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De Gruyter2012Theory of Interacting Quantum Fields
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Contents Preface v Notation viii 0 Introduction 1 I Symmetry Groups of Elementary Particles 1 Lorentz Group 8 1.1 Euclidean and Minkowski Spaces. Relativistic Notation . . . . . . . 8 1.2 HomogeneousLorentzGroup . . . . . . . . . . . . . . . . . . . . . 11 1.3 Inhomogeneous Lorentz Group–Poincaré Group . . . . . . . . . . . 14 1.4 ComplexLorentzTransformations . . . . . . . . . . . . . . . . . . 15 1.5 Representations of the Lorentz and Poincaré Groups, Field Functions, and Physical States . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5.1 Representation D.0;0/ . . . . . . . . . . . . . . . . . . . . 19 1.5.2 Representations D. 1 2 ;0/ and D.0; 1 2 / . . . . . . . . . . . . . 20 1.5.3 Representation D. 1 2 ; 1 2 / . . . . . . . . . . . . . . . . . . . . 20 2 Groups of Internal Symmetries 23 2.1 Abelian Unitary Group U.1/ . . . . . . . . . . . . . . . . . . . . . 23 2.2 Charge Conjugation C . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3 Special Unitary Group SU.n/ . . . . . . . . . . . . . . . . . . . . 24 2.3.1 SU.2/Symmetry . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.2 SU.3/Symmetry . . . . . . . . . . . . . . . . . . . . . . . 27 2.4 Groups of Local Transformations. Gauge Group . . . . . . . . . . . 29 3 Problems to Part I 35 II Classical Theory of the Free Fields 4 Lagrangian and Hamiltonian Formalisms of the Classical Field Theory 39 4.1 Variational Principle and Canonical Formalism of Classical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . 39 4.1.1 LagrangianEquations . . . . . . . . . . . . . . . . . . . . 39 xii Contents 4.1.2 Canonical Variables. Hamiltonian Equations . . . . . . . . 41 4.1.3 PoissonBrackets. Integrals ofMotion . . . . . . . . . . . . 42 4.1.4 Canonical Formalism in the Presence of Constraints . . . . 43 4.2 From Classical to Quantum Mechanics. Primary Quantization . . . 48 4.3 General Requirements to the Lagrangians of the Field Theory . . . . 52 4.4 Lagrange–EulerEquations . . . . . . . . . . . . . . . . . . . . . . 53 4.5 Noether’sTheoremandDynamic Invariants . . . . . . . . . . . . . 54 4.6 Vector of Energy-Momentum . . . . . . . . . . . . . . . . . . . . . 56 4.7 Tensors ofAngularMomentumandSpin . . . . . . . . . . . . . . . 57 4.8 Charge and the Vector of Current . . . . . . . . . . . . . . . . . . . 59 4.9 Canonical Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5 Classical Theory of Free Scalar Fields 61 5.1 Klein–Fock–Gordon Equation . . . . . . . . . . . . . . . . . . . . 61 5.2 Relativistic Invariance of the Klein–Fock–Gordon Equation . . . . . 62 5.3 Solutions of the Klein–Fock–Gordon Equation . . . . . . . . . . . . 64 5.4 Interpretation of Solutions. Hilbert Space of States . . . . . . . . . 66 5.5 bC, bP, and bT Transformations . . . . . . . . . . . . . . . . . . . . . 70 5.5.1 Transformation of Charge Conjugation bC . . . . . . . . . . 70 5.5.2 Space Reflection bP . . . . . . . . . . . . . . . . . . . . . . 72 5.5.3 Time Reversal bT . . . . . . . . . . . . . . . . . . . . . . . 73 5.5.4 bC bPbT-Invariance . . . . . . . . . . . . . . . . . . . . . . . 73 5.6 Representations of the Lorentz Group in the Space of States . . . . . 74 5.7 Lagrangian Formalism of the Scalar Field. Dynamic Invariants . . . 78 6 Spinor Field 82 6.1 Dirac Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 6.1.1 Construction of the Dirac Equation . . . . . . . . . . . . . 82 6.1.2 Properties of Dirac Matrices. Conjugate Equation . . . . . . 83 6.2 Relativistic Invariance . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.2.1 Transformation Properties of the Spinor Field . . . . . . . . 87 6.2.2 On Reducible and Irreducible Spinor Representations . . . . 91 6.2.3 Transformation Properties of Bilinear Forms NO . . . . . 92 6.3 Solutions of the Dirac Equation . . . . . . . . . . . . . . . . . . . . 94 6.3.1 Structure ofSolutions in theMomentumSpace . . . . . . . 94 6.3.2 Classification of Solutions. Helicity . . . . . . . . . . . . . 97 6.3.3 Relations Between Spinors . . . . . . . . . . . . . . . . . . 102 6.3.4 Wave Functions of the Electron and Positron. Charge Conjugation . . . . . . . . . . . . . . . . . . . . . 104 Contents xiii 6.3.5 bC bPbT -Transformation . . . . . . . . . . . . . . . . . . . . 107 6.4 Lagrangian Formalism . . . . . . . . . . . . . . . . . . . . . . . . 110 6.5 Representations of the Lorentz Group . . . . . . . . . . . . . . . . 115 6.5.1 Hilbert Space of States . . . . . . . . . . . . . . . . . . . . 115 6.5.2 Representations of the Lorentz Group in the Space of States 117 6.6 Applications of the Dirac Equation . . . . . . . . . . . . . . . . . . 118 6.6.1 Dirac Equation in the Presence of External Fields . . . . . . 118 6.7 Massless Spinor Field . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.7.1 Two-component Massless Spinor Field . . . . . . . . . . . 121 6.7.2 Relativistic Invariance . . . . . . . . . . . . . . . . . . . . 123 6.7.3 Are There Actual Particles Corresponding to the Massless Spinor Fields? Physical Interpretation ofSolutions. Neutrino . . . . . . . . . . . . . . . . . . . . 123 6.7.4 Lagrangian andDynamic Invariants . . . . . . . . . . . . . 125 6.7.5 On theMass ofNeutrino andMajoranaSpinors . . . . . . . 126 7 Vector Fields 128 7.1 Lagrangian Formalism . . . . . . . . . . . . . . . . . . . . . . . . 128 7.2 Representations in the Momentum Space . . . . . . . . . . . . . . . 131 7.3 Decomposition into the Longitudinal and Transverse Components . 131 7.4 bP;bT ; bC-Transformations . . . . . . . . . . . . . . . . . . . . . . . 133 8 Electromagnetic Field 135 8.1 MaxwellEquations . . . . . . . . . . . . . . . . . . . . . . . . . . 135 8.2 Potential of the Electromagnetic Field . . . . . . . . . . . . . . . . 136 8.3 Gradient Transformations and the Lorentz Condition: Transversality Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 8.4 Lagrangian Formalism for Electromagnetic Fields . . . . . . . . . . 139 8.5 Transversal, Longitudinal, and Time Components of the Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . 141 8.6 Quantum-Mechanical Characteristics of Photons . . . . . . . . . . . 143 8.7 bC; bP;bT-Transformations . . . . . . . . . . . . . . . . . . . . . . . 146 8.8 Consistency of the Lorentz and Gauge Transformations. VariousTypes ofGauges . . . . . . . . . . . . . . . . . . . . . . . 146 9 Equations for Fields with Higher Spins 149 9.1 Fields with Spin 3=2 . . . . . . . . . . . . . . . . . . . . . . . . . 149 9.2 Particles with Spin 2 . . . . . . . . . . . . . . . . . . . . . . . . . 151 10 Problems to Part II 152 xiv Contents III Classical Theory of Interacting Fields 11 Gauge Theory of the Electromagnetic Interaction 158 11.1 Principle ofGauge Invariance in theMaxwellTheory . . . . . . . . 158 11.2 Schrödinger Equation and Gradient (Gauge) Invariance . . . . . . . 159 11.3 Gauge Principle as the Dynamical Principle of Interaction between the Electromagnetic and Electron-Positron Fields . . . . . . . . . . 161 12 Classical Theory of Yang–Mills Fields 164 12.1 Gauge Principle and the Lagrangian of the Yang–Mills Fields . . . . 164 12.2 Equations of Motion for the Free Yang–Mills fields . . . . . . . . . 167 12.3 Yang–Mills Fields for Arbitrary Representations of the Group SU.N/ . . . . . . . . . . . . . . . . . . . . . . . . . 169 13 Masses of Particles and Spontaneous Breaking of Symmetry 171 13.1 SpontaneousBreaking ofSymmetry . . . . . . . . . . . . . . . . . 172 13.2 Higgs Mechanism for the Local U.1/Symmetry . . . . . . . . . . . 174 13.3 Higgs Mechanism for the Local SU.2/ symmetry . . . . . . . . . . 176 13.4 Generation of the Masses of Fermions . . . . . . . . . . . . . . . . 179 14 On the Construction of the General Lagrangian of Interacting Fields 181 14.1 Lagrangian of theQCD . . . . . . . . . . . . . . . . . . . . . . . . 183 14.2 Lagrangian of Weak Interactions . . . . . . . . . . . . . . . . . . . 184 14.3 On the Electroweak Interactions . . . . . . . . . . . . . . . . . . . 188 14.4 On the Lagrangian of Great Unification . . . . . . . . . . . . . . . 189 15 Solutions of the Equations for Classical Fields: Solitary Waves, Solitons, Instantons 191 16 Problems to Part III 197 IV Second Quantization of Fields 17 Axioms and General Principles of Quantization 201 17.1 Why Do We Need the Procedure of Second Quantization? Operator Nature of the Field Functions . . . . . . . . . . . . . . . . . . . . . 201 17.2 Schrödinger, Heisenberg, and Interaction Pictures . . . . . . . . . . 202 17.3 Axioms of Quantization . . . . . . . . . . . . . . . . . . . . . . . . 204 17.4 Relativistic Heisenberg Equation for Quantized Fields . . . . . . . . 213 17.4.1 Heisenberg Equation for a Free Scalar Field . . . . . . . . . 214 17.4.2 Heisenberg Equation for a Free Electron-Positron Field . . . 215 Contents xv 17.5 Physical Content of Positive- and Negative-Frequency Solutions of Equations for Free-Field Operators . . . . . . . . . . . . . . . . . . 217 18 Quantization of the Free Scalar Field 218 18.1 Commutation Relations. Commutator Functions . . . . . . . . . . . 218 18.2 Complex Scalar Field . . . . . . . . . . . . . . . . . . . . . . . . . 220 18.3 Operator Relations for Dynamic Invariants . . . . . . . . . . . . . . 221 19 Quantization of the Free Spinor Field 222 19.1 Commutator Functions of Fermi Fields . . . . . . . . . . . . . . . . 222 19.2 Dynamic Invariants of a Free Spinor Field . . . . . . . . . . . . . . 224 20 Quantization of the Vector and Electromagnetic Fields. Specific Features of the Quantization of Gauge Fields 225 20.1 Quantization of the Complex Vector Field . . . . . . . . . . . . . . 225 20.2 Quantization of an Electromagnetic Field . . . . . . . . . . . . . . 229 20.2.1 Specific Features and Difficulties of the Quantization of an Electromagnetic Field . . . . . . . . . . . . . . . . . . . . 229 20.2.2 Gupta–Bleuler Formalism . . . . . . . . . . . . . . . . . . 232 20.2.3 Canonical Method of Quantization . . . . . . . . . . . . . . 236 20.3 On the Quantization of Gauge Fields . . . . . . . . . . . . . . . . . 238 21 CPT . Spin and Statistics 240 21.1 The Transformation of Charge Conjugation . . . . . . . . . . . . . 241 21.2 The Transformation of Space Reflection . . . . . . . . . . . . . . . 242 21.3 The Transformation of Time Reversal . . . . . . . . . . . . . . . . 243 21.4 CPT -Theorem and the Connection of Spin and Statistics . . . . . . 246 21.5 Proofof thePauliTheorem . . . . . . . . . . . . . . . . . . . . . . 248 22 Representations of Commutation and Anticommutation Relations 250 22.1 GeneralStructure of theFockSpace . . . . . . . . . . . . . . . . . 250 22.2 Representations of Commutation Relations for a Free Real Scalar Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 22.2.1 The Fock Space of Free Scalar Bosons . . . . . . . . . . . . 252 22.2.2 Operators of Creation and Annihilation in the Fock Space. Momentum Representation . . . . . . . . . . . . . . . . . . 252 22.2.3 Vacuum State of Free Particle. Cyclicity of Vacuum. Set of Exponential Vectors . . . . . . . . . . . . . . . . . . . . . 256 22.2.4 Construction of Representations of Commutation Relations for a Complex Scalar Field . . . . . . . . . . . . . . . . . . 259 xvi Contents 22.2.5 Construction of Representations of Commutation Relations in the Configuration Space. Relativistic Invariance of a Free Field . . . . . . . . . . . . . . . . . . . . . . . . . . 260 22.3 Representation of Anticommutation Relations of Spinor Fields . . . 262 22.3.1 Representation of Anticommutation Relations of the Operators of Creation and Annihilation of Fermions and Antifermions . . . . . . . . . . . . . . . . . . . . . . . . . 262 22.3.2 Representation of Anticommutation Relations in the Configuration Space . . . . . . . . . . . . . . . . . . . . . 265 22.4 Space of States of a Free Electromagnetic Field . . . . . . . . . . . 267 22.5 Space of Occupation Numbers . . . . . . . . . . . . . . . . . . . . 271 23 Green Functions 274 23.1 Green Functions of the Scalar Field . . . . . . . . . . . . . . . . . 274 23.2 The Green Functions of Spinor, Vector, and Electromagnetic Fields . 277 23.3 Time-Ordered Product and Green Functions . . . . . . . . . . . . . 278 23.4 WickTheorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 23.4.1 WickTheoremforNormalProducts . . . . . . . . . . . . . 279 23.4.2 Wick Theorem for a Time-Ordered Product . . . . . . . . . 281 23.4.3 Generalized Wick Theorem . . . . . . . . . . . . . . . . . 284 23.5 Operation of Multiplication and the Regularization of Distributions . 284 23.6 N-Point Green Functions of Free Fields . . . . . . . . . . . . . . . 285 24 Problems to Part IV 287 V Quantum Theory of Interacting Fields. General Problems 25 Construction of Quantum Interacting Fields and Problems of This Construction 291 25.1 Formal Construction of a Quantum Field . . . . . . . . . . . . . . . 291 25.2 Mathematical Problems of Construction of a Quantum Interacting Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 26 Scattering Theory. Scattering Matrix 298 26.1 Quantum Description of Scattering. Definition of Scattering Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 26.2 Formal Construction of the Scattering Operator by the Method of Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . 302 26.3 Main Properties of the S-Operator . . . . . . . . . . . . . . . . . . 305 26.3.1 Normal Form of the Operator S . . . . . . . . . . . . . . . 305 26.3.2 Invariance of the Scattering Matrix under Lorentz Transformations and Transformations of Charge Conjugation 309 Contents xvii 26.3.3 Unitarity of the Scattering Operator . . . . . . . . . . . . . 310 26.3.4 Law of Conservation of Energy . . . . . . . . . . . . . . . 311 26.3.5 Matrix Elements of the S-Operator and the Scattering Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 26.4 FeynmanDiagrams . . . . . . . . . . . . . . . . . . . . . . . . . . 316 26.4.1 Feynman Diagrams for the S-Operator . . . . . . . . . . . 317 26.4.2 Feynman Diagrams for Coefficient Functions of the S-Operator . . . . . . . . . . . . . . . . . . . . . . . 317 26.4.3 Feynman Diagrams for Matrix Elements of the S-Operator . 319 26.5 Effective Cross-Sections and Scattering Matrix . . . . . . . . . . . 322 26.5.1 Classical Picture . . . . . . . . . . . . . . . . . . . . . . . 323 26.5.2 Quantum Picture . . . . . . . . . . . . . . . . . . . . . . . 325 27 Equations for Coefficient Functions of the S-Matrix 327 27.1 Creation and Annihilation Operators of External Lines of Feynman Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 27.2 Equations of theResolventType . . . . . . . . . . . . . . . . . . . 331 27.3 Equations of theEvolutionType . . . . . . . . . . . . . . . . . . . 334 28 Green Functions and Scattering Matrix 336 28.1 Green Functions and the S-Matrix in the Interaction Picture . . . . . 336 28.2 Schwinger Equation for Green Functions . . . . . . . . . . . . . . . 338 28.3 On the Relationship between the Green Functions and the Coefficient Functions of the Scattering S-Operator . . . . . . . . . . . . . . . . 341 28.4 Equations for Green Functions in Terms of Functional Derivatives . 342 28.5 Equations for Truncated Green Functions . . . . . . . . . . . . . . 344 28.6 Equations for One-Particle Irreducible Green Functions. Dyson Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 28.7 Spectral Representation of the 2-Point Green Function (Källén–Lehmann Representation) . . . . . . . . . . . . . . . . . . 353 29 On Renormalization in Perturbation Theory 358 29.1 Primitively-Divergent Diagrams. Separation of Divergences by the Pauli–Villars Method . . . . . . . . . . . . . . . . . . . . . . . . . 358 29.2 Degree ofDivergence ofFeynmanDiagram . . . . . . . . . . . . . 365 29.3 Elimination of Divergences by the Method of Bogoliubov–Parasiuk R-Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 29.4 R-Operation and Counterterms of a Lagrangian . . . . . . . . . . . 374 29.5 Classification of Interactions: Renormalizable and Nonrenormalizable Theories . . . . . . . . . . . . . . . . . . . . . 379 xviii Contents 29.6 Relationship between Counterterms and the Renormalization of Main Constants of theTheory . . . . . . . . . . . . . . . . . . . . . . . . 380 29.7 Equivalent Types of Renormalizations . . . . . . . . . . . . . . . . 387 30 Method of Functional (Path) Integrals in Quantized Field Theory 391 30.1 Notion of Path Integration and Main Formulas . . . . . . . . . . . . 392 30.2 Formalism of Feynman Integrals (Path Integrals) in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 30.3 Formalism of Feynman Integrals for Systems with Constraints . . . 405 30.4 Path Integral Representation for Scalar Fields . . . . . . . . . . . . 409 30.5 Path Integral Representation for Fermi Fields . . . . . . . . . . . . 411 31 Problems to Part V 416 VI Axiomatic and Euclidean Field Theories 32 Wightman Axiomatics 423 32.1 Wightman Axioms for Real Scalar Fields . . . . . . . . . . . . . . 423 32.2 Wightman Functions and Their Properties . . . . . . . . . . . . . . 425 32.3 Reconstruction Theorem . . . . . . . . . . . . . . . . . . . . . . . 427 33 Other Axiomatic Approaches 429 33.1 Haag–Ruelle Scattering Theory (HRST) . . . . . . . . . . . . . . . 429 33.2 Lehmann–Symanzik–Zimmermann Axiomatics . . . . . . . . . . . 432 33.3 Bogoliubov–Medvedev–Polivanov (BMP) Axiomatic Approach . . 435 34 Euclidean Field Theory 439 34.1 Analytic Continuation of Feynman Amplitudes . . . . . . . . . . . 440 34.2 Operators of Free Euclidean Fields . . . . . . . . . . . . . . . . . . 442 34.2.1 Real scalar field . . . . . . . . . . . . . . . . . . . . . . . . 442 34.2.2 Euclidean Fermi fields . . . . . . . . . . . . . . . . . . . . 444 34.3 Euclidean Green Functions of a Free Scalar Field . . . . . . . . . . 446 34.4 Euclidean Green Functions of Interacting Fields . . . . . . . . . . . 447 35 Euclidean Axiomatics 453 35.1 Analytic Continuation of Generalized Wightman Functions . . . . . 453 35.2 Euclidean Green Functions. Osterwalder–Schrader Axioms . . . . . 455 35.3 Reconstruction of the Wightman Theory . . . . . . . . . . . . . . . 457 36 Problems to Part VI 460 Contents xix VII Quantum Theory of Gauge Fields 37 Quantum Electrodynamics (QED) 465 37.1 Quantization of Interacting Electromagnetic Fields . . . . . . . . . 466 37.1.1 Gupta–Bleuler Formalism for Interacting Electromagnetic Fields . . . . . . . . . . . . . . . . . . . . 466 37.1.2 Quantization of Interacting Electromagnetic Fields in theCoulombGauge . . . . . . . . . . . . . . . . . . . . 467 37.1.3 PhotonPropagator andGaugeConditions . . . . . . . . . . 469 37.2 S-Matrix inQED . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 37.2.1 Perturbation Theory. Feynman Diagrams . . . . . . . . . . 471 37.2.2 Coefficient Functions of the S-Matrix in Terms of Creation and Annihilation Operators of Lines of Feynman Diagrams . 474 37.2.3 FurryTheorem . . . . . . . . . . . . . . . . . . . . . . . . 476 37.2.4 Gauge Invariance for Coefficient Functions of the S-Operator . . . . . . . . . . . . . . . . . . . . . . . . . . 479 37.3 Equations for Green Functions and Coefficient Functions of the S-Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 37.3.1 Schwinger Equation . . . . . . . . . . . . . . . . . . . . . 480 37.3.2 System of Equations for Self-Energy and Vertex Parts ofGreenFunctions . . . . . . . . . . . . . . . . . . . . . . 482 37.4 Divergences in QED and Methods for Their Elimination . . . . . . 485 37.4.1 Primitively-Divergent Diagrams and Their Regularization . 485 37.4.2 Mass and Charge Renormalization of Electron (Positron) . . 490 37.5 Spectral Representations of 2-Point Green Functions . . . . . . . . 493 38 Quantization of Gauge Fields 498 38.1 Path Integral for Green Functions in QED (Coulomb Gauge) . . . . 499 38.2 Covariant Gauges: Popov–Faddeev–de Witt Method . . . . . . . . . 502 38.3 Covariant Quantization of Electromagnetic Interaction . . . . . . . 506 38.3.1 Connection between Different Gauges . . . . . . . . . . . . 507 38.3.2 Ward Identity . . . . . . . . . . . . . . . . . . . . . . . . . 508 38.4 Quantization of Yang–Mills Fields Interacting with Matter Fields . . 510 38.5 Faddeev–PopovGhosts . . . . . . . . . . . . . . . . . . . . . . . . 514 38.6 BRST-Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 39 Standard Models of Interactions 521 39.1 Renormalization of Gauge Theories . . . . . . . . . . . . . . . . . 521 39.2 On the Masses of Gluons and Spontaneous Symmetry Breakdown . 530 39.2.1 Connection of the Radius of Interaction and the Mass of ExchangeBosons . . . . . . . . . . . . . . . . . . . . . . . 530 xx Contents 39.2.2 Are Theories with Nonzero Mass of Exchange Bosons Renormalizable? . . . . . . . . . . . . . . . . . . . . . . . 532 39.2.3 Spontaneous Breakdown of the U.1/-Symmetry . . . . . . . 533 39.2.4 Spontaneous Breakdown of the Local SU.N/-Symmetry . . 535 39.3 Models of Interactions of Elementary Particles . . . . . . . . . . . . 537 39.3.1 Strong Interaction. Model of QCD . . . . . . . . . . . . . . 537 39.3.2 Weak and Electroweak Interactions . . . . . . . . . . . . . 539 40 Problems to Part VII 542 Appendix Hints for the Solution of Problems 544 Bibliography 549 Index 562 |
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2015-03-05 06:54
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2015-03-05 15:54
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2015-03-06 19:25
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五星好評 頂一下，感謝分享！
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2020-08-24 16:27
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