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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 iη-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 Cradient 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 Expansion 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-Crder 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 Functions 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 Functions 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 Temperature 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 Fonr 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-εTime 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 Effiective 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 Transformations 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 Oscillator 987

14.4.4 Angular Barrier near Sphere,and Rosen-Morse Potential 994

14.4.5 Angular Barrier near Four-Dimensional Sphere,and General 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 Potential 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 Entanglement 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 Coefficients 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