Quantum Field Theory I

by Zeidler, Eberhard

• ISBN13: 9783540347620
• ISBN10: 3540347623
• Format: Hardcover
• Copyright: 2006-10-04
• Publisher: Springer Nature

Summary

This is the first volume of a modern introduction to quantum field theory which addresses both mathematicians and physicists ranging from advanced undergraduate students to professional scientists. The book tries to bridge the existing gap between the different languages used by mathematicians and physicists. For students of mathematics it is shown that detailed knowledge of the physical background helps to motivate the mathematical subjects and to discover interesting interrelationships between quite different mathematical topics. For students of physics, fairly advanced mathematics is presented, which is beyond the usual curriculum in physics. It is the author's goal to present the state of the art of realizing Einstein's dream of a unified theory for the four fundamental forces in the universe (gravitational, electromagnetic, strong, and weak interaction).

Author Biography

Prof. Dr. Dr. h.c. Eberhard Zeidler works at the Max Planck Institute for Mathematics in the Sciences in Leipzig (Germany). In 1996 he was one of the founding directors of this institute. He is a member of the Academy of Natural Scientists Leopoldina. In 2006 he was awarded the "Alfried Krupp Wissenschaftspreis" of the Alfried Krupp von Bohlen und Halbach-Stiftung. The author wrote the following books.(a) E. Zeidler, Nonlinear Functional Analysis and its Applications, Vols. I-IV,Springer Verlag New York, 1984-1988 (third edition 1998).(b) E. Zeidler, Applied Functional Analysis, Vol. 1: Applications to Mathematical Physics, 2nd edition, 1997, Springer Verlag, New York.(c) E. Zeidler, Applied Functional Analysis, Vol. 2: Main Principles and Their Applications, Springer-Verlag, New York, 1995.(d) E. Zeidler, Oxford Users' Guide to Mathematics, Oxford University Press, 2004(translated from German).

Table of Contents

Part I. Introduction
Prologue	1	(208)
1. Historical Introduction	21	(58)
1.1 The Revolution of Physics	22	(5)
1.2 Quantization in a Nutshell	27	(33)
1.2.1 Basic Formulas	30	(16)
1.2.2 The Fundamental Role of the Harmonic Oscillator in Quantum Field Theory	46	(6)
1.2.3 Quantum Fields and Second Quantization	52	(5)
1.2.4 The Importance of Functional Integrals	57	(3)
1.3 The Role of Göttingen	60	(7)
1.4 The Göttingen Tragedy	67	(2)
1.5 Highlights in the Sciences	69	(6)
1.5.1 The Nobel Prize	69	(2)
1.5.2 The Fields Medal in Mathematics	71	(1)
1.5.3 The Nevanlinna Prize in Computer Sciences	72	(1)
1.5.4 The Wolf Prize in Physics	73	(1)
1.5.5 The Wolf Prize in Mathematics	73	(2)
1.5.6 The Abel Prize in Mathematics	75	(1)
1.6 The Emergence of Physical Mathematics — a New Dimension of Mathematics	75	(2)
1.7 The Seven Millennium Prize Problems of the Clay Mathematics Institute	77	(2)
2. Phenomenology of the Standard Model for Elementary Particles	79	(108)
2.1 The System of Units	80	(1)
2.2 Waves in Physics	81	(16)
2.2.1 Harmonic Waves	81	(1)
2.2.2 Wave Packets	82	(2)
2.2.3 Standing Waves	84	(1)
2.2.4 Electromagnetic Waves	85	(1)
2.2.5 Superposition of Waves and the Fourier Transform	86	(3)
2.2.6 Damped Waves, the Laplace Transform, and Dispersion Relations	89	(5)
2.2.7 The Response Function, the Feynman Propagator, and Causality	94	(3)
2.3 Historical Background	97	(30)
2.3.1 Planck's Radiation Law	101	(5)
2.3.2 The Boltzmann Statistics and Planck's Quantum Hypothesis	106	(3)
2.3.3 Einstein's Theory of Special Relativity	109	(2)
2.3.4 Einstein's Theory of General Relativity	111	(1)
2.3.5 Einstein's Light Particle Hypothesis	112	(1)
2.3.6 Rutherford's Particle Scattering	113	(2)
2.3.7 The Cross Section for Compton Scattering	115	(5)
2.3.8 Bohr's Model of the Hydrogen Atom	120	(4)
2.3.9 Einstein's Radiation Law and Laser Beams	124	(2)
2.3.10 Quantum Computers	126	(1)
2.4 The Standard Model in Particle Physics	127	(13)
2.4.1 The Four Fundamental Forces in Nature	127	(3)
2.4.2 The Fundamental Particles in Nature	130	(10)
2.5 Magic Formulas	140	(3)
2.6 Quantum Numbers of Elementary Particles	143	(19)
2.6.1 The Spin	144	(10)
2.6.2 Conservation of Quantum Numbers	154	(8)
2.7 The Fundamental Role of Symmetry in Physics	162	(16)
2.7.1 Classical Symmetries	168	(2)
2.7.2 The CPT Symmetry Principle for Elementary Particles	170	(4)
2.7.3 Local Gauge Symmetry	174	(2)
2.7.4 Permutations and Pauli's Exclusion Principle	176	(1)
2.7.5 Crossing Symmetry	176	(1)
2.7.6 Forbidden Spectral Lines in Molecules	177	(1)
2.8 Symmetry Breaking	178	(5)
2.8.1 Parity Violation and CP Violation	178	(1)
2.8.2 Irreversibility	179	(1)
2.8.3 Splitting of Spectral Lines in Molecules	179	(1)
2.8.4 Spontaneous Symmetry Breaking and Particles	180	(2)
2.8.5 Bifurcation and Phase Transitions	182	(1)
2.9 The Structure of Interactions in Nature	183	(4)
2.9.1 The Electromagnetic Field as Generalized Curvature	183	(1)
2.9.2 Physics and Modern Differential Geometry	184	(3)
3. The Challenge of Different Scales in Nature	187	(22)
3.1 The Trouble with Scale Changes	187	(2)
3.2 Wilson's Renormalization Group Theory in Physics	189	(17)
3.2.1 A New Paradigm in Physics	191	(2)
3.2.2 Screening of the Coulomb Field and the Renormalization Group of Lie Type	193	(8)
3.2.3 The Running Coupling Constant and the Asymptotic Freedom of Quarks	201	(3)
3.2.4 The Quark Confinement	204	(1)
3.2.5 Proton Decay and Supersymmetric Grand Unification	205	(1)
3.2.6 The Adler—Bell—Jackiw Anomaly	205	(1)
3.3 Stable and Unstable Manifolds	206	(1)
3.4 A Glance at Conformal Field Theories	207	(2)
Part II. Basic Techniques in Mathematics
4. Analyticity	209	(18)
4.1 Power Series Expansion	210	(2)
4.2 Deformation Invariance of Integrals	212	(1)
4.3 Cauchy's Integral Formula	212	(1)
4.4 Cauchy's Residue Formula and Topological Charges	213	(1)
4.5 The Winding Number	214	(1)
4.6 Gauss' Fundamental Theorem of Algebra	215	(2)
4.7 Compactification of the Complex Plane	217	(1)
4.8 Analytic Continuation and the Local-Global Principle	218	(1)
4.9 Integrals and Riemann Surfaces	219	(4)
4.10 Domains of Holomorphy	223	(1)
4.11 A Glance at Analytic S-Matrix Theory	224	(1)
4.12 Important Applications	225	(2)
5. A Glance at Topology	227	(50)
5.1 Local and Global Properties of the Universe	227	(1)
5.2 Bolzano's Existence Principle	228	(2)
5.3 Elementary Geometric Notions	230	(4)
5.4 Manifolds and Diffeomorphisms	234	(1)
5.5 Topological Spaces, Homeomorphisms, and Deformations	235	(6)
5.6 Topological Quantum Numbers	241	(24)
5.6.1 The Genus of a Surface	241	(1)
5.6.2 The Euler Characteristic	242	(2)
5.6.3 Platonic Solids and Fullerenes	244	(1)
5.6.4 The Poincaré—Hopf Theorem for Velocity Fields	245	(1)
5.6.5 The Gauss—Bonnet Theorem	246	(4)
5.6.6 The Morse Theorem on Critical Points of Energy Functions	250	(1)
5.6.7 Magnetic Fields, the Gauss Integral, and the Linking Number	251	(2)
5.6.8 Electric Fields, the Kronecker Integral, and the Mapping Degree	253	(4)
5.6.9 The Heat Kernel and the Atiyah—Singer Index Theorem	257	(6)
5.6.10 Knots and Topological Quantum Field Theory	263	(2)
5.7 Quantum States	265	(10)
5.7.1 The Topological Character of the Electron Spin	265	(3)
5.7.2 The Hopf Fibration of the 3-Dimensional Sphere	268	(3)
5.7.3 The Homotopy Functor	271	(3)
5.7.4 Grassmann Manifolds and Projective Geometry	274	(1)
5.8 Perspectives	275	(2)
6. Many-Particle Systems in Mathematics and Physics	277	(48)
6.1 Partition Function in Statistical Physics	279	(4)
6.2 Euler's Partition Function	283	(2)
6.3 Discrete Laplace Transformation	285	(4)
6.4 Integral Transformations	289	(2)
6.5 The Riemann Zeta Function	291	(8)
6.5.1 The Prime Number Theorem — a Pearl of Mathematics	291	(5)
6.5.2 The Riemann Hypothesis	296	(1)
6.5.3 Dirichlet's L-Function	296	(3)
6.6 The Casimir Effect in Quantum Field Theory and the Epstein Zeta Function	299	(6)
6.7 Appendix: The Mellin Transformation and Other Useful Analytic Techniques by Don Zagier	305	(20)
6.7.1 The Generalized Mellin Transformation	305	(4)
6.7.2 Dirichlet Series and their Special Values	309	(3)
6.7.3 Application: the Casimir Effect	312	(5)
6.7.4 Asymptotics of Series of the Form Σƒ (nt)	317	(8)
7. Rigorous Finite-Dimensional Magic Formulas of Quantum Field Theory	325	(172)
7.1 Geometrization of Physics	325	(1)
7.2 Ariadne's Thread in Quantum Field Theory	326	(2)
7.3 Linear Spaces	328	(7)
7.4 Finite-Dimensional Hilbert Spaces	335	(5)
7.5 Groups	340	(2)
7.6 Lie Algebras	342	(3)
7.7 Lie's Logarithmic Trick for Matrix Groups	345	(2)
7.8 Lie Groups	347	(2)
7.9 Basic Notions in Quantum Physics	349	(6)
7.9.1 States, Costates, and Observables	350	(4)
7.9.2 Observers and Coordinates	354	(1)
7.10 Fourier Series	355	(4)
7.11 Dirac Calculus in Finite-Dimensional Hilbert Spaces	359	(4)
7.12 The Trace of a Linear Operator	363	(3)
7.13 Banach Spaces	366	(2)
7.14 Probability and Hilbert's Spectral Family of an Observable	368	(2)
7.15 Transition Probabilities, S-Matrix, and Unitary Operators	370	(2)
7.16 The Magic Formulas for the Green's Operator	372	(9)
7.16.1 Non-Resonance and Resonance	373	(4)
7.16.2 Causality and the Laplace Transform	377	(4)
7.17 The Magic Dyson Formula for the Retarded Propagator	381	(9)
7.17.1 Lagrange's Variation of the Parameter	383	(2)
7.17.2 Duhamel's Principle	385	(1)
7.17.3 The Volterra Integral Equation	386	(2)
7.17.4 The Dyson Series	388	(2)
7.18 The Magic Dyson Formula for the S-Matrix	390	(2)
7.19 Canonical Transformations	392	(3)
7.19.1 The Schrödinger Picture	392	(1)
7.19.2 The Heisenberg Picture	393	(1)
7.19.3 The Dirac Interaction Picture	394	(1)
7.20 Functional Calculus	395	(21)
7.20.1 Functional Derivatives	396	(5)
7.20.2 Partial Functional Derivatives	401	(8)
7.20.3 Infinitesimal Transformations	409	(7)
7.20.4 Functional Integration	416	(1)
7.21 The Discrete Feynman Path Integral	416	(8)
7.21.1 The Magic Feynman Propagator Formula	417	(5)
7.21.2 The Magic Formula for Time-Ordered Products	422	(1)
7.21.3 The Trace Formula	423	(1)
7.22 Causal Correlation Functions	424	(4)
7.22.1 The Wick Rotation Trick for Vacuum Expectation Values	425	(2)
7.22.2 The Magic Gell-Mann–Low Reduction Formula	427	(1)
7.23 The Magic Gaussian Integral	428	(10)
7.23.1 The One-Dimensional Prototype	428	(6)
7.23.2 The Determinant Trick	434	(1)
7.23.3 The Zeta Function Trick	434	(1)
7.23.4 The Moment Trick	435	(1)
7.23.5 The Method of Stationary Phase	435	(3)
7.24 The Rigorous Response Approach to Finite Quantum Fields	438	(21)
7.24.1 Basic Ideas	439	(2)
7.24.2 Discrete Space-Time Manifold	441	(4)
7.24.3 The Principle of Critical Action	445	(1)
7.24.4 The Response Function	446	(1)
7.24.5 The Global Quantum Action Principle	447	(1)
7.24.6 The Magic Quantum Action Reduction Formula for Correlation Functions	448	(1)
7.24.7 The Magic LSZ Reduction Formula for Scattering Functions	449	(3)
7.24.8 The Local Quantum Action Principle	452	(2)
7.24.9 Simplifying the Computation of Quantum Effects	454	(1)
7.24.10 Reduced Correlation Functions	455	(1)
7.24.11 The Mean Field Approximation	456	(1)
7.24.12 Vertex Functions and the Effective Action	457	(2)
7.25 The Discrete ρ4-Model and Feynman Diagrams	459	(18)
7.26 The Extended Response Approach	477	(6)
7.27 Complex-Valued Fields	483	(4)
7.28 The Method of Lagrange Multipliers	487	(5)
7.29 The Formal Continuum Limit	492	(5)
8. Rigorous Finite-Dimensional Perturbation Theory	497	(18)
8.1 Renormalization	497	(9)
8.1.1 Non-Resonance	497	(2)
8.1.2 Resonance, Regularizing Term, and Bifurcation	499	(3)
8.1.3 The Renormalization Group	502	(1)
8.1.4 The Main Bifurcation Theorem	503	(3)
8.2 The Rellich Theorem	506	(1)
8.3 The Trotter Product Formula	507	(1)
8.4 The Magic Baker—Campbell—Hausdorff Formula	508	(1)
8.5 Regularizing Terms	509	(6)
8.5.1 The Weierstrass Product Theorem	509	(1)
8.5.2 The Mittag—Leffler Theorem	510	(1)
8.5.3 Regularization of Divergent Integrals	511	(2)
8.5.4 The Polchinski Equation	513	(2)
9. Fermions and the Calculus for Grassmann Variables	515	(6)
9.1 The Grassmann Product	515	(1)
9.2 Differential Forms	516	(1)
9.3 Calculus for One Grassmann Variable	516	(1)
9.4 Calculus for Several Grassmann Variables	517	(1)
9.5 The Determinant Trick	518	(1)
9.6 The Method of Stationary Phase	519	(1)
9.7 The Fermionic Response Model	519	(2)
10. Infinite-Dimensional Hilbert Spaces	521	(54)
10.1 The Importance of Infinite Dimensions in Quantum Physics	521	(4)
10.1.1 The Uncertainty Relation	521	(3)
10.1.2 The Trouble with the Continuous Spectrum	524	(1)
10.2 The Hilbert Space L2 (Ω)	525	(7)
10.2.1 Measure and Integral	527	(2)
10.2.2 Dirac Measure and Dirac Integral	529	(1)
10.2.3 Lebesgue Measure and Lebesgue Integral	530	(1)
10.2.4 The Fischer—Riesz Theorem	531	(1)
10.3 Harmonic Analysis	532	(8)
10.3.1 Gauss' Method of Least Squares	532	(1)
10.3.2 Discrete Fourier Transform	533	(2)
10.3.3 Continuous Fourier Transform	535	(5)
10.4 The Dirichlet Problem in Electrostatics as a Paradigm	540	(39)
10.4.1 The Variational Lemma	542	(2)
10.4.2 Integration by Parts	544	(3)
10.4.3 The Variational Problem	547	(2)
10.4.4 Weierstrass' Counterexample	549	(3)
10.4.5 Typical Difficulties	552	(3)
10.4.6 The Functional Analytic Existence Theorem	555	(3)
10.4.7 Regularity of the Solution	558	(2)
10.4.8 The Beauty of the Green's Function	560	(4)
10.4.9 The Method of Orthogonal Projection	564	(3)
10.4.10 The Power of Ideas in Mathematics	567	(1)
10.4.11 The Ritz Method	568	(1)
10.4.12 The Main Existence Principle	569	(6)
11. Distributions and Green's Functions	575	(94)
11.1 Rigorous Basic Ideas	579	(10)
11.1.1 The Discrete Dirac Delta Function	580	(1)
11.1.2 Prototypes of Green's Functions	581	(5)
11.1.3 The Heat Equation and the Heat Kernel	586	(1)
11.1.4 The Diffusion Equation	587	(1)
11.1.5 The Schrödinger Equation and the Euclidean Approach	588	(1)
11.2 Dirac's Formal Approach	589	(18)
11.2.1 Dirac's Delta Function	590	(1)
11.2.2 Density of a Mass Distribution	591	(1)
11.2.3 Local Functional Derivative	591	(4)
11.2.4 The Substitution Rule	595	(1)
11.2.5 Formal Dirac Calculus and the Fourier Transform	596	(10)
11.2.6 Formal Construction of the Heat Kernel	606	(1)
11.3 Laurent Schwartz's Rigorous Approach	607	(11)
11.3.1 Physical Measurements and the Idea of Averaging	607	(1)
11.3.2 Distributions	608	(6)
11.3.3 Tempered Distributions	614	(3)
11.3.4 The Fourier Transform	617	(1)
11.4 Hadamard's Regularization of Integrals	618	(7)
11.4.1 Regularization of Divergent Integrals	618	(1)
11.4.2 The Sokhotski Formula	619	(1)
11.4.3 Steinmann's Renormalization Theorem	620	(2)
11.4.4 Regularization Terms	622	(3)
11.5 Renormalization of the Anharmonic Oscillator	625	(9)
11.5.1 Renormalization in a Nutshell	625	(1)
11.5.2 The Linearized Problem	625	(4)
11.5.3 The Nonlinear Problem and Non-Resonance	629	(1)
11.5.4 The Nonlinear Problem, Resonance, and Bifurcation	630	(2)
11.5.5 The Importance of the Renormalized Green's Function	632	(1)
11.5.6 The Renormalization Group	633	(1)
11.6 The Importance of Algebraic Feynman Integrals	634	(10)
11.6.1 Wick Rotation and Cut-Off	634	(2)
11.6.2 Dimensional Regularization	636	(2)
11.6.3 Weinberg's Power-Counting Theorem	638	(2)
11.6.4 Integration Tricks	640	(4)
11.7 Fundamental Solutions of Differential Equations	644	(7)
11.7.1 The Newtonian Potential	646	(1)
11.7.2 The Existence Theorem	646	(1)
11.7.3 The Beauty of Hironaka's Theorem	647	(4)
11.8 Functional Integrals	651	(9)
11.8.1 The Feynman Path Integral for the Heat Equation	651	(3)
11.8.2 Diffusion, Brownian Motion, and the Wiener Integral	654	(1)
11.8.3 The Method of Quantum Fluctuations	655	(2)
11.8.4 Infinite-Dimensional Gaussian Integrals and Zeta Function Regularization	657	(1)
11.8.5 The Euclidean Trick and the Feynman Path Integral for the Schrödinger Equation	658	(2)
11.9 A Glance at Harmonic Analysis	660	(6)
11.9.1 The Fourier–Laplace Transform	660	(2)
11.9.2 The Riemann–Hilbert Problem	662	(1)
11.9.3 The Hilbert Transform	663	(1)
11.9.4 Symmetry and Special Functions	664	(1)
11.9.5 Tempered Distributions as Boundary Values of Analytic Functions	665	(1)
11.10 The Trouble With the Euclidean Trick	666	(3)
12. Distributions and Physics	669	(70)
12.1 The Discrete Dirac Calculus	669	(6)
12.1.1 Lattices	669	(1)
12.1.2 The Four-Dimensional Discrete Dirac Delta Function	670	(3)
12.1.3 Rigorous Discrete Dirac Calculus	673	(1)
12.1.4 The Formal Continuum Limit	673	(2)
12.2 Rigorous General Dirac Calculus	675	(7)
12.2.1 Eigendistributions	675	(2)
12.2.2 Self-Adjoint Operators	677	(1)
12.2.3 The von Neumann Spectral Theorem	678	(1)
12.2.4 The Gelfand–Kostyuchenko Spectral Theorem	679	(1)
12.2.5 The Duality Map	679	(1)
12.2.6 Dirac's Notation	680	(1)
12.2.7 The Schwartz Kernel Theorem	681	(1)
12.3 Fundamental Limits in Physics	682	(8)
12.3.1 High-Energy Limit	682	(1)
12.3.2 Thermodynamic Limit and Phase Transitions	682	(3)
12.3.3 Adiabatic Limit	685	(4)
12.3.4 Singular Limit	689	(1)
12.4 Duality in Physics	690	(13)
12.4.1 Particles and de Broglie's Matter Waves	690	(2)
12.4.2 Time and Frequency	692	(1)
12.4.3 Time and Energy	692	(1)
12.4.4 Position and Momentum	692	(3)
12.4.5 Causality and Analyticity	695	(7)
12.4.6 Strong and Weak Interaction	702	(1)
12.5 Microlocal Analysis	703	(26)
12.5.1 Singular Support of a Distribution	704	(2)
12.5.2 Wave Front Set	706	(8)
12.5.3 The Method of Stationary Phase	714	(4)
12.5.4 Short-Wave Asymptotics for Electromagnetic Waves	718	(6)
12.5.5 Diffraction of Light	724	(4)
12.5.6 Pseudo-Differential Operators	728	(1)
12.5.7 Fourier Integral Operators	728	(1)
12.6 Multiplication of Distributions	729	(13)
12.6.1 Laurent Schwartz's Counterexample	730	(2)
12.6.2 Hörmander's Causal Product	732	(7)
Part III. Heuristic Magic Formulas of Quantum Field Theory
13. Basic Strategies in Quantum Field Theory	739	(26)
13.1 The Method of Moments and Correlation Functions	742	(3)
13.2 The Power of the S-Matrix	745	(1)
13.3 The Relation Between the S-Matrix and the Correlation Functions	746	(1)
13.4 Perturbation Theory and Feynman Diagrams	747	(1)
13.5 The Trouble with Interacting Quantum Fields	748	(1)
13.6 External Sources and the Generating Functional	749	(2)
13.7 The Beauty of Functional Integrals	751	(6)
13.7.1 The Principle of Critical Action	752	(1)
13.7.2 The Magic Feynman Representation Formula	753	(1)
13.7.3 Perturbation Theory	754	(1)
13.7.4 Renormalization	754	(1)
13.7.5 Transition Amplitudes	755	(1)
13.7.6 The Magic Trace Formula	756	(1)
13.8 Quantum Field Theory at Finite Temperature	757	(13)
13.8.1 The Partition Function	757	(3)
13.8.2 The Classical Hamiltonian Approach	760	(1)
13.8.3 The Magic Feynman Functional Integral for the Partition Function	761	(2)
13.8.4 The Thermodynamic Limit	763	(2)
14. The Response Approach	765	(48)
14.1 The Fourier—Minkowski Transform	770	(3)
14.2 The ρ4-Model	773	(16)
14.2.1 The Classical Principle of Critical Action	774	(1)
14.2.2 The Response Function and the Feynman Propagator	774	(8)
14.2.3 The Extended Quantum Action Functional	782	(1)
14.2.4 The Magic Quantum Action Reduction Formula for Correlation Functions	782	(3)
14.2.5 The Magic LSZ Reduction Formula for the S-Matrix	785	(2)
14.2.6 The Local Quantum Action Principle	787	(1)
14.2.7 The Mnemonic Functional Integral	787	(1)
14.2.8 Bose—Einstein Condensation of Dilute Gases	788	(1)
14.3 A Glance at Quantum Electrodynamics	789	(25)
14.3.1 The Equations of Motion	791	(1)
14.3.2 The Principle of Critical Action	792	(2)
14.3.3 The Gauge Field Approach	794	(3)
14.3.4 The Extended Action Functional with Source Term	797	(2)
14.3.5 The Response Function for Photons	799	(1)
14.3.6 The Response Function for Electrons	800	(1)
14.3.7 The Extended Quantum Action Functional	801	(2)
14.3.8 The Magic Quantum Action Reduction Formula	803	(1)
14.3.9 The Magic LSZ Reduction Formula	803	(1)
14.3.10 The Mnemonic Functional Integral	804	(9)
15. The Operator Approach	813	(64)
15.1 The ρ4-Model	814	(32)
15.1.1 The Lattice Approximation	815	(2)
15.1.2 Fourier Quantization	817	(3)
15.1.3 The Free 2-Point Green's Function	820	(2)
15.1.4 The Magic Dyson Formula for the S-Matrix	822	(2)
15.1.5 The Main Wick Theorem	824	(5)
15.1.6 Transition Amplitude	829	(8)
15.1.7 Transition Probability	837	(2)
15.1.8 Scattering Cross Section	839	(4)
15.1.9 General Feynman Rules for Particle Scattering	843	(2)
15.1.10 The Magic Gell-Mann-Low Reduction Formula for Green's Functions	845	(1)
15.2 A Glance at Quantum Electrodynamics	846	(1)
15.3 The Role of Effective Quantities in Physics	847	(1)
15.4 A Glance at Renormalization	848	(12)
15.4.1 The Trouble with the Continuum Limit	850	(1)
15.4.2 Basic Ideas of Renormalization	850	(3)
15.4.3 The BPHZ Renormalization	853	(1)
15.4.4 The Epstein—Glaser Approach	854	(4)
15.4.5 Algebraic Renormalization	858	(1)
15.4.6 The Importance of Hopf Algebras	859	(1)
15.5 The Convergence Problem in Quantum Field Theory	860	(2)
15.5.1 Dyson's No-Go Argument	860	(1)
15.5.2 The Power of the Classical Ritt Theorem in Quantum Field Theory	861	(1)
15.6 Rigorous Perspectives	862	(15)
15.6.1 Axiomatic Quantum Field Theory	866	(4)
15.6.2 The Euclidean Strategy in Constructive Quantum Field Theory	870	(2)
15.6.3 The Renormalization Group Method	872	(5)
16. Peculiarities of Gauge Theories	877	(30)
16.1 Basic Difficulties	877	(1)
16.2 The Principle of Critical Action	878	(6)
16.3 The Language of Physicists	884	(2)
16.4 The Importance of the Higgs Particle	886	(1)
16.5 Integration over Orbit Spaces	886	(2)
16.6 The Magic Faddeev—Popov Formula and Ghosts	888	(2)
16.7 The BRST Symmetry	890	(1)
16.8 The Power of Cohomology	891	(12)
16.8.1 Physical States, Unphysical States, and Cohomology	893	(1)
16.8.2 Forces and Potentials	894	(2)
16.8.3 The Cohomology of Geometric Objects	896	(3)
16.8.4 The Spectra of Atoms and Cohomology	899	(1)
16.8.5 BRST Symmetry and the Cohomology of Lie Groups	900	(3)
16.9 The Batalin—Vilkovisky Formalism	903	(1)
16.10 A Glance at Quantum Symmetries	904	(3)
17. A Panorama of the Literature	907	(24)
17.1 Introduction to Quantum Field Theory	907	(3)
17.2 Standard Literature in Quantum Field Theory	910	(1)
17.3 Rigorous Approaches to Quantum Field Theory	911	(2)
17.4 The Fascinating Interplay between Modern Physics and Mathematics	913	(6)
17.5 The Monster Group, Vertex Algebras, and Physics	919	(5)
17.6 Historical Development of Quantum Field Theory	924	(1)
17.7 General Literature in Mathematics and Physics	925	(1)
17.8 Encyclopedias	926	(1)
17.9 Highlights of Physics in the 20th Century	926	(2)
17.10 Actual Information	928	(3)
Appendix	931	(24)
A.1 Notation	931	(3)
A.2 The International System of Units	934	(2)
A.3 The Planck System	936	(6)
A.4 The Energetic System	942	(2)
A.5 The Beauty of Dimensional Analysis	944	(2)
A.6 The Similarity Principle in Physics	946	(9)
Epilogue	955	(4)
References	959	(32)
List of Symbols	991	(4)
Index	995