Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets 4
Contents 16
Preface 8
Preface to the Second Edition 10
Preface to the First Edition 12
1 FundamentaIs 38
1.1 Classical Mechanics 38
1.2 Relativistic Mechanics in Curved Spacetime 48
1.3 Quantum Mechanics 49
1.4 Dirac\u2019s Bra-Ket Formalism 55
1.5 Observables 63
1.6 Quantum Mechanics of General Lagrangian Systems 68
1.7 Particle on the Surface of a Sphere 74
1.8 Spinning Top 76
1.9 Time Evolution Operator 83
1.10 Properties of Time Evolution Operator 86
1.11 Scattering 93
1.11.1 Scattering Matrix 93
1.11.2 Cross Section 94
1.11.3 Born Approximation 96
1.11.4 Partial Wave Expansion and Eikonal Approximation 96
1.11.5 Scattering Amplitude from Time Evolution Amplitude 98
1.11.6 Lippmann-Schwinger Equation 99
1.12 Heisenberg Picture of Quantum Mechanics 102
1.13 Classical and Quantum Statistics 105
1.13.1 Canonical Ensemble 106
1.13.2 Grand-Canonical Ensemble 107
Appendix 1A The Asymmetric Top 111
Notes and References 113
2 Path Integrals \u2014 Elementary Properties and Simple Solutions 114
2.1 Path Integral Representation of Time Evolution Amplitudes 114
2.2 Exact Solution for Free Particle 124
2.3 Finite Slicing Properties of Free-Particle Amplitude 133
2.4 Exact Solution for Harmonic Oscillator 134
2.5 Useful Fluctuation Formulas 139
2.6 Oscillator Amplitude on Finite Time Lattice 141
2.7 Gelfand-Yaglom Formula 143
2.7.1 Recursive Calculation of Fluctuation Determinant 143
2.7.2 Examples 144
2.7.3 Calculation on Unsliced Time Axis 146
2.7.4 D\u2019Alembert\u2019s Construction 147
2.7.5 Another Simple Formula 148
2.7.6 Generalization to D Dimensions 150
2.8 Path Integral for Harmonic Oscillator with Arbitrary Time-Dependent Frequency 151
2.8.1 Coordinate Space 151
2.8.2 Momentum Space 154
2.9 Free-Particle and Oscillator Wave Functions 156
2.10 Path Integrals and Quantum Statistics 158
2.11 Density Matrix 160
2.12 Quantum Statistics of Harmonic Oscillator 165
2.13 Time-Dependent Harmonic Potential 171
2.14 Functional Measure in Fourier Space 175
2.15 Classical Limit 178
2.16 Calculation Techniques on Sliced Time Axis. Poisson Formula 179
2.17 Field-Theoretic Definition of Harmonic Path Integral by Analytic Regularization 184
2.17.1 Zero-Temperature Evaluation of Frequency Sum 185
2.17.2 Finite-Temperature Evaluation of Frequency Sum 188
2.17.3 Duality Transformation and Low-Temperature Expansion 190
2.18 Finite-N Behavior of Thermodynamic Quantities 196
2.19 Time Evolution Amplitude of Freely Falling Particle 198
2.20 Charged Particle in Magnetic Field 200
2.21 Charged Particle in Magnetic Field and Harmonic Potential 205
2.22 Gauge Invariance and Alternative Path Integral Representation 208
2.23 Velocity Path Integral 209
2.24 Path Integral Representation of Scattering Matrix 211
2.24.1 General Development 211
2.24.2 Improved Formulation 214
2.24.3 Eikonal Approximation to Scattering Amplitude 215
Appendix 2A Derivation of Baker-Campbell-Hausdorff and Magnus Formulas 216
Appendix 2B Direct Calculation of Time-Sliced Oscillator Amplitude 219
Appendix 2C Derivation of Mehler Formula 221
Notes and References 221
3 External Sources, Correlations, and Perturbation Theory 224
3.1 External Sources 224
3.2 Green Function of Harmonic Oscillator 227
3.2.1 Wronski Construction 228
Constant Frequency 228
Time-Dependent Frequency 229
3.2.2 Spectral Representation 232
Constant Frequency 232
Time-Dependent Frequency 233
3.3 Green Functions of First-Order Differential Equation 233
3.3.1 Time-Independent Frequency 234
3.3.2 Time-Dependent Frequency 241
3.4 Summing Spectral Representation of Green Function 242
3.5 Wronski Construction for Periodic and Antiperiodic Green Functions 244
3.6 Time Evolution Amplitude in Presence of Source Term 245
3.7 External Source in Quantum-Statistical Path Integral 250
3.7.1 Continuation of Real-Time Result 250
3.7.2 Calculation at Imaginary Time 254
3.8 Lattice Green Function 261
3.9 Correlation Functions, Generating Functional, and Wick Expansion 262
3.10 Correlation Functions of Charged Particle in Magnetic Field 265
3.11 Correlation Functions in Canonical Path Integral 266
3.11.1 Harmonic Correlation Functions 267
3.11.2 Relations between Various Amplitudes 270
3.11.3 Harmonic Generating Functionals 271
3.12 Particle in Heat Bath 274
3.13 Particle in Heat Bath of Photons 278
3.14 Harmonic Oscillator in Heat Bath 280
3.15 Perturbation Expansion of Anharmonic Systems 283
3.16 Calculation of Perturbation Series with Feynman Diagrams 286
3.17 Field-Theoretic Definition of Anharmonic Path Integral 290
3.18 Generating Functional of Connected Correlation Functions 291
3.18.1 Connectedness Structure of Correlation Functions 292
3.18.2 Decomposition of Correlation Functions into Connected Correlation Functions 295
3.18.3 Functional Generation of Vacuum Diagrams 297
3.18.4 Correlation Functions from Vacuum Diagrams 301
3.18.5 Generating Functional for Vertex Functions. Effective Action 303
3.18.6 Ginzburg-Landau Approximation to Generating Functional 308
3.18.7 Composite Fields 309
3.19 Path Integral Calculation of Effective Action by Loop Expansion 310
3.19.1 General Formalism 310
3.19.2 Quadratic Fluctuations 315
3.19.3 Effective Action to Second Order in h 318
3.19.4 Background Field Method for Effective Action 321
3.20 Nambu-Goldstone Theorem 324
3.21 Effective Classical Potential 326
3.21.1 Effective Classical Boltzmann Factor 328
3.21.2 High- and Low-Temperature Behavior 330
3.21.3 Alternative Candidate for Effective Classical Potential 332
3.21.4 Harmonic Correlation Function without Zero Mode 333
3.21.5 Perturbation Expansion 333
3.21.6 First-Order Perturbative Result 335
3.22 Perturbative Calculation of Scattering Amplitude 337
3.22.1 Generating Functional 337
3.22.2 Application to Scattering Amplitude 338
3.22.3 First Correction to Eikonal Approximation 339
3.23 Rayleigh-Schrodinger Perturbation Expansion 340
3.23.1 Energy Levels 340
3.23.2 Scattering Amplitudes 345
3.24 Functional Determinants from Green Functions 346
Appendix 3A Feynman Integrals for T 0 352
Appendix 3B Energy Shifts for gx4/4-Interaction 355
Appendix 3C Matrix Elements for General Potential 357
Appendix 3D Level-Shifts from Schrodinger Equation 359
Appendix 3E Recursion Relations for Perturbation Coefficients 361
3E.1 One-Dimensional Interaction x4 361
3E.2 Interaction r4 in D-Dimensional Radial Oscillator 365
3E.3 Interaction r2q in D Dimensions 366
3E.4 Polynomial Interaction in D Dimensions 366
Notes and References 366
4 Semiclassical Time Evolution Amplitude 369
4.1 The Wentzel-Kramers-Brillouin (WKB) Approximation 369
4.2 Saddle Point Approximation 373
4.2.1 Ordinary Integrals 374
4.2.2 Path Integrals 376
4.3 Van Vleck-Pauli-Morette Determinant 382
4.4 Fundamental Composition Law for Semiclassical Time Evolution Amplitude 386
4.5 Semiclassical Fixed-Energy Amplitude 388
4.6 Semiclassical Amplitude in Momentum Space 390
4.7 Semiclassical Quantum-Mechanical Partition Function 392
4.8 Multi-Dimensional Systems 397
4.9 Quantum Corrections to Classical Density of States 402
4.10 Thomas-Fermi Model of Neutral Atoms 407
4.10.1 Semiclassical Limit 407
4.10.2 Quantum Correction Near Origin 415
4.10.3 Exchange Energy 417
4.10.4 Higher Quantum Corrections to Thomas-Fermi Energies 419
4.11 Classical Action of Coulomb System 424
4.12 Semiclassical Scattering 433
4.12.1 General Formulation 433
4.12.2 Semiclassical Cross Section of Mott Scattering 436
Notes and References 437
5 Variational Perturbation Theory 440
5.1 Variational Approach to Effective Classical Partition Function 440
5.2 Local Harmonic Trial Partition Function 441
5.3 Optimal Upper Bound 446
5.4 Accuracy of Variational Approximation 447
5.5 Weakly Bound Ground State Energy in Finite-Range Potential Well 449
5.6 Possible Direct Generalizations 451
5.7 Effective Classical Potential for Anharmonic Oscillator 452
5.8 Particle Densities 458
5.9 Extension to D Dimensions 461
5.10 Application to Coulomb and Yukawa Potentials 463
5.11 Hydrogen Atom in Strong Magnetic Field 466
5.11.1 Weak-Field Behavior 470
5.11.2 Effective Classical Potential 470
5.12 Effective Potential and Magnetization Curves 473
5.13 Variational Approach to Excitation Energies 475
5.14 Systematic Improvement of Feynman-Kleinert Approximation 480
5.15 Applications of Variational Perturbation Expansion 483
5.15.1 Anharmonic Oscillator at T = 0 483
5.15.2 Anharmonic Oscillator for T > 0 485
5.16 Convergence of Variational Perturbation Expansion 489
5.17 Variational Perturbation Theory for Strong-Coupling Expansion 496
5.18 General Strong-Coupling Expansions 499
5.19 Variational Interpolation between Weak and Strong-Coupling Expansions 502
5.20 Systematic Improvement of Excited Energies 504
5.21 Variational Treatment of Double-Well Potential 505
5.22 Higher-Order Effective Classical Potential for Nonpolynomial Interactions 507
5.22.1 Evaluation of Path Integrals 508
5.22.2 Higher-Order Smearing Formula in D Dimensions 509
5.22.3 Isotropic Second-Order Approximation to Coulomb Problem 511
5.22.4 Anisotropic Second-Order Approximation to Coulomb Problem 513
5.22.5 Zero-Temperature Limit 515
5.23 Polarons 519
5.23.1 Partition Function 521
5.23.2 Harmonic Trial System 523
5.23.3 Effective Mass 529
5.23.4 Second-Order Correction 529
5.23.5 Polaron in Magnetic Field, Bipolarons, etc. 530
5.23.6 Variational Interpolation for Polaron Energy and Mass 531
5.24 Density Matrices 534
5.24.1 Harmonic Oscillator 535
5.24.2 Variational Perturbation Theory for Density Matrices 536
5.24.3 Smearing Formula for Density Matrices 538
5.24.4 First-Order Variational Results 540
Alternative First-Order Smearing Formula 541
a) Classical Limit of Effective Classical Potential 542
b) Zero-Temperature Limit 543
5.24.5 Smearing Formula in Higher Spatial Dimensions 545
a) Isotropic Approximation 545
b) Anisotropic Approximation 546
5.24.6 Applications 547
a) Double-Well 547
1) First-Order Approximation 547
b) Hydrogen Atom 554
Appendix 5A Feynman Integrals for T 0 without Zero Frequency 558
Appendix 5B Proof of Scaling Relation for Extrema of WN 560
Appendix 5C Second-Order Shift of Polaron Energy 563
Notes and References 564
6 Path Integrals with Topological Constraints 569
6.1 Point Particle on Circle 569
6.2 Infinite Wall 573
6.3 Point Particle in Box 578
6.4 Strong-Coupling Theory for Particle in Box 580
6.4.1 Partition Function 581
6.4.2 Perturbation Expansion 581
6.4.3 Variational Strong-Coupling Approximations 584
6.4.4 Special Properties of Expansion 586
6.4.5 Exponentially Fast Convergence 587
Notes and References 588
7 Many Particle Orbits \u2014 Statistics and Second Quantization 590
7.1 Ensembles of Bose and Fermi Particle Orbits 591
7.2 Bose-Einstein Condensation 598
7.2.1 Free Bose Gas 598
7.2.2 Effect of Interactions 607
7.2.3 Bose-Einstein Condensation in Harmonic Trap 612
7.2.4 Interactions in Harmonic Trap 621
7.3 Gas of Free Fermions 625
7.4 Statistics Interaction 630
7.5 Fractional Statistics 635
7.6 Second-Quantized Bose Fields 636
7.7 Fluctuating Bose Fields 639
7.8 Coherent States 645
7.9 Dimensional Regularization of Functional Determinants 649
7.10 Second-Quantized Fermi Fields 652
7.11 Fluctuating Fermi Fields 653
7.11.1 Grassmann Variables 653
7.11.2 Fermionic Functional Determinant 656
7.11.3 Coherent States for Fermions 660
7.12 Hilbert Space of Quantized Grassmann Variable 661
7.12.1 Single Real Grassmann Variable 662
7.12.2 Quantizing Harmonic Oscillator with Grassmann Variables 665
7.12.3 Spin System with Grassmann Variables 666
Pauli Algebra 666
Dirac Algebra 668
Relation between Harmonic Oscillator, Pauli and Dirac Algebra 669
7.13 External Sources in a*, a -Path Integral 670
7.14 Generalization to Pair Terms 672
7.15 Spatial Degrees of Freedom 674
7.15.1 Grand-Canonical Ensemble of Particle Orbits from Free Fluctuating Field 674
7.15.2 First versus Second Quantization 675
7.15.3 Interacting Fields 676
7.15.4 Effective Classical Field Theory 677
Notes and References 679
8 Path Integrals in Spherical Coordinates 683
8.1 Angular Decomposition in Two Dimensions 683
8.2 Trouble with Feynman\u2019s Path Integral Formula in Radial Coordinates 686
8.3 Cautionary Remarks 690
8.4 Time Slicing Corrections 693
8.5 Angular Decomposition in Three and More Dimensions 698
8.5.1 Three Dimensions 698
8.5.2 D Dimensions 701
8.6 Radial Path Integral for Harmonic Oscillator and Free Particle 706
8.7 Particle near the Surface of a Sphere in D Dimensions 707
8.8 Angular Barriers near the Surface of a Sphere 710
8.8.1 Angular Barriers in Three Dimensions 710
8.8.2 Angular Barriers in Four Dimensions 715
8.9 Motion on a Sphere in D Dimensions 720
8.10 Path Integrals on Group Spaces 724
8.11 Path Integral of a Spinning Top 727
Notes and References 728
9 Fixed-Energy Amplitude and Wave Functions 730
9.1 General Relations 730
9.2 Free Particle in D Dimensions 733
9.3 Harmonic Oscillator in D Dimensions 736
9.4 Free Particle from 0 -Limit of Oscillator 742
9.5 Charged Particle in Uniform Magnetic Field 744
Notes and References 751
10 Spaces with Curvature and Torsion 752
10.1 Einstein\u2019s Equivalence Principle 753
10.2 Classical Motion of Mass Point in General Metric-Affine Space 754
10.2.1 Equations of Motion 754
10.2.2 Nonholonomic Mapping to Spaces with Torsion 757
10.2.3 New Equivalence Principle 762
10.2.4 Classical Action Principle for Spaces with Curvature and Torsion 763
10.3 Path Integral in Metric-Affine Space 767
10.3.1 Nonholonomic Transformation of Action 768
10.3.2 Measure of Path Integration 772
10.4 Completing Solution of Path Integral on Surface of Sphere 778
10.5 External Potentials and Vector Potentials 779
10.6 Perturbative Calculation of Path Integrals in Curved Space 782
10.6.1 Free and Interacting Parts of Action 782
10.6.2 Zero Temperature 784
10.7 Model Study of Coordinate Invariance 787
10.7.1 Diagrammatic Expansion 788
10.7.2 Diagrammatic Expansion in d Time Dimensions 790
10.8 Calculating Loop Diagrams 791
10.8.1 Reformulation in Configuration Space 798
10.8.2 Integrals over Products of Two Distributions 799
10.8.3 Integrals over Products of Four Distributions 800
10.9 Distributions as Limits of Bessel Function 803
10.9.1 Correlation Function and Derivatives 803
10.9.2 Integrals over Products of Two Distributions 804
10.9.3 Integrals over Products of Four Distributions 806
10.10 Simple Rules for Calculating Singular Integrals 808
10.11 Perturbative Calculation on Finite Time Intervals 813
10.11.1 Diagrammatic Elements 814
10.11.2 Cumulant Expansion of D-Dimensional Free-Particle Amplitude in Curvilinear Coordinates 815
10.11.3 Propagator in 1 \u2013 e Time Dimensions 817
10.11.4 Coordinate Independence for Dirichlet Boundary Conditions 819
10.11.5 Time Evolution Amplitude in Curved Space 826
10.11.6 Covariant Results for Arbitrary Coordinates 832
10.12 Effective Classical Potential in Curved Space 839
10.12.1 Covariant Fluctuation Expansion 840
10.12.2 Arbitrariness of q 842
10.12.3 Zero-Mode Properties 844
10.12.4 Covariant Perturbation Expansion 847
10.12.5 Covariant Result from Noncovariant Expansion 848
10.12.6 Particle on Unit Sphere 851
10.13 Covariant Effective Action for Quantum Particle with Coordinate-Dependent Mass 854
10.13.1 Formulating the Problem 854
10.13.2 Derivative Expansion 857
Appendix 10A Nonholonomic Gauge Transformations in Electromagnetism 860
10A.1 Gradient Representation of Magnetic Field of Current Loop 861
10A.2 Generating Magnetic Field by Multivalued Gauge Transformations 864
10A.3 Magnetic Monopoles 865
10A.4 Minimal Magnetic Coupling of Particles from Multivalued Gauge Transformations 866
10A.5 Gauge Field Representation of Current Loops 868
Appendix 10B Difference between Multivalued Basis Tetrads and Vierbein Fields 870
Appendix 10C Cancellation of Powers of (0 ) 872
Notes and References 875
11 Schrodinger Equation in General Metric-Affine Spaces 880
11.1 Integral Equation for Time Evolution Amplitude 880
11.1.1 From the Recursion Relation to Schrodinger's Equation 881
11.1.2 Alternative Evaluation 884
11.2 Equivalent Path Integral Representations 887
11.3 Potentials and Vector Potentials 891
11.4 Unitarity Problem 892
11.5 Alternative Attempts 894
11.6 DeWitt-Seeley Expansion of Time Evolution Amplitude 895
Appendix 11A Cancellations in Effective Potential 899
Appendix 11B DeWitt\u2019s Amplitude 902
Notes and References 902
12 New Path Integral Formula for Singular PotentiaIs 904
12.1 Path Collapse in Feynman\u2019s formula for the Coulomb System 904
12.2 Stable Path Integral with Singular Potentials 907
12.3 Time-Dependent Regularization 912
12.4 Relation with Schrodinger Theory. Wave Functions 914
Notes and References 916
13 Path Integral of Coulomb System 917
13.1 Pseudotime Evolution Amplitude 917
13.2 Solution for the Two-Dimensional Coulomb System 919
13.3 Absence of Time Slicing Corrections for D = 2 924
13.4 Solution for the Three-Dimensional Coulomb System 930
13.5 Absence of Time Slicing Corrections for D = 3 936
13.6 Geometric Argument for Absence of Time Slicing Corrections 940
13.7 Comparison with Schrodinger Theory 941
13.8 Angular Decomposition of Amplitude, and Radial Wave Functions 946
13.9 Remarks on Geometry of Four-Dimensional u -Space 950
13.10 Solution in Momentum Space 952
13.10.1 Gauge-Invariant Canonical Path Integral 952
13.10.2 Another Form of Action 955
13.10.3 Absence of Extra R-Term 956
Appendix 13A Group-Theoretic Aspects of Coulomb States 956
Notes and References 961
14 Solution of Further Path Integrals by the Duru-Kleinert Method 962
14.1 One-Dimensional Systems 962
14.2 Derivation of the Effective Potential 965
14.3 Comparison with Schrodinger Quantum Mechanics 970
14.4 Applications 970
14.4.1 Radial Harmonic Oscillator and Morse System 971
14.4.2 Radial Coulomb System and Morse System 973
14.4.3 Equivalence of Radial Coulomb System and Radial Oscillator 974
14.4.4 Angular Barrier near Sphere, and Rosen-Morse Potential 982
14.4.5 Angular Barrier near Four-Dimensional Sphere, and General Rosen-Morse Potential 985
14.4.6 Hulthen Potential and General Rosen-Morse Potential 988
14.4.7 Extended Hulthen Potential and General Rosen-Morse Potential 990
14.5 D-Dimensional Systems 991
14.6 Path Integral of the Dionium Atom 992
14.6.1 Formal Solution 993
14.6.2 Absence of Time Slicing Corrections 996
14.7 Time-Dependent Duru-Kleinert Transformation 1000
Appendix 14A Affine Connection of Dionium Atom 1004
Appendix 14B Algebraic Aspects of Dionium States 1004
Notes and References 1005
15 Path Integrals in Polymer Physics 1006
15.1 Polymers and Ideal Random Chains 1006
15.2 Moments of End-to-End Distribution 1008
15.3 Exact End-to-End Distribution in Three Dimensions 1011
15.4 Short-Distance Expansion for a Long Polymer 1013
15.5 Saddle Point Approximation to Three-Dimensional End-to-End Distribution 1015
15.6 Path Integral for Continuous Gaussian Distribution 1016
15.7 Stiff Polymers 1018
15.7.1 Path Integral 1020
15.7.2 Moments of End-to-End Distribution 1021
15.8 Schrodinger Equation and Recursive Solution for Moments 1025
15.8.1 Recursive Solution of Schrodinger Equation. 1026
15.8.2 Approximation of End- to-End Distribution 1029
Large-Stiffness Approximation 1029
15.8.3 From Moments to End-to-End Distribution for D = 3 1034
15.9 Excluded-Volume Effects 1036
15.10 Flory's Argument 1043
15.11 Polymer Field Theory 1044
15.12 Fermi Fields for Self-Avoiding Lines 1052
Notes and References 1052
16 Polymers and Particle Orbits in Multiply Connected Spaces 1055
16.1 Simple Model for Entangled Polymers 1055
16.2 Entangled Fluctuating Particle Orbit: Aharonov-Bohm Effect 1058
16.3 Aharonov-Bohm Effect and Fractional Statistics 1069
16.4 Self-Entanglement of Polymer 1074
16.5 The Gauss Invariant of Two Curves 1088
16.6 Bound States of Polymers \u2014 Ribbons 1091
16.7 Chern-Simons Theory of Entanglements 1097
16.8 Entangled Pair of Polymers 1100
16.8.1 Polymer Field Theory for Probabilities 1103
16.8.2 Calculation of Partition Function 1105
16.8.3 Calculation of Numerator in Second Moment 1107
16.8.4 First Diagram in Fig. 16.23 1108
16.8.5 Second and Third Diagrams in Fig. 16.23 1110
16.8.6 Fourth Diagram in Fig. 16.23 1111
16.8.7 Second Topological Moment 1112
16.9 Chern-Simons Theory of Statistical Interaction 1113
16.10 Second-Quantized Anyon Fields 1115
16.11 Fractional Quantum Hall Effect 1119
16.12 Anyonic Superconductivity 1122
16.13 Non-Abelian Chern-Simons Theory 1124
Appendix 16A Calculation of Feynman Diagrams for Polymer Entanglement 1127
Appendix 16B Kauffman and BLM/Ho Polynomials 1128
Appendix 16C Skein Relation between Wilson Loop Integrals 1129
Appendix 16D London Equations 1132
Appendix 16E Hall Effect in Electron Gas 1134
Notes and References 1134
17 Tunneling 1140
17.1 Double-Well Potential 1140
17.2 Classical Solutions \u2014 Kinks and Antikinks 1143
17.3 Quadratic Fluctuations 1147
17.3.1 Zero-Eigenvalue Mode 1153
17.3.2 Continuum Part of Fluctuation Factor 1156
17.4 General Formula for Eigenvalue Ratios 1159
17.5 Fluctuation Determinant from Classical Solution 1161
17.6 Wave Functions of Double-Well 1164
17.7 Gas of Kinks and Antikinks and Level Splitting Formula 1165
17.8 Fluctuation Correction to Level Splitting 1170
17.9 Tunneling and Decay 1175
17.10 Large-Order Behavior of Perturbation Expansions 1184
17.10.1 Growth Properties of Expansion Coefficients 1185
17.10.2 Semiclassical Large-Order Behavior 1188
17.10.3 Fluctuation Correction to the Imaginary Part and Large-Order Behavior 1193
17.10.4 Variational Approach to Tunneling. Perturbation Coefficients to All Orders 1196
17.10.5 Convergence of Variational Perturbation Expansion 1204
17.11 Decay of Supercurrent in Thin Closed Wire 1213
17.12 Decay of Metastable Thermodynamic Phases 1224
17.13 Decay of Metastable Vacuum State in Quantum Field Theory 1231
17.14 Crossover from Quantum Tunneling to Thermally Driven Decay 1233
Appendix 17A Feynman Integrals for Fluctuation Correction 1234
Notes and References 1237
18 Nonequilibrium Quantum Statistics 1240
18.1 Linear Response and Time-Dependent Green Functions for T 0 1240
18.2 Spectral Representations of T 0 Green Functions 1243
18.3 Other Important Green Functions 1246
18.4 Hermitian Adjoint Operators 1249
18.5 Harmonic Oscillator Green Functions for T 0 1250
18.5.1 Creation Annihilation Operators 1250
18.5.2 Real Field Operators 1253
18.6 Nonequilibrium Green Functions 1255
18.7 Perturbation Theory for Nonequilibrium Green Functions 1264
18.8 Path Integral Coupled to Thermal Reservoir 1267
18.9 Fokker-Planck Equation 1272
18.9.1 Canonical Path Integral for Probability Distribution 1273
18.9.2 Solving the Operator Ordering Problem 1275
18.9.3 Strong Damping 1281
18.10 Langevin Equations 1284
18.11 Stochastic Calculus 1287
18.12 Supersymmetry 1293
18.13 Stochastic Quantum Liouville Equation 1295
18.14 Relation to Quantum Langevin Equation 1298
18.15 Electromagnetic Dissipation and Decoherence 1298
18.15.1 Forward\u2013Backward Path Integral 1299
18.16 Master Equation for Time Evolution 1303
18.17 Line Width 1306
18.18 Lamb shift 1307
18.19 Langevin Equations 1311
18.20 Fokker-Planck Equation in Spaces with Curvature and Torsion 1312
18.21 Stochastic Interpretation of Quantum-Mechanical Amplitudes 1314
18.22 Stochastic Equation for Schrodinger Wave Function 1316
18.23 Real Stochastic and Deterministic Equation for Schrodinger Wave Function 1318
18.23.1 Stochastic Differential Equation 1318
18.23.2 Equation for Noise Average 1319
18.23.3 Harmonic Oscillator 1320
18.23.4 General Potential 1320
18.23.5 Deterministic Equation 1321
18.24 Heisenberg Picture for Probability Evolution 1322
Appendix 18A Inequalities for Diagonal Green Functions 1326
Appendix 18B General Generating Functional 1330
Appendix 18C Wick Decomposition of Operator Products 1335
Notes and References 1337
19 Relativistic Particle Orbits 1340
19.1 Special Features of Relativistic Path Integrals 1342
19.2 Proper Action for Fluctuating Relativistic Particle Orbits 1345
19.2.1 Gauge-Invariant Formulation 1345
19.2.2 Simplest Gauge Fixing 1347
19.2.3 Partition Function of Ensemble of Closed Particle Loops 1348
19.2.4 Fixed-Energy Amplitude 1350
19.3 Relativistic Coulomb System 1350
19.4 Relativistic Particle in Electromagnetic Field 1354
19.4.1 Action and Partition Function 1354
19.4.2 Perturbation Expansion 1355
19.4.3 Lowest-Order Vacuum Polarization 1357
19.5 Path Integral for Spin-1/2 Particle 1361
19.5.1 Dirac Theory 1361
19.5.2 Path Integral 1365
19.5.3 Amplitude with Electromagnetic Interaction 1367
19.5.4 Effective Action in Electromagnetic Field 1370
19.5.5 Perturbation Expansion 1371
19.5.6 Vacuum Polarization 1372
19.6 Supersymmetry 1374
19.6.1 Global Invariance 1374
19.6.2 Local Invariance 1375
Notes and References 1377
20 Path Integrals and Financial Markets 1379
20.1 Fluctuation Properties of Financial Assets 1379
20.1.1 Harmonic Approximation to Fluctuations 1381
20.1.2 Levy Distributions 1383
20.1.3 Truncated Levy Distributions 1384
20.1.4 Asymmetric Truncated Levy Distributions 1389
20.1.5 Meixner Distributions 1392
20.1.6 Other Non-Gaussian Distributions 1393
20.1.7 Levy-Khintchine Formula 1397
20.1.8 Debye-Waller Factor for Non-Gaussian Fluctuations 1398
20.1.9 Path Integral for Non-Gaussian Distribution 1398
20.1.10 Fokker-Planck-Type Equation 1400
20.2 Martingales 1405
20.2.1 Gaussian Martingales 1405
20.2.2 Non-Gaussian Martingales 1406
Natural Martingales 1406
Esscher Martingales 1406
Other Non-Gaussian Martingales 1408
20.3 Origin of Heavy Tails 1408
20.3.1 Pair of Stochastic Differential Equations 1408
20.3.2 Fokker-Planck Equation 1409
20.3.3 Solution of Fokker-Planck Equation 1412
20.3.4 Pure x-Distribution 1413
20.3.5 Long-Time Behavior 1415
20.3.6 Tail Behavior for all Times 1419
20.3.7 Path Integral Calculation 1421
20.3.8 Natural Martingales 1422
20.4 Option Pricing 1423
20.4.1 Black-Scholes Option Pricing Model 1424
20.4.2 Evolution Equations of Portfolios with Options 1426
20.4.3 Option Pricing for Gaussian Fluctuations 1428
20.4.4 Option Pricing for Non-Gaussian Fluctuations 1432
20.4.5 Option Pricing for Fluctuating Variance 1435
20.4.6 Perturbation Expansion and Smile 1437
Appendix 20A Large-x Behavior of Truncated Levy Distribution 1440
Appendix 20B Gaussian Weight 1444
Appendix 20C Comparison with Dow-Jones Data 1445
Notes and References 1446
Index 1454