What is a quantum field theory?: a first introduction for mathematicians
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge, United Kingdom ; New York, NY, USA ; Port Melbourne, Australia ; New Delhi, India ; Singapore
Cambridge University Press
2022
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| Ausgabe: | First published |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverzeichnis Seite 733-737 |
| Beschreibung: | xv, 741 Seiten Illustrationen, Diagramme |
| ISBN: | 9781316510278 |
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| 245 | 1 | 0 | |a What is a quantum field theory? |b a first introduction for mathematicians |c Michel Talagrand |
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| 264 | 1 | |a Cambridge, United Kingdom ; New York, NY, USA ; Port Melbourne, Australia ; New Delhi, India ; Singapore |b Cambridge University Press |c 2022 | |
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Contents Introduction Part I Basics page 1 7 1 Preliminaries 1.1 Dimension 1.2 Notation 1.3 Distributions 1.4 The Delta Function 1.5 The Fourier Transform 9 9 10 12 14 17 2 Basics of Non-relativistic Quantum Mechanics 2.1 Basic Setting 2.2 Measuring Two Different Observables on the Same System 2.3 Uncertainty 2.4 Finite versus Continuous Models 2.5 Position State Space for a Particle 2.6 Unitary Operators 2.7 Momentum State Space for a Particle 2.8 Dirac’s Formalism 2.9 Why Are Unitary Transformations Ubiquitous? 2.10 Unitary Representations of Groups 2.11 Projective versus True Unitary Representations 2.12 Mathematicians Lookat Projective Representations 2.13 Projective Representations of R 2.14 One-parameter Unitary Groups and Stone’s Theorem 2.15 Time-evolution 21 22 27 28 30 31 38 39 40 46 47 49 50 51 52 59 vii
viii Contents 2.16 Schrödinger and Heisenberg Pictures 2.17 A First Contact with Creation and Annihilation Operators 2.18 The Harmonie Oscillator 62 64 66 3 Non-relativistic Quantum Fields 3.1 Tensor Products 3.2 Symmetrie Tensors 3.3 Creation and Annihilation Operators 3.4 Boson Fock Space 3.5 Unitary Evolution in the Boson Fock Space 3.6 Boson Fock Space and Collections of Harmonie Oscillators 3.7 Explicit Formulas: Position Space 3.8 Explicit Formulas: Momentum Space 3.9 Universe in a Box 3.10 Quantum Fields: Quantizing Spaces of Functions 73 73 76 78 82 84 86 88 92 93 94 4 The Lorentz Group and the Poincaré Group 4.1 Notation and Basics 4.2 Rotations 4.3 Pure Boosts 4.4 The Mass Shell and Its Invariant Measure 4.5 More about Unitary Representations 4.6 Group Actions and Representations 4.7 Quantum Mechanics, Special Relativity and the Poincaré Group 4.8 A Fundamental Representation of the Poincaré Group 4.9 Particles and Representations 4.10 The States |p) and |p) 4.11 The Physicists’Way 102 102 107 108 111 115 118 5 The Massive Scalar Free Field 5.1 Intrinsic Definition 5.2 Explicit Formulas 5.3 Time-evolution 5.4 Lorentz Invariant Formulas 132 132 140 142 143 6 Quantization 6.1 The Klein-Gordon Equation 6.2 Naive Quantization of the Klein-Gordon Field 6.3 Road Map 6.4 Lagrangian Mechanics 6.5 From Lagrangian Mechanics to Hamiltonian Mechanics 145 146 147 150 151 156 120 122 125 128 129
Contents 6.6 6.7 6.8 6.9 6.10 6.11 7 Canonical Quantization and Quadratic Potentials Quantization through the Hamiltonian Ultraviolet Divergences Quantization through Equal-time Commutation Relations Caveat Hamiltonian The Casimir Effect 7.1 Vacuum Energy 7.2 Regularization ix 161 163 164 165 172 173 176 176 177 Part II Spin 181 8 Representations of theOrthogonal and the Lorentz Group 8.1 The Groups SU(2) and SL(2, C) 8.2 A Fundamental Family of Representations of SU(2) 8.3 Tensor Products of Representations 8.4 SL(2, C) as a Universal Cover of the Lorentz Group 8.5 An Intrinsically Projective Representation 8.6 Deprojectivization 8.7 A Brief Introduction to Spin 8.8 Spin as an Observable 8.9 Parity and the Double Cover SL+(2, C) of O+(l, 3) 8.10 The Parity Operator and the Dirac Matrices 183 183 187 190 192 195 199 199 200 201 204 9 Representations of the Poincaré Group 9.1 The Physicists’Way 208 209 9.2 The Group P* 211 9.3 Road Map 9.3.1 How to Construct Representations? 9.3.2 Surviving the Formulas 9.3.3 Classifying the Representations 9.3.4 Massive Particles 9.3.5 Massless Particles 9.3.6 Massless Particles and Parity Elementary Construction of Induced Representations Variegated Formulas Fundamental Representations 9.6.1 Massive Particles 9.6.2 Massless Particles Particles, Spin, Representations 212 213 213 214 214 214 215 215 217 223 223 223 228 9.4 9.5 9.6 9.7
Contents X Abstract Presentation of Induced Representations Particles and Parity Dirac Equation History of the Dirac Equation Parity and Massless Particles Photons 232 235 236 238 240 245 Basic Free Fields 10.1 Charged Particles and Anti-particles 10.2 Lorentz Covariant Families of Fields 10.3 Road Map I 10.4 Form of the Annihilation Part of the Fields 10.5 Explicit Formulas 10.6 Creation Part of the Fields 10.7 Microcausality 10.8 Road Map II 10.9 The Simplest Case (A = 1) 10.10 A Very Simple Case (A = 4) 10.11 The Massive Vector Field (A = 4) 10.12 The Classical Massive Vector Field 10.13 Massive Weyl Spinors, First Attempt (A = 2) 10.14 Fermion Fock Space 10.15 Massive Weyl Spinors, Second Attempt 10.16 Equation of Motion for the Massive Weyl Spinor 10.17 Massless Weyl Spinors 10.18 Parity 10.19 Dirac Field 10.20 Dirac Field and Classical Mechanics 10.21 Majorana Field 10.22 Lack of a Suitable Field for Photons 250 251 253 255 256ί 260 262 264 267 268 268 269 271 273 275 279 281 283 284 285 288 293 293 9.8 9.9 9.10 9.11 9.12 9.13 10 Partlll Interactions 11 Perturbation Theory 11.1 Time-independent Perturbation Theory 11.2 Time-dependent Perturbation Theory and the Interaction Picture 11.3 Transition Rates 11.4 A Side Story: Oscillating Interactions 11.5 Interaction of a Particle with a Field: A Toy Model 297 299 299 303 307 310 312
Contents xi 12 Scattering, the Scattering Matrix and Cross-Sections 12.1 Heuristics in a Simple Case of Classical Mechanics 12.2 Non-relativistic Quantum Scattering by a Potential 12.3 The Scattering Matrix in Non-relativistic Quantum Scattering 12.4 The Scattering Matrix and Cross-Sections, I 12.5 Scattering Matrix in Quantum Field Theory 12.6 Scattering Matrix and Cross-Sections, II 13 The Scattering Matrix in Perturbation Theory 13.1 The Scattering Matrix and the Dyson Series 13.2 Prologue: The Born Approximation in Scattering by a Potential 13.3 Interaction Terms in Hamiltonians 13.4 Prickliness of the Interaction Picture 13.5 Admissible Hamiltonian Densities 13.6 Simple Models for Interacting Particles 13.7 A Computation at the First Order 13.8 Wick’s Theorem 13.9 Interlude: Summing the Dyson Series 13.10 The Feynman Propagator 13.11 Redefining the Incoming and Outgoing States 13.12 A Computation at Order Two with Trees 13.13 Feynman Diagrams and Symmetry Factors 13.14 The ^ 4 Model 13.15 A Closer Look at Symmetry Factors 13.16 A Computation at Order Two with One Loop 13.17 One Loop: A Simple Case of Renormalization 13.18 Wick Rotation and Feynman Parameters 13.19 Explicit Formulas 13.20 Counter-terms, I 13.21 Two Loops: Toward the Central Issues 13.22 Analysis of Diagrams 13.23 Cancellation of Infinities 13.24 Counter-terms, II 14 322 323 324 330 333 343 345 351 351 353 354 355 357 359 361 365 367 369 373 373 379 384 387 389 392 395 401 403 404 406 409 414 Interacting Quantum Fields 420 14.1 Interacting Quantum Fields and Particles 421 14.2 Road Map I 422 14.3 The
Gell-Mann—Low Formula and Theorem 423 14.4 Adiabatic Switching of the Interaction 430 14.5 Diagrammatic Interpretation of the Gell-Mann—Low Theorem 436
Contents xii 14.6 14.7 14.8 14.9 14.10 14.11 14.12 14.13 14.14 Road Map II Green Functions and S-matrix The Dressed Propagator in the Kâllén-Lehmann Representation Diagrammatic Computation of the Dressed Propagator Mass Renormalization Difficult Reconciliation Field Renormalization Putting It All Together Conclusions 440 441 447 453 457 460 462 467 469 Part IV Renormalization 471 15 Prologue: Power Counting 15.1 What Is Power Counting? 15.2 Weinberg’s Power Counting Theorem 15.3 The Fundamental Space ker £ 15.4 Power Counting in Feynman Diagrams 15.5 Proof of Theorem 15.3.1 15.6 A Side Story: Loops 15.7 Parameterization of Diagram Integrals 15.8 Parameterization of Diagram Integrals by Loops 473 473 480 483 484 489 490 492 494 16 The Bogoliubov-Parasiuk-Hepp-Zimmermann Scheme 16.1 Overall Approach 16.2 Simple Examples 16.3 Canonical Flow and the Taylor Operation 16.4 Subdiagrams 16.5 Forests 16.6 Renormalizing the Integrand: The Forest Formula 16.7 Diagrams That Need Not Be 1-PI 16.8 Interpretation 16.9 Specificity of the Parameterization 496 497 498 500 503 504 506 510 510 512 17 Counter-terms 17.1 What Is the Counter-term Method? 17.2 A Very Simple Case: Coupling Constant Renormalization 17.3 Mass and Field Renormalization: Diagrammatics 17.4 The BPHZ Renormalization Prescription 17.5 Cancelling Divergences with Counter-terms 17.6 Determining the Counter-terms from BPHZ 17.7 From BPHZ to the Counter-term Method 514 515 516 518 524 525 527 531
Contents 17.8 17.9 What Happened to Subdiagrams? Field Renormalization, II xiii 535 538 18 Controlling Singularities 18.1 Basic Principle 18.2 Zimmermann’s Theorem 18.3 Proof of Proposition 18.2.12 18.4 A Side Story: Feynman Diagrams and Wick Rotations 542 542 546 556 560 19 Proof of Convergence of the BPHZ Scheme 19.1 Proof of Theorem 16.1.1 19.2 Simple Facts 19.3 Grouping the Terms 19.4 Bringing Forward Cancellation 19.5 Regular Rational Functions 19.6 Controlling the Degree 563 563 565 567 575 578 583 Part V Complements 591 Appendix A Complements on Representations A.l Projective Unitary Representations of R A.2 Continuous Projective Unitary Representations A.3 Projective Finite-dimensional Representations A.4 Induced Representations for Finite Groups A.5 Representations of Finite Semidirect Products A.6 Representations of Compact Groups 593 593 596 598 600 604 608 Appendix E1 End of Proof of Stone’s Theorem 612 Appendix C’ Canonical Commutation Relations C.l First Manipulations C.2 Coherent States for the Harmonic Oscillator C.3 The Stone-von Neumann Theorem C.4 Non-equivalent Unitary Representations C.5 Orthogonal Ground States! 616 616 618 621 627 632 Appendix D A Crash Course on Lie Algebras D.l Basic Properties and so(3) D.2 Group Representations and Lie Algebra Representations D.3 Angular Momentum D.4 SU(2) = SO(3)! 635 635 639 641 642
Contents xiv From Lie Algebra Homomorphisms to Lie Group Homomorphisms D.6 Irreducible Representations of SU(2) D.7 Decomposition of Unitary Representations of SU(2) into Irreducibles D.8 Spherical Harmonics D.9 SO(1,3) = slc(2)! D.10 Irreducible Representations of SL (2, C) D. 11 QFT Is Not for the Meek D.12 Some Tensor Representations of 50^(1,3) D.5 644 646 650 652 654 656 658 660 Appendix E Special Relativity E.l Energy-Momentum E.2 Electromagnetism 664 664 666 Appendix F Does a Position Operator Exist? 668 Appendix G More on the Representations of the Poincaré Group G.l A Fun Formula G.2 Higher Spin: Bargmann-Wigner and Rarita-Schwinger 671 671 672 Appendix H Hamiltonian Formalism for Classical Fields H. 1 Hamiltonian for the Massive Vector Field H.2 From Hamiltonians to Lagrangians H.3 Functional Derivatives H.4 Two Examples H.5 Poisson Brackets 677 677 678 679 681 682 Appendix I Quantization of the Electromagnetic Field through the Gupta-Bleuler Approach 685 Appendix J Lippmann-Schwinger Equations and Scattering States 692 Appendix К Functions on Surfaces and Distributions 697 Appendix L What Is a Tempered Distribution Really? L.l Test Functions L.2 Tempered Distributions L.3 Adding and Removing Variables L.4 Fourier Transforms of Distributions 698 698 699 701 703 Appendix Μ Wightman Axioms and Haag’s Theorem Μ. 1 The Wightman Axioms M.2 Statement of Haag’s Theorem 704 704 710
Contents M.3 M.4 Easy Steps Wightman Functions XV 711 714 Appendix N Feynman Propagator and Klein-Gordon Equation N. 1 Contour Integrals N.2 Fundamental Solutions of Differential Equations 721 721 723 Appendix О Yukawa Potential 726 Appendix P Principal Values and Delta Functions 729 Solutions to Selected Exercises Reading Suggestions References Index 731 732 733 738 |
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| spelling | Talagrand, Michel 1952- Verfasser (DE-588)112924379 aut What is a quantum field theory? a first introduction for mathematicians Michel Talagrand First published Cambridge, United Kingdom ; New York, NY, USA ; Port Melbourne, Australia ; New Delhi, India ; Singapore Cambridge University Press 2022 xv, 741 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Literaturverzeichnis Seite 733-737 Quantenfeldtheorie (DE-588)4047984-5 gnd rswk-swf Mathematische Physik (DE-588)4037952-8 gnd rswk-swf Quantum field theory SCIENCE / Physics / Mathematical & Computational Quantenfeldtheorie (DE-588)4047984-5 s Mathematische Physik (DE-588)4037952-8 s DE-604 Erscheint auch als Online-Ausgabe, EPUB 978-1-108-22514-4 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032874068&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
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| title | What is a quantum field theory? a first introduction for mathematicians |
| title_auth | What is a quantum field theory? a first introduction for mathematicians |
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| title_full | What is a quantum field theory? a first introduction for mathematicians Michel Talagrand |
| title_fullStr | What is a quantum field theory? a first introduction for mathematicians Michel Talagrand |
| title_full_unstemmed | What is a quantum field theory? a first introduction for mathematicians Michel Talagrand |
| title_short | What is a quantum field theory? |
| title_sort | what is a quantum field theory a first introduction for mathematicians |
| title_sub | a first introduction for mathematicians |
| topic | Quantenfeldtheorie (DE-588)4047984-5 gnd Mathematische Physik (DE-588)4037952-8 gnd |
| topic_facet | Quantenfeldtheorie Mathematische Physik |
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