Quantum field theory 1. Basics in mathematics and physics

Contents......Page 12
Prologue......Page 24
1. Historical Introduction......Page 44
1.1 The Revolution of Physics......Page 45
1.2 Quantization in a Nutshell......Page 50
1.3 The Role of Göttingen......Page 83
1.4 The Göttingen Tragedy......Page 90
1.5 Highlights in the Sciences......Page 92
1.6 The Emergence of Physical Mathematics – a New Dimension of Mathematics......Page 98
1.7 The Seven Millennium Prize Problems of the Clay Mathematics Institute......Page 100
2. Phenomenology of the Standard Model for Elementary Particles......Page 102
2.1 The System of Units......Page 103
2.2 Waves in Physics......Page 104
2.3 Historical Background......Page 120
2.4 The Standard Model in Particle Physics......Page 150
2.5 Magic Formulas......Page 163
2.6 Quantum Numbers of Elementary Particles......Page 166
2.7 The Fundamental Role of Symmetry in Physics......Page 185
2.8 Symmetry Breaking......Page 201
2.9 The Structure of Interactions in Nature......Page 206
3.1 The Trouble with Scale Changes......Page 209
3.2 Wilson\'s Renormalization Group Theory in Physics......Page 211
3.3 Stable and Unstable Manifolds......Page 228
3.4 A Glance at Conformal Field Theories......Page 229
4. Analyticity......Page 230
4.1 Power Series Expansion......Page 231
4.3 Cauchy\'s Integral Formula......Page 233
4.4 Cauchy\'s Residue Formula and Topological Charges......Page 234
4.5 The Winding Number......Page 235
4.6 Gauss\' Fundamental Theorem of Algebra......Page 236
4.7 Compactification of the Complex Plane......Page 238
4.8 Analytic Continuation and the Local-Global Principle......Page 239
4.9 Integrals and Riemann Surfaces......Page 240
4.10 Domains of Holomorphy......Page 244
4.11 A Glance at Analytic S-Matrix Theory......Page 245
4.12 Important Applications......Page 246
5.1 Local and Global Properties of the Universe......Page 247
5.2 Bolzano\'s Existence Principle......Page 248
5.3 Elementary Geometric Notions......Page 250
5.4 Manifolds and Diffeomorphisms......Page 254
5.5 Topological Spaces, Homeomorphisms, and Deformations......Page 255
5.6 Topological Quantum Numbers......Page 261
5.7 Quantum States......Page 285
5.8 Perspectives......Page 295
6. Many-Particle Systems in Mathematics and Physics......Page 296
6.1 Partition Function in Statistical Physics......Page 298
6.2 Euler\'s Partition Function......Page 302
6.3 Discrete Laplace Transformation......Page 304
6.4 Integral Transformations......Page 308
6.5 The Riemann Zeta Function......Page 310
6.6 The Casimir Effect in Quantum Field Theory and the Epstein Zeta Function......Page 318
6.7 Appendix: The Mellin Transformation and Other Useful Analytic Techniques by Don Zagier......Page 324
7.1 Geometrization of Physics......Page 343
7.2 Ariadne\'s Thread in Quantum Field Theory......Page 344
7.3 Linear Spaces......Page 346
7.4 Finite-Dimensional Hilbert Spaces......Page 353
7.5 Groups......Page 358
7.6 Lie Algebras......Page 360
7.7 Lie\'s Logarithmic Trick for Matrix Groups......Page 363
7.8 Lie Groups......Page 365
7.9 Basic Notions in Quantum Physics......Page 367
7.10 Fourier Series......Page 373
7.11 Dirac Calculus in Finite-Dimensional Hilbert Spaces......Page 377
7.12 The Trace of a Linear Operator......Page 381
7.13 Banach Spaces......Page 384
7.14 Probability and Hilbert\'s Spectral Family of an Observable......Page 386
7.15 Transition Probabilities, S-Matrix, and Unitary Operators......Page 388
7.16 The Magic Formulas for the Green\'s Operator......Page 390
7.17 The Magic Dyson Formula for the Retarded Propagator......Page 399
7.18 The Magic Dyson Formula for the S-Matrix......Page 408
7.19 Canonical Transformations......Page 410
7.20 Functional Calculus......Page 413
7.21 The Discrete Feynman Path Integral......Page 434
7.22 Causal Correlation Functions......Page 442
7.23 The Magic Gaussian Integral......Page 446
7.24 The Rigorous Response Approach to Finite Quantum Fields......Page 456
7.25 The Discrete φ[sup(4)]-Model and Feynman Diagrams......Page 477
7.26 The Extended Response Approach......Page 495
7.27 Complex-Valued Fields......Page 501
7.28 The Method of Lagrange Multipliers......Page 505
7.29 The Formal Continuum Limit......Page 510
8.1 Renormalization......Page 514
8.2 The Rellich Theorem......Page 523
8.3 The Trotter Product Formula......Page 524
8.4 The Magic Baker–Campbell–Hausdorff Formula......Page 525
8.5 Regularizing Terms......Page 526
9.1 The Grassmann Product......Page 531
9.3 Calculus for One Grassmann Variable......Page 532
9.4 Calculus for Several Grassmann Variables......Page 533
9.5 The Determinant Trick......Page 534
9.7 The Fermionic Response Model......Page 535
10.1 The Importance of Infinite Dimensions in Quantum Physics......Page 537
10.2 The Hilbert Space L[sub(2)](Ω)......Page 541
10.3 Harmonic Analysis......Page 548
10.4 The Dirichlet Problem in Electrostatics as a Paradigm......Page 556
11. Distributions and Green\'s Functions......Page 590
11.1 Rigorous Basic Ideas......Page 594
11.2 Dirac\'s Formal Approach......Page 604
11.3 Laurent Schwartz\'s Rigorous Approach......Page 622
11.4 Hadamard\'s Regularization of Integrals......Page 633
11.5 Renormalization of the Anharmonic Oscillator......Page 640
11.6 The Importance of Algebraic Feynman Integrals......Page 649
11.7 Fundamental Solutions of Differential Equations......Page 659
11.8 Functional Integrals......Page 666
11.9 A Glance at Harmonic Analysis......Page 675
11.10 The Trouble with the Euclidean Trick......Page 681
12.1 The Discrete Dirac Calculus......Page 683
12.2 Rigorous General Dirac Calculus......Page 689
12.3 Fundamental Limits in Physics......Page 696
12.4 Duality in Physics......Page 704
12.5 Microlocal Analysis......Page 717
12.6 Multiplication of Distributions......Page 743
13. Basic Strategies in Quantum Field Theory......Page 752
13.1 The Method of Moments and Correlation Functions......Page 755
13.2 The Power of the S-Matrix......Page 758
13.3 The Relation Between the S-Matrix and the Correlation Functions......Page 759
13.4 Perturbation Theory and Feynman Diagrams......Page 760
13.5 The Trouble with Interacting Quantum Fields......Page 761
13.6 External Sources and the Generating Functional......Page 762
13.7 The Beauty of Functional Integrals......Page 764
13.8 Quantum Field Theory at Finite Temperature......Page 770
14. The Response Approach......Page 777
14.1 The Fourier–Minkowski Transform......Page 782
14.2 The φ[sup(4)]-Model......Page 785
14.3 A Glance at Quantum Electrodynamics......Page 801
15. The Operator Approach......Page 824
15.1 The φ[sup(4)]-Model......Page 825
15.2 A Glance at Quantum Electrodynamics......Page 857
15.3 The Role of Effective Quantities in Physics......Page 858
15.4 A Glance at Renormalization......Page 859
15.5 The Convergence Problem in Quantum Field Theory......Page 871
15.6 Rigorous Perspectives......Page 873
16.1 Basic Difficulties......Page 887
16.2 The Principle of Critical Action......Page 888
16.3 The Language of Physicists......Page 894
16.5 Integration over Orbit Spaces......Page 896
16.6 The Magic Faddeev–Popov Formula and Ghosts......Page 898
16.7 The BRST Symmetry......Page 900
16.8 The Power of Cohomology......Page 901
16.9 The Batalin–Vilkovisky Formalism......Page 913
16.10 A Glance at Quantum Symmetries......Page 914
17.1 Introduction to Quantum Field Theory......Page 916
17.2 Standard Literature in Quantum Field Theory......Page 919
17.3 Rigorous Approaches to Quantum Field Theory......Page 920
17.4 The Fascinating Interplay between Modern Physics and Mathematics......Page 922
17.5 The Monster Group, Vertex Algebras, and Physics......Page 928
17.6 Historical Development of Quantum Field Theory......Page 933
17.7 General Literature in Mathematics and Physics......Page 934
17.9 Highlights of Physics in the 20th Century......Page 935
17.10 Actual Information......Page 937
A.1 Notation......Page 940
A.2 The International System of Units......Page 943
A.3 The Planck System......Page 945
A.4 The Energetic System......Page 951
A.5 The Beauty of Dimensional Analysis......Page 953
A.6 The Similarity Principle in Physics......Page 955
Epilogue......Page 963
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List of Symbols......Page 999
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