دانلود کتاب Theory of Interacting Quantum Fields

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Preface\nNotation\n0 Introduction\nI Symmetry Groups of Elementary Particles\n 1 Lorentz Group\n 1.1 Euclidean and Minkowski Spaces. Relativistic Notation\n 1.2 Homogeneous Lorentz Group\n 1.3 Inhomogeneous Lorentz Group-Poincaré Group\n 1.4 Complex Lorentz Transformations\n 1.5 Representations of the Lorentz and Poincaré Groups, Field Functions, and Physical States\n 1.5.1 Representation D(0,0)\n 1.5.2 Representations D(1/2,0) and D(0,1/2)\n 1.5.3 Representation D(1/2,1/2)\n 2 Groups of Internal Symmetries\n 2.1 Abelian Unitary Group U(1)\n 2.2 Charge Conjugation C\n 2.3 Special Unitary Group SU(n)\n 2.3.1 SU(2) Symmetry\n 2.3.2 SU(3) Symmetry\n 2.4 Groups of Local Transformations. Gauge Group\n 3 Problems to Part I\nII Classical Theory of the Free Fields\n 4 Lagrangian and Hamiltonian Formalisms of the Classical Field Theory\n 4.1 Variational Principle and Canonical Formalism of Classical Mechanics\n 4.1.1 Lagrangian Equations\n 4.1.2 Canonical Variables. Hamiltonian Equations\n 4.1.3 Poisson Brackets. Integrals of Motion\n 4.1.4 Canonical Formalism in the Presence of Constraints\n 4.2 From Classical to Quantum Mechanics. Primary Quantization\n 4.3 General Requirements to the Lagrangians of the Field Theory\n 4.4 Lagrange–Euler Equations\n 4.5 Noether’s Theorem and Dynamic Invariants\n 4.6 Vector of Energy-Momentum\n 4.7 Tensors of Angular Momentum and Spin\n 4.8 Charge and the Vector of Current\n 4.9 Canonical Variables\n 5 Classical Theory of Free Scalar Fields\n 5.1 Klein–Fock–Gordon Equation\n 5.2 Relativistic Invariance of the Klein–Fock–Gordon Equation\n 5.3 Solutions of the Klein–Fock–Gordon Equation\n 5.4 Interpretation of Solutions. Hilbert Space of States\n 5.5 Ĉ, P̂, and T̂̂ Transformations\n 5.5.1 Transformation of Charge Conjugation Ĉ\n 5.5.2 Space Reflection P̂\n 5.5.3 Time Reversal T̂̂\n 5.5.4 ĈP̂T̂̂-Invariance\n 5.6 Representations of the Lorentz Group in the Space of States\n 5.7 Lagrangian Formalism of the Scalar Field. Dynamic Invariants\n 6 Spinor Field\n 6.1 Dirac Equation\n 6.1.1 Construction of the Dirac Equation\n 6.1.2 Properties of Dirac Matrices. Conjugate Equation\n 6.2 Relativistic Invariance\n 6.2.1 Transformation Properties of the Spinor Field\n 6.2.2 On Reducible and Irreducible Spinor Representations\n 6.2.3 Transformation Properties of Bilinear Forms ψ̄Oψ\n 6.3 Solutions of the Dirac Equation\n 6.3.1 Structure of Solutions in the Momentum Space\n 6.3.2 Classification of Solutions. Helicity\n 6.3.3 Relations Between Spinors\n 6.3.4 Wave Functions of the Electron and Positron. Charge Conjugation\n 6.3.5 ĈP̂T̂̂-Transformation\n 6.4 Lagrangian Formalism\n 6.5 Representations of the Lorentz Group\n 6.5.1 Hilbert Space of States\n 6.5.2 Representations of the Lorentz Group in the Space of States\n 6.6 Applications of the Dirac Equation\n 6.6.1 Dirac Equation in the Presence of External Fields\n 6.7 Massless Spinor Field\n 6.7.1 Two-component Massless Spinor Field\n 6.7.2 Relativistic Invariance\n 6.7.3 Are There Actual Particles Corresponding to the Massless Spinor Fields? Physical Interpretation of Solutions. Neutrino\n 6.7.4 Lagrangian and Dynamic Invariants\n 6.7.5 On the Mass of Neutrino and Majorana Spinors\n 7 Vector Fields\n 7.1 Lagrangian Formalism\n 7.2 Representations in the Momentum Space\n 7.3 Decomposition into the Longitudinal and Transverse Components\n 7.4 P̂, T̂̂, Ĉ-Transformations\n 8 Electromagnetic Field\n 8.1 Maxwell Equations\n 8.2 Potential of the Electromagnetic Field\n 8.3 Gradient Transformations and the Lorentz Condition: Transversality Condition\n 8.4 Lagrangian Formalism for Electromagnetic Fields\n 8.5 Transversal, Longitudinal, and Time Components of the Electromagnetic Field\n 8.6 Quantum-Mechanical Characteristics of Photons\n 8.7 Ĉ, P̂, T̂̂-Transformations\n 8.8 Consistency of the Lorentz and Gauge Transformations. Various Types of Gauges\n 9 Equations for Fields with Higher Spins\n 9.1 Fields with Spin 3/2\n 9.2 Particles with Spin 2\n 10 Problems to Part II\nIII Classical Theory of Interacting Fields\n 11 Gauge Theory of the Electromagnetic Interaction\n 11.1 Principle of Gauge Invariance in the Maxwell Theory\n 11.2 Schrödinger Equation and Gradient (Gauge) Invariance\n 11.3 Gauge Principle as the Dynamical Principle of Interaction between the Electromagnetic and Electron-Positron Fields\n 12 Classical Theory of Yang-Mills Fields\n 12.1 Gauge Principle and the Lagrangian of the Yang–Mills Fields\n 12.2 Equations of Motion for the Free Yang–Mills fields\n 12.3 Yang-Mills Fields for Arbitrary Representations of the Group SU(N)\n 13 Masses of Particles and Spontaneous Breaking of Symmetry\n 13.1 Spontaneous Breaking of Symmetry\n 13.2 Higgs Mechanism for the Local U(1) Symmetry\n 13.3 Higgs Mechanism for the Local SU(2) symmetry\n 13.4 Generation of the Masses of Fermions\n 14 On the Construction of the General Lagrangian of Interacting Fields\n 14.1 Lagrangian of the QCD\n 14.2 Lagrangian of Weak Interactions\n 14.3 On the Electroweak Interactions\n 14.4 On the Lagrangian of Great Unification\n 15 Solutions of the Equations for Classical Fields: Solitary Waves, Solitons, Instantons\n 16 Problems to Part III\nIV Second Quantization of Fields\n 17 Axioms and General Principles of Quantization\n 17.1 Why Do We Need the Procedure of Second Quantization? Operator Nature of the Field Functions\n 17.2 Schrödinger, Heisenberg, and Interaction Pictures\n 17.3 Axioms of Quantization\n 17.4 Relativistic Heisenberg Equation for Quantized Fields\n 17.4.1 Heisenberg Equation for a Free Scalar Field\n 17.4.2 Heisenberg Equation for a Free Electron-Positron Field\n 17.5 Physical Content of Positive- and Negative-Frequency Solutions of Equations for Free-Field Operators\n 18 Quantization of the Free Scalar Field\n 18.1 Commutation Relations. Commutator Functions\n 18.2 Complex Scalar Field\n 18.3 Operator Relations for Dynamic Invariants\n 19 Quantization of the Free Spinor Field\n 19.1 Commutator Functions of Fermi Fields\n 19.2 Dynamic Invariants of a Free Spinor Field\n 20 Quantization of the Vector and Electromagnetic Fields. Specific Features of the Quantization of Gauge Fields\n 20.1 Quantization of the Complex Vector Field\n 20.2 Quantization of an Electromagnetic Field\n 20.2.1 Specific Features and Difficulties of the Quantization of an Electromagnetic Field\n 20.2.2 Gupta–Bleuler Formalism\n 20.2.3 Canonical Method of Quantization\n 20.3 On the Quantization of Gauge Fields\n 21 CPT. Spin and Statistics\n 21.1 The Transformation of Charge Conjugation\n 21.2 The Transformation of Space Reflection\n 21.3 The Transformation of Time Reversal\n 21.4 CPT-Theorem and the Connection of Spin and Statistics\n 21.5 Proof of the Pauli Theorem\n 22 Representations of Commutation and Anticommutation Relations\n 22.1 General Structure of the Fock Space\n 22.2 Representations of Commutation Relations for a Free Real Scalar Field\n 22.2.1 The Fock Space of Free Scalar Bosons\n 22.2.2 Operators of Creation and Annihilation in the Fock Space. Momentum Representation\n 22.2.3 Vacuum State of Free Particle. Cyclicity of Vacuum. Set of Exponential Vectors\n 22.2.4 Construction of Representations of Commutation Relations for a Complex Scalar Field\n 22.2.5 Construction of Representations of Commutation Relations in the Configuration Space. Relativistic Invariance of a Free Field\n 22.3 Representation of Anticommutation Relations of Spinor Fields\n 22.3.1 Representation of Anticommutation Relations of the Operators of Creation and Annihilation of Fermions and Antifermions\n 22.3.2 Representation of Anticommutation Relations in the Configuration Space\n 22.4 Space of States of a Free Electromagnetic Field\n 22.5 Space of Occupation Numbers\n 23 Green Functions\n 23.1 Green Functions of the Scalar Field\n 23.2 The Green Functions of Spinor, Vector, and Electromagnetic Fields\n 23.3 Time-Ordered Product and Green Functions\n 23.4 Wick Theorems\n 23.4.1 Wick Theorem for Normal Products\n 23.4.2 Wick Theorem for a Time-Ordered Product\n 23.4.3 Generalized Wick Theorem\n 23.5 Operation of Multiplication and the Regularization of Distributions\n 23.6 N-Point Green Functions of Free Fields\n 24 Problems to Part IV\nV Quantum Theory of Interacting Fields. General Problems\n 25 Construction of Quantum Interacting Fields and Problems of This Construction\n 25.1 Formal Construction of a Quantum Field\n 25.2 Mathematical Problems of Construction of a Quantum Interacting Field\n 26 Scattering Theory. Scattering Matrix\n 26.1 Quantum Description of Scattering. Definition of Scattering Operator\n 26.2 Formal Construction of the Scattering Operator by the Method of Perturbation Theory\n 26.3 Main Properties of the S-Operator\n 26.3.1 Normal Form of the Operator S\n 26.3.2 Invariance of the Scattering Matrix under Lorentz Transformations and Transformations of Charge Conjugation\n 26.3.3 Unitarity of the Scattering Operator\n 26.3.4 Law of Conservation of Energy\n 26.3.5 Matrix Elements of the S-Operator and the Scattering Amplitude\n 26.4 Feynman Diagrams\n 26.4.1 Feynman Diagrams for the S-Operator\n 26.4.2 Feynman Diagrams for Coefficient Functions of the S-Operator\n 26.4.3 Feynman Diagrams for Matrix Elements of the S-Operator\n 26.5 Effective Cross-Sections and Scattering Matrix\n 26.5.1 Classical Picture\n 26.5.2 Quantum Picture\n 27 Equations for Coefficient Functions of the S-Matrix\n 27.1 Creation and Annihilation Operators of External Lines of Feynman Diagrams\n 27.2 Equations of the Resolvent Type\n 27.3 Equations of the Evolution Type\n 28 Green Functions and Scattering Matrix\n 28.1 Green Functions and the S-Matrix in the Interaction Picture\n 28.2 Schwinger Equation for Green Functions\n 28.3 On the Relationship between the Green Functions and the Coefficient Functions of the Scattering S-Operator\n 28.4 Equations for Green Functions in Terms of Functional Derivatives\n 28.5 Equations for Truncated Green Functions\n 28.6 Equations for One-Particle Irreducible Green Functions. Dyson Equation\n 28.7 Spectral Representation of the 2-Point Green Function (Källén-Lehmann Representation)\n 29 On Renormalization in Perturbation Theory\n 29.1 Primitively-Divergent Diagrams. Separation of Divergences by the Pauli-Villars Method\n 29.2 Degree of Divergence of Feynman Diagram\n 29.3 Elimination of Divergences by the Method of Bogoliubov-Parasiuk R-Operation\n 29.4 R-Operation and Counterterms of a Lagrangian\n 29.5 Classification of Interactions: Renormalizable and Nonrenormalizable Theories\n 29.6 Relationship between Counterterms and the Renormalization of Main Constants of the Theory\n 29.7 Equivalent Types of Renormalizations\n 30 Method of Functional (Path) Integrals in Quantized Field Theory\n 30.1 Notion of Path Integration and Main Formulas\n 30.2 Formalism of Feynman Integrals (Path Integrals) in Quantum Mechanics\n 30.3 Formalism of Feynman Integrals for Systems with Constraints\n 30.4 Path Integral Representation for Scalar Fields\n 30.5 Path Integral Representation for Fermi Fields\n 31 Problems to Part V\nVI Axiomatic and Euclidean Field Theories\n 32 Wightman Axiomatics\n 32.1 Wightman Axioms for Real Scalar Fields\n 32.2 Wightman Functions and Their Properties\n 32.3 Reconstruction Theorem\n 33 Other Axiomatic Approaches\n 33.1 Haag-Ruelle Scattering Theory (HRST)\n 33.2 Lehmann–Symanzik–Zimmermann Axiomatics\n 33.3 Bogoliubov–Medvedev–Polivanov (BMP) Axiomatic Approach\n 34 Euclidean Field Theory\n 34.1 Analytic Continuation of Feynman Amplitudes\n 34.2 Operators of Free Euclidean Fields\n 34.2.1 Real scalar field\n 34.2.2 Euclidean Fermi fields\n 34.3 Euclidean Green Functions of a Free Scalar Field\n 34.4 Euclidean Green Functions of Interacting Fields\n 35 Euclidean Axiomatics\n 35.1 Analytic Continuation of Generalized Wightman Functions\n 35.2 Euclidean Green Functions. Osterwalder–Schrader Axioms\n 35.3 Reconstruction of the Wightman Theory\n 36 Problems to Part VI\nVII Quantum Theory of Gauge Fields\n 37 Quantum Electrodynamics (QED)\n 37.1 Quantization of Interacting Electromagnetic Fields\n 37.1.1 Gupta–Bleuler Formalism for Interacting Electromagnetic Fields\n 37.1.2 Quantization of Interacting Electromagnetic Fields in the Coulomb Gauge\n 37.1.3 Photon Propagator and Gauge Conditions\n 37.2 S-Matrix in QED\n 37.2.1 Perturbation Theory. Feynman Diagrams\n 37.2.2 Coefficient Functions of the S-Matrix in Terms of Creation and Annihilation Operators of Lines of Feynman Diagrams\n 37.2.3 Furry Theorem\n 37.2.4 Gauge Invariance for Coefficient Functions of the S-Operator\n 37.3 Equations for Green Functions and Coefficient Functions of the S-Matrix\n 37.3.1 Schwinger Equation\n 37.3.2 System of Equations for Self-Energy and Vertex Parts of Green Functions\n 37.4 Divergences in QED and Methods for Their Elimination\n 37.4.1 Primitively-Divergent Diagrams and Their Regularization\n 37.4.2 Mass and Charge Renormalization of Electron (Positron)\n 37.5 Spectral Representations of 2-Point Green Functions\n 38 Quantization of Gauge Fields\n 38.1 Path Integral for Green Functions in QED (Coulomb Gauge)\n 38.2 Covariant Gauges: Popov–Faddeev–de Witt Method\n 38.3 Covariant Quantization of Electromagnetic Interaction\n 38.3.1 Connection between Different Gauges\n 38.3.2 Ward Identity\n 38.4 Quantization of Yang-Mills Fields Interacting with Matter Fields\n 38.5 Faddeev-Popov Ghosts\n 38.6 BRST-Invariance\n 39 Standard Models of Interactions\n 39.1 Renormalization of Gauge Theories\n 39.2 On the Masses of Gluons and Spontaneous Symmetry Breakdown\n 39.2.1 Connection of the Radius of Interaction and the Mass of Exchange Bosons\n 39.2.2 Are Theories with Nonzero Mass of Exchange Bosons Renormalizable?\n 39.2.3 Spontaneous Breakdown of the U(1)-Symmetry\n 39.2.4 Spontaneous Breakdown of the Local SU(N)-Symmetry.\n 39.3 Models of Interactions of Elementary Particles\n 39.3.1 Strong Interaction. Model of QCD\n 39.3.2 Weak and Electroweak Interactions\n 40 Problems to Part VII\nAppendix Hints for the Solution of Problems\nBibliography\nIndex