Nonlocal Quantum Field Theory and Stochastic Quantum Mechanics

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.