Quantum Field Theory and Critical Phenomena (Fourth Edition, 2002)

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