From classical to quantum fields:
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| Main Authors: | , , |
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| Format: | Book |
| Language: | English |
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Oxford
Oxford University Press
2017
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| Edition: | First edition |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xvii, 931 Seiten Illustrationen, Diagramme (teilweise farbig) |
| ISBN: | 9780198788393 9780198788409 |
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MARC
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| 245 | 1 | 0 | |a From classical to quantum fields |c Laurent Baulieu (CNRS and Sorbonne Universités), John Iliopoulos (CNRS and École Normale Supérieure), Roland Sénéor (CNRS and École Polytechnique) |
| 250 | |a First edition | ||
| 264 | 1 | |a Oxford |b Oxford University Press |c 2017 | |
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Record in the Search Index
| _version_ | 1804176801897906176 |
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| adam_text | Contents Prologue xviii 1 Introduction 1.1 1.2 1.3 The Descriptive Layers of Physical Reality Units and Notations Hamiltonian and Lagrangian Mechanics 1.3.1 Review of Variational Calculus 1.3.2 Noether’s Theorem 1.3.3 Applications of Noether’s Theorem 2 Relativistic Invariance 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 Introduction The Three-Dimensional Rotation Group Three-Dimensional Spinors Three-Dimensional Spinorial Tensors The Lorentz Group Generators and Lie Algebra of the Lorentz Group The Group SL(2, C) The Four-Dimensional Spinors Space Inversion and Bispinors Finite-Dimensional Representations of SU(2) and SL(2, C) Problems 3 The Electromagnetic Field Introduction Tensor Formulation of Maxwell’s Equations Maxwell’s Equations and Differential Forms Choice of a Gauge Invariance under Change of Coordinates Lagrangian Formulation 3.6.1 The Euler-Lagrange Equations and Noether’s Theorem 3.6.2 Examples of Noether Currents 3.6.3 Application to Electromagnetism 3.7 Interaction with a Charged Particle 3.8 Green Functions 3.8.1 The Green Functions of the Klein-Gordon Equation 3.8.2 The Green Functions of the Electromagnetic Field 3.1 3.2 3.3 3.4 3.5 3.6 1 1 3 4 4 6 7 9 9 11 14 18 20 23 25 27 30 33 35 37 37 38 40 43 45 47 48 51 53 56 58 59 63
viii Contents 3.9 Applications 3.9.1 The Liénard-Wiechert Potential 3.9.2 The Larmor Formula 3.9.3 The Thomson Formula 3.9.4 The Limits of Classical Electromagnetism 65 65 69 70 71 4 General Relativity: A Field Theory of Gravitation 73 4.1 The Equivalence Principle 4.1.1 Introduction 4.1.2 The Principle 4.1.3 Deflection of Light by a Gravitational Field 4.1.4 Influence of Gravity on Clock Synchronisation 4.2 Curved Geometry 4.2.1 Introduction 4.2.2 Tensorial Calculus for the Reparametrisation Symmetry 4.2.3 Affine Connection and Covariant Derivation 4.2.4 Parallel Transport and Christoffel Coefficients 4.2.5 Geodesics 4.2.6 The Curvature Tensor 4.3 Reparametrization Gauge Symmetry and Einstein’s General Relativity 4.3.1 Reparametrisation Invariance as a Gauge Symmetry 4.3.2 Reparametrisation Invariance and Energy-Momentum Tensor 4.3.3 The Einstein-Hilbert Equation 4.4 The Limits of Our Perception of Space and Time 4.4.1 Direct Measurements 4.4.2 Possible Large Defects 5 The Physical States 5.1 5.2 Introduction The Principles 5.2.1 Relativistic Invariance and Physical States 5.3 The Poincaré Group 5.3.1 The Irreducible Representations of thePoincaré Group 5.3.2 The Generatorsof the Poincaré Group 5.4 The Space of the Physical States 5.4.1 The One-Particle States 5.4.2 The Two-or More Particle States without Interaction 5.4.3 The Fock Space 5.4.4 Introducing Interactions 5.5 Problems 73 73 74 76 76 77 77 78 80 83 85 88 90 90 93 95 98 99 100 103 103 104 105 107 107 110 114 114 115 116 117 118
Contents 6 Relativistic Wave Equations 6.1 Introduction 6.2 The Klein-Gordon Equation 6.3 The Dirac Equation 6.3.1 The y Matrices 6.3.2 The Conjugate Equation 6.3.3 The Relativistic Invariance 6.3.4 The Current 6.3.5 The Hamiltonian 6.3.6 The Standard Representation 6.3.7 The Spin 6.3.8 The Plane Wave Solutions 6.3.9 The Coupling with the Electromagnetic Field 6.3.10 The Constants of Motion 6.3.11 Lagrangian and Green Functions 6.4 Relativistic Equations for Vector Fields 7 Towards a Relativistic Quantum Mechanics 7.1 Introduction 7.2 The Klein-Gordon Equation 7.3 The Dirac Equation 7.3.1 The Non-relativistic Limit of the Dirac Equation 7.3.2 Charge Conjugation 7.3.3 PCT Symmetry 7.3.4 The Massless Case 7.3.5 Weyl and Majorana Spinors 7.3.6 Hydrogenoid Systems 7.4 Problems 8 Functional Integrals and Probabilistic Amplitudes Introduction Brief Historical Comments The Physical Approach The Reconstruction of Quantum Mechanics 8.4.1 The Quantum Mechanics of a Free Particle 8.4.2 A Particle in a Potential 8.4.3 The Schrôdinger Equation 8.5 The Feynman Formula 8.5.1 The Representations of Quantum Mechanics 8.5.2 The Feynman Formula for Systems with One Degree of Freedom 8.6 The Harmonic Oscillator 8.1 8.2 8.3 8.4 ix 120 120 120 123 126 127 128 129 129 129 131 132 135 136 137 138 142 142 142 144 144 146 149 150 152 153 159 162 162 163 165 168 169 170 170 173 173 176 180
x Contents 8.7 The Bargmann Representation 8.7.1 The Coherent States 8.7.2 The Path Integral Formulain the Bargmann Space 8.8 Problems 9 Functional Integrals and Quantum Mechanics: Formal Developments 9.1 9.2 9.3 9.4 9.5 T-Products 9.1.1 General D efinition 9.1.2 Application to the Harmonic Oscillator S-Matrix and T-Products 9.2.1 Three Examples Elements of Perturbation Theory Generalizations 9.4.1 Three-Dimensional Quantum Mechanics 9.4.2 The Free Scalar Field Problems 10 The Euclidean Functional Integrals 10.1 Introduction 10.1.1 The Wiener Measure 10.2 The Gaussian Measures in Euclidean Field Theories 10.2.1 Definition 10.2.2 The Integration by Parts Formula 10.2.3 The Wick Ordering 10.3 Application to Interacting Fields 10.3.1 The 2-Point Function 10.3.2 The 4-Point Function 10.3.3 The General Feynman Rules 10.4 Problems 11 Fermions and Functional Formalism Introduction The Grassmann Algebras 11.2.1 The Derivative 11.2.2 The Integration 11.3 The Clifford Algebras 11.4 Fermions in Quantum Mechanics 11.4.1 Quantum Mechanics and Fermionic Oscillators 11.4.2 The Free Fermion Fields 11.5 The Path Integrals 11.5.1 The Case of Quantum Mechanics 11.5.2 The Case of Field Theory 11.1 11.2 186 186 191 194 196 196 196 197 201 204 206 211 211 212 217 218 218 220 224 225 228 229 230 231 236 238 239 240 240 243 243 244 249 250 250 253 254 254 257
Contents 12 Relativistic Quantum Fields 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.10 12.11 Introduction Relativistic Field Theories 12.2.1 The Axiomatic Field Theory The Asymptotic States 12.3.1 Introduction 12.3.2 The Fock Space 12.3.3 Existence of Asymptotic States The Reduction Formulae 12.4.1 The Feynman Diagrams The Case of the Maxwell Field 12.5.1 The Classical Maxwell Field 12.5.2 The Quantum Field: I. The Functional Integral 12.5.3 The Quantum Field: II. The Particle Concept 12.5.4 The Casimir Effect Quantization of a Massive Field of Spin-1 The Reduction Formulae for Photons The Reduction Formulae for Fermions Quantum Electrodynamics 12.9.1 The Feynman Rules A Formal Expression for the S-Matrix Problems 13 Applications 13.1 13.2 On Cross Sections Formal Theory of Scattering in Quantum Mechanics 13.2.1 An Integral Equation for the Green Function 13.2.2 The Cross Section in Quantum Mechanics 13.3 Scattering in Field Theories 13.3.1 The Case of Two Initial Particles 13.3.2 The Case of One Initial Particle 13.4 Applications 13.5 The Feynman Rules for the S-Matrix 13.5.1 Feynman Rules for Other Theories 13.6 Problems 14 Geometry and Quantum Dynamics 14.1 14.2 14.3 14.4 Introduction. QED Revisited Non-Abelian Gauge Invariance and Yang-MillsTheories Field Theories of Vector Fields Gauge Fixing and BRST Invariance 14.4.1 Introduction 14.4.2 The Traditional Faddeev-Popov Method xi 260 260 260 261 279 279 281 282 286 292 296 296 298 300 304 305 308 309 310 310 311 319 320 320 324 325 335 340 341 343 344 353 354 358 360 360 362 365 369 369 369
xii Contents 14.5 14.6 14.7 14.8 14.9 14.4.3 Graded Notation for the Classical and Ghost Yang-Mills Fields 14.4.4 Determination of the BRST Symmetry as the Extension of the Gauge Symmetry for the Classical and Ghost Fields 14.4.5 General BRST Invariant Action for the Yang-Mills Theory Feynman Rules for the BRST Invariant Yang-Mills Action BRST Quantization of Gravity Seen as a Gauge Theory The Gribov Ambiguity: The Failure of the Gauge-Fixing Process beyond Perturbation Theory 14.7.1 A Simple Example 14.7.2 The Gribov Question in a Broader Framework Historical Notes Problems 15 Broken Symmetries 15.1 Introduction 15.2 Global Symmetries 15.2.1 An Example from Classical Mechanics 15.2.2 Spontaneous SymmetryBreaking in Non-relativistic Quantum Mechanics 15.2.3 A Simple Field Theory Model 15.2.4 The Linear σ-Model 15.2.5 The Non-linear σ-Model 15.2.6 Goldstone Theorem 15.3 Gauge Symmetries 15.3.1 The Abelian Model 15.3.2 The Non-Abelian Case 15.4 Problems 16 Quantum Field Theory at Higher Orders 16.1 Existence of Divergences in Loop Diagrams. Discussion 16.2 Connected and 1 -PI Diagrams 16.3 Power Counting. Definition of Super-Renormalisable, Renormalizable, and Non-renormalisable Quantum Field Theories 16.4 Régularisation 16.5 Renormalisation 16.5.1 1-Loop Diagrams 16.5.2 Some 2-Loop Examples 16.5.3 All Orders 16.5.4 An‘Almost’Renormalisable Theory 16.5.5 Composite Operators 16.6 The Renormalisation Group 376 378 382 385 386 389 389 390 393 396 398 398 399 399 400 403 405 408 410 412 413 416 418 420 420 421 425 428 433 433 441 443 446 448 450
Contents 16.7 16.6.1 General Discussion 16.6.2 The Renormalisation Group in Dimensional Régularisation 16.6.3 Dependence of the ß and γ Functions on the Renormalization Scheme Problems 17 A First Glance at Renormalisation and Symmetry 17.1 17.2 17.3 Introduction Global Symmetries Gauge Symmetries: Examples 17.3.1 The Adler-Bell-Jackiw Anomaly 17.3.2 A Path Integral Derivation 17.3.3 The Axial Anomaly and Renormalisation 17.3.4 A Consistency Condition for Anomalies 17.4 The Breaking of Conformal Invariance 17.5 A Non-Perturbative Anomaly 17.6 Problems 18 Renormalisation of Yang-Mills Theory and BRST Symmetry 18.1 18.2 Introduction Generating Functional of BRST Covariant Green Functions 18.2.1 BRST Ward Identities in a Functional Form 18.3 Anomaly Condition 18.3.1 General Solution for the Anomalies of the Ward Identities 18.3.2 The Possible Anomalous Vertices and the Anomaly Vanishing Condition 18.4 Dimensional Régularisation and Multiplicative Renormalisation 18.4.1 Introduction 18.4.2 Linear Gauges and Ward Identities for the BRST Symmetry and Ghost Equations of Motion 18.4.3 Inverting the Ward Identities in Linear Gauges for a Local Field and Source Functional 18.4.4 The Structure of the Counter-terms within the Dimensional Régularisation Method 18.5 Observables 18.6 Problems 19 Some Consequences of the Renormalisation Group 19.1 19.2 19.3 19.4 19.5 Introduction The Asymptotic Behaviour of Green Functions Stability and the Renormalization Group Dimensional Transmutation Problems xiii 450 453 456 457 462 462 463 468 468 471 476 476 478 482 485 486 486 487 489 490 491 495
497 497 498 499 501 504 505 507 507 508 511 514 518
xiv Contents 20 Analyticity Properties of Feynman Diagrams 20.1 20.2 20.3 20.4 20.5 20.6 20.7 Introduction Singularities of Tree Diagrams Loop Diagrams Unstable Particles Cutkosky Unitarity Relations The Analytic S-Matrix Theory Problems 21 Infrared Singularities 21.1 21.2 21.3 21.4 21.5 Introduction. Physical Origin The Example of Quantum Electrodynamics General Discussion Infrared Singularities in Other Theories Problems 22 Coherent States and Classical Limit of Quantum Electrodynamics 22.1 Introduction 22.2 The Definition of Coherent States 22.3 Fluctuations 22.3.1 Time Evolution of Coherent States 22.3.2 Dispersion of Coherent States 22.4 Coherent States and the Classical Limit of QED towards Maxwell Theory 22.5 Squeezed States 22.6 Problems 521 521 522 524 528 532 534 543 544 544 545 550 551 554 555 555 556 559 560 560 561 563 565 23 Quantum Field Theories with a Large Number of Fields 566 Introduction Vector Models Fields in the Adjoint Representation The Large N Limit as a Classical Field Theory Problems 566 567 571 574 578 23.1 23.2 23.3 23.4 23.5 24 The Existence of Field Theories beyond the Perturbation Expansion Introduction The Equivalence between Relativistic and Euclidean Field Theories Construction of Field Theories The Zero-Dimensional λφ4 Model 24.4.1 The Divergence of the Perturbation Series 24.4.2 The Borel Summability 24.5 General Facts about Scalar Field Theories in d = 2 or d = 3 Dimensions 24.6 The λφ4 Theory in d = 2 Dimensions 24.1 24.2 24.3 24.4 580 580 582 584 590 591 592 594 597
Contents 24.6.1 The Divergence of the λφ^ Perturbation Series 24.6.2 The Existence of the λ02 Theory 24.6.3 The Cluster Expansion 24.6.4 The Mayer Expansion 24.6.5 The Infinite Volume Limit of λφ^ 24.6.6 The Borel Summability of the ՃՓշ Theory 24.6.7 The Mass Gap for ф^ in a Strong External Field 24.7 The λίφ^ Theory in d = 3 Dimensions 24.7.1 The Expansion: Definition 24.7.2 The Expansion Completed 24.7.3 The Results 24.8 The Massive Gross-Neveu Model in d = 2 Dimensions 24.8.1Definition of the Model 24.8.2The Infinite Volume Limit 24.8.3 The Removal of the Ultraviolet Cut-off 24.8.4 The Behaviour of the Effective Constants and the Approximate Renormalization Group Flow 24.9 The Yang-Mills Field Theory in d = 4 24.9.1 A Physical Problem 24.9.2 Many Technical Problems 25 Fundamental Interactions 25.1 Introduction. What Is an ‘Elementary Particle’? 25.2 The Four Interactions 25.3 The Standard Model of Weak and Electromagnetic Interactions 25.3.1 A Brief Summary of the Phenomenology 25.3.2 Model Building 25.3.3 The Lepton World 25.3.4 Extension to Hadrons 25.3.5 The Neutrino Masses 25.3.6 Some Sample Calculations 25.3.7 Anomalies in the Standard Model 25.4 A Gauge Theory for Strong Interactions 25.4.1 Are Strong Interactions Simple? 25.4.2 Quantum Chromodynamics 25.4.3 Quantum Chromodynamics in Perturbation Theory 25.4.4 Quantum Chromodynamics on a Space-Time Lattice 25.4.5 Instantons 25.5 Problems 26 Beyond the Standard Model 26.1 Why 26.1.1 The Standard Model Has Been Enormously Successful xv 597 601 607 617 621 622 623 625 630 632 634 635 635 639 639 640 644 644 645 648
648 649 651 651 655 655 660 668 671 676 680 680 683 692 712 729 740 746 747 747
xvi Contents 26.1.2 Predictions for New Physics 26.1.3 Unsolved Problems of the Standard Model 26.2 Grand Unified Theories 26.2.1 Generalities 26.2.2 The Simplest GUT: SU(5) 26.2.3 Dynamics of GUTs 26.2.4 Other Grand Unified Theories 26.2.5 Magnetic Monopoles 26.3 The Trial of Scalars 27 Supersymmetry, or the Defence of Scalars 27.1 Introduction 27.2 The Supersymmetry Algebra 27.3 Why This Particular Algebra; or All Possible Supersymmetries of the S Matrix 27.4 Representations in Terms of One-Particle States 27.4.1 Massive Case 27.4.2 Massless Case 27.5 Representations in Terms of Field Operators:Superspace 27.6 A Simple Field Theory Model 27.7 Supersymmetry and Gauge Invariance 27.7.1 The Abelian Case 27.7.2 The Non-Abelian Case 27.7.3 Extended Supersymmetries 27.8 Spontaneous Symmetry Breaking and Supersymmetry 27.8.1 Goldstone and BEH Phenomena in the Presence of Supersymmetry 27.8.2 Spontaneous Supersymmetry Breaking in Perturbation Theory 27.8.3 Dynamical Breaking of Supersymmetry 27.9 Dualities in Supersymmetric Gauge Theories 27.10 Twisted Supersymmetry and Topological Field Theories 27.10.1 Introduction 27.10.2 A Quantum Mechanical Toy Model 27.10.3 Yang-Mills TQFT 27.11 Supersymmetry and Particle Physics 27.11.1 Supersymmetry and the Standard Model 27.11.2 Supersymmetry and Grand Unified Theories 27.11.3 The Minimal Supersymmetric Standard Model 27.12 Gauge Supersymmetry 27.12.1 N=1 Supergravity 27.12.2 Μ = 8 Supergravity 27.13 Problems 748 752 752 752 754 757 764 769 780 784 784 785 786 787 787 789 791 797 803 803 806 808 811 812 818 821 823 833 833 837 839 848
851 854 855 858 859 861 862
Contents xvii Appendix A Tensor Calculus 863 A. 1 Algebraic Theory of Tensors A. 1.1 Definitions A. 1.2 Examples A. 1.3 Algebraic Properties of Tensors A. 1.4 Bases A. 2 Manifolds and Tensors A.2.1 Manifolds, Tangent, and Cotangent Bundles A.2.2 Differential of a Mapping A.2.3 Vector Fields A.2.4 Cotangent Bundle A.2.5 Tensors A.2.6 Lie Derivative A.2.7 Riemannian Structure 863 863 864 866 866 868 868 869 870 871 871 875 877 Appendix В Differential Calculus 879 B.l Differential Form B.2 Exterior Differential B.2.1 Integration 879 883 884 Appendix C Groups and Lie Algebras 889 Lie Groups C.l.l Definitions C.1.2 Representations C.1.3 Lie Groups C.l.4 One Parameter Subgroup. Tangent Space C.2 Lie Algebras C.2.1 Definition C.2.2 Matrix Lie Algebras 889 889 890 892 893 896 896 898 C.l Appendix D A Collection of Useful Formulae D. 1 D.2 D.3 D.4 902 903 906 910 Units and Notations Free Fields Feynman RulesforScattering Amplitudes Examples Appendix E Extract from Maxwell’s A Treatise on Electricity and Magnetism Index 902 912 915
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| indexdate | 2024-07-10T07:37:58Z |
| institution | BVB |
| isbn | 9780198788393 9780198788409 |
| language | English |
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| physical | xvii, 931 Seiten Illustrationen, Diagramme (teilweise farbig) |
| publishDate | 2017 |
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| publisher | Oxford University Press |
| record_format | marc |
| spelling | Baulieu, Laurent Verfasser (DE-588)1078752370 aut From classical to quantum fields Laurent Baulieu (CNRS and Sorbonne Universités), John Iliopoulos (CNRS and École Normale Supérieure), Roland Sénéor (CNRS and École Polytechnique) First edition Oxford Oxford University Press 2017 xvii, 931 Seiten Illustrationen, Diagramme (teilweise farbig) txt rdacontent n rdamedia nc rdacarrier Quantenfeldtheorie (DE-588)4047984-5 gnd rswk-swf Quantum field theory / Textbooks Quantenfeldtheorie (DE-588)4047984-5 s DE-604 Iliopoulos, Jean 1940- Verfasser (DE-588)1120028825 aut Seneor, Roland Verfasser (DE-588)1120029244 aut Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029308453&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Baulieu, Laurent Iliopoulos, Jean 1940- Seneor, Roland From classical to quantum fields Quantenfeldtheorie (DE-588)4047984-5 gnd |
| subject_GND | (DE-588)4047984-5 |
| title | From classical to quantum fields |
| title_auth | From classical to quantum fields |
| title_exact_search | From classical to quantum fields |
| title_full | From classical to quantum fields Laurent Baulieu (CNRS and Sorbonne Universités), John Iliopoulos (CNRS and École Normale Supérieure), Roland Sénéor (CNRS and École Polytechnique) |
| title_fullStr | From classical to quantum fields Laurent Baulieu (CNRS and Sorbonne Universités), John Iliopoulos (CNRS and École Normale Supérieure), Roland Sénéor (CNRS and École Polytechnique) |
| title_full_unstemmed | From classical to quantum fields Laurent Baulieu (CNRS and Sorbonne Universités), John Iliopoulos (CNRS and École Normale Supérieure), Roland Sénéor (CNRS and École Polytechnique) |
| title_short | From classical to quantum fields |
| title_sort | from classical to quantum fields |
| topic | Quantenfeldtheorie (DE-588)4047984-5 gnd |
| topic_facet | Quantenfeldtheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029308453&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT baulieulaurent fromclassicaltoquantumfields AT iliopoulosjean fromclassicaltoquantumfields AT seneorroland fromclassicaltoquantumfields |