1 Fundamentals 1
1.1 Classical Mechanics 1
1.2 Relativistic Mechanics in Curved Spacetime 10
1.3 Quantum Mechanics 11
1.3.1 Bragg Reflections and Interference 12
1.3.2 Matter Waves 13
1.3.3 Schr？dinger Equation 15
1.3.4 Particle Current Conservation 17
1.4 Dirac's Bra-Ket Formalism 18
1.4.1 Basis Transformations 18
1.4.2 Bracket Notation 20
1.4.3 Continuum Limit 22
1.4.4 Generalized Functions 23
1.4.5 Schr？dinger Equation in Dirac Notation 25
1.4.6 Momentum States 26
1.4.7 Incompleteness and Poisson's Summation Formula 28
1.5 Observables 31
1.5.1 Uncertainty Relation 32
1.5.2 Density Matrix and Wigner Function 33
1.5.3 Generalization to Many Particles 34
1.6 Time Evolution Operator 34
1.7 Properties of the Time Evolution Operator 37
1.8 Heisenberg Picture of Quantum Mechanics 39
1.9 Interaction Picture and Perturbation Expansion 42
1.10 Time Evolution Amplitude 43
1.11 Fixed-Energy Amplitude 45
1.12 Free-Particle Amplitudes 47
1.13 Quantum Mechanics of General Lagrangian Systems 51
1.14 Particle on the Surface of a Sphere 57
1.15 Spinning Top 59
1.16 Scattering 67
1.16.1 Scattering Matrix 67
1.16.2 Cross Section 68
1.16.3 Born Approximation 70
1.16.4 Partial Wave Expansion and Eikonal Approximation 70
1.16.5 Scattering Amplitude from Time Evolution Amplitude 72
1.16.6 Lippmann-Schwinger Equation 72
1.17 Classical and Quantum Statistics 76
1.17.1 Canonical Ensemble 77
1.17.2 Grand-Canonical Ensemble 77
1.18 Density of States and Tracelog 82
Appendix 1A Simple Time Evolution Operator 84
Appendix 1B Convergence of the Fresnel Integral 84
Appendix 1C The Asymmetric Top 85
Notes and References 87
2 Path Integrals—Elementary Properties and Simple Solutions 89
2.1 Path Integral Representation of Time Evolution Amplitudes 89
2.1.1 Sliced Time Evolution Amplitude 89
2.1.2 Zero-Hamiltonian Path Integral 91
2.1.3 Schr？dinger Equation for Time Evolution Amplitude 92
2.1.4 Convergence of of the Time-Sliced Evolution Amplitude 93
2.1.5 Time Evolution Amplitude in Momentum Space 94
2.1.6 Quantum-Mechanical Partition Function 96
2.1.7 Feynman's Configuration Space Path Integral 97
2.2 Exact Solution for the Free Particle 101
2.2.1 Direct Solution 101
2.2.2 Fluctuations around the Classical Path 102
2.2.3 Fluctuation Factor 104
2.2.4 Finite Slicing Properties of Free-Particle Amplitude 111
2.3 Exact Solution for Harmonic Oscillator 112
2.3.1 Fluctuations around the Classical Path 112
2.3.2 Fluctuation Factor 114
2.3.3 The in-Prescription and Maslov-Morse Index 115
2.3.4 Continuum Limit 116
2.3.5 Useful Fluctuation Formulas 117
2.3.6 Oscillator Amplitude on Finite Time Lattice 119
2.4 Gelfand-Yaglom Formula 120
2.4.1 Recursive Calculation of Fluctuation Determinant 121
2.4.2 Examples 121
2.4.3 Calculation on Unsliced Time Axis 123
2.4.4 D'Alembert's Construction 124
2.4.5 Another Simple Formula 125
2.4.6 Generalization to D Dimensions 127
2.5 Harmonic Oscillator with Time-Dependent Frequency 127
2.5.1 Coordinate Space 128
2.5.2 Momentum Space 130
2.6 Free-Particle and Oscillator Wave Functions 132
2.7 General Time-Dependent Harmonic Action 134
2.8 Path Integrals and Quantum Statistics 135
2.9 Density Matrix 138
2.10 Quantum Statistics of the Harmonic Oscillator 143
2.11 Time-Dependent Harmonic Potential 148
2.12 Functional Measure in Fourier Space 151
2.13 Classical Limit 154
2.14 Calculation Techniques on Sliced Time Axis via the Poisson Formula 155
2.15 Field-Theoretic Definition of Harmonic Path Integrals by Analytic Regularization 158
2.15.1 Zero-Temperature Evaluation of the Frequency Sum 159
2.15.2 Finite-Temperature Evaluation of the Frequency Sum 162
2.15.3 Quantum-Mechanical Harmonic Oscillator 164
2.15.4 Tracelog of the First-Order Differential Operator 165
2.15.5 Gradient Expansion of the One-Dimensional Tracelog 167
2.15.6 Duality Transformation and Low-Temperature Expansion 168
2.16 Finite-N Behavior of Thermodynamic Quantities 175
2.17 Time Evolution Amplitude of Freely Falling Particle 177
2.18 Charged Particle in Magnetic Field 179
2.18.1 Action 179
2.18.2 Gauge Properties 182
2.18.3 Time-Sliced Path Integration 182
2.18.4 Classical Action 184
2.18.5 Translational Invariance 185
2.19 Charged Particle in Magnetic Field plus Harmonic Potential 186
2.20 Gauge Invariance and Alternative Path Integral Representation 188
2.21 Velocity Path Integral 189
2.22 Path Integral Representation of the Scattering Matrix 190
2.22.1 General Development 190
2.22.2 Improved Formulation 193
2.22.3 Eikonal Approximation to the Scattering Amplitude 194
2.23 Heisenberg Operator Approach to Time Evolution Amplitude 194
2.23.1 Free Particle 195
2.23.2 Harmonic Oscillator 197
2.23.3 Charged Particle in Magnetic Field 197
Appendix 2A Baker-Campbell-Hausdorff Formula and Magnus Expan-sion 201
Appendix 2B Direct Calculation of the Time-Sliced Oscillator Amplitude 204
Appendix 2C Derivation of Mehler Formula 205
Notes and References 206
3 External Sources，Correlations，and Perturbation Theory 209
3.1 External Sources 209
3.2 Green Function of Harmonic Oscillator 213
3.2.1 Wronski Construction 213
3.2.2 Spectral Representation 217
3.3 Green Functions of First-Order Differential Equation 219
3.3.1 Time-Independent Frequency 219
3.3.2 Time-Dependent Frequency 226
3.4 Summing Spectral Representation of Green Function 229
3.5 Wronski Construction for Periodic and Antiperiodic Green Func-tions 231
3.6 Time Evolution Amplitude in Presence of Source Term 232
3.7 Time Evolution Amplitude at Fixed Path Average 236
3.8 External Source in Quantum-Statistical Path Integral 237
3.8.1 Continuation of Real-Time Result 238
3.8.2 Calculation at Imaginary Time 242
3.9 Lattice Green Function 249
3.10 Correlation Functions，Generating Functional，and Wick Expansion 249
3.10.1 Real-Time Correlation Functions 252
3.11 Correlation Functions of Charged Particle in Magnetic Field 254
3.12 Correlation Functions in Canonical Path Integral 255
3.12.1 Harmonic Correlation Functions 256
3.12.2 Relations between Various Amplitudes 258
3.12.3 Harmonic Generating Functionals 259
3.13 Particle in Heat Bath 262
3.14 Heat Bath of Photons 266
3.15 Harmonic Oscillator in Ohmic Heat Bath 268
3.16 Harmonic Oscillator in Photon Heat Bath 271
3.17 Perturbation Expansion of Anharmonic Systems 272
3.18 Rayleigh-Schr？dinger and Brillouin-Wigner Perturbation Expansion 276
3.19 Level-Shifts and Perturbed Wave Functions from Schr？dinger Equation 280
3.20 Calculation of Perturbation Series via Feynman Diagrams 282
3.21 Perturbative Definition of Interacting Path Integrals 287
3.22 Generating Functional of Connected Correlation Functions 288
3.22.1 Connectedness Structure of Correlation Functions 289
3.22.2 Correlation Functions versus Connected Correlation Func-tions 292
3.22.3 Functional Generation of Vacuum Diagrams 294
3.22.4 Correlation Functions from Vacuum Diagrams 298
3.22.5 Generating Functional for Vertex Functions.Effective Action 300
3.22.6 Ginzburg-Landau Approximation to Generating Functional 305
3.22.7 Composite Fields 306
3.23 Path Integral Calculation of Effective Action by Loop Expansion 307
3.23.1 General Formalism 307
3.23.2 Mean-Field Approximation 308
3.23.3 Corrections from Quadratic Fluctuations 312
3.23.4 Effective Action to Second Order in ？ 315
3.23.5 Finite-Temperature Two-Loop Effective Action 319
3.23.6 Background Field Method for Effective Action 321
3.24 Nambu-Goldstone Theorem 324
3.25 Effective Classical Potential 326
3.25.1 Effective Classical Boltzmann Factor 327
3.25.2 Effective Classical Hamiltonian 330
3.25.3 High- and Low-Temperature Behavior 331
3.25.4 Alternative Candidate for Effective Classical Potential 332
3.25.5 Harmonic Correlation Function without Zero Mode 333
3.25.6 Perturbation Expansion 334
3.25.7 Effective Potential and Magnetization Curves 336
3.25.8 First-Order Perturbative Result 338
3.26 Perturbative Approach to Scattering Amplitude 340
3.26.1 Generating Functional 340
3.26.2 Application to Scattering Amplitude 341
3.26.3 First Correction to Eikonal Approximation 341
3.26.4 Rayleigh-Schr？dinger Expansion of Scattering Amplitude 342
3.27 Functional Determinants from Green Functions 344
Appendix 3A Matrix Elements for General Potential 350
Appendix 3B Energy Shifts for gx4/4-Interaction 351
Appendix 3C Recursion Relations for Perturbation Coefficients 353
3C.1 One-Dimensional Interaction x4 353
3C.2 General One-Dimensional Interaction 356
3C.3 Cumulative Treatment of Interactions x4 and x3 356
3C.4 Ground-State Energy with External Current 358
3C.5 Recursion Relation for Effective Potential 360
3C.6 Interaction r4 in D-Dimensional Radial Oscillator 363
3C.7 Interaction r2q in D Dimensions 364
3C.8 Polynomial Interaction in D Dimensions 364
Appendix 3D Feynman Integrals for T≠0 364
Notes and References 367
4 Semiclassical Time Evolution Amplitude 369
4.1 Wentzel-Kramers-Brillouin（WKB）Approximation 369
4.2 Saddle Point Approximation 376
4.2.1 Ordinary Integrals 376
4.2.2 Path Integrals 379
4.3 Van Vleck-Pauli-Morette Determinant 385
4.4 Fundamental Composition Law for Semiclassical Time Evolution Amplitude 389
4.5 Semiclassical Fixed-Energy Amplitude 391
4.6 Semiclassical Amplitude in Momentum Space 393
4.7 Semiclassical Quantum-Mechanical Partition Function 395
4.8 Multi-Dimensional Systems 400
4.9 Quantum Corrections to Classical Density of States 405
4.9.1 One-Dimensional Case 406
4.9.2 Arbitrary Dimensions 408
4.9.3 Bilocal Density of States 409
4.9.4 Gradient Expansion of Tracelog of Hamiltonian Operator 411
4.9.5 Local Density of States on Circle 415
4.9.6 Quantum Corrections to Bohr-Sommerfeld Approximation 416
4.10 Thomas-Fermi Model of Neutral Atoms 419
4.10.1 Semiclassical Limit 419
4.10.2 Self-Consistent Field Equation 421
4.10.3 Energy Functional of Thomas-Fermi Atom 423
4.10.4 Calculation of Energies 424
4.10.5 Virial Theorem 427
4.10.6 Exchange Energy 428
4.10.7 Quantum Correction Near Origin 429
4.10.8 Systematic Quantum Corrections to Thomas-Fermi Energies 432
4.11 Classical Action of Coulomb System 436
4.12 Semiclassical Scattering 444
4.12.1 General Formulation 444
4.12.2 Semiclassical Cross Section of Mott Scattering 448
Appendix 4A Semiclassical Quantization for Pure Power Potentials 449
Appendix 4B Derivation of Semiclassical Time Evolution Amplitude 451
Notes and References 455
5 Variational Perturbation Theory 458
5.1 Variational Approach to Effective Classical Partition Function 458
5.2 Local Harmonic Trial Partition Function 459
5.3 Optimal Upper Bound 464
5.4 Accuracy of Variational Approximation 465
5.5 Weakly Bound Ground State Energy in Finite-Range Potential Well 468
5.6 Possible Direct Generalizations 469
5.7 Effective Classical Potential for Anharmonic Oscillator 470
5.8 Particle Densities 475
5.9 Extension to D Dimensions 479
5.10 Application to Coulomb and Yukawa Potentials 481
5.11 Hydrogen Atom in Strong Magnetic Field 484
5.11.1 Weak-Field Behavior 488
5.11.2 Effective Classical Hamiltonian 488
5.12 Variational Approach to Excitation Energies 492
5.13 Systematic Improvement of Feynman-Kleinert Approximation 496
5.14 Applications of Variational Perturbation Expansion 498
5.14.1 Anharmonic Oscillator at T=0 499
5.14.2 Anharmonic Oscillator for T>0 501
5.15 Convergence of Variational Perturbation Expansion 505
5.16 Variational Perturbation Theory for Strong-Coupling Expansion 512
5.17 General Strong-Coupling Expansions 515
5.18 Variational Interpolation between Weak and Strong-Coupling Ex-pansions 518
5.19 Systematic Improvement of Excited Energies 520
5.20 Variational Treatment of Double-Well Potential 521
5.21 Higher-Order Effective Classical Potential for Nonpolynomial In-teractions 523
5.21.1 Evaluation of Path Integrals 524
5.21.2 Higher-Order Smearing Formula in D Dimensions 525
5.21.3 Isotropic Second-Order Approximation to Coulomb Problem 527
5.21.4 Anisotropic Second-Order Approximation to Coulomb Prob-lem 528
5.21.5 Zero-Temperature Limit 529
5.22 Polarons 533
5.22.1 Partition Function 535
5.22.2 Harmonic Trial System 537
5.22.3 Effective Mass 542
5.22.4 Second-Order Correction 543
5.22.5 Polaron in Magnetic Field，Bipolarons，etc 544
5.22.6 Variational Interpolation for Polaron Energy and Mass 545
5.23 Density Matrices 548
5.23.1 Harmonic Oscillator 548
5.23.2 Variational Perturbation Theory for Density Matrices 550
5.23.3 Smearing Formula for Density Matrices 552
5.23.4 First-Order Variational Approximation 554
5.23.5 Smearing Formula in Higher Spatial Dimensions 558
Appendix 5A Feynman Integrals for T≠0 without Zero Frequency 560
Appendix 5B Proof of Scaling Relation for the Extrema of WN 562
Appendix 5C Second-Order Shift of Polaron Energy 564
Notes and References 565
6 Path Integrals with Topological Constraints 571
6.1 Point Particle on Circle 571
6.2 Infinite Wall 575
6.3 Point Particle in Box 579
6.4 Strong-Coupling Theory for Particle in Box 582
6.4.1 Partition Function 583
6.4.2 Perturbation Expansion 583
6.4.3 Variational Strong-Coupling Approximations 585
6.4.4 Special Properties of Expansion 587
6.4.5 Exponentially Fast Convergence 588
Notes and References 589
7 Many Particle Orbits—Statistics and Second Quantization 591
7.1 Ensembles of Bose and Fermi Particle Orbits 592
7.2 Bose-Einstein Condensation 599
7.2.1 Free Bose Gas 599
7.2.2 Bose Gas in Finite Box 607
7.2.3 Effect of Interactions 609
7.2.4 Bose-Einstein Condensation in Harmonic Trap 615
7.2.5 Thermodynamic Functions 615
7.2.6 Critical Temperature 617
7.2.7 More General Anisotropic Trap 620
7.2.8 Rotating Bose-Einstein Gas 621
7.2.9 Finite-Size Corrections 622
7.2.10 Entropy and Specific Heat 623
7.2.11 Interactions in Harmonic Trap 626
7.3 Gas of Free Fermions 630
7.4 Statistics Interaction 635
7.5 Fractional Statistics 640
7.6 Second-Quantized Bose Fields 641
7.7 Fluctuating Bose Fields 644
7.8 Coherent States 650
7.9 Second-Quantized Fermi Fields 654
7.10 Fluctuating Fermi Fields 654
7.10.1 Grassmann Variables 654
7.10.2 Fermionic Functional Determinant 657
7.10.3 Coherent States for Fermions 661
7.11 Hilbert Space of Quantized Grassmann Variable 663
7.11.1 Single Real Grassmann Variable 663
7.11.2 Quantizing Harmonic Oscillator with Grassmann Variables 666
7.11.3 Spin System with Grassmann Variables 667
7.12 External Sources in a，a-Path Integral 672
7.13 Generalization to Pair Terms 674
7.14 Spatial Degrees of Freedom 676
7.14.1 Grand-Canonical Ensemble of Particle Orbits from Free Fluctuating Field 676
7.14.2 First versus Second Quantization 678
7.14.3 Interacting Fields 678
7.14.4 Effective Classical Field Theory 679
7.15 Bosonization 681
7.15.1 Collective Field 682
7.15.2 Bosonized versus Original Theory 684
Appendix 7A Treatment of Singularities in Zeta-Function 686
7A.1 Finite Box 687
7A.2 Harmonic Trap 689
Appendix 7B Experimental versus Theoretical Would-be Critical Tem-perature 691
Notes and References 692
8 Path Integrals in Polar and Spherical Coordinates 697
8.1 Angular Decomposition in Two Dimensions 697
8.2 Trouble with Feynman's Path Integral Formula in Radial Coordi-nates 700
8.3 Cautionary Remarks 704
8.4 Time Slicing Corrections 707
8.5 Angular Decomposition in Three and More Dimensions 711
8.5.1 Three Dimensions 712
8.5.2 D Dimensions 714
8.6 Radial Path Integral for Harmonic Oscillator and Free Particle 720
8.7 Particle near the Surface of a Sphere in D Dimensions 721
8.8 Angular Barriers near the Surface of a Sphere 724
8.8.1 Angular Barriers in Three Dimensions 725
8.8.2 Angular Barriers in Four Dimensions 730
8.9 Motion on a Sphere in D Dimensions 734
8.10 Path Integrals on Group Spaces 739
8.11 Path Integral of Spinning Top 741
8.12 Path Integral of Spinning Particle 743
8.13 Berry Phase 748
8.14 Spin Precession 748
Notes and References 750
9 Wave Functions 752
9.1 Free Particle in D Dimensions 752
9.2 Harmonic Oscillator in D Dimensions 755
9.3 Free Particle from ω→0-Limit of Oscillator 761
9.4 Charged Particle in Uniform Magnetic Field 763
9.5 Dirac δ-Function Potential 770
Notes and References 772
10 Spaces with Curvature and Torsion 773
10.1 Einstein's Equivalence Principle 774
10.2 Classical Motion of Mass Point in General Metric-Affine Space 775
10.2.1 Equations of Motion 775
10.2.2 Nonholonomic Mapping to Spaces with Torsion 778
10.2.3 New Equivalence Principle 784
10.2.4 Classical Action Principle for Spaces with Curvature and Torsion 784
10.3 Path Integral in Metric-Affine Space 789
10.3.1 Nonholonomic Transformation of Action 789
10.3.2 Measure of Path Integration 794
10.4 Completing the Solution of Path Integral on Surface of Sphere 800
10.5 External Potentials and Vector Potentials 802
10.6 Perturbative Calculation of Path Integrals in Curved Space 804
10.6.1 Free and Interacting Parts of Action 804
10.6.2 Zero Temperature 807
10.7 Model Study of Coordinate Invariance 809
10.7.1 Diagrammatic Expansion 811
10.7.2 Diagrammatic Expansion in d Time Dimensions 813
10.8 Calculating Loop Diagrams 814
10.8.1 Reformulation in Configuration Space 821
10.8.2 Integrals over Products of Two Distributions 822
10.8.3 Integrals over Products of Four Distributions 823
10.9 Distributions as Limits of Bessel Function 825
10.9.1 Correlation Function and Derivatives 825
10.9.2 Integrals over Products of Two Distributions 827
10.9.3 Integrals over Products of Four Distributions 828
10.10 Simple Rules for Calculating Singular Integrals 830
10.11 Perturbative Calculation on Finite Time Intervals 835
10.11.1 Diagrammatic Elements 836
10.11.2 Cumulant Expansion of D-Dimensional Free-Particle Am-plitude in Curvilinear Coordinates 837
10.11.3 Propagator in 1-ε Time Dimensions 839
10.11.4 Coordinate Independence for Dirichlet Boundary Conditions 840
10.11.5 Time Evolution Amplitude in Curved Space 846
10.11.6 Covariant Results for Arbitrary Coordinates 852
10.12 Effective Classical Potential in Curved Space 857
10.12.1 Covariant Fluctuation Expansion 858
10.12.2 Arbitrariness of qμ0 861
10.12.3 Zero-Mode Properties 862
10.12.4 Covariant Perturbation Expansion 865
10.12.5 Covariant Result from Noncovariant Expansion 866
10.12.6 Particle on Unit Sphere 869
10.13 Covariant Effective Action for Quantum Particle with Coordinate-Dependent Mass 871
10.13.1 Formulating the Problem 872
10.13.2 Gradient Expansion 875
Appendix 10A Nonholonomic Gauge Transformations in Electromag-netism 875
10A.1 Gradient Representation of Magnetic Field of Current Loops 876
10A.2 Generating Magnetic Fields by Multivalued Gauge Trans-formations 880
10A.3 Magnetic Monopoles 881
10A.4 Minimal Magnetic Coupling of Particles from Multivalued Gauge Transformations 883
10A.5 Gauge Field Representation of Current Loops and Monopoles 884
Appendix 10B Comparison of Multivalued Basis Tetrads with Vierbein Fields 886
Appendix 10C Cancellation of Powers of δ（0） 888
Notes and References 890
11 Schr？dinger Equation in General Metric-Affine Spaces 894
11.1 Integral Equation for Time Evolution Amplitude 894
11.1.1 From Recursion Relation to Schr？dinger Equation 895
11.1.2 Alternative Evaluation 898
11.2 Equivalent Path Integral Representations 901
11.3 Potentials and Vector Potentials 905
11.4 Unitarity Problem 906
11.5 Alternative Attempts 909
11.6 DeWitt-Seeley Expansion of Time Evolution Amplitude 910
Appendix 11A Cancellations in Effective Potential 914
Appendix 11B DeWitt's Amplitude 916
Notes and References 917
12 New Path Integral Formula for Singular Potentials 918
12.1 Path Collapse in Feynman's formula for the Coulomb System 918
12.2 Stable Path Integral with Singular Potentials 921
12.3 Time-Dependent Regularization 926
12.4 Relation to Schr？dinger Theory.Wave Functions 928
Notes and References 930
13 Path Integral of Coulomb System 931
13.1 Pseudotime Evolution Amplitude 931
13.2 Solution for the Two-Dimensional Coulomb System 933
13.3 Absence of Time Slicing Corrections for D=2 938
13.4 Solution for the Three-Dimensional Coulomb System 943
13.5 Absence of Time Slicing Corrections for D=3 949
13.6 Geometric Argument for Absence of Time Slicing Corrections 951
13.7 Comparison with Schr？dinger Theory 952
13.8 Angular Decomposition of Amplitude，and Radial Wave Functions 957
13.9 Remarks on Geometry of Four-Dimensional uμ-Space 961
13.10 Runge-Lenz-Pauli Group of Degeneracy 963
13.11 Solution in Momentum Space 964
13.11.1 Another Form of Action 968
Appendix 13A Dynamical Group of Coulomb States 969
Notes and References 972
14 Solution of Further Path Integrals by Duru-Kleinert Method 974
14.1 One-Dimensional Systems 974
14.2 Derivation of the Effective Potential 978
14.3 Comparison with Schr？dinger Quantum Mechanics 982
14.4 Applications 983
14.4.1 Radial Harmonic Oscillator and Morse System 983
14.4.2 Radial Coulomb System and Morse System 985
14.4.3 Equivalence of Radial Coulomb System and Radial Oscilla-tor 987
14.4.4 Angular Barrier near Sphere，and Rosen-Morse Potential 994
14.4.5 Angular Barrier near Four-Dimensional Sphere，and Gen-eral Rosen-Morse Potential 997
14.4.6 Hulthén Potential and General Rosen-Morse Potential 1000
14.4.7 Extended Hulthén Potential and General Rosen-Morse Po-tential 1002
14.5 D-Dimensional Systems 1003
14.6 Path Integral of the Dionium Atom 1004
14.6.1 Formal Solution 1005
14.6.2 Absence of Time Slicing Corrections 1009
14.7 Time-Dependent Duru-Kleinert Transformation 1012
Appendix 14A Affine Connection of Dionium Atom 1015
Appendix 14B Algebraic Aspects of Dionium States 1016
Notes and References 1016
15 Path Integrals in Polymer Physics 1019
15.1 Polymers and Ideal Random Chains 1019
15.2 Moments of End-to-End Distribution 1021
15.3 Exact End-to-End Distribution in Three Dimensions 1024
15.4 Short-Distance Expansion for Long Polymer 1026
15.5 Saddle Point Approximation to Three-Dimensional End-to-End Distribution 1028
15.6 Path Integral for Continuous Gaussian Distribution 1029
15.7 Stiff Polymers 1032
15.7.1 Sliced Path Integral 1034
15.7.2 Relation to Classical Heisenberg Model 1035
15.7.3 End-to-End Distribution 1037
15.7.4 Moments of End-to-End Distribution 1037
15.8 Continuum Formulation 1038
15.8.1 Path Integral 1038
15.8.2 Correlation Functions and Moments 1039
15.9 Schr？dinger Equation and Recursive Solution for Moments 1043
15.9.1 Setting up the Schr？dinger Equation 1043
15.9.2 Recursive Solution of Schr？dinger Equation 1044
15.9.3 From Moments to End-to-End Distribution for D=3 1047
15.9.4 Large-Stiffness Approximation to End-to-End Distribution 1049
15.9.5 Higher Loop Corrections 1054
15.10 Excluded-Volume Effects 1062
15.11 Flory's Argument 1069
15.12 Polymer Field Theory 1070
15.13 Fermi Fields for Self-Avoiding Lines 1077
Appendix 15A Basic Integrals 1078
Appendix 15B Loop Integrals 1079
Appendix 15C Integrals Involving Modified Green Function 1080
Notes and References 1081
16 Polymers and Particle Orbits in Multiply Connected Spaces 1084
16.1 Simple Model for Entangled Polymers 1084
16.2 Entangled Fluctuating Particle Orbit:Aharonov-Bohm Effect 1088
16.3 Aharonov-Bohm Effect and Fractional Statistics 1096
16.4 Self-Entanglement of Polymer 1101
16.5 The Gauss Invariant of Two Curves 1115
16.6 Bound States of Polymers and Ribbons 1117
16.7 Chern-Simons Theory of Entanglements 1124
16.8 Entangled Pair of Polymers 1127
16.8.1 Polymer Field Theory for Probabilities 1129
16.8.2 Calculation of Partition Function 1130
16.8.3 Calculation of Numerator in Second Moment 1132
16.8.4 First Diagram in Fig.16.23 1134
16.8.5 Second and Third Diagrams in Fig.16.23 1135
16.8.6 Fourth Diagram in Fig.16.23 1136
16.8.7 Second Topological Moment 1137
16.9 Chern-Simons Theory of Statistical Interaction 1137
16.10 Second-Quantized Anyon Fields 1140
16.11 Fractional Quantum Hall Effect 1143
16.12 Anyonic Superconductivity 1147
16.13 Non-Abelian Chern-Simons Theory 1149
Appendix 16A Calculation of Feynman Diagrams in Polymer Entangle-ment 1151
Appendix 16B Kauffman and BLM/Ho polynomials 1153
Appendix 16C Skein Relation between Wilson Loop Integrals 1153
Appendix 16D London Equations 1156
Appendix 16E Hall Effect in Electron Gas 1158
Notes and References 1158
17 Tunneling 1164
17.1 Double-Well Potential 1164
17.2 Classical Solutions—Kinks and Antikinks 1167
17.3 Quadratic Fluctuations 1171
17.3.1 Zero-Eigenvalue Mode 1177
17.3.2 Continuum Part of Fluctuation Factor 1181
17.4 General Formula for Eigenvalue Ratios 1183
17.5 Fluctuation Determinant from Classical Solution 1185
17.6 Wave Functions of Double-Well 1189
17.7 Gas of Kinks and Antikinks and Level Splitting Formula 1190
17.8 Fluctuation Correction to Level Splitting 1194
17.9 Tunneling and Decay 1199
17.10 Large-Order Behavior of Perturbation Expansions 1207
17.10.1 Growth Properties of Expansion Coefficients 1208
17.10.2 Semiclassical Large-Order Behavior 1211
17.10.3 Fluctuation Correction to the Imaginary Part and Large-Order Behavior 1216
17.10.4 Variational Approach to Tunneling.Perturbation Coeffi-cients to All Orders 1219
17.10.5 Convergence of Variational Perturbation Expansion 1227
17.11 Decay of Supercurrent in Thin Closed Wire 1235
17.12 Decay of Metastable Thermodynamic Phases 1247
17.13 Decay of Metastable Vacuum State in Quantum Field Theory 1254
17.14 Crossover from Quantum Tunneling to Thermally Driven Decay 1255
Appendix 17A Feynman Integrals for Fluctuation Correction 1257
Notes and References 1259
18 Nonequilibrium Quantum Statistics 1262
18.1 Linear Response and Time-Dependent Green Functions for T≠0 1262
18.2 Spectral Representations of Green Functions for T≠0 1265
18.3 Other Important Green Functions 1268
18.4 Hermitian Adjoint Operators 1271
18.5 Harmonic Oscillator Green Functions for T≠0 1272
18.5.1 Creation Annihilation Operators 1272
18.5.2 Real Field Operators 1275
18.6 Nonequilibrium Green Functions 1277
18.7 Perturbation Theory for Nonequilibrium Green Functions 1286
18.8 Path Integral Coupled to Thermal Reservoir 1289
18.9 Fokker-Planck Equation 1295
18.9.1 Canonical Path Integral for Probability Distribution 1296
18.9.2 Solving the Operator Ordering Problem 1298
18.9.3 Strong Damping 1303
18.10 Langevin Equations 1307
18.11 Path Integral Solution of Klein-Kramers Equation 1311
18.12 Stochastic Quantization 1312
18.13 Stochastic Calculus 1316
18.13.1 Kubo's stochastic Liouville equation 1316
18.13.2 From Kubo's to Fokker-Planck Equations 1317
18.13.3 Itō's Lemma 1320
18.14 Solving the Langevin Equation 1323
18.15 Heisenberg Picture for Probability Evolution 1327
18.16 Supersymmetry 1330
18.17 Stochastic Quantum Liouville Equation 1332
18.18 Master Equation for Time Evolution 1334
18.19 Relation to Quantum Langevin Equation 1336
18.20 Electromagnetic Dissipation and Decoherence 1337
18.20.1 Forward-Backward Path Integral 1337
18.20.2 Master Equation for Time Evolution in Photon Bath 1340
18.20.3 Line Width 1341
18.20.4 Lamb shift 1342
18.20.5 Langevin Equations 1346
18.21 Fokker-Planck Equation in Spaces with Curvature and Torsion 1347
18.22 Stochastic Interpretation of Quantum-Mechanical Amplitudes 1348
18.23 Stochastic Equation for Schr？dinger Wave Function 1350
18.24 Real Stochastic and Deterministic Equation for Schr？dinger Wave Function 1351
18.24.1 Stochastic Differential Equation 1352
18.24.2 Equation for Noise Average 1352
18.24.3 Harmonic Oscillator 1353
18.24.4 General Potential 1353
18.24.5 Deterministic Equation 1354
Appendix 18A Inequalities for Diagonal Green Functions 1355
Appendix 18B General Generating Functional 1359
Appendix 18C Wick Decomposition of Operator Products 1363
Notes and References 1364
19 Relativistic Particle Orbits 1368
19.1 Special Features of Relativistic Path Integrals 1370
19.1.1 Simplest Gauge Fixing 1373
19.1.2 Partition Function of Ensemble of Closed Particle Loops 1375
19.1.3 Fixed-Energy Amplitude 1376
19.2 Tunneling in Relativistic Physics 1377
19.2.1 Decay Rate of Vacuum in Electric Field 1377
19.2.2 Birth of Universe 1386
19.2.3 Friedmann Model 1392
19.2.4 Tunneling of Expanding Universe 1396
19.3 Relativistic Coulomb System 1397
19.4 Relativistic Particle in Electromagnetic Field 1400
19.4.1 Action and Partition Function 1401
19.4.2 Perturbation Expansion 1401
19.4.3 Lowest-Order Vacuum Polarization 1404
19.5 Path Integral for Spin-1/2 Particle 1408
19.5.1 Dirac Theory 1408
19.5.2 Path Integral 1412
19.5.3 Amplitude with Electromagnetic Interaction 1414
19.5.4 Effective Action in Electromagnetic Field 1417
19.5.5 Perturbation Expansion 1418
19.5.6 Vacuum Polarization 1419
19.6 Supersymmetry 1421
19.6.1 Global Invariance 1421
19.6.2 Local Invariance 1422
Appendix 19A Proof of Same Quantum Physics of Modified Action 1424
Notes and References 1426
20 Path Integrals and Financial Markets 1428
20.1 Fluctuation Properties of Financial Assets 1428
20.1.1 Harmonic Approximation to Fluctuations 1430
20.1.2 Lévy Distributions 1432
20.1.3 Truncated Lévy Distributions 1434
20.1.4 Asymmetric Truncated Lévy Distributions 1439
20.1.5 Gamma Distribution 1442
20.1.6 Boltzmann Distribution 1443
20.1.7 Student or Tsallis Distribution 1446
20.1.8 Tsallis Distribution in Momentum Space 1448
20.1.9 Relativistic Particle Boltzmann Distribution 1449
20.1.10 Meixner Distributions 1450
20.1.11 Generalized Hyperbolic Distributions 1451
20.1.12 Debye-Waller Factor for Non-Gaussian Fluctuations 1454
20.1.13 Path Integral for Non-Gaussian Distribution 1454
20.1.14 Time Evolution of Distribution 1457
20.1.15 Central Limit Theorem 1457
20.1.16 Additivity Property of Noises and Hamiltonians 1459
20.1.17 Lévy-Khintchine Formula 1460
20.1.18 Semigroup Property of Asset Distributions 1461
20.1.19 Time Evolution of Moments of Distribution 1463
20.1.20 Boltzmann Distribution 1464
20.1.21 Fourier-Transformed Tsallis Distribution 1467
20.1.22 Superposition of Gaussian Distributions 1468
20.1.23 Fokker-Planck-Type Equation 1470
20.1.24 Kramers-Moyal Equation 1471
20.2 Itō-like Formula for Non-Gaussian Distributions 1473
20.2.1 Continuous Time 1473
20.2.2 Discrete Times 1476
20.3 Martingales 1477
20.3.1 Gaussian Martingales 1477
20.3.2 Non-Gaussian Martingale Distributions 1479
20.4 Origin of Semi-Heavy Tails 1481
20.4.1 Pair of Stochastic Differential Equations 1482
20.4.2 Fokker-Planck Equation 1482
20.4.3 Solution of Fokker-Planck Equation 1485
20.4.4 Pure x-Distribution 1487
20.4.5 Long-Time Behavior 1488
20.4.6 Tail Behavior for all Times 1492
20.4.7 Path Integral Calculation 1494
20.4.8 Natural Martingale Distribution 1495
20.5 Time Series 1496
20.6 Spectral Decomposition of Power Behaviors 1497
20.7 Option Pricing 1498
20.7.1 Black-Scholes Option Pricing Model 1499
20.7.2 Evolution Equations of Portfolios with Options 1501
20.7.3 Option Pricing for Gaussian Fluctuations 1505
20.7.4 Option Pricing for Boltzmann Distribution 1509
20.7.5 Option Pricing for General Non-Gaussian Fluctuations 1509
20.7.6 Option Pricing for Fluctuating Variance 1512
20.7.7 Perturbation Expansion and Smile 1514
Appendix 20A Large-x Behavior of Truncated Lévy Distribution 1517
Appendix 20B Gaussian Weight 1520
Appendix 20C Comparison with Dow-Jones Data 1521
Notes and References 1522
Index 1529