Markov Processes with Non-Linear Drift -- 12.5 Revival of a Markov Process -- 12.6 Construction of Branching Markov Processes -- Appendix: -- a.1 Fรฉnyesโ{128}{153} โ{128}{156}Equation of Motionโ{128}{157} of Probability Densities -- a.2 Stochastic Mechanics -- a.3 Segregation of a Population -- a.4 Euclidean Quantum Mechanics -- a.5 Remarks -- a.6 Bohmian Mechanics -- References


Mathematics Probabilities Mathematics Probability Theory and Stochastic Processes



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