Author | Nagasawa, Masao. author |
---|---|
Title | Stochastic Processes in Quantum Physics [electronic resource] / by Masao Nagasawa |
Imprint | Basel : Birkhรคuser Basel : Imprint: Birkhรคuser, 2000 |
Connect to | http://dx.doi.org/10.1007/978-3-0348-8383-2 |
Descript | VII, 598 p. online resource |
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 Itรด 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 Schrรถdingerโs 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 Schrรถdinger Equation -- 3.8 State Preparations and Measurements -- 3.9 Diffusion or Schrรถdinger Equations ? -- 3.10 The First Technical Convention -- IV Stationary Schrรถdinger 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 Schrรถdinger 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 Lรฉvy Processes -- 6.5 Stable Processes -- 6.6 Bochnerโs 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 Schrรถdinger Equation for a Spinless Paticle -- 7.2 Equation of Motion for Relativistic Quantum Particles -- 7.3 Stationary States of the Relativistic Schrรถdinger 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 Lรฉvy Processes and the Light Cone -- 14.1 The Vertex Process of a Lรฉvy 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 Gudderโs Hidden-Variable Theory -- 12.6 Spin-Correlations in Gudderโs 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 Schrรถdinger 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 ltรด Calculus -- 16.1 The Itรด Integral -- 16.2 Martingales -- 16.3 The Itรด Integral with Local Martingales -- 16.4 Itรดโs formula -- 16.5 Stochastic Differential Equations -- 16.6 Stochastic Differential Calculus -- References