Methods of Bosonic and Fermionic Path Integrals Representations : Continuum Random Geometry in Quantum Field Theory.

Intro -- METHODS OF BOSONIC AND FERMIONIC PATH INTEGRALS REPRESENTATIONS: CONTINUUM RANDOM GEOMETRY IN QUANTUM FIELD THEORY -- Contents -- About This Monograph (ForewordI) -- Loop Space Path Integrals Representations for Euclidean Quantum Fields Path Integrals and the Covariant Path Integral -- 1.1. Introduction -- 1.2. The Bosonic Loop Space Formulation of the O(N)-Scalar Field Theory -- 1.3. A Fermionic Loop Space for QCD -- 1.4. Invariant Path Integration and the Covariant Functional Measure for Einstein Gravitation Theory -- References -- Appendix A. -- Appendix B. -- Appendix C. -- Appendix D. -- Appendix E. -- Path Integrals Evaluations in Bosonic Random Loop Geometry-Abelian Wilson Loops -- 2.1. Introduction -- 2.2. Abelian Wilson Loop Interaction at Finite Temperature -- 2.3. The Static Confining Potential for Q.C.D. in the Mandel-stam Model through Path Integrals -- Path-Integrals on Quantum Magnetic Monopoles -- References -- The Triviality-Quantum Decoherence of Quantum Chromodynamics SU(∞) in the Presence of an External Strong White-Noise Eletromagnetic Field -- 3.1. Introduction -- 3.2. The Triviality-Quantum Decoherence Analysis -- 3.3. Random Surface Dynamical Factor in the Analytical Regularization Scheme -- 3.4. The Non-relativistic Case -- 3.5. The Static Confining Potential in a Tensor Axion Model -- 3.6. The Confining Potential on the Axion-String Model in the Axion Higher-Energy Region -- 3.7.A λφ4 String Field Theory as a Dynamics of Self Avoiding Random Surfaces -- Appendix A. -- Appendix B. -- References -- The Confining Behaviour and Asymptotic Freedom for QCD(SU(∞))- A Constant Gauge Field Path Integral Analysis -- 4.1. Introduction -- 4.2. The Model and Its Confining Behavior -- 4.3. The Path-Integral Triviality Argument for the Thirring Model at SU(∞) -- 4.4. The Loop Space Argument for the Thirring Model Triviality.

References -- Triviality-Quantum Decoherence of Fermionic Quantum Chromodynamics SU(Nc) in the Presence of an External Strong U(∞) Flavored Constant noise Field -- 5.1. Introduction -- 5.2. The Triviality-Quantum Decoherence Analysis for Quantum Chromodynamics -- Appendix A. -- Appendix B. -- References -- Fermions on the Lattice by Means of Mandelstam-Wilson Phase Factors: A Bosonic Lattice Path-Integral Framework -- 6.1. Introduction -- 6.2. The Framework -- References -- A Connection between Fermionic Strings and Quantum Gravity States-A Loop Space Approach -- 7.1. Introduction -- 7.2. The Loop Space Approach for Quantum Gravity -- 7.3. The Wheeler-De Witt Geometrodynamical Propagator -- 7.4. A λφ4 Geometrodynamical Field Theory for Quantum Gravity -- Appendix A -- References -- A Fermionic Loop Wave Equation for Quantum Chromodynamics at Nc=+∞ -- 8.1. Introduction -- 8.2. The Fermionic Loop Wave Equation -- References -- String Wave Equations in Polyakov's Path Integral Framework -- 9.1. Introduction -- 9.2. The Wave Equation in Covariant Particle Dynamics -- 9.3. The Wave Equation in the Covariant Bosonic String Dynamics -- 9.4. A String Solution for the QCD [SU(∞)] Bosonic Contour Average Equation -- 9.5. The Neveu-Schwarz String Wave Equation -- Appendix A. -- Appendix B. -- Appendix C. -- Appendix D. -- Appendix E. -- References -- A Random Surface Membrane Wave Equation for Bosonic Q.C.D.(SU(∞)) -- 10.1. Introduction -- 10.2. The Random Surface Wave Functional -- 10.3. A Connection with Q.C.D (SU(∞)) -- Appendix A. -- Appendix B. -- References -- Covariant Functional Diffusion Equation for Polyakov's Bosonic String -- 11.1. Introduction -- 11.2. The Covariant Equation -- 11.3. The Wheeler-De Witt Equation as a Functional Diffusion Equation -- References -- Covariant Path Integral for Nambu-Goto String Theory -- 12.1. Introduction.

12.2. The Nambu-Goto Full Path Integral -- References -- Topological Fermionic String Representation for Chern-Simons Non-Abelian Gauge Theories -- 13.1. Introduction -- 13.2. The Fermionic String Representation -- References -- Fermionic String Representation for the Three-Dimensional Ising Model -- 14.1. Introduction -- 14.2. The Proposed String Theory -- References -- A Polyakov Fermionic String as a Quantum State of Einstein Theory of Gravitation -- 15.1. Introduction -- 15.2. The Quantum Gravity String -- References -- A Scattering Amplitude in the Quantum Geometry of Fermionic Strings -- 16.1. Introduction -- 16.2. The Scattering Amplitude -- References -- Path-Integral Bosonization for the Thirring Model on a Riemann Surface -- 17.1. Introduction -- 17.2. The Path-Integral Bosonization on a Riemann Surface -- References -- A Path-Integral Approach for Bosonic Effective Theories for Fermion Fields in Four and Three Dimensions -- 18.1. Introduction -- 18.2. The Bosonic High-Energy Effective Theory -- 18.3. The Bosonic Low-Energy Effective Theory -- 18.4. Polyakov's Fermi-Bose Transmutation in 3D Abelian-Thirring Model -- 18.5. Effective Four-Dimensional Bosonic Actions- Some Comments -- 18.6. The Triviality of the Abelian-Thirring Quantum Field Model -- Appendix A -- References -- Domains of Bosonic Functional Integrals and Some Applications to the Mathematical Physics of Path Integrals and String Theory -- 19.1. Introduction -- 19.2. The Euclidean Schwinger Generating Functional as a Functional Fourier Transform -- 19.3. The Support of Functional Measures- The Minlos Theorem -- 19.4. Some Rigorous Quantum Field Path Integral in the Analytical Regularization Scheme -- 19.5. Remarks on the Theory of Integration of Functionals on Distributional Spaces and Hilbert-Banach Spaces -- Appendix A -- Appendix B -- References.

Non-linear Diffusionin RD and in Hilbert Spaces, a Path Integral Study -- 20.1. Introduction -- 20.2. The Non-linear Diffusion -- 20.3. The Linear Diffusion in the Space L2(Ω) -- Appendix A. -- Appendix B. -- References -- Basics Integrals Representations in Mathematical Analysis of Euclidean Functional Integrals -- 21.1. On the Riesz-Markov Theorem -- 21.2.The L.Schwartz Representation Theorem on C∞(Ω)(Distribution Theory) -- 21.3. Equivalence of Gaussian Measures in Hilbert Spaces and Functional Jacobians -- SupplementaryAppendixes -- Appendix 22. A. -- Appendix 22. B. -- Appendix 22. C. -- Appendix 22. D. -- Appendix 22. E. -- Index.

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