Zeidler Quantum Field Theory I: Basics in Mathematics and Physics

A Bridge between Mathematicians and Physicists
1. Auflage 2007
ISBN: 978-3-540-34764-4
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 1051 Seiten

1 Daß ich erkenne, was die Welt im Innersten zusammenh¨ alt. Faust Concepts without intuition are empty, intuition without concepts is blind. Immanuel Kant (1724–1804) The greatest mathematicians like Archimedes, Newton, and Gauss have always been able to combine theory and applications into one. Felix Klein (1849–1925) The present comprehensive introduction to the mathematical and physical aspects of quantum ?eld theory consists of the following six volumes: Volume I: Basics in Mathematics and Physics Volume II: Quantum Electrodynamics Volume III: Gauge Theory Volume IV: Quantum Mathematics Volume V: The Physics of the Standard Model Volume VI: Quantum Gravity and String Theory. Since ancient times, both physicists and mathematicians have tried to und- stand the forces acting in nature. Nowadays we know that there exist four fundamental forces in nature: • Newton’s gravitational force, • Maxwell’s electromagnetic force, • the strong force between elementary particles, and • the weak force between elementary particles (e.g., the force responsible for the radioactive decay of atoms). In the 20th century, physicists established two basic models, namely, • the Standard Model in cosmology based on Einstein’s theory of general relativity, and • the Standard Model in elementary particle physics based on gauge theory. 1 So that I may perceive whatever holds the world together in its inmost folds.

Autoren/Hrsg.

Zeidler, Eberhard

Inhaltsverzeichnis

1;Preface;6
2;Contents;12
3;Prologue;24
4;1. Historical Introduction;44
4.1;1.1 The Revolution of Physics;45
4.2;1.2 Quantization in a Nutshell;50
4.3;1.3 The Role of Göttingen;83
4.4;1.4 The Göttingen Tragedy;90
4.5;1.5 Highlights in the Sciences;92
4.6;1.6 The Emergence of Physical Mathematics – a New Dimension of Mathematics;98
4.7;1.7 The Seven Millennium Prize Problems of the Clay Mathematics Institute;100
5;2. Phenomenology of the Standard Model for Elementary Particles;102
5.1;2.1 The System of Units;103
5.2;2.2 Waves in Physics;104
5.3;2.3 Historical Background;120
5.4;2.4 The Standard Model in Particle Physics;150
5.5;2.5 Magic Formulas;163
5.6;2.6 Quantum Numbers of Elementary Particles;166
5.7;2.7 The Fundamental Role of Symmetry in Physics;185
5.8;2.8 Symmetry Breaking;201
5.9;2.9 The Structure of Interactions in Nature;206
6;3. The Challenge of Different Scales in Nature;209
6.1;3.1 The Trouble with Scale Changes;209
6.2;3.2 Wilson’s Renormalization Group Theory in Physics;211
6.3;3.3 Stable and Unstable Manifolds;228
6.4;3.4 A Glance at Conformal Field Theories;229
7;4. Analyticity;230
7.1;4.1 Power Series Expansion;231
7.2;4.2 Deformation Invariance of Integrals;233
7.3;4.3 Cauchy’s Integral Formula;233
7.4;4.4 Cauchy’s Residue Formula and Topological Charges;234
7.5;4.5 The Winding Number;235
7.6;4.6 Gauss’ Fundamental Theorem of Algebra;236
7.7;4.7 Compacti.cation of the Complex Plane;238
7.8;4.8 Analytic Continuation and the Local-Global Principle;239
7.9;4.9 Integrals and Riemann Surfaces;240
7.10;4.10 Domains of Holomorphy;244
7.11;4.11 A Glance at Analytic S-Matrix Theory;245
7.12;4.12 Important Applications;246
8;5. A Glance at Topology;247
8.1;5.1 Local and Global Properties of the Universe;247
8.2;5.2 Bolzano’s Existence Principle;248
8.3;5.3 Elementary Geometric Notions;250
8.4;5.4 Manifolds and Diffeomorphisms;254
8.5;5.5 Topological Spaces, Homeomorphisms, and Deformations;255
8.6;5.6 Topological Quantum Numbers;261
8.7;5.7 Quantum States;285
8.8;5.8 Perspectives;295
9;6. Many-Particle Systems in Mathematics and Physics;296
9.1;6.1 Partition Function in Statistical Physics;298
9.2;6.2 Euler’s Partition Function;302
9.3;6.3 Discrete Laplace Transformation;304
9.4;6.4 Integral Transformations;308
9.5;6.5 The Riemann Zeta Function;310
9.6;6.6 The Casimir Effect in Quantum Field Theory and the Epstein Zeta Function;318
9.7;6.7 Appendix: The Mellin Transformation and Other Useful Analytic Techniques by Don Zagier;324
10;7. Rigorous Finite-Dimensional Magic Formulas of Quantum Field Theory;343
10.1;7.1 Geometrization of Physics;343
10.2;7.2 Ariadne’s Thread in Quantum Field Theory;344
10.3;7.3 Linear Spaces;346
10.4;7.4 Finite-Dimensional Hilbert Spaces;353
10.5;7.5 Groups;358
10.6;7.6 Lie Algebras;360
10.7;7.7 Lie’s Logarithmic Trick for Matrix Groups;363
10.8;7.8 Lie Groups;365
10.9;7.9 Basic Notions in Quantum Physics;367
10.10;7.10 Fourier Series;373
10.11;7.11 Dirac Calculus in Finite-Dimensional Hilbert Spaces;377
10.12;7.12 The Trace of a Linear Operator;381
10.13;7.13 Banach Spaces;384
10.14;7.14 Probability and Hilbert’s Spectral Family of an Observable;386
10.15;7.15 Transition Probabilities, S-Matrix, and Unitary Operators;388
10.16;7.16 The Magic Formulas for the Green’s Operator;390
10.17;7.17 The Magic Dyson Formula for the Retarded Propagator;399
10.18;7.18 The Magic Dyson Formula for the S-Matrix;408
10.19;7.19 Canonical Transformations;410
10.20;7.20 Functional Calculus;413
10.21;7.21 The Discrete Feynman Path Integral;434
10.22;7.22 Causal Correlation Functions;442
10.23;7.23 The Magic Gaussian Integral;446
10.24;7.24 The Rigorous Response Approach to Finite Quantum Fields;456
10.25;7.25 The Discrete .4-Model and Feynman Diagrams;477
10.26;7.26 The Extended Response Approach;495
10.27;7.27 Complex-Valued Fields;501
10.28;7.28 The Method of Lagrange Multipliers;505
10.29;7.29 The Formal Continuum Limit;510
10.30;Problems;511
11;8. Rigorous Finite-Dimensional Perturbation Theory;514
11.1;8.1 Renormalization;514
11.2;8.2 The Rellich Theorem;523
11.3;8.3 The Trotter Product Formula;524
11.4;8.4 The Magic Baker–Campbell–Hausdorff Formula;525
11.5;8.5 Regularizing Terms;526
12;9. Fermions and the Calculus for Grassmann Variables;531
12.1;9.1 The Grassmann Product;531
12.2;9.2 Differential Forms;532
12.3;9.3 Calculus for One Grassmann Variable;532
12.4;9.4 Calculus for Several Grassmann Variables;533
12.5;9.5 The Determinant Trick;534
12.6;9.6 The Method of Stationary Phase;535
12.7;9.7 The Fermionic Response Model;535
13;10. Infinite-Dimensional Hilbert Spaces;537
13.1;10.1 The Importance of Infinite Dimensions in Quantum Physics;537
13.2;10.2 The Hilbert Space;541
13.3;10.3 Harmonic Analysis;548
13.4;10.4 The Dirichlet Problem in Electrostatics as a Paradigm;556
13.5;Problems;587
14;11. Distributions and Green’s Functions;590
14.1;11.1 Rigorous Basic Ideas;594
14.2;11.2 Dirac’s Formal Approach;604
14.3;11.3 Laurent Schwartz’s Rigorous Approach;622
14.4;11.4 Hadamard’s Regularization of Integrals;633
14.5;11.5 Renormalization of the Anharmonic Oscillator;640
14.6;11.6 The Importance of Algebraic Feynman Integrals;649
14.7;11.7 Fundamental Solutions of Differential Equations;659
14.8;11.8 Functional Integrals;666
14.9;11.9 A Glance at Harmonic Analysis;675
14.10;11.10 The Trouble with the Euclidean Trick;681
15;12. Distributions and Physics;683
15.1;12.1 The Discrete Dirac Calculus;683
15.2;12.2 Rigorous General Dirac Calculus;689
15.3;12.3 Fundamental Limits in Physics;696
15.4;12.4 Duality in Physics;704
15.5;12.5 Microlocal Analysis;717
15.6;12.6 Multiplication of Distributions;743
15.7;Problems;746
16;13. Basic Strategies in Quantum Field Theory;752
16.1;13.1 The Method of Moments and Correlation Functions;755
16.2;13.2 The Power of the S-Matrix;758
16.3;13.3 The Relation Between the S-Matrix and the Correlation Functions;759
16.4;13.4 Perturbation Theory and Feynman Diagrams;760
16.5;13.5 The Trouble with Interacting Quantum Fields;761
16.6;13.6 External Sources and the Generating Functional;762
16.7;13.7 The Beauty of Functional Integrals;764
16.8;13.8 Quantum Field Theory at Finite Temperature;770
17;14. The Response Approach;777
17.1;14.1 The Fourier–Minkowski Transform;782
17.2;14.2 The .4-Model;785
17.3;14.3 A Glance at Quantum Electrodynamics;801
17.4;Problems;816
18;15. The Operator Approach;824
18.1;15.1 The .4-Model;825
18.2;15.2 A Glance at Quantum Electrodynamics;857
18.3;15.3 The Role of Effective Quantities in Physics;858
18.4;15.4 A Glance at Renormalization;859
18.5;15.5 The Convergence Problem in Quantum Field Theory;871
18.6;15.6 Rigorous Perspectives;873
19;16. Peculiarities of Gauge Theories;887
19.1;16.1 Basic Difficulties;887
19.2;16.2 The Principle of Critical Action;888
19.3;16.3 The Language of Physicists;894
19.4;16.4 The Importance of the Higgs Particle;896
19.5;16.5 Integration over Orbit Spaces;896
19.6;16.6 The Magic Faddeev–Popov Formula and Ghosts;898
19.7;16.7 The BRST Symmetry;900
19.8;16.8 The Power of Cohomology;901
19.9;16.9 The Batalin–Vilkovisky Formalism;913
19.10;16.10 A Glance at Quantum Symmetries;914
20;17. A Panorama of the Literature;916
20.1;17.1 Introduction to Quantum Field Theory;916
20.2;17.2 Standard Literature in Quantum Field Theory;919
20.3;17.3 Rigorous Approaches to Quantum Field Theory;920
20.4;17.4 The Fascinating Interplay between Modern Physics and Mathematics;922
20.5;17.5 The Monster Group, Vertex Algebras, and Physics;928
20.6;17.6 Historical Development of Quantum Field Theory;933
20.7;17.7 General Literature in Mathematics and Physics;934
20.8;17.8 Encyclopedias;935
20.9;17.9 Highlights of Physics in the 20th Century;935
20.10;17.10 Actual Information;937
21;Appendix;940
21.1;A.1 Notation;940
21.2;A.2 The International System of Units;943
21.3;A.3 The Planck System;945
21.4;A.4 The Energetic System;951
21.5;A.5 The Beauty of Dimensional Analysis;953
21.6;A.6 The Similarity Principle in Physics;955
22;Epilogue;963
23;References;967
24;List of Symbols;999
25;Index;1003