Stochastic Processes in Quantum Physics

by Nagasawa, Masao

• ISBN13: 9783764362089
• ISBN10: 3764362081
• Format: Hardcover
• Copyright: 2000-01-01
• Publisher: Birkhauser

Summary

"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 particular 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.

Table of Contents

Preface
Markov Processes
Classical Mechanics	1	(2)
Movement of a Particle with Noise	3	(4)
Transition Probability and the Markov Property	7	(5)
Diffusion Equations	12	(4)
Brownian Motions	16	(7)
The Ito formula	23	(4)
Appendix. Monotone Lemmas	26	(1)
Time Reversal and Duality
Time Reversal of Markov Processes and Duality	27	(10)
Space-Time Markov Processes and Space-Time Duality	37	(7)
Time Reversal and Schrodinger's Representation	44	(9)
Non-Relativistic Quantum Theory
Non-Relativistic Equation of Motion	53	(5)
Stationary States and Eigenvalue Problem	58	(4)
Time Reversal of Diffusion Processes	62	(2)
Duality Relation of Diffusion Processes	64	(5)
Equation of Motion in General Cases	69	(5)
Principle of Superposition of Markov Processes	74	(7)
Non-Relativistic Schrodinger Equation	81	(4)
State Preparations and Measurements	85	(8)
Diffusion or Schrodinger Equations?	93	(3)
The First Technical Convention	96	(9)
Stationary Schrodinger Processes
Stationary States	105	(1)
One-Dimensional Harmonic Oscillator	106	(3)
An Example in Two-Dimension	109	(4)
Superposition of Eigenfunctions	113	(5)
Further Excited States	118	(4)
Hydrogen Atom	122	(17)
Construction of the Schrodinger Processes
The Feynman-Kac Formula	139	(4)
Solving the Equation of Motion	143	(10)
Transformation by Multiplicative Functionals	153	(6)
Renormalization	159	(3)
A Variational Method	162	(6)
The Maruyama-Girsanov Formula	168	(7)
A lagrangian Formulation	175	(4)
The Second Technical Convention	179	(6)
Markov Processes with Jumps
Poisson and Compound Poisson Processes	185	(6)
Poisson Random measures and Point Processes	191	(3)
Stochastic Integrals with Poisson Point Processes	194	(7)
Levy Processes	201	(9)
Stable Processes	210	(2)
Bochner's Subordination	212	(9)
Duality of Subordinate Semi-Groups	221	(4)
Harmonic Transformation of Subordinate Semi-Groups	225	(3)
Duality of Fractional Powers of Time-Dependent Operators	228	(3)
Relativistic Quantum Particles
A Relativistic Schrodinger Equation for a Spinless Paticle	231	(3)
Equation of Motion for Relativistic Quantum Particles	234	(13)
Stationary States of the Relativistic Schrodinger Equation	247	(4)
Stochastic Processes for Relativistic Spinless Particles	251	(6)
Non-Relativistic Limit	257	(3)
A Diffusion Approximation	260	(3)
Stochastic Differential Equations of Pure-Jumps
Markov Processes with the Generators of Fractional Power	263	(2)
Stochastic Differential Equations of Pure-Jumps	265	(5)
The Case with no Potential Term	270	(6)
To Solve the Stochastic Differential Equations of Pure-Jumps	276	(5)
To Construct Pure-Jump Markov Processes	281	(3)
A Remark on the Integrability Condition	284	(3)
Variational Principle for Relativistic Quantum Particles
Absolute Continuity	287	(2)
Pure-Jump Markov Processes	289	(5)
A Multiplicative Functional	294	(13)
Renormalization and Variational Principle	307	(8)
Time Dependent Subordination and Markov Processes with Jumps
Time-Inhomogeneous Subordination	315	(4)
Lemmas	319	(8)
Stochastic Differential Equation with Jumps	327	(7)
A Formula of Feynman-Kac Type	334	(16)
Markov Processes with Jumps	350	(5)
Appendix. Integration by Parts Formulae	353	(2)
Concave Majorants of Levy Processes and the Light Cone
The Vertex Process of a Levy Process	355	(4)
Propositions on Random walks	359	(2)
Proof of Propositions on Random Walks	361	(11)
Proof of the main Theorems	372	(10)
Examples	382	(4)
The light Cone	386	(3)
The Locality in Quantum Physics
Historical Overview	389	(1)
Hidden-Variable Theories	390	(9)
Locality of Hidden-Variable Theories	399	(9)
Spin-Correlation of Three Particles	408	(9)
Gudder's Hidden-Variable Theory	417	(4)
Spin-Correlations in Gudder's Theory	421
Some Remarks	252	(185)
Micro Statistical Theory
The Source of the Noise	437	(5)
Large Deviations of the Renormalized Processes	442	(2)
The Propagation of Chaos	444	(3)
Micro Statistical Mechanics	447	(2)
Propagation of Chaos of Pure-Jump Processes	449	(2)
Superposition of Movements	451	(4)
A Remark on the Gibbs Distribution	455	(6)
Processes on Open Time Intervals
Diffusion Processes on Open Time Intervals	461	(2)
Time-Reversed Schrodinger Processes	463	(7)
A Theorem of Jeulin-Yor	470	(3)
Reflecting Brownian Motion	473	(4)
Two-Sided Skorokhod Type Problem	477	(6)
Skorokhod Problem with Singular Drift	483	(4)
The Minimum and Maximum Solutions	487	(2)
The Uniqueness and Non-Uniqueness of Solutions	489	(4)
An Application: The Origin of Universes	493	(8)
Creation and Killing of Particles
Non-Linear Differential Equations	501	(4)
Branching Markov Processes	505	(6)
The Expected Number of Particles	511	(6)
Killing	517	(4)
The Ito Calculus
The Ito Integral	521	(9)
Martingales	530	(16)
The Ito Integral with Local Martingales	546	(7)
Ito's formula	553	(8)
Stochastic Differential Equations	561	(6)
Stochastic Differential Calculus	567	(6)
References	573	(18)
Index	591