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Stochastic Processes in Quantum Physics

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概要 "Stochastic Processes in Quantum Physics" addresses the question 'What is the mathematics needed for describing the movement of quantum particles', and shows that it is the theory of stochastic (in pa...rticular Markov) processes and that a relativistic quantum particle has pure-jump sample paths while sample paths of a non-relativistic quantum particle are continuous. Together with known techniques, some new stochastic methods are applied in solving the equation of motion and the equation of dynamics of relativistic quantum particles. The problem of the origin of universes is discussed as an application of the theory. The text is almost self-contained and requires only an elementary knowledge of probability theory at the graduate level, and some selected chapters can be used as (sub-)textbooks for advanced courses on stochastic processes, quantum theory and theoretical chemistry.続きを見る
目次 I Markov Processes
1.1 Classical Mechanics
1.2 Movement of a Particle with Noise
1.3 Transition Probability and the Markov Property
1.4 Diffusion Equations
1.5 Brownian Motions
1.6 The Itô formula
Appendix. Monotone Lemmas
II Time Reversal and Duality
2.1 Time Reversal of Markov Processes and Duality
2.2 Space-Time Markov Processes and Space-Time Duality
2.3 Time Reversal and Schrödinger's Representation
III Non-Relativistic Quantum Theory
3.1 Non-Relativistic Equation of Motion
3.2 Stationary States and Eigenvalue Problem
3.3 Time Reversal of Diffusion Processes
3.4 Duality Relation of Diffusion Processes
3.5 Equation of Motion in General Cases
3.6 Principle of Superposition of Markov Processes
3.7 Non-Relativistic Schrödinger Equation
3.8 State Preparations and Measurements
3.9 Diffusion or Schrödinger Equations ?
3.10 The First Technical Convention
IV Stationary Schrödinger Processes
4.1 Stationary States
4.2 One-Dimensional Harmonic Oscillator
4.3 An Example in Two-Dimension
4.4 Superposition of Eigenfunctions
4.5 Further Excited States
4.6 Hydrogen Atom
V Construction of the Schrödinger Processes
5.1 The Feynman-Kac Formula
5.2 Solving the Equation of Motion
5.3 Transformation by Multiplicative Functionals
5.4 Renormalization
5.5 A Variational Method
5.6 The Maruyama-Girsanov Formula
5.7 A Lagrangian Formulation
5.8 The Second Technical Convention
VI Markov Processes with Jumps
6.1 Poisson and Compound Poisson Processes
6.2 Poisson Random measures and Point Processes
6.3 Stochastic Integrals with Poisson Point Processes
6.4 Lévy Processes
6.5 Stable Processes
6.6 Bochner's Subordination
6.7 Duality of Subordinate Semi-Groups
6.8 Harmonic Transformation of Subordinate Semi-Groups
6.9 Duality of Fractional Powers of Time-Dependent Operators
VII Relativistic Quantum Particles
7.1 A Relativistic Schrödinger Equation for a Spinless Paticle
7.2 Equation of Motion for Relativistic Quantum Particles
7.3 Stationary States of the Relativistic Schrödinger Equation
7.4 Stochastic Processes for Relativistic Spinless Particles
7.5 Non-Relativistic Limit
7.6 A Diffusion Approximation
VIII Stochastic Differential Equations of Pure-Jumps
8.1 Markov Processes with the Generators of Fractional Power
8.2 Stochastic Differential Equations of Pure-Jumps
8.3 The Case with no Potential Term
8.4 To Solve the Stochastic Differential Equations of Pure-Jumps
8.5 To Construct Pure-Jump Markov Processes
8.6 A Remark on the Integrability Condition
IX Variational Principle for Relativistic Quantum Particles
9.1 Absolute Continuity
9.2 Pure-Jump Markov Processes
9.3 A Multiplicative Functional
9.4 Renormalization and Variational Principle
X Time Dependent Subordination and Markov Processes with Jumps
10.1 Time-Inhomogeneous Subordination
10.2 Lemmas
10.3 Stochastic Differential Equation with Jumps
10.4 A Formula of Feynman-Kac Type
10.5 Markov Processes with Jumps
Appendix. Integration by Parts Formulae
XI Concave Majorants of Lévy Processes and the Light Cone
14.1 The Vertex Process of a Lévy Process
14.2 Propositions on Random walks
14.3 Proof of Propositions on Random Walks
14.4 Proof of the main Theorems
14.5 Examples
14.6 The light Cone
XII The Locality in Quantum Physics
12.1 Historical Overview
12.2 Hidden-Variable Theories
12.3 Locality of Hidden-Variable Theories
12.4 Spin-Correlation of Three Particles
12.5 Gudder's Hidden-Variable Theory
12.6 Spin-Correlations in Gudder's Theory
12.7 Some Remarks
XIII Micro Statistical Theory
13.1 The Source of the Noise
13.2 Large Deviations of the Renormalized Processes
13.3 The Propagation of Chaos
13.4 Micro Statistical Mechanics
13.5 Propagation of Chaos of Pure-Jump Processes
13.6 Superposition of Movements
12.7 A Remark on the Gibbs Distribution
XIV Processes on Open Time Intervals
14.1 Diffusion Processes on Open Time Intervals
14.2 Time-Reversed Schrödinger Processes
14.3 A Theorem of Jeulin-Yor
14.4 Reflecting Brownian Motion
14.5 Two-Sided Skorokhod Type Problem
14.6 Skorokhod Problem with Singular Drift
14.7 The Minimum and Maximum Solutions
14.8 The Uniqueness and Non-Uniqueness of Solutions
14.9 An Application: The Origin of Universes
XV Creation and Killing of Particles
15.1 Non-Linear Differential Equations
15.2 Branching Markov Processes
15.3 The Expected Number of Particles
15.4 Killing
XVI The ltô Calculus
16.1 The Itô Integral
16.2 Martingales
16.3 The Itô Integral with Local Martingales
16.4 Itô's formula
16.5 Stochastic Differential Equations
16.6 Stochastic Differential Calculus
References.
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登録日 2020.06.27
更新日 2020.06.28