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Fractional quantum field theory, path integral, and stochastic differential equation for strongly interacting many-particle systems

Hagen M. Kleinert

Published Jan 1, 2012• 15 citations
Publisher

Abstract

While free and weakly interacting particles are described by a second- quantized nonlinear Schrödinger field, or relativistic versions of it, the fields of strongly interacting particles are governed by effective actions, whose quadratic terms are extremized by fractional wave equations. Their particle orbits perform universal Lévy walks rather than Gaussian random walks with perturbations.

DOI
10.17169/refubium-21365

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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.

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While free and weakly interacting particles are described by a second- quantized nonlinear Schrödinger field, or relativistic versions of it, the fields of strongly interacting particles are governed by effective actions, whose quadratic terms are extremized by fractional wave equations. Their particle orbits perform universal Lévy walks rather than Gaussian random walks with perturbations.
Fractional quantum mechanics in the non-relativistic case is described by the space-fractional Schrödinger equation containing the fractional Riesz operator, which is a non-local operator. The non-locality makes it difficult to solve the fractional Schrödinger equation in the local potentials. In this paper, we study the solution of the fractional Schrödinger equation of a free particle moving on a circle by use of the Lévy path integrals approach. We present the Lévy path integrals propagator of a particle on circle and then use it to get the wave functions and energy eigenvalues of the free particle case. In addition, the Laplace transform, energy-time transform, and momentum representation of the free particle kernel are also obtained. The results of this paper contain the ones in standard quantum mechanics as special cases. The problem of a particle on circle is a fundamental problem of path integrals with topological constraints in quantum systems, and the method of Lévy path integrals can be generalized to study more complex local potentials.
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.