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# Nonlocal Quantum Field Theory and Stochastic Quantum Mechanics

## byK.H. Namsrai

### Paperback | January 19, 2012

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over this stochastic space-time leads to the non local fields considered by G. V. Efimov. In other words, stochasticity of space-time (after being averaged on a large scale) as a self-memory makes the theory nonlocal. This allows one to consider in a unified way the effect of stochasticity (or nonlocality) in all physical processes. Moreover, the universal character of this hypothesis of space-time at small distances enables us to re-interpret the dynamics of stochastic particles and to study some important problems of the theory of stochastic processes [such as the relativistic description of diffusion, Feynman type processes, and the problem of the origin of self-turbulence in the motion of free particles within nonlinear (stochastic) mechanics]. In this direction our approach (Part II) may be useful in recent developments of the stochastic interpretation of quantum mechanics and fields due to E. Nelson, D. Kershaw, I. Fenyes, F. Guerra, de la Pena-Auerbach, J. -P. Vigier, M. Davidson, and others. In particular, as shown by N. Cufaro Petroni and J. -P. Vigier, within the discussed approach, a causal action-at-distance interpretation of a series of experiments by A. Aspect and his co-workers indicating a possible non locality property of quantum mechanics, may also be obtained. Aspect's results have recently inspired a great interest in different nonlocal theories and models devoted to an understanding of the implications of this nonlocality. This book consists of two parts.

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Title:Nonlocal Quantum Field Theory and Stochastic Quantum MechanicsFormat:PaperbackDimensions:426 pages, 23.5 × 15.5 × 0.02 inPublished:January 19, 2012Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:9401085137

ISBN - 13:9789401085137

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Table of Contents

I: Nonlocal Quantum Field Theory.- I/Foundation of the Nonlocal Model of Quantized Fields.- 1.1. Introduction.- 1.2. Stochastic Space-Time.- 1.3. The Method of Averaging in Stochastic Space-Time and Nonlocality.- 1.4. The Class of Test Functions and Generalized Functions.- 1.4.1. Introduction.- 1.4.2. Space of Test Functions.- 1.4.3. Linear Functional and Generalized Functions.- 1.4.3a. General Definition.- 1.4.3b. Transformation of the Arguments and Differentiation of the Generalized Functions.- 1.4.3c. The Fourier Transform of Generalized Functions.- 1.4.3d. Multiplication of the Generalized Functions by a Smooth Function and Their Convolution.- 1.4.4. Generalized Functions of Quantum Field Theory.- 1.4.5.The Class of Test Functions in the Nonlocal Case.- 1.4.6. The Class of Generalized Functions in the Nonlocal Case.- 2/The Basic Problems of Nonlocal Quantum Field Theory.- 2.1. Nonlocality and the Interaction Lagrangian.- 2.2. Quantization of Nonlocal Field Theory.- 2.2.1. Formulation of the Quantization Problem.- 2.2.2. Regularization Procedure.- 2.2.3. Quantization of the Regularized Equation.- 2.2.4. Green Functions of the Field ??(x).- 2.2.5. The Interacting System Before Removal of the Regularization.- 2.2.6. The Green Functions in the Limit ??0.- 2.3. The Physical Meaning of the Form Factors.- 2.4.The Causality Condition and Unitarity of the S-Matrix in Nonlocal Quantum Field Theory.- 2.4.1. Introduction.- 2.4.2. The Causality Condition.- 2.4.3. The Scheme of Proof of Unitarity of the S-Matrix in Perturbation Theory.- 2.4.4. An Intermediate Regularization Scheme.- 2.4.5. Proof of the Unitarity of the S-Matrix in a Functional Form.- 2.5. The Schrödinger Equation in Quantum Field Theory with Nonlocal Interactions.- 2.5.1. Introduction.- 2.5.2. The Field Operator at Imaginary Time.- 2.5.3. The State Space at Imaginary Time.- 2.5.4. The Interaction Hamiltonian and the Evolution Equation.- 2.5.5. Appendix A.- 3/Electromagnetic Interactions in Stochastic Space-Time.- 3.1. Introduction.- 3.2. Gauge Invariance of the Theory and Generalization of Kroll's Procedure.- 3.3. The Interaction Lagrangian and the Construction of the S-Matrix.- 3.4. Construction of a Perturbation Series for the S-Matrix in Quantum Electrodynamics.- 3.4.1. The Diagrams of Vacuum Polarization.- 3.4.2. The Diagram of Self-Energy.- 3.4.3. The Vertex Diagram and the Corrections to the Anomalous Magnetic Moment (AMM) of Leptons and to the Lamb Shift.- 3.5 The Electrodynamics of Particles with Spins 0 and 1.- 3.5.1. Introduction.- 3.5.2. The Diagrams of the Vacuum Polarization of Boson Fields.- 3.5.3. The Self-Energy of Bosons.- 4/Four-Fermion Weak Interactions in Stochastic Space-Time.- 4.1. Introduction.- 4.2. Gauge Invariance for the S-Matrix in Stochastic-Nonlocal Theory of Weak Interactions.- 4.3. Calculation of the 'Weak' Corrections to the Anomalous Magnetic Moment (AMM) of Leptons.- 4.4. Some Consequences of Neutrino Oscillations in Stochastic- Nonlocal Theory.- 4.4.1. Introduction.- 4.4.2. The $\mu\rightarrow 3e$ Decay.- 4.4.3. The $K_{L}^{0}\rightarrow\mu e$ Decay.- 4.5. Neutrino Electromagnetic Properties in the Stochastic-Nonlocal Theory of Weak Interactions.- 4.6. Studies of the Decay $K_{L}^{0}\rightarrow\mu^{+}\mu^{-}$ and $K_{L}^{0}$- and $K_{S}^{0}$-Meson Mass Difference.- 4.6.1. Introduction.- 4.6.2. The $K_{L}^{0}\rightarrow\mu^{+}\mu^{-}$ Decay.- 4.6.3. The Mass Difference of $K_{L}^{0}$- and $K_{S}^{0}$-Mesons.- 4.7. Appendix B. Calculation of the Contour Integral.- 5/Functional Integral Techniques in Quantum Field Theory.- 5.1. Mathematical Preliminaries.- 5.2. Historical Background of Path Integrals.- 5.3. Analysis on a Finite-Dimensional Grassmann Algebra.- 5.3.1. Definition.- 5.3.2. Derivatives.- 5.3.3. Integration over a Grassmann Algebra (Finite-Dimensional Case).- 5.4. Grassmann Algebra with an Infinite Number of Generators.- 5.4.1. Definition.- 5.4.2. Grassmann Algebra with Involution.- 5.4.3. Functional (or Variational) Derivatives.- 5.4.4. Continual (or Functional) Integrals over the Grassmann Algebra (Formal Definition).- 5.4.5. Examples.- 5.5. Functional Integral and the S-Matrix Theory.- 5.5.1. Introduction.- 5.5.2. Functional Integral over a Bose Field in the Case of Nonlocal-Stochastic Theory (Definition).- 5.5.2a. Definition of Functional Integral.- 5.5.2b. Upper and Lower Bounds of Vacuum Energy E(g) in Nonlocal Theory and in the Anharmonic Oscillator Case.- 5.5.3. Functional Integrals for Fermions in Quantum Field Theory.- II: Stochastic Quantum Mechanics and Fields.- 6/The Basic Concepts of Random Processes and Stochastic Calculus.- 6.1. Events.- 6.2. Probability.- 6.3. Random Variable.- 6.4. Expectation and Concept of Convergence over the Probability.- 6.5. Independence.- 6.6. Conditional Probability and Conditional (Mathematical) Expectation.- 6.7. Martingales.- 6.8. Definition of Random Processes and Gaussian Processes.- 6.9. Stochastic Processes with Independent Increments.- 6.10. Markov Processes.- 6.11. Wiener Processes.- 6.12. Functionals of Stochastic Processes and Stochastic Calculus.- 7/Basic Ideas of Stochastic Quantization.- 7.1. Introduction.- 7.2. The Hypothesis of Space-Time Stochasticity as the Origin of Stochasticity in Physics.- 7.3. Stochastic Space and Random Walk.- 7.4. The Main Prescriptions of Stochastic Quantization.- 7.5. Stochastic Field Theory and its Connection with Euclidean Field Theory.- 7.6. Euclidean Quantum Field Theory.- 8/Stochastic Mechanics.- 8.1. Introduction.- 8.2. Equations of Motion of a Nonrelativistic Particle.- 8.3. Relativistic Dynamics of Stochastic Particles.- 8.4. The Two-Body Problem in Stochastic Theory.- 8.4.1. The Nonrelativistic Case.- 8.4.2. The Relativistic Case.- 9/Selected Topics in Stochastic Mechanics.- 9.1. A Stochastic Derivation of the Sivashinsky Equation for the Self-Turbulent Motion of a Free Particle.- 9.2. Relativistic Feynman-Type Integrals.- 9.2.1. Diffusion Process in Real Time.- 9.2.2. 'Diffusion Process' in Complex Time.- 9.2.3. Introduction of Interactions into the Scheme.- 9.3. Discussion of the Equations of Motion in Stochastic Mechanics.- 9.4. Cauchy Problem for the Diffusion Equation.- 9.5. Position-Momentum Uncertainty Relations in Stochastic Mechanics.- 9.6. Appendix C. Concept of the 'Differential Form' and Directional Derivative.- 10 Further Developments in Stochastic Quantization.- 10.1. Introduction.- 10.2 Davidson's Model for Free Scalar Field Theory.- 10.3. The Electromagnetic Field as a Stochastic Process.- 10.4. Stochastic Quantization of the Gauge Theories.- 10.4.1. Introduction.- 10.4.2. Another Stochastic Quantization Scheme.- 10.5. Equivalence of Stochastic and Canonical Quantization in Perturbation Theory in the Case of Gauge Theories.- 10.6. The Mechanism of the Vacuum Tunneling Phenomena in the Framework of Stochastic Quantization.- 10.7. Stochastic Fluctuations of the Classical Yang-Mills Fields.- 10.8. Appendix D. Solutions to the Free Fokker-Planck Equation.- 11/Some Physical Consequences of the Hypothesis of Stochastic Space-Time and the Fundamental Length.- 11.1. Prologue.- 11.2. Nonlocal-Stochastic Model for Free Scalar Field Theory.- 11.3. Zero-Point Electromagnetic Field and the Connection Between the Value of the Fundamental Length and the Density of Matter.- 11.4. Hierarchical Scales and 'Family' of Black Holes.- 11.5. The Decay of the Proton and the Fundamental Length.- 11.6. A Hypothesis of Nonlocality of Space-Time Metric and its Consequences.- 11.7. On the Origin of Cosmic Rays and the Value of the Fundamental Length.- 11.8. Space-Time Structure near Particles and its Influence on Particle Behavior.- 11.8.1. Introduction.- 11.8.2. Stochastic Behavior of Particles and its Connection with Stochastic Mechanical Dynamics.- 11.8.3. Soliton-Like Behavior of Particles.