Description of quantum-mechanical motion by using the formalism of non-Markov stochastic processes

A. T. Gavrilin

Thomas Schuermann

Piotr Garbaczewski

Bose stochastic mechanics is sufficient for the probabilistic description of the quantised two-level system in terms of Gaussian Markov processes.

正雄 長沢

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 Ito 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 Schroedinger'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 Schroedinger Equation.- 3.8 State Preparations and Measurements.- 3.9 Diffusion or Schroedinger Equations ?.- 3.10 The First Technical Convention.- IV Stationary Schroedinger 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 Schroedinger 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 Levy 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 Schroedinger Equation for a Spinless Paticle.- 7.2 Equation of Motion for Relativistic Quantum Particles.- 7.3 Stationary States of the Relativistic Schroedinger 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 Levy Processes and the Light Cone.- 14.1 The Vertex Process of a Levy 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 Schroedinger 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 lto Calculus.- 16.1 The Ito Integral.- 16.2 Martingales.- 16.3 The Ito Integral with Local Martingales.- 16.4 Ito's formula.- 16.5 Stochastic Differential Equations.- 16.6 Stochastic Differential Calculus.- References.

Mikio Namiki, Satoshi Tanaka

Philip Taranto, Felix A. Pollock, Simon Milz +2 more

We formally extend the notion of Markov order to open quantum processes by accounting for the instruments used to probe the system of interest at different times. Our description recovers the classical property in the appropriate limit: when the stochastic process is classical and the instruments are noninvasive, i.e., restricted to orthogonal, projective measurements. We then prove that there do not exist non-Markovian quantum processes that have finite Markov order with respect to all possible instruments; the same process exhibits distinct memory effects when probed by different instruments. This naturally leads to a relaxed definition of quantum Markov order with respect to specified instrument sequences. The memory effects captured by different choices of instruments vary dramatically, providing a rich landscape for future exploration.