Contents. Part I. Introduction. Prologue... 1

Transcription

1 Contents Part I. Introduction Prologue Mathematical Principles of Modern Natural Philosophy BasicPrinciples The Infinitesimal Strategy and Differential Equations The Optimality Principle The Basic Notion of Action in Physics and the Idea of Quantization TheMethodoftheGreen sfunction HarmonicAnalysisandtheFourierMethod The Method of Averaging and the Theory of Distributions The Symbolic Method Gauge Theory Local Symmetry and the Description of InteractionsbyGaugeFields The Challenge of Dark Matter 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 Idea of 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... 64

2 XXVI Contents 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 in MinkowskiSpace Distribution-Valued Meromorphic Functions Application to Newton s Equation of Motion Hints for Further Reading FurtherRegularizationMethodsinMathematics 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 FunctionRegularization TroubleinMathematics Interchanging Limits The Ambiguity of Regularization Methods Pseudo-Convergence Ill-Posed Problems Mathemagics The Power of Combinatorics Algebras The Algebra of Multilinear Functionals Fusion,Splitting,andHopfAlgebras The Bialgebra of Linear Differential Operators The Definition of Hopf Algebras PowerSeriesExpansionandHopfAlgebras The Importance of Cancellations The Kepler Equation and the Lagrange InversionFormula The Composition Formula for Power Series The Faà di Bruno Hopf Algebra for the Formal DiffeomorphismGroupoftheComplexPlane The Generalized Zimmermann Forest Formula The Logarithmic Function and Schur Polynomials Correlation Functions in Quantum Field Theory

3 Contents XXVII Random Variables, Moments, and Cumulants SymmetryandHopfAlgebras The Strategy of Coordinatization in Mathematics andphysics The Coordinate Hopf Algebra of a Finite Group The Coordinate Hopf Algebra of an Operator Group The Tannaka Krein Duality for Compact Lie Groups RegularizationandRota BaxterAlgebras Regularization of the Laurent Series Projection Operators The q-integral The Volterra Spitzer Exponential Formula The Importance of the Exponential Function in MathematicsandPhysics Partially Ordered Sets and Combinatorics Incidence Algebras and the Zeta Function The MöbiusFunctionasanInverseFunction The Inclusion Exclusion Principle in Combinatorics Applications to Number Theory HintsforFurtherReading The Strategy of Equivalence Classes in Mathematics EquivalenceClassesinAlgebra The Gaussian Quotient Ring and the Quadratic ReciprocityLawinNumberTheory 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 EquivalenceClassesinGeometry The Basic Idea of Geometry Epitomized by Klein s ErlangenProgram 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

4 XXVIII Contents 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 TheStrategyofPartialOrdering 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 Part II. Basic Ideas in Classical Mechanics 5. Geometrical Optics Ariadne sthreadingeometricaloptics Fermat sprincipleofleasttime Huygens PrincipleonWaveFronts Carathéodory sroyalroadtogeometricaloptics The Duality between Light Rays and Wave Fronts From Wave Fronts to Light Rays From Light Rays to Wave Fronts TheJacobiApproachtoFocalPoints Lie scontactgeometry Basic Ideas Contact Manifolds and Contact Transformations Applications to Geometrical Optics Equilibrium Thermodynamics and Legendre Submanifolds LightRaysandNon-EuclideanGeometry Linear Symplectic Spaces The KählerFormofaComplexHilbertSpace The Refraction Index and Geodesics The Trick of Gauge Fixing

5 Contents XXIX 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 of the Sphere The Beltrami Model The Poincaré Model of Hyperbolic Geometry KählerGeometryandtheGaussianCurvature Kähler EinsteinGeometry Symplectic Geometry Riemannian Geometry Ariadne s Thread in Gauge Theory Parallel Transport of Physical Information the Key tomodernphysics The Phase Equation and Fiber Bundles Gauge Transformations and Gauge-Invariant DifferentialForms Perspectives Classification of Two-Dimensional Compact Manifolds The PoincaréConjectureandtheRicciFlow A Glance at Modern Optimization Theory Hints for Further Reading The Principle of Critical Action and the Harmonic Oscillator Ariadne s Thread in Classical Mechanics PrototypesofExtremalProblems TheMotionofaParticle NewtonianMechanics 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 Energy Dissipation Resonance and Small Divisors SymmetryandConservationLaws The Symmetries of the Harmonic Oscillator The Noether Theorem

6 XXX Contents 6.7 The Pendulum and Dynamical Systems The Equation of Motion Elliptic Integrals and Elliptic Functions The Phase Space of the Pendulum and Bundles HamiltonianMechanics The Canonical Equation The Hamiltonian Flow The Hamilton Jacobi Partial Differential Equation PoissonianMechanics 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(E 3 )ofrotations 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 Part III. Basic Ideas in Quantum Mechanics 7. Quantization of the Harmonic Oscillator Ariadne s Thread in Quantization CompleteOrthonormalSystems Bosonic Creation and Annihilation Operators Heisenberg squantummechanics 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

7 Contents XXXI 7.4 Schrödinger squantummechanics The SchrödingerEquation 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 UncertaintyRelation Schrödinger scoherentstates Feynman squantummechanics Main Ideas The Diffusion Kernel and the Euclidean Strategy in QuantumPhysics 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 of FunctionalAnalysis The Free Hamiltonian The Rescaled Fourier Transform The Quantized Harmonic Oscillator and the Maslov Index Ideal Gases and von Neumann s Density Operator TheFeynmanPathIntegral 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-DimensionalGaussianIntegrals Basic Formulas Free Moments, the Wick Theorem, and Feynman Diagrams 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.. 576

8 XXXII Contents 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 PropagatorKernel 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 Quantization Functor 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 aparadigm 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

9 Contents XXXIII 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 Quantum Particles on the Real Line Ariadne s Thread in Scattering Theory ClassicalDynamicsVersusQuantumDynamics The Stationary SchrödingerEquation One-Dimensional Quantum Motion in a Square-Well Potential Free Motion Scattering States and the S-Matrix Bound States Bound-State Energies and the Singularities of the S-Matrix The Energetic Riemann Surface, Resonances, and the Breit WignerFormula 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 TheMethodoftheGreen sfunctioninanutshell The Inhomogeneous Helmholtz Equation The Retarded Green s Function, and the Existence anduniquenesstheorem The Advanced Green s Function Perturbation of the Retarded and Advanced Green s Function Feynman s Regularized Fourier Method TheLippmann SchwingerIntegralEquation The Born Approximation The Existence and Uniqueness Theorem via Banach s FixedPointTheorem Hypoellipticity A Glance at General Scattering Theory TheFormalBasicIdea The Rigorous Time-Dependent Approach The Rigorous Time-Independent Approach

10 XXXIV Contents 9.4 ApplicationstoQuantumMechanics AGlanceatQuantumFieldTheory HintsforFurtherReading Part IV. Quantum Electrodynamics (QED) 10. 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 The Basic Equations in Quantum Electrodynamics The Classical Lagrangian The Gauge Condition 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 OperatorsinQED Gupta Bleuler Quantization The Interacting Quantum Field, and the Magic Dyson Series for the S-Matrix Dyson s Key Formula The Basic Strategy of Reduction Formulas The Wick Theorem Feynman Propagators

11 Contents XXXV Discrete Feynman Propagators for Photons and Electrons Regularized Discrete Propagators The Continuum Limit of Feynman Propagators Classical Wave Propagation versus Feynman Propagator 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 Scattering Processes The Crucial Limits Appendix: Table of Feynman Rules Applications to Physical Effects Compton Effect Duality between Light Waves and Light Particles inthehistoryofphysics The Trace Method for Computing Cross Sections Relativistic Invariance Asymptotically Free Electrons in an External ElectromagneticField 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 Electromagnetic Field The Spontaneous Emission of Photons by the Atom Motivation of the Key Formula Intensity of Spectral Lines Cherenkov Radiation Part V. Renormalization 16. The Continuum Limit The Fundamental Limits The Formal Limits Fail Basic Ideas of Renormalization

12 XXXVI Contents 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 Algebraic FeynmanIntegrals Multiplicative Renormalization The Theory of Approximation Schemes in Mathematics 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 A Glance at Renormalization to all Orders of Perturbation Theory One-Particle Irreducible Feynman Graphs and Divergences Overlapping Divergences and Manoukian s Equivalence Principle The Renormalizability of Quantum Electrodynamics Automated Multi-Loop Computations in Perturbation Theory Perspectives BPHZ Renormalization Bogoliubov s Iterative R-Method Zimmermann s Forest Formula The Classical BPHZ Method The Causal Epstein Glaser S-MatrixApproach Kreimer s Hopf Algebra Revolution The History of the Hopf Algebra Approach

13 Contents XXXVII Renormalization and the Iterative Birkhoff FactorizationforComplexLieGroups The Renormalization of Quantum Electrodynamics The Scope of the Riemann Hilbert Problem The Gaussian Hypergeometric Differential Equation The Confluent Hypergeometric Function and the SpectrumoftheHydrogenAtom 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-Valued AnalyticFunctions Computational Strategies The Renormalization Group Operator Product Expansions Binary Planar Graphs and the Renormalization ofquantumelectrodynamics The Master Ward Identity Trouble in Quantum Electrodynamics The Landau Inconsistency Problem in Quantum Electrodynamics The Lack of Asymptotic Freedom in Quantum Electrodynamics Hints for Further Reading Epilogue References List of Symbols Index

14