Theory of Interacting Quantum Fields (eBook)

lt;!doctype html public "-//w3c//dtd html 4.0 transitional//en"> Alexei L. Rebenko, Department of Mathematics,National Academy of Sciences of Ukraine, Kiev, Ukraine.

Preface 5
Notation 8
0 Introduction 21
I Symmetry Groups of Elementary Particles 25
1 Lorentz Group 28
1.1 Euclidean and Minkowski Spaces. Relativistic Notation 28
1.2 Homogeneous Lorentz Group 31
1.3 Inhomogeneous Lorentz Group-Poincaré Group 34
1.4 Complex Lorentz Transformations 35
1.5 Representations of the Lorentz and Poincaré Groups, Field Functions, and Physical States 36
1.5.1 Representation D(0,0) 39
1.5.2 Representations D(1/2,0) and D(0,1/2) 40
1.5.3 Representation D(1/2,1/2) 40
2 Groups of Internal Symmetries 43
2.1 Abelian Unitary Group U(1) 43
2.2 Charge Conjugation C 44
2.3 Special Unitary Group SU(n) 44
2.3.1 SU(2) Symmetry 45
2.3.2 SU(3) Symmetry 47
2.4 Groups of Local Transformations. Gauge Group 49
3 Problems to Part I 55
II Classical Theory of the Free Fields 57
4 Lagrangian and Hamiltonian Formalisms of the Classical Field Theory 59
4.1 Variational Principle and Canonical Formalism of Classical Mechanics 59
4.1.1 Lagrangian Equations 59
4.1.2 Canonical Variables. Hamiltonian Equations 61
4.1.3 Poisson Brackets. Integrals of Motion 62
4.1.4 Canonical Formalism in the Presence of Constraints 63
4.2 From Classical to Quantum Mechanics. Primary Quantization 68
4.3 General Requirements to the Lagrangians of the Field Theory 72
4.4 Lagrange–Euler Equations 73
4.5 Noether’s Theorem and Dynamic Invariants 74
4.6 Vector of Energy-Momentum 76
4.7 Tensors of Angular Momentum and Spin 77
4.8 Charge and the Vector of Current 79
4.9 Canonical Variables 80
5 Classical Theory of Free Scalar Fields 81
5.1 Klein–Fock–Gordon Equation 81
5.2 Relativistic Invariance of the Klein–Fock–Gordon Equation 82
5.3 Solutions of the Klein–Fock–Gordon Equation 84
5.4 Interpretation of Solutions. Hilbert Space of States 86
5.5 C^, P^, and T^^ Transformations 90
5.5.1 Transformation of Charge Conjugation C^ 90
5.5.2 Space Reflection P^ 92
5.5.3 Time Reversal T^^ 93
5.5.4 C^P^T^^-Invariance 93
5.6 Representations of the Lorentz Group in the Space of States 94
5.7 Lagrangian Formalism of the Scalar Field. Dynamic Invariants 98
6 Spinor Field 102
6.1 Dirac Equation 102
6.1.1 Construction of the Dirac Equation 102
6.1.2 Properties of Dirac Matrices. Conjugate Equation 103
6.2 Relativistic Invariance 106
6.2.1 Transformation Properties of the Spinor Field 107
6.2.2 On Reducible and Irreducible Spinor Representations 111
6.2.3 Transformation Properties of Bilinear Forms .¯O. 112
6.3 Solutions of the Dirac Equation 114
6.3.1 Structure of Solutions in the Momentum Space 114
6.3.2 Classification of Solutions. Helicity 117
6.3.3 Relations Between Spinors 122
6.3.4 Wave Functions of the Electron and Positron. Charge Conjugation 124
6.3.5 C^P^T^^-Transformation 127
6.4 Lagrangian Formalism 130
6.5 Representations of the Lorentz Group 135
6.5.1 Hilbert Space of States 135
6.5.2 Representations of the Lorentz Group in the Space of States 137
6.6 Applications of the Dirac Equation 138
6.6.1 Dirac Equation in the Presence of External Fields 138
6.7 Massless Spinor Field 141
6.7.1 Two-component Massless Spinor Field 141
6.7.2 Relativistic Invariance 143
6.7.3 Are There Actual Particles Corresponding to the Massless Spinor Fields? Physical Interpretation of Solutions. Neutrino 143
6.7.4 Lagrangian and Dynamic Invariants 145
6.7.5 On the Mass of Neutrino and Majorana Spinors 146
7 Vector Fields 148
7.1 Lagrangian Formalism 148
7.2 Representations in the Momentum Space 151
7.3 Decomposition into the Longitudinal and Transverse Components 151
7.4 P^, T^^, C^-Transformations 153
8 Electromagnetic Field 155
8.1 Maxwell Equations 155
8.2 Potential of the Electromagnetic Field 156
8.3 Gradient Transformations and the Lorentz Condition: Transversality Condition 157
8.4 Lagrangian Formalism for Electromagnetic Fields 159
8.5 Transversal, Longitudinal, and Time Components of the Electromagnetic Field 161
8.6 Quantum-Mechanical Characteristics of Photons 163
8.7 C^, P^, T^^-Transformations 166
8.8 Consistency of the Lorentz and Gauge Transformations. Various Types of Gauges 166
9 Equations for Fields with Higher Spins 169
9.1 Fields with Spin 3/2 169
9.2 Particles with Spin 2 171
10 Problems to Part II 172
III Classical Theory of Interacting Fields 175
11 Gauge Theory of the Electromagnetic Interaction 178
11.1 Principle of Gauge Invariance in the Maxwell Theory 178
11.2 Schrödinger Equation and Gradient (Gauge) Invariance 179
11.3 Gauge Principle as the Dynamical Principle of Interaction between the Electromagnetic and Electron-Positron Fields 181
12 Classical Theory of Yang-Mills Fields 184
12.1 Gauge Principle and the Lagrangian of the Yang–Mills Fields 184
12.2 Equations of Motion for the Free Yang–Mills fields 187
12.3 Yang-Mills Fields for Arbitrary Representations of the Group SU(N) 189
13 Masses of Particles and Spontaneous Breaking of Symmetry 191
13.1 Spontaneous Breaking of Symmetry 192
13.2 Higgs Mechanism for the Local U(1) Symmetry 194
13.3 Higgs Mechanism for the Local SU(2) symmetry 196
13.4 Generation of the Masses of Fermions 199
14 On the Construction of the General Lagrangian of Interacting Fields 201
14.1 Lagrangian of the QCD 203
14.2 Lagrangian of Weak Interactions 204
14.3 On the Electroweak Interactions 208
14.4 On the Lagrangian of Great Unification 209
15 Solutions of the Equations for Classical Fields: Solitary Waves, Solitons, Instantons 211
16 Problems to Part III 217
IV Second Quantization of Fields 219
17 Axioms and General Principles of Quantization 221
17.1 Why Do We Need the Procedure of Second Quantization? Operator Nature of the Field Functions 221
17.2 Schrödinger, Heisenberg, and Interaction Pictures 222
17.3 Axioms of Quantization 224
17.4 Relativistic Heisenberg Equation for Quantized Fields 233
17.4.1 Heisenberg Equation for a Free Scalar Field 234
17.4.2 Heisenberg Equation for a Free Electron-Positron Field 235
17.5 Physical Content of Positive- and Negative-Frequency Solutions of Equations for Free-Field Operators 237
18 Quantization of the Free Scalar Field 238
18.1 Commutation Relations. Commutator Functions 238
18.2 Complex Scalar Field 240
18.3 Operator Relations for Dynamic Invariants 241
19 Quantization of the Free Spinor Field 242
19.1 Commutator Functions of Fermi Fields 242
19.2 Dynamic Invariants of a Free Spinor Field 244
20 Quantization of the Vector and Electromagnetic Fields. Specific Features of the Quantization of Gauge Fields 245
20.1 Quantization of the Complex Vector Field 245
20.2 Quantization of an Electromagnetic Field 249
20.2.1 Specific Features and Difficulties of the Quantization of an Electromagnetic Field 249
20.2.2 Gupta–Bleuler Formalism 252
20.2.3 Canonical Method of Quantization 256
20.3 On the Quantization of Gauge Fields 258
21 CPT. Spin and Statistics 260
21.1 The Transformation of Charge Conjugation 261
21.2 The Transformation of Space Reflection 262
21.3 The Transformation of Time Reversal 263
21.4 CPT-Theorem and the Connection of Spin and Statistics 266
21.5 Proof of the Pauli Theorem 268
22 Representations of Commutation and Anticommutation Relations 270
22.1 General Structure of the Fock Space 270
22.2 Representations of Commutation Relations for a Free Real Scalar Field 272
22.2.1 The Fock Space of Free Scalar Bosons 272
22.2.2 Operators of Creation and Annihilation in the Fock Space. Momentum Representation 272
22.2.3 Vacuum State of Free Particle. Cyclicity of Vacuum. Set of Exponential Vectors 276
22.2.4 Construction of Representations of Commutation Relations for a Complex Scalar Field 279
22.2.5 Construction of Representations of Commutation Relations in the Configuration Space. Relativistic Invariance of a Free Field 280
22.3 Representation of Anticommutation Relations of Spinor Fields 282
22.3.1 Representation of Anticommutation Relations of the Operators of Creation and Annihilation of Fermions and Antifermions 282
22.3.2 Representation of Anticommutation Relations in the Configuration Space 285
22.4 Space of States of a Free Electromagnetic Field 287
22.5 Space of Occupation Numbers 291
23 Green Functions 294
23.1 Green Functions of the Scalar Field 294
23.2 The Green Functions of Spinor, Vector, and Electromagnetic Fields 297
23.3 Time-Ordered Product and Green Functions 298
23.4 Wick Theorems 299
23.4.1 Wick Theorem for Normal Products 299
23.4.2 Wick Theorem for a Time-Ordered Product 301
23.4.3 Generalized Wick Theorem 304
23.5 Operation of Multiplication and the Regularization of Distributions 304
23.6 N-Point Green Functions of Free Fields 305
24 Problems to Part IV 307
V Quantum Theory of Interacting Fields. General Problems 309
25 Construction of Quantum Interacting Fields and Problems of This Construction 311
25.1 Formal Construction of a Quantum Field 311
25.2 Mathematical Problems of Construction of a Quantum Interacting Field 314
26 Scattering Theory. Scattering Matrix 318
26.1 Quantum Description of Scattering. Definition of Scattering Operator 318
26.2 Formal Construction of the Scattering Operator by the Method of Perturbation Theory 322
26.3 Main Properties of the S-Operator 325
26.3.1 Normal Form of the Operator S 325
26.3.2 Invariance of the Scattering Matrix under Lorentz Transformations and Transformations of Charge Conjugation 329
26.3.3 Unitarity of the Scattering Operator 330
26.3.4 Law of Conservation of Energy 331
26.3.5 Matrix Elements of the S-Operator and the Scattering Amplitude 333
26.4 Feynman Diagrams 336
26.4.1 Feynman Diagrams for the S-Operator 337
26.4.2 Feynman Diagrams for Coefficient Functions of the S-Operator 337
26.4.3 Feynman Diagrams for Matrix Elements of the S-Operator 339
26.5 Effective Cross-Sections and Scattering Matrix 342
26.5.1 Classical Picture 343
26.5.2 Quantum Picture 345
27 Equations for Coefficient Functions of the S-Matrix 347
27.1 Creation and Annihilation Operators of External Lines of Feynman Diagrams 348
27.2 Equations of the Resolvent Type 351
27.3 Equations of the Evolution Type 354
28 Green Functions and Scattering Matrix 356
28.1 Green Functions and the S-Matrix in the Interaction Picture 356
28.2 Schwinger Equation for Green Functions 358
28.3 On the Relationship between the Green Functions and the Coefficient Functions of the Scattering S-Operator 361
28.4 Equations for Green Functions in Terms of Functional Derivatives 362
28.5 Equations for Truncated Green Functions 364
28.6 Equations for One-Particle Irreducible Green Functions. Dyson Equation 367
28.7 Spectral Representation of the 2-Point Green Function (Källén-Lehmann Representation) 373
29 On Renormalization in Perturbation Theory 378
29.1 Primitively-Divergent Diagrams. Separation of Divergences by the Pauli-Villars Method 378
29.2 Degree of Divergence of Feynman Diagram 385
29.3 Elimination of Divergences by the Method of Bogoliubov-Parasiuk R-Operation 388
29.4 R-Operation and Counterterms of a Lagrangian 394
29.5 Classification of Interactions: Renormalizable and Nonrenormalizable Theories 399
29.6 Relationship between Counterterms and the Renormalization of Main Constants of the Theory 400
29.7 Equivalent Types of Renormalizations 407
30 Method of Functional (Path) Integrals in Quantized Field Theory 411
30.1 Notion of Path Integration and Main Formulas 412
30.2 Formalism of Feynman Integrals (Path Integrals) in Quantum Mechanics 420
30.3 Formalism of Feynman Integrals for Systems with Constraints 425
30.4 Path Integral Representation for Scalar Fields 429
30.5 Path Integral Representation for Fermi Fields 431
31 Problems to Part V 436
VI Axiomatic and Euclidean Field Theories 439
32 Wightman Axiomatics 443
32.1 Wightman Axioms for Real Scalar Fields 443
32.2 Wightman Functions and Their Properties 445
32.3 Reconstruction Theorem 447
33 Other Axiomatic Approaches 449
33.1 Haag-Ruelle Scattering Theory (HRST) 449
33.2 Lehmann–Symanzik–Zimmermann Axiomatics 452
33.3 Bogoliubov–Medvedev–Polivanov (BMP) Axiomatic Approach 455
34 Euclidean Field Theory 459
34.1 Analytic Continuation of Feynman Amplitudes 460
34.2 Operators of Free Euclidean Fields 462
34.2.1 Real scalar field 462
34.2.2 Euclidean Fermi fields 464
34.3 Euclidean Green Functions of a Free Scalar Field 466
34.4 Euclidean Green Functions of Interacting Fields 467
35 Euclidean Axiomatics 473
35.1 Analytic Continuation of Generalized Wightman Functions 473
35.2 Euclidean Green Functions. Osterwalder–Schrader Axioms 475
35.3 Reconstruction of the Wightman Theory 477
36 Problems to Part VI 480
VII Quantum Theory of Gauge Fields 483
37 Quantum Electrodynamics (QED) 485
37.1 Quantization of Interacting Electromagnetic Fields 486
37.1.1 Gupta–Bleuler Formalism for Interacting Electromagnetic Fields 486
37.1.2 Quantization of Interacting Electromagnetic Fields in the Coulomb Gauge 487
37.1.3 Photon Propagator and Gauge Conditions 489
37.2 S-Matrix in QED 491
37.2.1 Perturbation Theory. Feynman Diagrams 491
37.2.2 Coefficient Functions of the S-Matrix in Terms of Creation and Annihilation Operators of Lines of Feynman Diagrams 494
37.2.3 Furry Theorem 496
37.2.4 Gauge Invariance for Coefficient Functions of the S-Operator 499
37.3 Equations for Green Functions and Coefficient Functions of the S-Matrix 500
37.3.1 Schwinger Equation 500
37.3.2 System of Equations for Self-Energy and Vertex Parts of Green Functions 502
37.4 Divergences in QED and Methods for Their Elimination 505
37.4.1 Primitively-Divergent Diagrams and Their Regularization 505
37.4.2 Mass and Charge Renormalization of Electron (Positron) 510
37.5 Spectral Representations of 2-Point Green Functions 513
38 Quantization of Gauge Fields 518
38.1 Path Integral for Green Functions in QED (Coulomb Gauge) 519
38.2 Covariant Gauges: Popov–Faddeev–de Witt Method 522
38.3 Covariant Quantization of Electromagnetic Interaction 526
38.3.1 Connection between Different Gauges 527
38.3.2 Ward Identity 528
38.4 Quantization of Yang-Mills Fields Interacting with Matter Fields 530
38.5 Faddeev-Popov Ghosts 534
38.6 BRST-Invariance 536
39 Standard Models of Interactions 541
39.1 Renormalization of Gauge Theories 541
39.2 On the Masses of Gluons and Spontaneous Symmetry Breakdown 550
39.2.1 Connection of the Radius of Interaction and the Mass of Exchange Bosons 550
39.2.2 Are Theories with Nonzero Mass of Exchange Bosons Renormalizable? 552
39.2.3 Spontaneous Breakdown of the U(1)-Symmetry 553
39.2.4 Spontaneous Breakdown of the Local SU(N)-Symmetry. 555
39.3 Models of Interactions of Elementary Particles 557
39.3.1 Strong Interaction. Model of QCD 557
39.3.2 Weak and Electroweak Interactions 559
40 Problems to Part VII 562
Appendix Hints for the Solution of Problems 564
Bibliography 569
Index 582

lt;P>"Overall, it seems to me that it [the book] strives to strike a balance between mathematics and physics. It is well written, and the presentation and the discussion of the individual arguments are good. In my opinion, this is indeed an excellent reference book. If you want to pick some specific argument, look at the index; you will find it and (most importantly) you will understand it. [...] The explanations are excellent: the teaching experience of the author is indubitable. [...] All in all, it is a good book." Mathematical Reviews

"This self-contained, comprehensive monograph presents a careful modern introduction to the basic methods and applications of quantum field theory. [...] This book should be useful as a reference for students and researchers alike with interests in QFT, particle physics, mathematics and related areas." Zentralblatt für Mathematik