Quantum Field Theory 1-3. Basics in Mathematics and Physics, Quantum Electrodynamics, Gauge Theory: A Bridge between Mathematicians and Physicists

Zeidler - Quantum Field Theory (vol. 1)
	Contents
	Part I. Introduction
		Prologue
		1. Historical Introduction
			1.1 The Revolution of Physics
			1.2 Quantization in a Nutshell
			1.3 The Role of Göttingen
			1.4 The Göttingen Tragedy
			1.5 Highlights in the Sciences
			1.6 The Emergence of Physical Mathematics – a New Dimension of Mathematics
			1.7 The Seven Millennium Prize Problems of the Clay Mathematics Institute
		2. Phenomenology of the Standard Model for Elementary Particles
			2.1 The System of Units
			2.2 Waves in Physics
			2.3 Historical Background
			2.4 The Standard Model in Particle Physics
			2.5 Magic Formulas
			2.6 Quantum Numbers of Elementary Particles
			2.7 The Fundamental Role of Symmetry in Physics
			2.8 Symmetry Breaking
			2.9 The Structure of Interactions in Nature
		3. The Challenge of Different Scales in Nature
			3.1 The Trouble with Scale Changes
			3.2 Wilson's Renormalization Group Theory in Physics
			3.3 Stable and Unstable Manifolds
			3.4 A Glance at Conformal Field Theories
	Part II. Basic Techniques in Mathematics
		4. Analyticity
			4.1 Power Series Expansion
			4.2 Deformation Invariance of Integrals
			4.3 Cauchy's Integral Formula
			4.4 Cauchy's Residue Formula and Topological Charges
			4.5 The Winding Number
			4.6 Gauss' Fundamental Theorem of Algebra
			4.7 Compactification of the Complex Plane
			4.8 Analytic Continuation and the Local-Global Principle
			4.9 Integrals and Riemann Surfaces
			4.10 Domains of Holomorphy
			4.11 A Glance at Analytic S-Matrix Theory
			4.12 Important Applications
		5. A Glance at Topology
			5.1 Local and Global Properties of the Universe
			5.2 Bolzano's Existence Principle
			5.3 Elementary Geometric Notions
			5.4 Manifolds and Diffeomorphisms
			5.5 Topological Spaces, Homeomorphisms, and Deformations
			5.6 Topological Quantum Numbers
			5.7 Quantum States
			5.8 Perspectives
		6. Many-Particle Systems in Mathematics and Physics
			6.1 Partition Function in Statistical Physics
			6.2 Euler's Partition Function
			6.3 Discrete Laplace Transformation
			6.4 Integral Transformations
			6.5 The Riemann Zeta Function
			6.6 The Casimir Effect in Quantum Field Theory and the Epstein Zeta Function
			6.7 Appendix: The Mellin Transformation and Other Useful Analytic Techniques by Don Zagier
		7. Rigorous Finite-Dimensional Magic Formulas of Quantum Field Theory
			7.1 Geometrization of Physics
			7.2 Ariadne's Thread in Quantum Field Theory
			7.3 Linear Spaces
			7.4 Finite-Dimensional Hilbert Spaces
			7.5 Groups
			7.6 Lie Algebras
			7.7 Lie's Logarithmic Trick for Matrix Groups
			7.8 Lie Groups
			7.9 Basic Notions in Quantum Physics
			7.10 Fourier Series
			7.11 Dirac Calculus in Finite-Dimensional Hilbert Spaces
			7.12 The Trace of a Linear Operator
			7.13 Banach Spaces
			7.14 Probability and Hilbert's Spectral Family of an Observable
			7.15 Transition Probabilities, S-Matrix, and Unitary Operators
			7.16 The Magic Formulas for the Green's Operator
			7.17 The Magic Dyson Formula for the Retarded Propagator
			7.18 The Magic Dyson Formula for the S-Matrix
			7.19 Canonical Transformations
			7.20 Functional Calculus
			7.21 The Discrete Feynman Path Integral
			7.22 Causal Correlation Functions
			7.23 The Magic Gaussian Integral
			7.24 The Rigorous Response Approach to Finite Quantum Fields
			7.25 The Discrete φ[sup(4)]-Model and Feynman Diagrams
			7.26 The Extended Response Approach
			7.27 Complex-Valued Fields
			7.28 The Method of Lagrange Multipliers
			7.29 The Formal Continuum Limit
		8. Rigorous Finite-Dimensional Perturbation Theory
			8.1 Renormalization
			8.2 The Rellich Theorem
			8.3 The Trotter Product Formula
			8.4 The Magic Baker–Campbell–Hausdorff Formula
			8.5 Regularizing Terms
		9. Fermions and the Calculus for Grassmann Variables
			9.1 The Grassmann Product
			9.2 Differential Forms
			9.3 Calculus for One Grassmann Variable
			9.4 Calculus for Several Grassmann Variables
			9.5 The Determinant Trick
			9.6 The Method of Stationary Phase
			9.7 The Fermionic Response Model
		10. Infinite-Dimensional Hilbert Spaces
			10.1 The Importance of Infinite Dimensions in Quantum Physics
			10.2 The Hilbert Space L[sub(2)](Ω)
			10.3 Harmonic Analysis
			10.4 The Dirichlet Problem in Electrostatics as a Paradigm
		11. Distributions and Green's Functions
			11.1 Rigorous Basic Ideas
			11.2 Dirac's Formal Approach
			11.3 Laurent Schwartz's Rigorous Approach
			11.4 Hadamard's Regularization of Integrals
			11.5 Renormalization of the Anharmonic Oscillator
			11.6 The Importance of Algebraic Feynman Integrals
			11.7 Fundamental Solutions of Differential Equations
			11.8 Functional Integrals
			11.9 A Glance at Harmonic Analysis
			11.10 The Trouble with the Euclidean Trick
		12. Distributions and Physics
			12.1 The Discrete Dirac Calculus
			12.2 Rigorous General Dirac Calculus
			12.3 Fundamental Limits in Physics
			12.4 Duality in Physics
			12.5 Microlocal Analysis
			12.6 Multiplication of Distributions
	Part III. Heuristic Magic Formulas of Quantum Field Theory
		13. Basic Strategies in Quantum Field Theory
			13.1 The Method of Moments and Correlation Functions
			13.2 The Power of the S-Matrix
			13.3 The Relation Between the S-Matrix and the Correlation Functions
			13.4 Perturbation Theory and Feynman Diagrams
			13.5 The Trouble with Interacting Quantum Fields
			13.6 External Sources and the Generating Functional
			13.7 The Beauty of Functional Integrals
			13.8 Quantum Field Theory at Finite Temperature
		14. The Response Approach
			14.1 The Fourier–Minkowski Transform
			14.2 The φ[sup(4)]-Model
			14.3 A Glance at Quantum Electrodynamics
		15. The Operator Approach
			15.1 The φ[sup(4)]-Model
			15.2 A Glance at Quantum Electrodynamics
			15.3 The Role of Effective Quantities in Physics
			15.4 A Glance at Renormalization
			15.5 The Convergence Problem in Quantum Field Theory
			15.6 Rigorous Perspectives
		16. Peculiarities of Gauge Theories
			16.1 Basic Difficulties
			16.2 The Principle of Critical Action
			16.3 The Language of Physicists
			16.4 The Importance of the Higgs Particle
			16.5 Integration over Orbit Spaces
			16.6 The Magic Faddeev–Popov Formula and Ghosts
			16.7 The BRST Symmetry
			16.8 The Power of Cohomology
			16.9 The Batalin–Vilkovisky Formalism
			16.10 A Glance at Quantum Symmetries
		17. A Panorama of the Literature
			17.1 Introduction to Quantum Field Theory
			17.2 Standard Literature in Quantum Field Theory
			17.3 Rigorous Approaches to Quantum Field Theory
			17.4 The Fascinating Interplay between Modern Physics and Mathematics
			17.5 The Monster Group, Vertex Algebras, and Physics
			17.6 Historical Development of Quantum Field Theory
			17.7 General Literature in Mathematics and Physics
			17.8 Encyclopedias
			17.9 Highlights of Physics in the 20th Century
			17.10 Actual Information
	Appendix
		A.1 Notation
		A.2 The International System of Units
		A.3 The Planck System
		A.4 The Energetic System
		A.5 The Beauty of Dimensional Analysis
		A.6 The Similarity Principle in Physics
	Epilogue
	References
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	List of Symbols
	Index
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Zeidler - Quantum Field Theory (vol. 2)
	cover-large.TIF
	front-matter.pdf
	00001.pdf
		Prologue
	00002.pdf
		Mathematical Principles of Modern Natural Philosophy
			Basic Principles
			The Infinitesimal Strategy and Differential Equations
			The Optimality Principle
			The Basic Notion of Action in Physics and the Idea ofQuantization
			The Method of the Green's Function
			Harmonic Analysis and the Fourier Method
			The Method of Averaging and the Theory of Distributions
			The Symbolic Method
			Gauge Theory -- Local Symmetry and the Description of Interactions by Gauge Fields
			The Challenge of Dark Matter
	00003.pdf
		The Basic Strategy of Extracting Finite Information from Infinities -- Ariadne's Thread in Renormalization Theory
			Renormalization Theory in a Nutshell
				Effective Frequency and Running Coupling Constant of an Anharmonic Oscillator
				The Zeta Function and Riemann's Idea of Analytic Continuation
				Meromorphic Functions and Mittag-Leffler's Ideaof Subtractions
				The Square of the Dirac Delta Function
			Regularization of Divergent Integrals in Baby Renormalization Theory
				Momentum Cut-off and the Method of Power-Counting
				The Choice of the Normalization Momentum
				The Method of Differentiating Parameter Integrals
				The Method of Taylor Subtraction
				Overlapping Divergences
				The Role of Counterterms
				Euler's Gamma Function
				Integration Tricks
				Dimensional Regularization via Analytic Continuation
				Pauli--Villars Regularization
				Analytic Regularization
				Application to Algebraic Feynman Integrals inMinkowski Space
				Distribution-Valued Meromorphic Functions
				Application to Newton's Equation of Motion
				Hints for Further Reading.
			Further Regularization Methods in Mathematics
				Euler's Philosophy
				Adiabatic Regularization of Divergent Series
				Adiabatic Regularization of Oscillating Integrals
				Regularization by Averaging
				Borel Regularization
				Hadamard's Finite Part of Divergent Integrals
				Infinite-Dimensional Gaussian Integrals and the Zeta Function Regularization
			Trouble in Mathematics
				Interchanging Limits
				The Ambiguity of Regularization Methods
				Pseudo-Convergence
				Ill-Posed Problems
			Mathemagics
	00004.pdf
		The Power of Combinatorics
			Algebras
			The Algebra of Multilinear Functionals
			Fusion, Splitting, and Hopf Algebras
				The Bialgebra of Linear Differential Operators
				The Definition of Hopf Algebras
			Power Series Expansion and Hopf Algebras
				The Importance of Cancellations
				The Kepler Equation and the LagrangeInversion Formula
				The Composition Formula for Power Series
				The Faà di Bruno Hopf Algebra for the FormalDiffeomorphism Group of the Complex Plane
				The Generalized Zimmermann Forest Formula
				The Logarithmic Function and Schur Polynomials
				Correlation Functions in Quantum Field Theory
				Random Variables, Moments, and Cumulants
			Symmetry and Hopf Algebras
				The Strategy of Coordinatization in Mathematics and Physics
				The Coordinate Hopf Algebra of a Finite Group
				The Coordinate Hopf Algebra of an Operator Group
				The Tannaka--Krein Duality for Compact Lie Groups
			Regularization and Rota--Baxter Algebras
				Regularization of the Laurent Series
				Projection Operators
				The q-Integral
				The Volterra--Spitzer Exponential Formula
				The Importance of the Exponential Function inMathematics and Physics
			Partially Ordered Sets and Combinatorics
				Incidence Algebras and the Zeta Function
				The Möbius Function as an Inverse Function
				The Inclusion--Exclusion Principle in Combinatorics
				Applications to Number Theory
			Hints for Further Reading
	00005.pdf
		The Strategy of Equivalence Classes in Mathematics
			Equivalence Classes in Algebra
				The Gaussian Quotient Ring and the QuadraticReciprocity Law in Number Theory
				Application of the Fermat--Euler Theorem in Coding Theory
				Quotient Rings, Quotient Groups, and Quotient Fields
				Linear Quotient Spaces
				Ideals and Quotient Algebras
			Superfunctions and the Heaviside Calculus in Electrical Engineering
			Equivalence Classes in Geometry
				The Basic Idea of Geometry Epitomized by Klein's Erlangen Program
				Symmetry Spaces, Orbit Spaces, and Homogeneous Spaces
				The Space of Quantum States
				Real Projective Spaces
				Complex Projective Spaces
				The Shape of the Universe
			Equivalence Classes in Topology
				Topological Quotient Spaces
				Physical Fields, Observers, Bundles, and Cocycles
				Generalized Physical Fields and Sheaves
				Deformations, Mapping Classes, and Topological Charges
				Poincaré's Fundamental Group
				Loop Spaces and Higher Homotopy Groups
				Homology, Cohomology, and Electrodynamics
				Bott's Periodicity Theorem
				K-Theory
				Application to Fredholm Operators
				Hints for Further Reading
			The Strategy of Partial Ordering
				Feynman Diagrams
				The Abstract Entropy Principle in Thermodynamics
				Convergence of Generalized Sequences
				Inductive and Projective Topologies
				Inductive and Projective Limits
				Classes, Sets, and Non-Sets
				The Fixed-Point Theorem of Bourbaki--Kneser
				Zorn's Lemma
			Leibniz's Infinitesimals and Non-Standard Analysis
				Filters and Ultrafilters
				The Full-Rigged Real Line
	00006.pdf
		Part II. Basic Ideas in Classical Mechanics
			Geometrical Optics
				Ariadne's Thread in Geometrical Optics
				Fermat's Principle of Least Time
				Huygens' Principle on Wave Fronts
				Carathéodory's Royal Road to Geometrical Optics
				The Duality between Light Rays and Wave Fronts
					From Wave Fronts to Light Rays
					From Light Rays to Wave Fronts
				The Jacobi Approach to Focal Points
				Lie's Contact Geometry
					Basic Ideas
					Contact Manifolds and Contact Transformations
					Applications to Geometrical Optics
					Equilibrium Thermodynamics and LegendreSubmanifolds
				Light Rays and Non-Euclidean Geometry
					Linear Symplectic Spaces
					The Kähler Form of a Complex Hilbert Space
					The Refraction Index and Geodesics
					The Trick of Gauge Fixing
					Geodesic Flow
					Hamilton's Duality Trick and Cogeodesic Flow
					The Principle of Minimal Geodesic Energy
				Spherical Geometry
					The Global Gauss--Bonnet Theorem
					De Rham Cohomology and the Chern Class ofthe Sphere
					The Beltrami Model
				The Poincaré Model of Hyperbolic Geometry
					Kähler Geometry and the Gaussian Curvature
					Kähler--Einstein Geometry
					Symplectic Geometry
					Riemannian Geometry
				Ariadne's Thread in Gauge Theory
					Parallel Transport of Physical Information -- the Key to Modern Physics
					The Phase Equation and Fiber Bundles
					Gauge Transformations and Gauge-InvariantDifferential Forms
					Perspectives
				Classification of Two-Dimensional Compact Manifolds
				The Poincaré Conjecture and the Ricci Flow
				A Glance at Modern Optimization Theory
				Hints for Further Reading
	00007.pdf
		The Principle of Critical Action and the HarmonicOscillator -- Ariadne's Thread in Classical Mechanics
			Prototypes of Extremal Problems
			The Motion of a Particle
			Newtonian Mechanics
			A Glance at the History of the Calculus of Variations
			Lagrangian Mechanics
				The Harmonic Oscillator
				The Euler--Lagrange Equation
				Jacobi's Accessory Eigenvalue Problem
				The Morse Index
				The Anharmonic Oscillator
				The Ginzburg--Landau Potential and the Higgs Potential
				Damped Oscillations, Stability, and EnergyDissipation
				Resonance and Small Divisors
			Symmetry and Conservation Laws
				The Symmetries of the Harmonic Oscillator
				The Noether Theorem
			The Pendulum and Dynamical Systems
				The Equation of Motion
				Elliptic Integrals and Elliptic Functions
				The Phase Space of the Pendulum and Bundles
			Hamiltonian Mechanics
				The Canonical Equation
				The Hamiltonian Flow
				The Hamilton--Jacobi Partial Differential Equation
			Poissonian Mechanics
				Poisson Brackets and the Equation of Motion
				Conservation Laws
			Symplectic Geometry
				The Canonical Equations
				Symplectic Transformations
			The Spherical Pendulum
				The Gaussian Principle of Critical Constraint
				The Lagrangian Approach
				The Hamiltonian Approach
				Geodesics of Shortest Length
			The Lie Group SU(E3) of Rotations
				Conservation of Angular Momentum
				Lie's Momentum Map
			Carathéodory's Royal Road to the Calculus of Variations
				The Fundamental Equation
				Lagrangian Submanifolds in Symplectic Geometry
				The Initial-Value Problem for the Hamilton--Jacobi Equation
				Solution of Carathéodory's Fundamental Equation
			Hints for Further Reading
	00008.pdf
		Part III. Basic Ideas in Quantum Mechanics
			Quantization of the Harmonic Oscillator -- Ariadne's Thread in Quantization
				Complete Orthonormal Systems
				Bosonic Creation and Annihilation Operators
				Heisenberg's Quantum Mechanics
					Heisenberg's Equation of Motion
					Heisenberg's Uncertainty Inequality for the Harmonic Oscillator
					Quantization of Energy
					The Transition Probabilities
					The Wightman Functions
					The Correlation Functions
				Schrödinger's Quantum Mechanics
					The Schrödinger Equation
					States, Observables, and Measurements
					The Free Motion of a Quantum Particle
					The Harmonic Oscillator
					The Passage to the Heisenberg Picture
					Heisenberg's Uncertainty Principle
					Unstable Quantum States and the Energy-Time Uncertainty Relation
					Schrödinger's Coherent States
				Feynman's Quantum Mechanics
					Main Ideas
					The Diffusion Kernel and the Euclidean Strategy in Quantum Physics
					Probability Amplitudes and the Formal Propagator Theory
				Von Neumann's Rigorous Approach
					The Prototype of the Operator Calculus
					The General Operator Calculus
					Rigorous Propagator Theory
					The Free Quantum Particle as a Paradigm ofFunctional Analysis
					The Free Hamiltonian
					The Rescaled Fourier Transform
					The Quantized Harmonic Oscillator and the Maslov Index
					Ideal Gases and von Neumann's Density Operator
				The Feynman Path Integral
					The Basic Strategy
					The Basic Definition
					Application to the Free Quantum Particle
					Application to the Harmonic Oscillator
					The Propagator Hypothesis
					Motivation of Feynman's Path Integral
				Finite-Dimensional Gaussian Integrals
					Basic Formulas
					Free Moments, the Wick Theorem, and FeynmanDiagrams
					Full Moments and Perturbation Theory
				Rigorous Infinite-Dimensional Gaussian Integrals
					The Infinite-Dimensional Dispersion Operator
					Zeta Function Regularization and Infinite-Dimensional Determinants
					Application to the Free Quantum Particle
					Application to the Quantized Harmonic Oscillator
					The Spectral Hypothesis
				The Semi-Classical WKB Method
				Brownian Motion
					The Macroscopic Diffusion Law
					Einstein's Key Formulas for the Brownian Motion
					The Random Walk of Particles
					The Rigorous Wiener Path Integral
					The Feynman--Kac Formula
				Weyl Quantization
					The Formal Moyal Star Product
					Deformation Quantization of the Harmonic Oscillator
					Weyl Ordering
					Operator Kernels
					The Formal Weyl Calculus
					The Rigorous Weyl Calculus
				Two Magic Formulas
					The Formal Feynman Path Integral for the Propagator Kernel
					The Relation between the Scattering Kernel and the Propagator Kernel
				The Poincaré--Wirtinger Calculus
				Bargmann's Holomorphic Quantization
				The Stone--Von Neumann Uniqueness Theorem
					The Prototype of the Weyl Relation
					The Main Theorem
					C*-Algebras
					Operator Ideals
					Symplectic Geometry and the Weyl QuantizationFunctor
				A Glance at the Algebraic Approach to Quantum Physics
					States and Observables
					Gleason's Extension Theorem -- the Main Theorem of Quantum Logic
					The Finite Standard Model in Statistical Physics as a Paradigm
					Information, Entropy, and the Measure of Disorder
					Semiclassical Statistical Physics
					The Classical Ideal Gas
					Bose--Einstein Statistics
					Fermi--Dirac Statistics
					Thermodynamic Equilibrium and KMS-States
					Quasi-Stationary Thermodynamic Processes and Irreversibility
					The Photon Mill on Earth
				Von Neumann Algebras
					Von Neumann's Bicommutant Theorem
					The Murray--von Neumann Classification of Factors
					The Tomita--Takesaki Theory and KMS-States
				Connes' Noncommutative Geometry
				Jordan Algebras
				The Supersymmetric Harmonic Oscillator
				Hints for Further Reading
	00009.pdf
		Quantum Particles on the Real Line -- Ariadne's Thread in Scattering Theory
			Classical Dynamics Versus Quantum Dynamics
			The Stationary Schrödinger Equation
			One-Dimensional Quantum Motion in a Square-WellPotential
				Free Motion
				Scattering States and the S-Matrix
				Bound States
				Bound-State Energies and the Singularities of theS-Matrix
				The Energetic Riemann Surface, Resonances, and the Breit--Wigner Formula
				The Jost Functions
				The Fourier--Stieltjes Transformation
				Generalized Eigenfunctions of the Hamiltonian
				Quantum Dynamics and the Scattering Operator
				The Feynman Propagator
			Tunnelling of Quantum Particles and Radioactive Decay
			The Method of the Green's Function in a Nutshell
				The Inhomogeneous Helmholtz Equation
				The Retarded Green's Function, and the Existence and Uniqueness Theorem
				The Advanced Green's Function
				Perturbation of the Retarded and Advanced Green's Function
				Feynman's Regularized Fourier Method
			The Lippmann--Schwinger Integral Equation
				The Born Approximation
				The Existence and Uniqueness Theorem via Banach's Fixed Point Theorem
				Hypoellipticity
	00010.pdf
		A Glance at General Scattering Theory
			The Formal Basic Idea
			The Rigorous Time-Dependent Approach
			The Rigorous Time-Independent Approach
			Applications to Quantum Mechanics
			A Glance at Quantum Field Theory
			Hints for Further Reading
	00011.pdf
		Part IV. Quantum Electrodynamics (QED)
			Creation and Annihilation Operators
				The Bosonic Fock Space
					The Particle Number Operator
					The Ground State
				The Fermionic Fock Space and the Pauli Principle
				General Construction
				The Main Strategy of Quantum Electrodynamics
	00012.pdf
		The Basic Equations in Quantum Electrodynamics
			The Classical Lagrangian
			The Gauge Condition
	00013.pdf
		The Free Quantum Fields of Electrons, Positrons,and Photons
			Classical Free Fields
				The Lattice Strategy in Quantum Electrodynamics
				The High-Energy Limit and the Low-Energy Limit
				The Free Electromagnetic Field
				The Free Electron Field
			Quantization
				The Free Photon Quantum Field
				The Free Electron Quantum Field and Antiparticles
				The Spin of Photons
			The Ground State Energy and the Normal Product
			The Importance of Mathematical Models
				The Trouble with Virtual Photons
				Indefinite Inner Product Spaces
				Representation of the Creation and Annihilation Operators in QED
				Gupta--Bleuler Quantization
	00014.pdf
		The Interacting Quantum Field, and the MagicDyson Series for the S-Matrix
			Dyson's Key Formula
			The Basic Strategy of Reduction Formulas
			The Wick Theorem
			Feynman Propagators
				Discrete Feynman Propagators for Photons and Electrons
				Regularized Discrete Propagators
				The Continuum Limit of Feynman Propagators
				Classical Wave Propagation versus Feynman Propagator
	00015.pdf
		The Beauty of Feynman Diagrams in QED
			Compton Effect and Feynman Rules in Position Space
			Symmetry Properties
			Summary of the Feynman Rules in Momentum Space
			Typical Examples
			The Formal Language of Physicists
			Transition Probabilities and Cross Sections of ScatteringProcesses
			The Crucial Limits
			Appendix: Table of Feynman Rules
	00016.pdf
		Applications to Physical Effects
			Compton Effect
				Duality between Light Waves and Light Particles in the History of Physics
				The Trace Method for Computing Cross Sections
				Relativistic Invariance
			Asymptotically Free Electrons in an ExternalElectromagnetic Field
				The Key Formula for the Cross Section
				Application to Yukawa Scattering
				Application to Coulomb Scattering
				Motivation of the Key Formula via S-Matrix
				Perspectives
			Bound Electrons in an External ElectromagneticField
				The Spontaneous Emission of Photons by the Atom
				Motivation of the Key Formula
				Intensity of Spectral Lines
			Cherenkov Radiation
	00017.pdf
		Part V. Renormalization
			The Continuum Limit
				The Fundamental Limits
				The Formal Limits Fail
				Basic Ideas of Renormalization
					The Effective Mass and the Effective Charge of the Electron
					The Counterterms of the Modified Lagrangian
					The Compensation Principle
					Fundamental Invariance Principles
					Dimensional Regularization of Discrete AlgebraicFeynman Integrals
					Multiplicative Renormalization
				The Theory of Approximation Schemes in Mathematics
	00018.pdf
		Radiative Corrections of Lowest Order
			Primitive Divergent Feynman Graphs
			Vacuum Polarization
			Radiative Corrections of the Propagators
				The Photon Propagator
				The Electron Propagator
				The Vertex Correction and the Ward Identity
			The Counterterms of the Lagrangian and the Compensation Principle
			Application to Physical Problems
				Radiative Correction of the Coulomb Potential
				The Anomalous Magnetic Moment of the Electron
				The Anomalous Magnetic Moment of the Muon
				The Lamb Shift
				Photon-Photon Scattering
	00019.pdf
		A Glance at Renormalization to all Orders ofPerturbation Theory
			One-Particle Irreducible Feynman Graphs andDivergences
			Overlapping Divergences and Manoukian's EquivalencePrinciple
			The Renormalizability of Quantum Electrodynamics
			Automated Multi-Loop Computations in PerturbationTheory
	00020.pdf
		Perspectives
			BPHZ Renormalization
				Bogoliubov's Iterative R-Method
				Zimmermann's Forest Formula
				The Classical BPHZ Method
			The Causal Epstein--Glaser S-Matrix Approach
			Kreimer's Hopf Algebra Revolution
				The History of the Hopf Algebra Approach
				Renormalization and the Iterative BirkhoffFactorization for Complex Lie Groups
				The Renormalization of QuantumElectrodynamics
			The Scope of the Riemann--Hilbert Problem
				The Gaussian Hypergeometric Differential Equation
				The Confluent Hypergeometric Function and theSpectrum of the Hydrogen Atom
				Hilbert's 21th Problem
				The Transport of Information in Nature
				Stable Transport of Energy and Solitons
				Ariadne's Thread in Soliton Theory
				Resonances
				The Role of Integrable Systems in Nature
			The BFFO Hopf Superalgebra Approach
			The BRST Approach and Algebraic Renormalization
			Analytic Renormalization and Distribution-ValuedAnalytic Functions
			Computational Strategies
				The Renormalization Group
				Operator Product Expansions
				Binary Planar Graphs and the Renormalizationof Quantum Electrodynamics
			The Master Ward Identity
			Trouble in Quantum Electrodynamics
				The Landau Inconsistency Problem in QuantumElectrodynamics
				The Lack of Asymptotic Freedom in QuantumElectrodynamics
			Hints for Further Reading
	back-matter.pdf
		Epilogue
		References
		List of Symbols
		Index
Zeidler - Quantum Field Theory (vol. 3)
	Cover
	Quantum Field Theory III: Gauge Theory
	ISBN 9783642224201
	Preface
	Contents
	Prologue
	1. The Euclidean Space E3 (Hilbert Space and Lie Algebra Structure)
		1.1 A Glance at History
		1.2 Algebraic Basic Ideas
			1.2.1 Symmetrization and Antisymmetrization
			1.2.2 Cramer's Rule for Systems of Linear Equations
			1.2.3 Determinants and the Inverse Matrix
			1.2.4 The Hilbert Space Structure
			1.2.5 Orthogonality and the Dirac Calculus
			1.2.6 The Lie Algebra Structure
			1.2.7 The Metric Tensor
			1.2.8 The Volume Form
			1.2.9 Grassmann's Alternating Product
			1.2.10 Perspectives
		1.3 The Skew-Field H of Quaternions
			1.3.1 The Field C of Complex Numbers
			1.3.2 The Galois Group Gal(C|R) and Galois Theory
			1.3.3 A Glance at the History of Hamilton's Quaternions
			1.3.4 Pauli's Spin Matrices and the Lie Algebras su(2) and sl(2, C)
			1.3.5 Cayley's Matrix Approach to Quaternions
			1.3.6 The Unit Sphere U(1,H) and the Electroweak Gauge Group SU(2)
			1.3.7 The Four-Dimensional Extension of the Euclidean Space E3
			1.3.8 Hamilton's Nabla Operator
			1.3.9 The Indefinite Hilbert Space H and the Minkowski Space M4
		1.4 Riesz Duality between Vectors and Covectors
		1.5 The Heisenberg Group, the Heisenberg Algebra, and Quantum Physics
		1.6 The Heisenberg Group Bundle and Gauge Transformations
	2. Algebras and Duality (Tensor Algebra, Grassmann Algebra, Clifford Algebra, Lie Algebra)
		2.1 Multilinear Functionals
			2.1.1 The Graded Algebra of Polynomials
			2.1.2 Products of Multilinear Functionals
			2.1.3 Tensor Algebra
			2.1.4 Grassmann Algebra (Alternating Algebra)
			2.1.5 Symmetric Tensor Algebra
			2.1.6 The Universal Property of the Tensor Product
			2.1.7 Diagram Chasing
		2.2 The Clifford Algebra (E1) of the One-Dimensional Euclidean Space E1
		2.3 Algebras of the Two-Dimensional Euclidean Space E2
			2.3.1 The Clifford Algebra (E2) and Quaternions
			2.3.2 The Cauchy–Riemann Differential Equations in Complex Function Theory
			2.3.3 The Grassmann Algebra (E2)
			2.3.4 The Grassmann Algebra (Ed2)
			2.3.5 The Symplectic Structure of E2
			2.3.6 The Tensor Algebra (E2)
			2.3.7 The Tensor Algebra (E2d)
		2.4 Algebras of the Three-Dimensional Euclidean Space E3
			2.4.1 Lie Algebra
			2.4.2 Tensor Algebra
			2.4.3 Grassmann Algebra
			2.4.4 Clifford Algebra
		2.5 Algebras of the Dual Euclidean Space E3d
			2.5.1 Tensor Algebra
			2.5.2 Grassmann Algebra
		2.6 The Mixed Tensor Algebra
		2.7 The Hilbert Space Structure of the Grassmann Algebra (Hodge Duality)
			2.7.1 The Hilbert Space (E3)
			2.7.2 The Hilbert Space (Ed3)
			2.7.3 Multivectors
		2.8 The Clifford Structure of the Grassmann Algebra (Exterior–Interior Kähler Algebra)
			2.8.1 The Kähler Algebra (E3)
			2.8.2 The Kähler Algebra (E3d)
		2.9 The C*-Algebra End (E3) of the Euclidean Space
		2.10 Linear Operator Equations
			2.10.1 The Prototype
			2.10.2 The Grassmann Theorem
			2.10.3 The Superposition Principle
			2.10.4 Duality and the Fredholm Alternative
			2.10.5 The Language of Matrices
			2.10.6 The Gaussian Elimination Method
		2.11 Changing the Basis and the Cobasis
			2.11.1 Similarity of Matrices
			2.11.2 Volume Functions
			2.11.3 The Determinant of a Linear Operator
			2.11.4 The Reciprocal Basis in Crystallography
			2.11.5 Dual Pairing
			2.11.6 The Trace of a Linear Operator
			2.11.7 The Dirac Calculus
		2.12 The Strategy of Quotient Algebras and Universal Properties
		2.13 A Glance at Division Algebras
			2.13.1 From Real Numbers to Cayley's Octonions
			2.13.2 Uniqueness Theorems
			2.13.3 The Fundamental Dimension Theorem
	3. Representations of Symmetries in Mathematics and Physics
		3.1 The Symmetric Group as a Prototype
		3.2 Incredible Cancellations
		3.3 The Symmetry Strategy in Mathematics and Physics
		3.4 Lie Groups and Lie Algebras
		3.5 Basic Notions of Representation Theory
			3.5.1 Linear Representations of Groups
			3.5.2 Linear Representations of Lie Algebras
		3.6 The Reflection Group Z2 as a Prototype
			3.6.1 Representations of Z2
			3.6.2 Parity of Elementary Particles
			3.6.3 Reflections and Chirality in Nature
			3.6.4 Parity Violation in Weak Interaction
			3.6.5 Helicity
		3.7 Permutation of Elementary Particles
			3.7.1 The Principle of Indistinguishability of Quantum Particles
			3.7.2 The Pauli Exclusion Principle
			3.7.3 Entangled Quantum States
		3.8 The Diagonalization of Linear Operators
			3.8.1 The Theorem of Principal Axes in Geometry and in Quantum Theory
			3.8.2 The Schur Lemma in Linear Representation Theory
			3.8.3 The Jordan Normal Form of Linear Operators
			3.8.4 The Standard Maximal Torus of the Lie Group SU(n) and the Standard Cartan Subalgebra of the Lie Algebra su(n)
			3.8.5 Eigenvalues and the Operator Strategy for Lie Algebras (Adjoint Representation)
		3.9 The Action of a Group on a Physical State Space, Orbits, and Gauge Theory
		3.10 The Intrinsic Symmetry of a Group
		3.11 Linear Representations of Finite Groups and the Hilbert Space of Functions on the Group
		3.12 The Tensor Product of Representations and Characters
		3.13 Applications to the Symmetric Group Sym(n)
			3.13.1 The Characters of the Symmetric Group Sym(2)
			3.13.2 The Characters of the Symmetric Group Sym(3)
			3.13.3 Partitions and Young Frames
			3.13.4 Young Tableaux and the Construction of a Complete System of Irreducible Representations
		3.14 Application to the Standard Model in Elementary Particle Physics
			3.14.1 Quarks and Baryons
			3.14.2 Antiquarks and Mesons
			3.14.3 The Method of Highest Weight for Composed Particles
			3.14.4 The Pauli Exclusion Principle and the Color of Quarks
		3.15 The Complexification of Lie Algebras
			3.15.1 Basic Ideas
			3.15.2 The Complex Lie Algebra slC(3, C) and Root Functionals
			3.15.3 Representations of the Complex Lie Algebra slC(3, C) and Weight Functionals
		3.16 Classification of Groups
			3.16.1 Simplicity
			3.16.2 Direct Product and Semisimplicity
			3.16.3 Solvablity
			3.16.4 Semidirect Product
		3.17 Classification of Lie Algebras
			3.17.1 The Classification of Complex Simple Lie Algebras
			3.17.2 Semisimple Lie Algebras
			3.17.3 Solvability and the Heisenberg Algebra in Quantum Mechanics
			3.17.4 Semidirect Product and the Levi Decomposition
			3.17.5 The Casimir Operators
		3.18 Symmetric and Antisymmetric Functions
			3.18.1 Symmetrization and Antisymmetrization
			3.18.2 Elementary Symmetric Polynomials
			3.18.3 Power Sums
			3.18.4 Completely Symmetric Polynomials
			3.18.5 Symmetric Schur Polynomials
			3.18.6 Raising Operators and the Creation and Annihilation of Particles
		3.19 Formal Power Series Expansions and Generating Functions
			3.19.1 The Fundamental Frobenius Character Formula
			3.19.2 The Pfaffian
		3.20 Frobenius Algebras and Frobenius Manifolds
		3.21 Historical Remarks
		3.22 Supersymmetry
			3.22.1 Graduation in Nature
			3.22.2 General Strategy in Mathematics
			3.22.3 The Super Lie Algebra of the Euclidean Space
		3.23 Artin's Braid Group
			3.23.1 The Braid Relation
			3.23.2 The Yang–Baxter Equation
			3.23.3 The Geometric Meaning of the Braid Group
			3.23.4 The Topology of the State Space of n Indistinguishable Particles in the Plane
		3.24 The HOMFLY Polynomials in Knot Theory
		3.25 Quantum Groups
			3.25.1 Quantum Mechanics as a Deformation
			3.25.2 Manin's Quantum Planes R2q and C2q
			3.25.3 The Coordinate Algebra of the Lie Group SL(2, C)
			3.25.4 The Quantum Group SLq(2, C)
			3.25.5 The Quantum Algebra slq(2,C)
			3.25.6 The Coaction of the Quantum Group SLq(2, C) on the Quantum Plane C2q
			3.25.7 Noncommutative Euclidean Geometry and Quantum Symmetry
		3.26 Additive Groups, Betti Numbers, Torsion Coefficients, and Homological Products
		3.27 Lattices and Modules
	4. The Euclidean Manifold E3
		4.1 Velocity Vectors and the Tangent Space
		4.2 Duality and Cotangent Spaces
		4.3 Parallel Transport and Acceleration
		4.4 Newton's Law of Motion
		4.5 Bundles Over the Euclidean Manifold
			4.5.1 The Tangent Bundle and Velocity Vector Fields
			4.5.2 The Cotangent Bundle and Covector Fields
			4.5.3 Tensor Bundles and Tensor Fields
			4.5.4 The Frame Bundle
		4.6 Historical Remarks
			4.6.1 Newton and Leibniz
			4.6.2 The Lebesgue Integral
			4.6.3 The Dirac Delta Function and Laurent Schwartz's Distributions
			4.6.4 The Algebraization of the Calculus
			4.6.5 Formal Power Series Expansions and the Ritt Theorem
			4.6.6 Differential Rings and Derivations
			4.6.7 The p-adic Numbers
			4.6.8 The Local–Global Principle in Mathematics
			4.6.9 The Global Adelic Ring
			4.6.10 Solenoids, Foliations, and Chaotic Dynamical Systems
			4.6.11 Period Three Implies Chaos
			4.6.12 Differential Calculi, Noncommutative Geometry, and the Standard Model in Particle Physics
			4.6.13 BRST-Symmetry, Cohomology, and the Quantization of Gauge Theories
			4.6.14 Itô's Stochastic Calculus
	5. The Lie Group U(1) as a Paradigm in Harmonic Analysis and Geometry
		5.1 Linearization and the Lie Algebra u(1)
		5.2 The Universal Covering Group of U(1)
		5.3 Left-Invariant Velocity Vector Fields on U(1)
			5.3.1 The Maurer–Cartan Form of U(1)
			5.3.2 The Maurer–Cartan Structural Equation
		5.4 The Riemannian Manifold U(1) and the Haar Measure
		5.5 The Discrete Fourier Transform
			5.5.1 The Hilbert Space L2(U(1))
			5.5.2 Pseudo–Differential Operators
			5.5.3 The Sobolev Space Wm2(U(1))
		5.6 The Group of Motions on the Gaussian Plane
		5.7 Rotations of the Euclidean Plane
		5.8 Pontryagin Duality for U(1) and Quantum Groups
	6. Infinitesimal Rotations and Constraints in Physics
		6.1 The Group U(E3) of Unitary Transformations
		6.2 Euler's Rotation Formula
		6.3 The Lie Algebra of Infinitesimal Rotations
		6.4 Constraints in Classical Physics
			6.4.1 Archimedes' Lever Principle
			6.4.2 d'Alembert's Principle of Virtual Power
			6.4.3 d'Alembert's Principle of Virtual Work
			6.4.4 The Gaussian Principle of Least Constraint and Constraining Forces
			6.4.5 Manifolds and Lagrange's Variational Principle
			6.4.6 The Method of Perturbation Theory
			6.4.7 Further Reading on Perturbation Theory and its Applications
		6.5 Application to the Motion of a Rigid Body
			6.5.1 The Center of Gravity
			6.5.2 Moving Orthonormal Frames and Infinitesimal Rotations
			6.5.3 Kinetic Energy and the Inertia Tensor
			6.5.4 The Equations of Motion – the Existence and Uniqueness Theorem
			6.5.5 Euler's Equation of the Spinning Top
			6.5.6 Equilibrium States and Torque
			6.5.7 The Principal Bundle R3 SO(3) – the Position Space of a Rigid Body
		6.6 A Glance at Constraints in Quantum Field Theory
			6.6.1 Gauge Transformations and Virtual Degrees of Freedom in Gauge Theory
			6.6.2 Elimination of Unphysical States (Ghosts)
			6.6.3 Degenerate Minimum Problems
			6.6.4 Variation of the Action Functional
			6.6.5 Degenerate Lagrangian and Constraints
			6.6.6 Degenerate Legendre Transformation
			6.6.7 Global and Local Symmetries
			6.6.8 Quantum Symmetries and Anomalies
		6.7 Perspectives
			6.7.1 Topological Constraints in Maxwell's Theory of Electromagnetism
			6.7.2 Constraints in Einstein's Theory of General Relativity
			6.7.3 Hilbert's Algebraic Theory of Relations (Syzygies)
		6.8 Further Reading
	7. Rotations, Quaternions, the Universal Covering Group, and the Electron Spin
		7.1 Quaternions and the Cayley–Hamilton Rotation Formula
		7.2 The Universal Covering Group SU(2)
		7.3 Irreducible Unitary Representations of the Group SU(2) and the Spin
			7.3.1 The Spin Quantum Numbers
			7.3.2 The Addition Theorem for the Spin
			7.3.3 The Model of Homogeneous Polynomials
			7.3.4 The Clebsch–Gordan Coefficients
		7.4 Heisenberg's Isospin
	8. Changing Observers – A Glance at Invariant Theory Based on the Principle of the Correct Index Picture
		8.1 A Glance at the History of Invariant Theory
		8.2 The Basic Philosophy
		8.3 The Mnemonic Principle of the Correct Index Picture
		8.4 Real-Valued Physical Fields
			8.4.1 The Chain Rule and the Key Duality Relation
			8.4.2 Linear Differential Operators
			8.4.3 Duality and Differentials
			8.4.4 Admissible Systems of Observers
			8.4.5 Tensorial Families and the Construction of Invariants via the Basic Trick of Index Killing
			8.4.6 Orientation, Pseudo-Tensorial Families, and the Levi-Civita Duality
		8.5 Differential Forms (Exterior Product)
			8.5.1 Cartan Families and the Cartan Differential
			8.5.2 Hodge Duality, the Hodge Codifferential, and the Laplacian (Hodge's Star Operator)
		8.6 The Kähler–Clifford Calculus and the Dirac Operator (Interior Product)
			8.6.1 The Exterior Differential Algebra
			8.6.2 The Interior Differential Algebra
			8.6.3 Kähler Duality
			8.6.4 Applications to Fundamental Differential Equations in Physics
			8.6.5 The Potential Equation and the Importance of the de Rham Cohomology
			8.6.6 Tensorial Differential Forms
		8.7 Integrals over Differential Forms
		8.8 Derivatives of Tensorial Families
			8.8.1 The Lie Algebra of Linear Differential Operators and the Lie Derivative
			8.8.2 The Inverse Index Principle
			8.8.3 The Covariant Derivative (Weyl's Affine Connection)
		8.9 The Riemann–Weyl Curvature Tensor
			8.9.1 Second-Order Covariant Partial Derivatives
			8.9.2 Local Flatness
			8.9.3 The Method of Differential Forms (Cartan's Structural Equations)
			8.9.4 The Operator Method
		8.10 The Riemann–Christoffel Curvature Tensor
			8.10.1 The Levi-Civita Metric Connection
			8.10.2 Levi-Civita's Parallel Transport
			8.10.3 Symmetry Properties of the Riemann–Christoffel Curvature Tensor
			8.10.4 The Ricci Curvature Tensor and the Einstein Tensor
			8.10.5 The Conformal Weyl Curvature Tensor
			8.10.6 The Hodge Codifferential and the Covariant Partial Derivative
			8.10.7 The Weitzenböck Formula for the Hodge Laplacian
			8.10.8 The One-Dimensional sigma-Model and Affine Geodesics
		8.11 The Beauty of Connection-Free Derivatives
			8.11.1 The Lie Derivative
			8.11.2 The Cartan Derivative
			8.11.3 The Weyl Derivative
		8.12 Global Analysis
		8.13 Summary of Notation
		8.14 Two Strategies in Invariant Theory
		8.15 Intrinsic Tangent Vectors and Derivations
		8.16 Further Reading on Symmetry and Invariants
	9. Applications of Invariant Theory to the Rotation Group
		9.1 The Method of Orthonormal Frames on the Euclidean Manifold
			9.1.1 Hamilton's Quaternionic Analysis
			9.1.2 Transformation of Orthonormal Frames
			9.1.3 The Coordinate-Dependent Approach (SO(3)-Tensor Calculus)
			9.1.4 The Coordinate-Free Approach
			9.1.5 Hamilton's Nabla Calculus
			9.1.6 Rotations and Cauchy's Invariant Functions
		9.2 Curvilinear Coordinates
			9.2.1 Local Observers
			9.2.2 The Metric Tensor
			9.2.3 The Volume Form
			9.2.4 Special Coordinates
		9.3 The Index Principle of Mathematical Physics
			9.3.1 The Basic Trick
			9.3.2 Applications to Vector Analysis
		9.4 The Euclidean Connection and Gauge Theory
			9.4.1 Covariant Partial Derivative
			9.4.2 Curves of Least Kinectic Energy (Affine Geodesics)
			9.4.3 Curves of Minimal Length
			9.4.4 The Gauss Equations of Moving Frames
			9.4.5 Parallel Transport of a Velocity Vector and Cartan's Propagator Equation
			9.4.6 The Dual Cartan Equations of Moving Frames
			9.4.7 Global Parallel Transport on Lie Groups and the Maurer–Cartan Form
			9.4.8 Cartan's Global Connection Form on the Frame Bundle of the Euclidean Manifold
			9.4.9 The Relation to Gauge Theory
			9.4.10 The Reduction of the Frame Bundle to the Orthonormal Frame Bundle
		9.5 The Sphere as a Paradigm in Riemannian Geometry and Gauge Theory
			9.5.1 The Newtonian Equation of Motion and Levi-Civita's Parallel Transport
			9.5.2 Geodesic Triangles and the Gaussian Curvature
			9.5.3 Geodesic Circles and the Gaussian Curvature
			9.5.4 The Spherical Pendulum
			9.5.5 Geodesics and Gauge Transformations
			9.5.6 The Local Hilbert Space Structure
			9.5.7 The Almost Complex Structure
			9.5.8 The Levi-Civita Connection on the Tangent Bundle and the Riemann Curvature Tensor
			9.5.9 The Components of the Riemann Curvature Tensor and Gauge Fixing
			9.5.10 Computing the Riemann Curvature Operator via Parallel Transport Along Loops
			9.5.11 The Connection on the Frame Bundle and Parallel Transport
			9.5.12 Poincaré's Topological No-Go Theorem for Velocity Vector Fields on a Sphere
		9.6 Gauss' Theorema Egregium
			9.6.1 The Natural Basis and Cobasis
			9.6.2 Intrinsic Metric Properties
			9.6.3 The Extrinsic Definition of the Gaussian Curvature
			9.6.4 The Gauss–Weingarten Equations for Moving Frames
			9.6.5 The Integrability Conditions and the Riemann Curvature Tensor
			9.6.6 The Intrinsic Characterization of the Gaussian Curvature (Theorema Egregium)
			9.6.7 Differential Invariants and the Existence and Uniqueness Theorem of Classical Surface Theory
			9.6.8 Gauss' Theorema Elegantissimum and the Gauss–Bonnet Theorem
			9.6.9 Gauss' Total Curvature and Topological Charges
			9.6.10 Cartan's Method of Moving Orthonormal Frames
		9.7 Parallel Transport in Physics
		9.8 Finsler Geometry
		9.9 Further Reading
	10. Temperature Fields on the Euclidean Manifold E3
		10.1 The Directional Derivative
		10.2 The Lie Derivative of a Temperature Field along the Flow of Fluid Particles
			10.2.1 The Flow
			10.2.2 The Linearized Flow
			10.2.3 The Lie Derivative
			10.2.4 Conservation Laws
		10.3 Higher Variations of a Temperature Field and the Taylor Expansion
		10.4 The Fréchet Derivative
		10.5 Global Linearization of Smooth Maps and the Tangent Bundle
		10.6 The Global Chain Rule
		10.7 The Transformation of Temperature Fields
	11. Velocity Vector Fields on the Euclidean Manifold E3
		11.1 The Transformation of Velocity Vector Fields
		11.2 The Lie Derivative of an Electric Field along the Flow of Fluid Particles
			11.2.1 The Lie Derivative
			11.2.2 Conservation Laws
			11.2.3 The Lie Algebra of Velocity Vector Fields
	12. Covector Fields and Cartan's Exterior Differential – the Beauty of Differential Forms
		12.1 Ariadne's Thread
			12.1.1 One Dimension
			12.1.2 Two Dimensions
			12.1.3 Three Dimensions
			12.1.4 Integration over Manifolds
			12.1.5 Integration over Singular Chains
		12.2 Applications to Physics
			12.2.1 Single-Valued Potentials and Gauge Transformations
			12.2.2 Multi-Valued Potentials and Riemann Surfaces
			12.2.3 The Electrostatic Coulomb Force and the Dirac Delta Distribution
			12.2.4 The Magic Green's Function and the Dirac Delta Distribution
			12.2.5 Conservation of Heat Energy – the Paradigm of Conservation Laws in Physics
			12.2.6 The Classical Predecessors of the Yang–Mills Equations in Gauge Theory (Fluid Dynamics and Electrodynamics)
			12.2.7 Thermodynamics and the Pfaff Problem
			12.2.8 Classical Mechanics and Symplectic Geometry
			12.2.9 The Universality of Differential Forms
			12.2.10 Cartan's Covariant Differential and the Four Fundamental Interactions in Nature
		12.3 Cartan's Algebra of Alternating Differential Forms
			12.3.1 The Geometric Approach
			12.3.2 The Grassmann Bundle
			12.3.3 The Tensor Bundle
			12.3.4 The Transformation of Covector Fields
		12.4 Cartan's Exterior Differential
			12.4.1 Invariant Definition via the Lie Algebra of Velocity Vector Fields
			12.4.2 The Supersymmetric Leibniz Rule
			12.4.3 The Poincaré Cohomology Rule
			12.4.4 The Axiomatic Approach
		12.5 The Lie Derivative of Differential Forms
			12.5.1 Invariant Definition via the Flow of Fluid Particles
			12.5.2 The Contraction Product between Velocity Vector Fields and Differential Forms
			12.5.3 Cartan's Magic Formula
			12.5.4 The Lie Derivative of the Volume Form
			12.5.5 The Lie Derivative of the Metric Tensor Field
			12.5.6 The Lie Derivative of Linear Operator Fields
		12.6 Diffeomorphisms and the Mechanics of Continua – the Prototype of an Effective Theory in Physics
			12.6.1 Linear Diffeomorphisms and Deformation Operators
			12.6.2 Local Diffeomorphisms
			12.6.3 Proper Maps and Hadamard's Theorem on Diffeomorphisms
			12.6.4 Monotone Operators and Diffeomorphisms
			12.6.5 Sard's Theorem on the Genericity of Regular Solution Sets
			12.6.6 The Strain Tensor and the Stress Tensor in Cauchy's Theory of Elasticity
			12.6.7 The Rate-of-Strain Tensor and the Stress Tensor in the Hydrodynamics of Viscous Fluids
			12.6.8 Vorticity Lines of a Fluid
			12.6.9 The Lie Derivative of the Covector Field
		12.7 The Generalized Stokes Theorem (Main Theorem of Calculus)
		12.8 Conservation Laws
			12.8.1 Infinitesimal Isometries (Metric Killing Vector Fields)
			12.8.2 Absolute Integral Invariants and Incompressible Fluids
			12.8.3 Relative Integral Invariants and the Vorticity Theorems for Fluids due to Thomson and Helmholtz
			12.8.4 The Transport Theorem
			12.8.5 The Noether Theorem – Symmetry Implies Conservation Laws in the Calculus of Variations
		12.9 The Hamiltonian Flow on the Euclidean Manifold – a Paradigm of Hamiltonian Mechanics
			12.9.1 Hamilton's Principle of Critical Action
			12.9.2 Basic Formulas
			12.9.3 The Poincaré–Cartan Integral Invariant
			12.9.4 Energy Conservation and the Liouville Integral Invariant
			12.9.5 Jacobi's Canonical Transformations, Lie's Contact Geometry, and Symplectic Geometry
			12.9.6 Hilbert's Invariant Integral
			12.9.7 Jacobi's Integration Method
			12.9.8 Legendre Transformation
			12.9.9 Carathéodory's Royal Road to the Calculus of Variations
			12.9.10 Geometrical Optics
		12.10 The Main Theorems in Classical Gauge Theory (Existence of Potentials)
			12.10.1 Contractible Manifolds (the Poincaré–Volterra Theorem)
			12.10.2 Non-Contractible Manifolds and Betti Numbers (De Rham's Theorem on Periods)
			12.10.3 The Main Theorem for Velocity Vector Fields
		12.11 Systems of Differential Forms
			12.11.1 Integrability Condition
			12.11.2 The Frobenius Theorem for Pfaff Systems
			12.11.3 The Dual Frobenius Theorem
			12.11.4 The Pfaff Normal Form and the Second Law of Thermodynamics
		12.12 Hodge Duality
			12.12.1 The Hodge Codifferential
			12.12.2 The Hodge Homology Rule
			12.12.3 The Relation between the Cartan–Hodge Calculus and Classical Vector Analysis via Riesz Duality
			12.12.4 The Classical Prototype of the Yang–Mills Equation in Gauge Theory
			12.12.5 The Hodge–Laplace Operator and Harmonic Forms
		12.13 Further Reading
		12.14 Historical Remarks
	13. The Commutative Weyl U(1)-Gauge Theory and the Electromagnetic Field
		13.1 Basic Ideas
		13.2 The Fundamental Principle of Local Symmetry Invariance in Modern Physics
			13.2.1 The Free Meson
			13.2.2 Local Symmetry and the Charged Meson in an Electromagnetic Field
		13.3 The Vector Bundle M4 C, Covariant Directional Derivative, and Curvature
		13.4 The Principal Bundle M4 U(1) and the Parallel Transport of the Local Phase Factor
		13.5 Parallel Transport of Physical Fields – the Propagator Approach
		13.6 The Wilson Loop and Holonomy
	14. Symmetry Breaking
		14.1 The Prototype in Mechanics
		14.2 The Goldstone-Particle Mechanism
		14.3 The Higgs-Particle Mechanism
		14.4 Dimensional Reduction and the Kaluza–Klein Approach
		14.5 Superconductivity and the Ginzburg–Landau Equation
		14.6 The Idea of Effective Theories in Physics
	15. The Noncommutative Yang–Mills SU(N)-Gauge Theory
		15.1 The Vector Bundle M4CN, Covariant Directional Derivative, and Curvature
		15.2 The Principal Bundle M4 G and the Parallel Transport of the Local Phase Factor
		15.3 Parallel Transport of Physical Fields – the Propagator Approach
		15.4 The Principle of Critical Action and the Yang–Mills Equations
		15.5 The Universal Extension Strategy via the Leibniz Rule
		15.6 Tensor Calculus on Vector Bundles
			15.6.1 Tensor Algebra
			15.6.2 Connection and Christoffel Symbols
			15.6.3 Covariant Differential for Differential Forms of Tensor Type
			15.6.4 Application to the Riemann Curvature Operator
	16. Cocycles and Observers
		16.1 Cocycles
		16.2 Physical Fields via the Cocycle Strategy
		16.3 Local Phase Factors via the Cocycle Strategy
	17. The Axiomatic Geometric Approach to Bundles
		17.1 Connection on a Vector Bundle
		17.2 Connection on a Principal Bundle
		17.3 The Philosophy of Parallel Transport
			17.3.1 Vector Bundles Associated to a Principal Bundle
			17.3.2 Horizontal Vector Fields on a Principal Bundle
			17.3.3 The Lifting of Curves in Fiber Bundles
		17.4 A Glance at the History of Gauge Theory
			17.4.1 From Weyl's Gauge Theory in Gravity to the Standard Model in Particle Physics
			17.4.2 From Gauss' Theorema Egregium to Modern Differential Geometry
			17.4.3 The Work of Hermann Weyl
	18. Inertial Systems and Einstein's Principle of Special Relativity
		18.1 The Principle of Special Relativity
			18.1.1 The Lorentz Boost
			18.1.2 The Transformation of Velocities
			18.1.3 Time Dilatation
			18.1.4 Length Contraction
			18.1.5 The Synchronization of Clocks
			18.1.6 General Change of Inertial Systems in Terms of Physics
		18.2 Matrix Groups
			18.2.1 The Group O(1,1)
			18.2.2 The Lorentz Group O(1,3)
		18.3 Infinitesimal Transformations
			18.3.1 The Lie Algebra o(1,3) of the Lorentz Group O(1,3)
			18.3.2 The Lie Algebra p(1,3) of the Poincaré Group P(1,3)
		18.4 The Minkowski Space M4
			18.4.1 Pseudo-Orthonormal Systems and Inertial Systems
			18.4.2 Orientation
			18.4.3 Proper Time and the Twin Paradox
			18.4.4 The Free Relativistic Particle and the Energy-Mass Equivalence
			18.4.5 The Photon
		18.5 The Minkowski Manifold M4
			18.5.1 Causality and the Maximal Signal Velocity
			18.5.2 Hodge Duality
			18.5.3 Arbitrary Local Coordinates
	19. The Relativistic Invariance of the Maxwell Equations
		19.1 Historical Background
			19.1.1 The Coulomb Force and the Gauss Law
			19.1.2 The Ampère Force and the Ampère Law
			19.1.3 Joule's Heat Energy Law
			19.1.4 Faraday's Induction Law
			19.1.5 Electric Dipoles
			19.1.6 Magnetic Dipoles
			19.1.7 The Electron Spin
			19.1.8 The Dirac Magnetic Monopole
			19.1.9 Vacuum Polarization in Quantum Electrodynamics
		19.2 The Maxwell Equations in a Vacuum
			19.2.1 The Global Maxwell Equations Based on Electric and Magnetic Flux
			19.2.2 The Local Maxwell Equations Formulated in Maxwell's Language of Vector Calculus
			19.2.3 Discrete Symmetries and CPT
		19.3 Invariant Formulation of the Maxwell Equations in a Vacuum
			19.3.1 Einstein's Language of Tensor Calculus
			19.3.2 The Language of Differential Forms and Hodge Duality
			19.3.3 De Rham Cohomology and the Four-Potential of the Electromagnetic Field
			19.3.4 The Language of Fiber Bundles
		19.4 The Transformation Law for the Electromagnetic Field
		19.5 Electromagnetic Waves
		19.6 Invariants of the Electromagnetic Field
			19.6.1 The Motion of a Charged Particle and the Lorentz Force
			19.6.2 The Energy Density and the Energy-Momentum Tensor
			19.6.3 Conservation Laws
		19.7 The Principle of Critical Action
			19.7.1 The Electromagnetic Field
			19.7.2 Motion of Charged Particles and Gauge Transformations
		19.8 Weyl Duality and the Maxwell Equations in Materials
			19.8.1 The Maxwell Equations in the Rest System
			19.8.2 Typical Examples of Constitutive Laws
			19.8.3 The Maxwell Equations in an Arbitrary Inertial System
		19.9 Physical Units
			19.9.1 The SI System
			19.9.2 The Universal Approach
		19.10 Further Reading
	20. The Relativistic Invariance of the Dirac Equation and the Electron Spin
		20.1 The Dirac Equation
		20.2 Changing the Inertial System
		20.3 The Electron Spin
	21. The Language of Exact Sequences
		21.1 Applications to Linear Algebra
		21.2 The Fredholm Alternative
		21.3 The Deviation from Exact Sequences and Cohomology
		21.4 Perspectives
	22. Electrical Circuits as a Paradigm in Homology and Cohomology
		22.1 Basic Equations
		22.2 Euler's Bridge Problem and the Kirchhoff Rules
		22.3 Weyl's Theorem on Electrical Circuits
		22.4 Homology and Cohomology in Electrical Circuits
		22.5 Euler Characteristic and Betti Numbers
		22.6 The Discrete de Rham Theory
	23. The Electromagnetic Field and the de Rham Cohomology
		23.1 The De Rham Cohomology Groups
			23.1.1 Elementary Examples
			23.1.2 Advanced Examples
			23.1.3 Topological Invariance of the de Rham Cohomology Groups
			23.1.4 Homotopical Invariance of the de Rham Cohomology Groups
		23.2 The Fundamental Potential Equation in Gauge Theory and the Analytic Meaning of the Betti Numbers
		23.3 Hodge Theory (Representing Cohomology Classes by Harmonic Forms)
		23.4 The Topology of the Electromagnetic Field and Potentials
		23.5 The Analysis of the Electromagnetic Field
			23.5.1 The Main Theorem of Electrostatics, the Dirichlet Principle, and Generalized Functions
			23.5.2 The Coulomb Gauge and the Main Theorem of Magnetostatics
			23.5.3 The Main Theorem of Electrodynamics
		23.6 Important Tools
			23.6.1 The Exact Mayer–Vietoris Sequence and the Computation of the de Rham Cohomology Groups
			23.6.2 The de Rham Cohomology Algebra
		23.7 The Beauty of Partial Differential Equations in Physics, Analysis, and Topology
		23.8 A Glance at Topological Quantum Field Theory (Statistics for Mathematical Structures)
		23.9 Further Reading
	Appendix
		A.1 Manifolds and Diffeomorphisms
			A.1.1 Manifolds without Boundary
			A.1.2 Manifolds with Boundary
			A.1.3 Submanifolds
			A.1.4 Partition of Unity and the Globalization of Physical Fields
		A.2 The Solution of Nonlinear Equations
			A.2.1 Linearization and the Rank Theorem
			A.2.2 Violation of the Rank Condition and Bifurcation
		A.3 Lie Matrix Groups
		A.4 The Main Theorem on the Global Structure of Lie Groups
	Epilogue
	References
	List of Symbols
	Index