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I: Nonlocal Quantum Field Theory. I/Foundation of the Nonlocal Model of Quantized Fields. 1.1. Introduction. 1.2. Shastic SpaceTime. 1.3. The Method of Averaging in Shastic SpaceTime 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 SMatrix 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 SMatrix in Perturbation Theory. 2.4.4. An Intermediate Regularization Scheme. 2.4.5. Proof of the Unitarity of the SMatrix 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 Shastic SpaceTime. 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 SMatrix. 3.4. Construction of a Perturbation Series for the SMatrix in Quantum Electrodynamics. 3.4.1. The Diagrams of Vacuum Polarization. 3.4.2. The Diagram of SelfEnergy. 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 SelfEnergy of Bosons. 4/FourFermion Weak Interactions in Shastic SpaceTime. 4.1. Introduction. 4.2. Gauge Invariance for the SMatrix in ShasticNonlocal 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 Shastic Nonlocal Theory. 4.4.1. Introduction. 4.4.2. The $\mu\rightarrow 3e$ Decay. 4.4.3. The $K_{L}sub{0}\rightarrow\mu e$ Decay. 4.5. Neutrino Electromagnetic Properties in the ShasticNonlocal Theory of Weak Interactions. 4.6. Studies of the Decay $K_{L}sub{0}\rightarrow\musub{+}\musub{}$ and $K_{L}sub{0}$ and $K_{S}sub{0}$Meson Mass Difference. 4.6.1. Introduction. 4.6.2. The $K_{L}sub{0}\rightarrow\musub{+}\musub{}$ Decay. 4.6.3. The Mass Difference of $K_{L}sub{0}$ and $K_{S}sub{0}$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 FiniteDimensional Grassmann Algebra. 5.3.1. Definition. 5.3.2. Derivatives. 5.3.3. Integration over a Grassmann Algebra (FiniteDimensional 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 SMatrix Theory. 5.5.1. Introduction. 5.5.2. Functional Integral over a Bose Field in the Case of NonlocalShastic 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: Shastic Quantum Mechanics and Fields. 6/The Basic Concepts of Random Processes and Shastic 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. Shastic Processes with Independent Increments. 6.10. Markov Processes. 6.11. Wiener Processes. 6.12. Functionals of Shastic Processes and Shastic Calculus. 7/Basic Ideas of Shastic Quantization. 7.1. Introduction. 7.2. The Hypothesis of SpaceTime Shasticity as the Origin of Shasticity in Physics. 7.3. Shastic Space and Random Walk. 7.4. The Main Prescriptions of Shastic Quantization. 7.5. Shastic Field Theory and its Connection with Euclidean Field Theory. 7.6. Euclidean Quantum Field Theory. 8/Shastic Mechanics. 8.1. Introduction. 8.2. Equations of Motion of a Nonrelativistic Particle. 8.3. Relativistic Dynamics of Shastic Particles. 8.4. The TwoBody Problem in Shastic Theory. 8.4.1. The Nonrelativistic Case. 8.4.2. The Relativistic Case. 9/Selected Topics in Shastic Mechanics. 9.1. A Shastic Derivation of the Sivashinsky Equation for the SelfTurbulent Motion of a Free Particle. 9.2. Relativistic FeynmanType 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 Shastic Mechanics. 9.4. Cauchy Problem for the Diffusion Equation. 9.5. PositionMomentum Uncertainty Relations in Shastic Mechanics. 9.6. Appendix C. Concept of the ‘Differential Form’ and Directional Derivative. 10 Further Developments in Shastic Quantization. 10.1. Introduction. 10.2 Davidson’s Model for Free Scalar Field Theory. 10.3. The Electromagnetic Field as a Shastic Process. 10.4. Shastic Quantization of the Gauge Theories. 10.4.1. Introduction. 10.4.2. Another Shastic Quantization Scheme. 10.5. Equivalence of Shastic 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 Shastic Quantization. 10.7. Shastic 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 Shastic SpaceTime and the Fundamental Length. 11.1. Prologue. 11.2. NonlocalShastic Model for Free Scalar Field Theory. 11.3. ZeroPoint 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 SpaceTime Metric and its Consequences. 11.7. On the Origin of Cosmic Rays and the Value of the Fundamental Length. 11.8. SpaceTime Structure near Particles and its Influence on Particle Behavior. 11.8.1. Introduction. 11.8.2. Shastic Behavior of Particles and its Connection with Shastic Mechanical Dynamics. 11.8.3. SolitonLike Behavior of Particles.