Quantum Field Theory and Critical Phenomena (The International Series of Monographs on Physics, 113)

The book is an introduction to quantum field theory and renormalization group. It shows that these frameworks are essential for the understanding of phenomena belonging to many different areas of physics, which range from phase transitions in macroscopic systems to the theory of fundamental interactions. This advanced new edition is based on graduate courses and summer schools given by the author over a number of years. Although there are several good textbooks on QFT, this is the first to emphasize the common aspects of particle physics and the theory of critical phenomena in a unified framework. The book has been fully updated, with about 50% new material added. Three new chapters have been included: an introduction to non-relativistic quantum statistical physics; a chapter on critical phenomena in non-magnetic systems, polymers, liquid-vapour, and helium superfluid transitions; and a chapter on finite temperature relativistic quantum field theory. The book can be roughly divided into four parts: chapters 1-12 deal with general field theory, functional integrals, and functional methods. In chapters 13-21, renormalization properties of theories with symmetries are studied and specific applications to particle physics are emphasized. Chapters 23-37 are devoted to critical phenomena. Chapters 39-43 describe the role of instantons in quantum mechanics and field theory. Title ......Page 1 Copyright ......Page 2 Preface ......Page 3 Acknowledgements ......Page 7 General References ......Page 8 Contents ......Page 9 1.1 Gaussian Integrals ......Page 19 1.2 Perturbation Theory. Connected Contributions. Steepest Descent ......Page 21 1.3 Complex Structures ......Page 23 1.4 Grassmann Algebras. Differential Forms ......Page 24 1.5 Differentiation in Grassmann Algebras ......Page 26 1.6 Integration in Grassmann Algebras ......Page 27 1.7 Gaussian Integrals with Grassmann Variables ......Page 31 1.8 Legendre Transformation ......Page 34 Bibliographical Notes ......Page 36 2 Euclidean Path Integrals in Quantum Mechanics ......Page 37 2.1 Path Integrals: The General Idea ......Page 38 2.2 Path Integral Representation: Special Hamiltonians ......Page 39 2.3 Explicit Evaluation of a Path Integral: The Harmonic Oscillator ......Page 42 2.4 Partition Function. Correlation Functions ......Page 44 2.5 Generating Functional of Correlation Functions. Perturbative Expansion ......Page 47 2.6 Semi-Classical Expansion ......Page 53 Bibliographical Notes ......Page 57 A2.1 The Two-Point Function: Spectral Representation ......Page 58 3.1 General Hamiltonians: Phase Space Path Integral ......Page 60 3.2 Hamiltonians Quadratic in Momentum Variables ......Page 63 3.3 The Spectrum of the O(2) Symmetric Rigid Rotator ......Page 68 3.4 The Spectrum of the O(N) Symmetric Rigid Rotator ......Page 69 Bibliographical Notes ......Page 72 A3.1 Symplectic Form and Quantization: General Remarks ......Page 73 A3.2 Spin Dynamics and Quantization ......Page 74 A3.3 The Magnetic Monopole ......Page 77 4.1 The Langevin Equation ......Page 78 4.2 A Simple Example: The Linear Langevin Equation ......Page 79 4.3 The Fokker-Planck Equation ......Page 81 4.4 Equilibrium Distribution. Correlation Functions ......Page 83 4.5 A Special Case: The Dissipative Langevin Equation ......Page 85 4.6 Path Integral Representation ......Page 87 4.7 General Discretized Langevin Equation ......Page 89 4.8 Brownian Motion on Riemannian Manifolds ......Page 91 Bibliographical Notes ......Page 96 A4.1 The Spectrum of the Transition Matrix ......Page 97 A4.2 Detailed Balance ......Page 99 A4.3 Stochastic Process with Prescribed Equilibrium Distribution ......Page 100 5.1 Quantum Mechanics: Holomorphic Formalism ......Page 101 5.2 Holomorphic Path Integral ......Page 104 5.3 Path Integrals with Fermions ......Page 108 5.5 The Bose Gas. Functional Integrals ......Page 116 5.6 The Fermi Gas. Functional Integrals ......Page 124 Bibliographical Notes ......Page 127 6 Quantum Evolution: From Particles to Fields ......Page 128 6.1 Time Evolution and Scattering Matrix in Quantum Mechanics ......Page 129 6.2 Path Integral and S-Matrix: Perturbation Theory ......Page 131 6.3 Path Integral and 5-Matrix: Semi-Classical Expansions ......Page 133 6.4 S-Matrix and Holomorphic Formalism ......Page 137 6.5 Fermi Gas: Evolution Operator ......Page 141 6.6 Relativistic Quantum Field Theory: The Scalar Field ......Page 142 6.7 The S-Matrix ......Page 146 6.8 S-Matrix and Field Asymptotic Conditions ......Page 149 6.9 Field Renormalization ......Page 153 6.10 S-matrix and Correlation Functions ......Page 155 6.11 The Non-Relativistic Limit ......Page 158 Bibliographical Notes ......Page 159 A6.1 Time-Ordered Products of Operators ......Page 160 A6.2 Perturbation Theory in the Operator Formalism ......Page 161 7 Quantum Field Theory: Functional Methods and Perturbation Theory ......Page 163 7.1 Functional Integrals. Correlation Functions ......Page 164 7.2 Perturbative Expansion. Wick's Theorem and Feynman Diagrams ......Page 167 7.3 Algebraic Properties of Functional Integrals. Field Equations ......Page 171 7.4 Connected Correlation Functions. Cluster Properties ......Page 177 7.5 Legendre Transformation. Proper Vertices ......Page 182 7.6 Momentum Representation ......Page 185 7.7 Semi-Classical or Loop Expansion ......Page 186 7.8 Legendre Transformation and 1-Irreducibility ......Page 190 7.9 Loop Expansion at Higher Orders ......Page 191 7.10 Statistical and Quantum Interpretation of the IPI Functional ......Page 193 Bibliographical Notes ......Page 196 A7.1 Two-Loop Calculation ......Page 197 A7.2 The Background Field Method ......Page 198 A7.3 Cluster Properties of Connected Feynman Diagrams ......Page 199 8.1 Massive Dirac Fermions ......Page 202 8.2 Free Euclidean Relativistic Fermions ......Page 208 8.3 Partition Function. Correlations ......Page 212 8.4 Generating Functional ......Page 215 8.5 Connection between Spin and Statistics ......Page 216 Bibliographical Notes ......Page 218 A8.1 Spin Group. Dirac Matrices ......Page 219 A8.3 Traces of Products of Matrices ......Page 227 A8.4 The Fierz Transformation ......Page 229 9 Quantum Field Theory: Divergences and Regularization ......Page 230 9.1 Divergences at One-Loop Order: The 3 Field Theory ......Page 231 9.2 Divergences General Analysis and Power Counting ......Page 234 9.3 Classification of Renormalizable Field Theories ......Page 238 9.4 Operator Insertions: Generating Functional, Power Counting ......Page 240 9.5 Momentum Cut-Off and Regulator Fields ......Page 242 9.6 Lattice Regularization ......Page 246 9.6 Dimensional Regularization ......Page 250 Bibliographical Notes ......Page 255 A9.1 Schwinger's Proper Time Representation ......Page 257 A9.2 One-Loop Divergences ......Page 258 10.1 Power Counting. Dimensional Analysis ......Page 261 10.2 Bare and Renormalized Field Theory. Operator 2 Insertions ......Page 263 10.3 One-Loop Divergences ......Page 266 10.4 Divergences Beyond One-Loop: Skeleton Diagrams ......Page 268 10.5 Callan-Symanzik Equations ......Page 271 10.6 Inductive Proof of Renormalizability ......Page 274 10.8 The Renormalized Action: General Construction ......Page 278 10.9 The Massless Theory ......Page 280 10.10 Homogeneous RG Equations: Massive Theory ......Page 284 10.11 Covariance of RG Functions ......Page 285 Bibliographical Notes ......Page 286 A10.1 Large Momentum Mode Integration and General RG Equations ......Page 288 A10.2 Super-Renormalizable Field Theories: The Normal-Ordered Product ......Page 290 11.1 Renormalization Group (RG) Functions ......Page 293 11.2 Dimensional Regularization: The Form of Renormalization Constants ......Page 294 11.3 Minimal Subtraction Scheme ......Page 295 11.4 The Massless Theory ......Page 298 11.5 RG Functions at Two-Loop Order in the 4 Field Theory ......Page 299 11.6 Generalization to Several Component Fields ......Page 304 11.7 One-Loop RG Functions in a Theory with Scalar Bosons and Fermions ......Page 307 Bibliographical Notes ......Page 312 A11.1 Feynman Parameters ......Page 313 12.1 Renormalization of Operator Insertions ......Page 314 12.2 Quantum Field Equations ......Page 318 12.3 Short Distance Expansion (SDE) of Operator Products ......Page 322 12.4 Large Momentum Expansion of the SDE Coefficients: CS Equations ......Page 326 12.5 SDE Beyond Leading Order. General Operator Product ......Page 328 12.6 Light Cone Expansion (LCE) of Operator Products ......Page 329 Bibliographical Notes ......Page 330 13.1 Preliminary Remarks ......Page 331 13.2 Linear Global Symmetries ......Page 333 13.3 Linear Symmetry Breaking ......Page 336 13.4 Spontaneous Symmetry Breaking ......Page 340 13.5 Quadratic Symmetry Breaking ......Page 343 13.6 Chiral Symmetry Breaking in Strong Interactions ......Page 348 Bibliographical Notes ......Page 355 A13.1 Currents in Classical Field Theory ......Page 357 A13.2 Euclidean Quantum Field Theory ......Page 358 A13.3 The Energy-Momentum Tensor ......Page 359 A13.4 Energy-Momentum Tensor and Euclidean Field Theory ......Page 361 A13.5 Dilatation and Conformal Invariance ......Page 362 14.1 The Non-Linear -Model: Definition ......Page 364 14.2 Perturbation Theory. Power Counting ......Page 366 14.3 Regularization ......Page 368 14.4 Infrared (IR) Divergences ......Page 370 14.5 WT Identities and Master Equation ......Page 371 14.6 Renormalization ......Page 374 14.7 The Renormalized Action: Solution to the Master Equation ......Page 377 14.9 A Linear Representation ......Page 380 Bibliographical Notes ......Page 382 15.1 Homogeneous Spaces and Goldstone Modes ......Page 383 15.2 WT Identities and Renormalization in Linear Coordinates ......Page 386 15.3 Renormalization in Arbitrary Coordinates, BRS Symmetry ......Page 390 15.4 Symmetric Spaces: Definition ......Page 394 15.5 The Classical Action. Conservation Laws ......Page 395 15.6 Quantum Theory: Perturbation Theory and RG Functions ......Page 398 15.7 Generalizations ......Page 403 Bibliographical Notes ......Page 404 A15.2 Metric and Curvature in Homogeneous Spaces ......Page 406 A15.3 Explicit Expressions for the Metric ......Page 408 A15.4 Symmetric Spaces: Classification ......Page 409 16 ST and BRS Symmetries, Stochastic Field Equations ......Page 414 16.1 Slavnov-Taylor (ST) Symmetry ......Page 415 16.2 Constraints and BRS Symmetry ......Page 417 16.3 Grassmann Coordinates, Gradient Equations ......Page 419 16.4 BRS Symmetry and Compatibility Condition, Group Manifolds ......Page 420 16.5 Stochastic Equations ......Page 422 16.6 Application: Stochastic Field Equations ......Page 427 16.7 Langevin and Fokker-Planck Equations ......Page 430 16.8 Time-Dependent Correlation Functions and Equilibrium ......Page 431 16.9 Renormalization and BRS Symmetry ......Page 434 Bibliographical Notes ......Page 436 17 From Langevin Equation to Supersymmetry ......Page 437 17.1 The Purely Dissipative Langevin Equation ......Page 438 17.2 Supersymmetry and Equilibrium Correlation Functions ......Page 441 17.3 Stochastic Quantization of Two-Dimensional Chiral Models ......Page 443 17.4 Langevin Equation and Riemannian Manifolds ......Page 445 17.5 Scalar Supersymmetric Fields Below Four Dimensions ......Page 448 17.6 Supersymmetry in Four Dimensions ......Page 454 Bibliographical Notes ......Page 461 A17.1 Extension of BRS Symmetries: Supersymmetry ......Page 463 A17.2 Supersymmetry: The Random Field Ising Model ......Page 464 18.1 The Massive Vector Field ......Page 466 18.2 Action with Fermion Matter ......Page 470 18.3 Massless Vector Field: Abelian Gauge Symmetry ......Page 471 18.4 Canonical Quantization and Gauge Invariance ......Page 473 18.5 Perturbation Theory, Regularization ......Page 479 18.6 WT Identities, Renormalization ......Page 483 18.7 Gauge Dependence ......Page 484 18.8 Renormalization Group Equations ......Page 487 18.9 The One-Loop -Function ......Page 488 18.10 The Abelian Higgs Model ......Page 491 18.11 Quantization of the Higgs Model ......Page 492 18.13 Stochastic Quantization: The Example of Gauge Theories ......Page 495 Bibliographical Notes ......Page 497 A18.1 Vacuum Energy and Casimir Effect ......Page 499 A18.2 Gauge Dependence ......Page 502 A18.3 Divergences at One-Loop with Schwinger's Representation ......Page 503 19.1 Geometric Construction ......Page 505 19.3 Hamiltonian Formalism. Quantization ......Page 508 19.4 Perturbation Theory, Regularization ......Page 514 19.5 The Non-Abelian Higgs Mechanism ......Page 516 Bibliographical Notes ......Page 520 A19 Massive Yang-Mills Fields ......Page 521 20.1 The Standard Model of Weak-Electromagnetic Interactions ......Page 523 20.2 Quantum Chromodynamics: Renormalization Group ......Page 531 20.3 The Abelian Anomaly ......Page 536 20.4 Non-Abelian Anomaly ......Page 545 20.5 Physical Applications ......Page 547 Bibliographical Notes ......Page 549 21.1 Notation and Geometric Structure ......Page 550 21.2 Quantization ......Page 552 21.3 BRS Symmetry ......Page 553 21.4 WT Identities and Master Equation ......Page 554 21.5 Renormalization: General Considerations ......Page 556 21.6 The Renormalized Action ......Page 559 21.7 Gauge Independence ......Page 564 Bibliographical Notes ......Page 565 22.1 Change of Coordinates. Tensors ......Page 567 22.2 Parallel Transport: Connection, Covariant Derivative ......Page 570 22.3 The Metric Tensor ......Page 573 22.4 The Curvature (Riemann) Tensor ......Page 574 22.5 Covariant Volume Element ......Page 578 22.6 Fermions, Vielbein, Spin Connection ......Page 579 22.7 Classical Gravity. Equations of Motion ......Page 581 22.8 Quantization in the Temporal Gauge: Pure Gravity ......Page 584 Bibliographical Notes ......Page 587 A22.1 Quantum 2D Euclidean Gravity ......Page 588 A22.2 The One-Matrix Model ......Page 589 A22.3 The Method of Orthogonal Polynomials ......Page 592 23 Critical Phenomena: General Considerations ......Page 595 23.1 Phase Transitions and Transfer Matrix ......Page 597 23.2 The Infinite Transverse Size Limit: Ising-Like Systems ......Page 599 23.3 Order Parameter and Cluster Properties ......Page 602 23.4 Stochastic Processes and Phase Transitions ......Page 604 23.5 Continuous Symmetries ......Page 605 Bibliographical Notes ......Page 606 A23 Quenched Averages ......Page 607 24.1 Ising-like Ferromagnetic Systems ......Page 610 24.2 High Temperature Expansion ......Page 612 24.3 Mean Field Approximation ......Page 613 24.4 Universality within Mean Field Approximation ......Page 616 24.5 Beyond Mean Field Approximation ......Page 620 24.6 Power Counting and the Role of Dimension 4 ......Page 624 Bibliographical Notes ......Page 626 A24.1 Mean Field Approximation ......Page 628 A24.2 Mean Field Expansion ......Page 631 A24.3 High, Low Temperature and Mean Field Approximations ......Page 632 25 General Renormalization Group. The Critical Theory near Dimension Four ......Page 634 25.1 Renormalization Group: The General Idea ......Page 635 25.2 The Gaussian Fixed Point ......Page 640 25.3 Critical Behaviour: The Effective 4 Field Theory ......Page 643 25.4 Renormalization Group Equations near Four Dimensions ......Page 644 25.5 Solution of the RG Equations: The -Expansion ......Page 647 25.6 Critical Correlation Functions with 2(x) Insertions ......Page 650 Bibliographical Notes ......Page 653 26 Scaling Behaviour in the Critical Domain ......Page 654 26.1 Strong Scaling above Tc: The Renormalized Theory ......Page 655 26.3 Scaling Laws above Tc ......Page 659 26.4 Correlation Functions with 2 Insertions ......Page 661 26.5 Scaling Laws in a Magnetic Field and Below Tc ......Page 663 26.6 The N-Vector Model ......Page 666 26.7 Asymptotic Expansion of the Two-Point Function ......Page 672 Bibliographical Notes ......Page 674 A26 The Specific Heat for a = 0 ......Page 676 27.1 Corrections to Scaling: Generic Dimensions ......Page 678 27.2 Logarithmic Corrections at the Upper-Critical Dimension ......Page 680 27.3 Irrelevant Operators and the Question of Universality ......Page 683 27.4 Corrections Coming from Irrelevant Operators. Improved Action ......Page 685 27.5 Application: Uniaxial Systems with Strong Dipolar Forces ......Page 687 Bibliographical Notes ......Page 692 28.1 Statistics of Self-Repelling Chains, Approximations ......Page 693 28.2 Liquid-Vapour Phase Transition and Field Theory ......Page 698 28.3 Superfluid Transition ......Page 703 Bibliographical Notes ......Page 707 29.1 The -Expansion ......Page 708 29.2 The Perturbative Expansion at Fixed Dimension ......Page 716 29.3 The Series Summation ......Page 718 29.4 Numerical Estimates of Critical Exponents ......Page 720 29.5 Comparison with Lattice Model Estimates ......Page 722 29.6 Critical Exponents from Experiments ......Page 723 29.7 Amplitude Ratios ......Page 724 Bibliographical Notes ......Page 725 30.1 The Large N Action ......Page 728 30.2 Large N Limit: Saddle Point Equations, Critical Domain ......Page 730 30.3 RG Functions and Leading Corrections to Scaling ......Page 736 30.4 Small Coupling Constant, Large Momentum Expansions for d < 4 ......Page 738 30.5 Dimension 4: Triviality, Higgs Mass ......Page 739 30.6 The Non-Linear -Model in the Large N Limit ......Page 741 30.7 The 1/N-Expansion: An Alternative Field Theory ......Page 745 30.8 Explicit Calculations: Critical Exponents ......Page 747 Bibliographical Notes ......Page 749 31 Phase Transitions near Two Dimensions ......Page 751 31.1 ( 2)2 Field Theory and Non-Linear -Model ......Page 752 31.2 The Non-Linear cr-Model: Symmetry Breaking, RG Equations ......Page 754 31.3 RG Equations: Discussion ......Page 757 31.4 Results Beyond One-Loop ......Page 761 31.5 The Dimension 2 ......Page 764 31.7 The Gross-Neveu Model ......Page 765 31.8 The Gross-Neveu-Yukawa Model ......Page 768 31.9 GNY and GN Models in the Large N Limit ......Page 770 31.10 The Large N Expansion ......Page 774 Bibliographical Notes ......Page 776 32.1 The Free Massless Scalar Field ......Page 777 32.2 The Free Massless Dirac Fermion ......Page 781 32.3 The Sine-Gordon Model ......Page 787 32.4 The Schwinger Model ......Page 789 32.5 The Massive Thirring Model ......Page 792 32.6 A Two-Fermion Model ......Page 795 Bibliographical Notes ......Page 799 A32.1 The Schwinger Model ......Page 800 A32.2 The SU(N) Thirring Model ......Page 801 A32.3 Solitons in the Sine-Gordon Model ......Page 804 33 The O(2) Classical Spin Model in Two Dimensions ......Page 805 33.1 The Spin Correlation Functions ......Page 806 33.2 Correlation Functions in a Field ......Page 808 33.3 The Coulomb Gas in Two Dimensions ......Page 809 33.4 O(2) Spin Model and Coulomb Gas ......Page 813 33.5 The Critical Two-Point Function in the O(2) Model ......Page 815 Bibliographical Notes ......Page 817 34.1 Gauge Invariance on the Lattice ......Page 818 34.2 The Pure Gauge Theory ......Page 820 34.3 Wilson's Loop and Confinement ......Page 823 34.4 Mean Field Approximation ......Page 828 Bibliographical Notes ......Page 831 A34 Gauge Theory and Confinement in Two Dimensions ......Page 833 35.1 The ( 2)2 Field Theory: Large Momentum Behaviour and Triviality ......Page 835 35.2 General 4-like Field Theories: d = 4 ......Page 840 35.3 Theories with Scalar Bosons and Fermions ......Page 842 35.4 Gauge Theories ......Page 844 35.5 Applications: The Theory of Strong Interactions ......Page 846 Bibliographical Notes ......Page 849 36 Critical Dynamics ......Page 850 36.1 Dissipative Case: RG Equations near Four Dimensions ......Page 851 36.2 Dissipative Case: RG Equations Near Two Dimensions ......Page 855 36.3 Conserved Order Parameter ......Page 857 36.4 Relaxational Model with Energy Conservation ......Page 858 36.5 A Non-Relaxational Model ......Page 861 Bibliographical Notes ......Page 863 A36.1 The ( 2)2 Field Theory: Dynamic Exponent ......Page 865 A36.2 The Non-Linear -Model ......Page 867 37.1 Renormalization Group in Finite Geometries ......Page 870 37.2 Momentum Quantization ......Page 874 37.3 The 4 Field Theory in a Periodic Hypercube ......Page 876 37.4 The 4 Field Theory: The Cylindrical Geometry ......Page 882 37.5 Finite Size Effects in the Non-Linear -Model ......Page 886 37.6 Finite Size Effects and Dynamics ......Page 892 Bibliographical Notes ......Page 896 A37.1 Discrete Symmetries and Finite Size Effects ......Page 898 A37.2 Perturbation Theory in a Finite Volume ......Page 902 38.1 Finite (and High) Temperature Field Theory ......Page 903 38.2 The Example of the 4
,d-1 Field Theory ......Page 907 38.3 High Temperature and Critical Limits ......Page 912 38.4 The Non-Linear -Model in the Large N Limit ......Page 916 38.5 The Non-Linear -Model: Dimensional Reduction ......Page 920 38.6 The Gross-Neveu in the Large N Expansion ......Page 927 38.7 Abelian Gauge Theories ......Page 933 38.8 Non-Abelian Gauge Theories ......Page 941 Bibliographical Notes ......Page 944 A38.1 One-Loop Calculations ......Page 946 A38.2 Group Measure ......Page 948 39 Instantons in Quantum Mechanics ......Page 949 39.1 The Quartic Anharmonic Oscillator for Negative Coupling ......Page 950 39.2 A Toy Model: A Simple Integral ......Page 951 39.3 Quantum Mechanics: Instantons ......Page 953 39.4 Instanton Contribution at Leading Order ......Page 954 39.5 General Potentials: Instanton Contributions ......Page 959 39.6 Gaussian Integration: The Shifting Method ......Page 960 39.7 Low Temperature Evaluation ......Page 966 Bibliographical Notes ......Page 967 A39.1 Classical Equations of Motion ......Page 968 A39.2 The WKB Method ......Page 970 A39.3 The Average Action in Path Integrals ......Page 973 40 Unstable Vacua in Quantum Field Theory ......Page 974 40.1 The 4 Field Theory for Negative Coupling ......Page 975 40.2 General Potentials: Instanton Contributions ......Page 980 40.3 The 4 Field Theory in Dimension 4 ......Page 981 40.4 Instanton Contributions at Leading Order ......Page 983 40.5 Coupling Constant Renormalization ......Page 986 40.6 The Imaginary Part of the n-Point Function ......Page 988 40.7 The Massive Theory ......Page 989 40.8 Cosmology: The Decay of the False Vacuum ......Page 990 Bibliographical Notes ......Page 991 A40.1 Virial Theorem ......Page 992 A40.2 Sobolev Inequalities ......Page 993 A40.3 Instantons and RG Equations ......Page 995 A40.4 Conformal Invariance ......Page 996 41.1 The Double-Well Potential ......Page 998 41.2 The Periodic Cosine Potential ......Page 1000 41.3 Instantons and Stochastic Dynamics ......Page 1003 41.4 Instantons in Stable Boson Field Theories: General Remarks ......Page 1006 41.5 Instantons in CP(N - 1) Models ......Page 1008 41.6 Instantons in the SU(2) Gauge Theory ......Page 1010 Bibliographical Notes ......Page 1013 A41.1 Trace Formula for Periodic Potentials ......Page 1014 42.1 Quantum Mechanics ......Page 1015 42.2 Scalar Field Theory ......Page 1018 42.3 The 4 Field Theory in Four Dimensions ......Page 1019 42.4 Field Theories with Fermions ......Page 1023 42.5 Divergent Series, Borel Summability ......Page 1028 42.6 Large Order Behaviour and Borel Summability ......Page 1029 42.7 Practical Summation Methods ......Page 1030 Bibliographical Notes ......Page 1034 A42.2 Non-Loop Expansions ......Page 1036 A42.3 Linear Differential Approximants ......Page 1037 43 Multi-lnstantons in Quantum Mechanics ......Page 1038 43.1 The Double-Well Potential ......Page 1039 43.2 The Periodic Cosine Potential ......Page 1046 43.3 General Potentials with Degenerate Minima ......Page 1050 43.4 The O(v) Symmetric Anharmonic Oscillator ......Page 1053 43.5 Generalized Bohr-Sommerfeld Quantization Formula ......Page 1055 Bibliographical Notes ......Page 1056 A43.1 Multi-lnstantons: The Determinant ......Page 1057 A43.2 The Instanton Interaction ......Page 1058 A43.3 A Simple Example of Non-Borel Summability ......Page 1060 A43.4 Multi-lnstantons and WKB Approximation ......Page 1062 Index ......Page 1065