Schrödinger equations and diffusion theory
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Bibliographic Information
Schrödinger equations and diffusion theory
（Monographs in mathematics, v. 86）
Birkhäuser Verlag, 1993
- : sz
- : us
Available at / 66 libraries
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Hokkaido University, Library, Graduate School of Science, Faculty of Science and School of Science図書
est.Baseldc20:519.2/n1312070259116
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Includes bibliographical references and index
Description and Table of Contents
Description
Schroedinger Equations and Diffusion Theory addresses the question "What is the Schroedinger equation?" in terms of diffusion processes, and shows that the Schroedinger equation and diffusion equations in duality are equivalent. In turn, Schroedinger's conjecture of 1931 is solved. The theory of diffusion processes for the Schroedinger 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 Schroedinger equations.
The theory of large deviations and the propagation of chaos of interacting diffusion particles reveal the statistical mechanical nature of the Schroedinger equation, namely, quantum mechanics.
The text is practically self-contained and requires only an elementary knowledge of probability theory at the graduate level.
Table of Contents
I Introduction and Motivation.- 1.1 Quantization.- 1.2 Schroedinger Equation.- 1.3 Quantum Mechanics and Diffusion Processes.- 1.4 Equivalence of Schroedinger and Diffusion Equations.- 1.5 Time Reversal and Duality.- 1.6 QED and Quantum Field Theory.- 1.7 What is the Schroedinger 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 Kac's Semi-Group and its Renormalization.- 2.8 Time Change.- 2.9 Dirichlet Problem.- 2.10 Feller's One-Dimensional Diffusion Processes.- 2.11 Feller's Test.- III Duality and Time Reversal of Diffusion Processes.- 3.1 Kolmogoroff's Duality.- 3.2 Time Reversal of Diffusion Processes.- 3.3 Duality of Time-Inhomogeneous Diffusion Processes.- 3.4 Schroedinger's and Kolmogoroff s Representations.- 3.5 Some Remarks.- IV Equivalence of Diffusion and Schroedinger Equations.- 4.1 Change of Variable Formulae.- 4.2 Equivalence Theorem.- 4.3 Discussion of the Non-Linear Dependence.- 4.4 A Solution to Schroedinger'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 Csiszar's Projection Theorem.- 5.3 Reference Processes.- 5.4 Diffusion Processes in Schroedinger'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 Kac'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 Schroedinger Equations with Singular Potentials.- VIII The Schroedinger 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 Schroedinger Equations.- 8.4 Some Comments.- IX Applications of the Statistical Model for Schroedinger 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 Csiszar's Projection.- 10.1 Relative Entropy.- 10.2 Csiszar'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 Fenyes' "Equation of Motion" 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.
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