量子力学、统计学、聚合物物理学和金融市场中的路径积分 第2分册 第5版 英文PDF|Epub|txt|kindle电子书版本网盘下载

量子力学、统计学、聚合物物理学和金融市场中的路径积分 第2分册 第5版 英文PDF格式电子书版下载

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1 Fundamentals1

1.1 Classical Mechanics1

1.2 Relativistic Mechanics in Curved Spacetime10

1.3 Quantum Mechanics11

1.3.1 Bragg Reflections and Interference12

1.3.2 Matter Waves13

1.3.3 Schr？dinger Equation15

1.3.4 Particle Current Conservation17

1.4 Dirac's Bra-Ket Formalism18

1.4.1 Basis Transformations18

1.4.2 Bracket Notation20

1.4.3 Continuum Limit22

1.4.4 Generalized Functions23

1.4.5 Schr？dinger Equation in Dirac Notation25

1.4.6 Momentum States26

1.4.7 Incompleteness and Poisson's Summation Formula28

1.5 Observables31

1.5.1 Uncertainty Relation32

1.5.2 Density Matrix and Wigner Function33

1.5.3 Generalization to Many Particles34

1.6 Time Evolution Operator34

1.7 Properties of the Time Evolution Operator37

1.8 Heisenberg Picture of Quantum Mechanics39

1.9 Interaction Picture and Perturbation Expansion42

1.10 Time Evolution Amplitude43

1.11 Fixed-Energy Amplitude45

1.12 Free-Particle Amplitudes47

1.13 Quantum Mechanics of General Lagrangian Systems51

1.14 Particle on the Surface of a Sphere57

1.15 Spinning Top59

1.16 Scattering67

1.16.1 Scattering Matrix67

1.16.2 Cross Section68

1.16.3 Born Approximation70

1.16.4 Partial Wave Expansion and Eikonal Approximation70

1.16.5 Scattering Amplitude from Time Evolution Amplitude72

1.16.6 Lippmann-Schwinger Equation72

1.17 Classical and Quantum Statistics76

1.17.1 Canonical Ensemble77

1.17.2 Grand-Canonical Ensemble77

1.18 Density of States and Tracelog82

Appendix 1A Simple Time Evolution Operator84

Appendix 1B Convergence of the Fresnel Integral84

Appendix 1C The Asymmetric Top85

Notes and References87

2 Path Integrals—Elementary Properties and Simple Solutions89

2.1 Path Integral Representation of Time Evolution Amplitudes89

2.1.1 Sliced Time Evolution Amplitude89

2.1.2 Zero-Hamiltonian Path Integral91

2.1.3 Schr？dinger Equation for Time Evolution Amplitude92

2.1.4 Convergence of of the Time-Sliced Evolution Amplitude93

2.1.5 Time Evolution Amplitude in Momentum Space94

2.1.6 Quantum-Mechanical Partition Function96

2.1.7 Feynman's Configuration Space Path Integral97

2.2 Exact Solution for the Free Particle101

2.2.1 Direct Solution101

2.2.2 Fluctuations around the Classical Path102

2.2.3 Fluctuation Factor104

2.2.4 Finite Slicing Properties of Free-Particle Amplitude111

2.3 Exact Solution for Harmonic Oscillator112

2.3.1 Fluctuations around the Classical Path112

2.3.2 Fluctuation Factor114

2.3.3 The iη-Prescription and Maslov-Morse Index115

2.3.4 Continuum Limit116

2.3.5 Useful Fluctuation Formulas117

2.3.6 Oscillator Amplitude on Finite Time Lattice119

2.4 Gelfand-Yaglom Formula120

2.4.1 Recursive Calculation of Fluctuation Determinant121

2.4.2 Examples121

2.4.3 Calculation on Unsliced Time Axis123

2.4.4 D'Alembert's Construction124

2.4.5 Another Simple Formula125

2.4.6 Generalization to D Dimensions127

2.5 Harmonic Oscillator with Time-Dependent Frequency127

2.5.1 Coordinate Space128

2.5.2 Momentum Space130

2.6 Free-Particle and Oscillator Wave Functions132

2.7 General Time-Dependent Harmonic Action134

2.8 Path Integrals and Quantum Statistics135

2.9 Density Matrix138

2.10 Quantum Statistics of the Harmonic Oscillator143

2.11 Time-Dependent Harmonic Potential148

2.12 Functional Measure in Fourier Space151

2.13 Classical Limit154

2.14 Calculation Techniques on Sliced Time Axis via the Poisson Formula155

2.15 Field-Theoretic Definition of Harmonic Path Integrals by Analytic Regularization158

2.15.1 Zero-Temperature Evaluation of the Frequency Sum159

2.15.2 Finite-Temperature Evaluation of the Frequency Sum162

2.15.3 Quantum-Mechanical Harmonic Oscillator164

2.15.4 Tracelog of the First-Order Differential Operator165

2.15.5 Cradient Expansion of the One-Dimensional Tracelog167

2.15.6 Duality Transformation and Low-Temperature Expansion168

2.16 Finite-N Behavior of Thermodynamic Quantities175

2.17 Time Evolution Amplitude of Freely Falling Particle177

2.18 Charged Particle in Magnetic Field179

2.18.1 Action179

2.18.2 Gauge Properties182

2.18.3 Time-Sliced Path Integration182

2.18.4 Classical Action184

2.18.5 Translational Invariance185

2.19 Charged Particle in Magnetic Field plus Harmonic Potential186

2.20 Gauge Invariance and Alternative Path Integral Representation188

2.21 Velocity Path Integral189

2.22 Path Integral Representation of the Scattering Matrix190

2.22.1 General Development190

2.22.2 Improved Formulation193

2.22.3 Eikonal Approximation to the Scattering Amplitude194

2.23 Heisenberg Operator Approach to Time Evolution Amplitude194

2.23.1 Free Particle195

2.23.2 Harmonic Oscillator197

2.23.3 Charged Particle in Magnetic Field197

Appendix 2A Baker-Campbell-Hausdorff Formula and Magnus Expansion201

Appendix 2B Direct Calculation of the Time-Sliced Oscillator Amplitude204

Appendix 2C Derivation of Mehler Formula205

Notes and References206

3 External Sources,Correlations,and Perturbation Theory209

3.1 External Sources209

3.2 Green Function of Harmonic Oscillator213

3.2.1 Wronski Construction213

3.2.2 Spectral Representation217

3.3 Green Functions of First-Crder Differential Equation219

3.3.1 Time-Independent Frequency219

3.3.2 Time-Dependent Frequency226

3.4 Summing Spectral Representation of Green Function229

3.5 Wronski Construction for Periodic and Antiperiodic Green Functions231

3.6 Time Evolution Amplitude in Presence of Source Term232

3.7 Time Evolution Amplitude at Fixed Path Average236

3.8 External Source in Quantum-Statistical Path Integral237

3.8.1 Continuation of Real-Time Result238

3.8.2 Calculation at Imaginary Time242

3.9 Lattice Green Function249

3.10 Correlation Functions,Generating Functional,and Wick Expansion249

3.10.1 Real-Time Correlation Functions252

3.11 Correlation Functions of Charged Particle in Magnetic Field254

3.12 Correlation Functions in Canonical Path Integral255

3.12.1 Harmonic Correlation Functions256

3.12.2 Relations between Various Amplitudes258

3.12.3 Harmonic Generating Functionals259

3.13 Particle in Heat Bath262

3.14 Heat Bath of Photons266

3.15 Harmonic Oscillator in Ohmic Heat Bath268

3.16 Harmonic Oscillator in Photon Heat Bath271

3.17 Perturbation Expansion of Anharmonic Systems272

3.18 Rayleigh-Schr？dinger and Brillouin-Wigner Perturbation Expansion276

3.19 Level-Shifts and Perturbed Wave Functions from Schr？dinger Equation280

3.20 Calculation of Perturbation Series via Feynman Diagrams282

3.21 Perturbative Definition of Interacting Path Integrals287

3.22 Generating Functional of Connected Correlation Functions288

3.22.1 Connectedness Structure of Correlation Functions289

3.22.2 Correlation Functions versus Connected Correlation Functions292

3.22.3 Functional Generation of Vacuum Diagrams294

3.22.4 Correlation Functions from Vacuum Diagrams298

3.22.5 Generating Functional for Vertex Functions.Effective Action300

3.22.6 Ginzburg-Landau Approximation to Generating Functional305

3.22.7 Composite Fields306

3.23 Path Integral Calculation of Effective Action by Loop Expansion307

3.23.1 General Formalism307

3.23.2 Mean-Field Approximation308

3.23.3 Corrections from Quadratic Fluctuations312

3.23.4 Effective Action to Second Order in ？315

3.23.5 Finite-Temperature Two-Loop Effective Action319

3.23.6 Background Field Method for Effective Action321

3.24 Nambu-Goldstone Theorem324

3.25 Effective Classical Potential326

3.25.1 Effective Classical Boltzmann Factor327

3.25.2 Effective Classical Hamiltonian330

3.25.3 High-and Low-Temperature Behavior331

3.25.4 Alternative Candidate for Effective Classical Potential332

3.25.5 Harmonic Correlation Function without Zero Mode333

3.25.6 Perturbation Expansion334

3.25.7 Effective Potential and Magnetization Curves336

3.25.8 First-Order Perturbative Result338

3.26 Perturbative Approach to Scattering Amplitude340

3.26.1 Generating Functional340

3.26.2 Application to Scattering Amplitude341

3.26.3 First Correction to Eikonal Approximation341

3.26.4 Rayleigh-Schr？dinger Expansion of Scattering Amplitude342

3.27 Functional Determinants from Green Functions344

Appendix 3A Matrix Elements for General Potential350

Appendix 3B Energy Shifts for gx4/4-Interaction351

Appendix 3C Recursion Relations for Perturbation Coefficients353

3C.1 One-Dimensional Interaction x4353

3C.2 General One-Dimensional Interaction356

3C.3 Cumulative Treatment of Interactions x4 and x3356

3C.4 Ground-State Energy with External Current358

3C.5 Recursion Relation for Effective Potential360

3C.6 Interaction r4 in D-Dimensional Radial Oscillator363

3C.7 Interaction r2q in D Dimensions364

3C.8 Polynomial Interaction in D Dimensions364

Appendix 3D Feynman Integrals for T≠0364

Notes and References367

4 Semiclassical Time Evolution Amplitude369

4.1 Wentzel-Kramers-Brillouin (WKB) Approximation369

4.2 Saddle Point Approximation376

4.2.1 Ordinary Integrals376

4.2.2 Path Integrals379

4.3 Van Vleck-Pauli-Morette Determinant385

4.4 Fundamental Composition Law for Semiclassical Time Evolution Amplitude389

4.5 Semiclassical Fixed-Energy Amplitude391

4.6 Semiclassical Amplitude in Momentum Space393

4.7 Semiclassical Quantum-Mechanical Partition Function395

4.8 Multi-Dimensional Systems400

4.9 Quantum Corrections to Classical Density of States405

4.9.1 One-Dimensional Case406

4.9.2 Arbitrary Dimensions408

4.9.3 Bilocal Density of States409

4.9.4 Gradient Expansion of Tracelog of Hamiltonian Operator411

4.9.5 Local Density of States on Circle415

4.9.6 Quantum Corrections to Bohr-Sommerfeld Approximation416

4.10 Thomas-Fermi Model of Neutral Atoms419

4.10.1 Semiclassical Limit419

4.10.2 Self-Consistent Field Equation421

4.10.3 Energy Functional of Thomas-Fermi Atom423

4.10.4 Calculation of Energies424

4.10.5 Virial Theorem427

4.10.6 Exchange Energy428

4.10.7 Quantum Correction Near Origin429

4.10.8 Systematic Quantum Corrections to Thomas-Fermi Energies432

4.11 Classical Action of Coulomb System436

4.12 Semiclassical Scattering444

4.12.1 General Formulation444

4.12.2 Semiclassical Cross Section of Mott Scattering448

Appendix 4A Semiclassical Quantization for Pure Power Potentials449

Appendix 4B Derivation of Semiclassical Time Evolution Amplitude451

Notes and References455

5 Variational Perturbation Theory458

5.1 Variational Approach to Effective Classical Partition Function458

5.2 Local Harmonic Trial Partition Function459

5.3 Optimal Upper Bound464

5.4 Accuracy of Variational Approximation465

5.5 Weakly Bound Ground State Energy in Finite-Range Potential Well468

5.6 Possible Direct Generalizations469

5.7 Effective Classical Potential for Anharmonic Oscillator470

5.8 Particle Densities475

5.9 Extension to D Dimensions479

5.10 Application to Coulomb and Yukawa Potentials481

5.11 Hydrogen Atom in Strong Magnetic Field484

5.11.1 Weak-Field Behavior488

5.11.2 Effective Classical Hamiltonian488

5.12 Variational Approach to Excitation Energies492

5.13 Systematic Improvement of Feynman-Kleinert Approximation496

5.14 Applications of Variational Perturbation Expansion498

5.14.1 Anharmonic Oscillator at T=0499

5.14.2 Anharmonic Oscillator for T＞0501

5.15 Convergence of Variational Perturbation Expansion505

5.16 Variational Perturbation Theory for Strong-Coupling Expansion512

5.17 General Strong-Coupling Expansions515

5.18 Variational Interpolation between Weak and Strong-Coupling Ex-pansions518

5.19 Systematic Improvement of Excited Energies520

5.20 Variational Treatment of Double-Well Potential521

5.21 Higher-Order Effective Classical Potential for Nonpolynomial In-teractions523

5.21.1 Evaluation of Path Integrals524

5.21.2 Higher-Order Smearing Formula in D Dimensions525

5.21.3 Isotropic Second-Order Approximation to Coulomb Problem527

5.21.4 Anisotropic Second-Order Approximation to Coulomb Prob-lem528

5.21.5 Zero-Temperature Limit529

5.22 Polarons533

5.22.1 Partition Function535

5.22.2 Harmonic Trial System537

5.22.3 Effective Mass542

5.22.4 Second-Order Correction543

5.22.5 Polaron in Magnetic Field,Bipolarons,etc544

5.22.6 Variational Interpolation for Polaron Energy and Mass545

5.23 Density Matrices548

5.23.1 Harmonic Oscillator548

5.23.2 Variational Perturbation Theory for Density Matrices550

5.23.3 Smearing Formula for Density Matrices552

5.23.4 First-Order Variational Approximation554

5.23.5 Smearing Formula in Higher Spatial Dimensions558

Appendix 5A Feynman Integrals for T≠0 without Zero Frequency560

Appendix 5B Proof of Scaling Relation for the Extrema of WN562

Appendix 5C Second-Order Shift of Polaron Energy564

Notes and References565

6 Path Integrals with Topological Constraints571

6.1 Point Particle on Circle571

6.2 Infinite Wall575

6.3 Point Particle in Box579

6.4 Strong-Coupling Theory for Particle in Box582

6.4.1 Partition Function583

6.4.2 Perturbation Expansion583

6.4.3 Variational Strong-Coupling Approximations585

6.4.4 Special Properties of Expansion587

6.4.5 Exponentially Fast Convergence588

Notes and References589

7 Many Particle Orbits—Statistics and Second Quantization591

7.1 Ensembles of Bose and Fermi Particle Orbits592

7.2 Bose-Einstein Condensation599

7.2.1 Free Bose Gas599

7.2.2 Bose Gas in Finite Box607

7.2.3 Effect of Interactions609

7.2.4 Bose-Einstein Condensation in Harmonic Trap615

7.2.5 Thermodynamic Functions615

7.2.6 Critical Temperature617

7.2.7 More General Anisotropic Trap620

7.2.8 Rotating Bose-Einstein Gas621

7.2.9 Finite-Size Corrections622

7.2.10 Entropy and Specific Heat623

7.2.11 Interactions in Harmonic Trap626

7.3 Gas of Free Fermions630

7.4 Statistics Interaction635

7.5 Fractional Statistics640

7.6 Second-Quantized Bose Fields641

7.7 Fluctuating Bose Fields644

7.8 Coherent States650

7.9 Second-Quantized Fermi Fields654

7.10 Fluctuating Fermi Fields654

7.10.1 Grassmann Variables654

7.10.2 Fermionic Functional Determinant657

7.10.3 Coherent States for Fermions661

7.11 Hilbert Space of Quantized Grassmann Variable663

7.11.1 Single Real Grassmann Variable663

7.11.2 Quantizing Harmonic Oscillator with Grassmann Variables666

7.11.3 Spin System with Grassmann Variables667

7.12 External Sources in a＊,a-Path Integral672

7.13 Generalization to Pair Terms674

7.14 Spatial Degrees of Freedom676

7.14.1 Grand-Canonical Ensemble of Particle Orbits from Free Fluctuating Field676

7.14.2 First versus Second Quantization678

7.14.3 Interacting Fields678

7.14.4 Effective Classical Field Theory679

7.15 Bosonization681

7.15.1 Collective Field682

7.15.2 Bosonized versus Original Theory684

Appendix 7A Treatment of Singularities in Zeta-Function686

7A.1 Finite Box687

7A.2 Harmonic Trap689

Appendix 7B Experimental versus Theoretical Would-be Critical Temperature691

Notes and References692

8 Path Integrals in Polar and Spherical Coordinates697

8.1 Angular Decomposition in Two Dimensions697

8.2 Trouble with Feynman's Path Integral Formula in Radial Coordi-nates700

8.3 Cautionary Remarks704

8.4 Time Slicing Corrections707

8.5 Angular Decomposition in Three and More Dimensions711

8.5.1 Three Dimensions712

8.5.2 D Dimensions714

8.6 Radial Path Integral for Harmonic Oscillator and Free Particle720

8.7 Particle near the Surface of a Sphere in D Dimensions721

8.8 Angular Barriers near the Surface of a Sphere724

8.8.1 Angular Barriers in Three Dimensions725

8.8.2 Angular Barriers in Four Dimensions730

8.9 Motion on a Sphere in D Dimensions734

8.10 Path Integrals on Group Spaces739

8.11 Path Integral of Spinning Top741

8.12 Path Integral of Spinning Particle743

8.13 Berry Phase748

8.14 Spin Precession748

Notes and References750

9 Wave Functions752

9.1 Free Particle in D Dimensions752

9.2 Harmonic Oscillator in D Dimensions755

9.3 Free Particle from ω→0-Limit of Oscillator761

9.4 Charged Particle in Uniform Magnetic Field763

9.5 Dirac δ-Function Potential770

Notes and References772

10 Spaces with Curvature and Torsion773

10.1 Einstein's Equivalence Principle774

10.2 Classical Motion of Mass Point in General Metric-Affine Space775

10.2.1 Equations of Motion775

10.2.2 Nonholonomic Mapping to Spaces with Torsion778

10.2.3 New Equivalence Principle784

10.2.4 Classical Action Principle for Spaces with Curvature and Torsion784

10.3 Path Integral in Metric-Affine Space789

10.3.1 Nonholonomic Transformation of Action789

10.3.2 Measure of Path Integration794

10.4 Completing the Solution of Path Integral on Surface of Sphere800

10.5 External Potentials and Vector Potentials802

10.6 Perturbative Calculation of Path Integrals in Curved Space804

10.6.1 Free and Interacting Parts of Action804

10.6.2 Zero Temperature807

10.7 Model Study of Coordinate Invariance809

10.7.1 Diagrammatic Expansion811

10.7.2 Diagrammatic Expansion in d Time Dimensions813

10.8 Calculating Loop Diagrams814

10.8.1 Reformulation in Configuration Space821

10.8.2 Integrals over Products of Two Distributions822

10.8.3 Integrals over Products of Fonr Distributions823

10.9 Distributions as Limits of Bessel Function825

10.9.1 Correlation Function and Derivatives825

10.9.2 Integrals over Products of Two Distributions827

10.9.3 Integrals over Products of Four Distributions828

10.10 Simple Rules for Calculating Singular Integrals830

10.11 Perturbative Calculation on Finite Time Intervals835

10.11.1 Diagrammatic Elements836

10.11.2 Cumulant Expansion of D-Dimensional Free-Particle Am-plitude in Curvilinear Coordinates837

10.11.3 Propagator in 1-εTime Dimensions839

10.11.4 Coordinate Independence for Dirichlet Boundary Conditions840

10.11.5 Time Evolution Amplitude in Curved Space846

10.11.6 Covariant Results for Arbitrary Coordinates852

10.12 Effective Classical Potential in Curved Space857

10.12.1 Covariant Fluctuation Expansion858

10.12.2 Arbitrariness of qμ0861

10.12.3 Zero-Mode Properties862

10.12.4 Covariant Perturbation Expansion865

10.12.5 Covariant Result from Noncovariant Expansion866

10.12.6 Particle on Unit Sphere869

10.13 Covariant Effiective Action for Quantum Particle with Coordinate-Dependent Mass871

10.13.1 Formulating the Problem872

10.13.2 Gradient Expansion875

Appendix 10A Nonholonomic Gauge Transformations in Electromag-netism875

10A.1 Gradient Representation of Magnetic Field of Current Loops876

10A.2 Generating Magnetic Fields by Multivalued Gauge Transformations880

10A.3 Magnetic Monopoles881

10A.4 Minimal Magnetic Coupling of Particles from Multivalued Gauge Transformations883

10A.5 Gauge Field Representation of Current Loops and Monopoles884

Appendix 10B Comparison of Multivalued Basis Tetrads with Vierbein Fields886

Appendix 10C Cancellation of Powers of δ(0)888

Notes and References890

11 Schr？dinger Equation in General Metric-Affine Spaces894

11.1 Integral Equation for Time Evolution Amplitude894

11.1.1 From Recursion Relation to Schr？dinger Equation895

11.1.2 Alternative Evaluation898

11.2 Equivalent Path Integral Representations901

11.3 Potentials and Vector Potentials905

11.4 Unitarity Problem906

11.5 Alternative Attempts909

11.6 DeWitt-Seeley Expansion of Time Evolution Amplitude910

Appendix 11A Cancellations in Effective Potential914

Appendix 11B DeWitt's Amplitude916

Notes and References917

12 New Path Integral Formula for Singular Potentials918

12.1 Path Collapse in Feynman's formula for the Coulomb System918

12.2 Stable Path Integral with Singular Potentials921

12.3 Time-Dependent Regularization926

12.4 Relation to Schr？dinger Theory.Wave Functions928

Notes and References930

13 Path Integral of Coulomb System931

13.1 Pseudotime Evolution Amplitude931

13.2 Solution for the Two-Dimensional Coulomb System933

13.3 Absence of Time Slicing Corrections for D=2938

13.4 Solution for the Three-Dimensional Coulomb System943

13.5 Absence of Time Slicing Corrections for D=3949

13.6 Geometric Argument for Absence of Time Slicing Corrections951

13.7 Comparison with Schr？dinger Theory952

13.8 Angular Decomposition of Amplitude,and Radial Wave Functions957

13.9 Remarks on Geometry of Four-Dimensional uμ-Space961

13.10 Runge-Lenz-Pauli Group of Degeneracy963

13.11 Solution in Momentum Space964

13.11.1 Another Form of Action968

Appendix 13A Dynamical Group of Coulomb States969

Notes and References972

14 Solution of Further Path Integrals by Duru-Kleinert Method974

14.1 One-Dimensional Systems974

14.2 Derivation of the Effective Potential978

14.3 Comparison with Schr？dinger Quantum Mechanics982

14.4 Applications983

14.4.1 Radial Harmonic Oscillator and Morse System983

14.4.2 Radial Coulomb System and Morse System985

14.4.3 Equivalence of Radial Coulomb System and Radial Oscillator987

14.4.4 Angular Barrier near Sphere,and Rosen-Morse Potential994

14.4.5 Angular Barrier near Four-Dimensional Sphere,and General Rosen-Morse Potential997

14.4.6 Hulthén Potential and General Rosen-Morse Potential1000

14.4.7 Extended Hulthén Potential and General Rosen-Morse Potential1002

14.5 D-Dimensional Systems1003

14.6 Path Integral of the Dionium Atom1004

14.6.1 Formal Solution1005

14.6.2 Absence of Time Slicing Corrections1009

14.7 Time-Dependent Duru-Kleinert Transformation1012

Appendix 14A Affine Connection of Dionium Atom1015

Appendix 14B Algebraic Aspects of Dionium States1016

Notes and References1016

15 Path Integrals in Polymer Physics1019

15.1 Polymers and Ideal Random Chains1019

15.2 Moments of End-to-End Distribution1021

15.3 Exact End-to-End Distribution in Three Dimensions1024

15.4 Short-Distance Expansion for Long Polymer1026

15.5 Saddle Point Approximation to Three-Dimensional End-to-End Distribution1028

15.6 Path Integral for Continuous Gaussian Distribution1029

15.7 Stiff Polymers1032

15.7.1 Sliced Path Integral1034

15.7.2 Relation to Classical Heisenberg Model1035

15.7.3 End-to-End Distribution1037

15.7.4 Moments of End-to-End Distribution1037

15.8 Continuum Formulation1038

15.8.1 Path Integral1038

15.8.2 Correlation Functions and Moments1039

15.9 Schr？dinger Equation and Recursive Solution for Moments1043

15.9.1 Setting up the Schr？dinger Equation1043

15.9.2 Recursive Solution of Schr？dinger Equation1044

15.9.3 From Moments to End-to-End Distribution for D=31047

15.9.4 Large-Stiffness Approximation to End-to-End Distribution1049

15.9.5 Higher Loop Corrections1054

15.10 Excluded-Volume Effects1062

15.11 Flory's Argument1069

15.12 Polymer Field Theory1070

15.13 Fermi Fields for Self-Avoiding Lines1077

Appendix 15A Basic Integrals1078

Appendix 15B Loop Integrals1079

Appendix 15C Integrals Involving Modified Green Function1080

Notes and References1081

16 Polymers and Particle Orbits in Multiply Connected Spaces1084

16.1 Simple Model for Entangled Polymers1084

16.2 Entangled Fluctuating Particle Orbit:Aharonov-Bohm Effect1088

16.3 Aharonov-Bohm Effect and Fractional Statistics1096

16.4 Self-Entanglement of Polymer1101

16.5 The Gauss Invariant of Two Curves1115

16.6 Bound States of Polymers and Ribbons1117

16.7 Chern-Simons Theory of Entanglements1124

16.8 Entangled Pair of Polymers1127

16.8.1 Polymer Field Theory for Probabilities1129

16.8.2 Calculation of Partition Function1130

16.8.3 Calculation of Numerator in Second Moment1132

16.8.4 First Diagram in Fig.16.231134

16.8.5 Second and Third Diagrams in Fig.16.231135

16.8.6 Fourth Diagram in Fig.16.231136

16.8.7 Second Topological Moment1137

16.9 Chern-Simons Theory of Statistical Interaction1137

16.10 Second-Quantized Anyon Fields1140

16.11 Fractional Quantum Hall Effect1143

16.12 Anyonic Superconductivity1147

16.13 Non-Abelian Chern-Simons Theory1149

Appendix 16A Calculation of Feynman Diagrams in Polymer Entanglement1151

Appendix 16B Kauffman and BLM/Ho polynomials1153

Appendix 16C Skein Relation between Wilson Loop Integrals1153

Appendix 16D London Equations1156

Appendix 16E Hall Effect in Electron Gas1158

Notes and References1158

17 Tunneling1164

17.1 Double-Well Potential1164

17.2 Classical Solutions—Kinks and Antikinks1167

17.3 Quadratic Fluctuations1171

17.3.1 Zero-Eigenvalue Mode1177

17.3.2 Continuum Part of Fluctuation Factor1181

17.4 General Formula for Eigenvalue Ratios1183

17.5 Fluctuation Determinant from Classical Solution1185

17.6 Wave Functions of Double-Well1189

17.7 Gas of Kinks and Antikinks and Level Splitting Formula1190

17.8 Fluctuation Correction to Level Splitting1194

17.9 Tunneling and Decay1199

17.10 Large-Order Behavior of Perturbation Expansions1207

17.10.1 Growth Properties of Expansion Coefficients1208

17.10.2 Semiclassical Large-Order Behavior1211

17.10.3 Fluctuation Correction to the Imaginary Part and Large-Order Behavior1216

17.10.4 Variational Approach to Tunneling.Perturbation Coefficients to All Orders1219

17.10.5 Convergence of Variational Perturbation Expansion1227

17.11 Decay of Supercurrent in Thin Closed Wire1235

17.12 Decay of Metastable Thermodynamic Phases1247

17.13 Decay of Metastable Vacuum State in Quantum Field Theory1254

17.14 Crossover from Quantum Tunneling to Thermally Driven Decay1255

Appendix 17A Feynman Integrals for Fluctuation Correction1257

Notes and References1259

18 Nonequilibrium Quantum Statistics1262

18.1 Linear Response and Time-Dependent Green Functions for T≠01262

18.2 Spectral Representations of Green Functions for T≠01265

18.3 Other Important Green Functions1268

18.4 Hermitian Adjoint Operators1271

18.5 Harmonic Oscillator Green Functions for T≠01272

18.5.1 Creation Annihilation Operators1272

18.5.2 Real Field Operators1275

18.6 Nonequilibrium Green Functions1277

18.7 Perturbation Theory for Nonequilibrium Green Functions1286

18.8 Path Integral Coupled to Thermal Reservoir1289

18.9 Fokker-Planck Equation1295

18.9.1 Canonical Path Integral for Probability Distribution1296

18.9.2 Solving the Operator Ordering Problem1298

18.9.3 Strong Damping1303

18.10 Langevin Equations1307

18.11 Path Integral Solution of Klein-Kramers Equation1311

18.12 Stochastic Quantization1312

18.13 Stochastic Calculus1316

18.13.1 Kubo's stochastic Liouville equation1316

18.13.2 From Kubo's to Fokker-Planck Equations1317

18.13.3 Itō's Lemma1320

18.14 Solving the Langevin Equation1323

18.15 Heisenberg Picture for Probability Evolution1327

18.16 Supersymmetry1330

18.17 Stochastic Quantum Liouville Equation1332

18.18 Master Equation for Time Evolution1334

18.19 Relation to Quantum Langevin Equation1336

18.20 Electromagnetic Dissipation and Decoherence1337

18.20.1 Forward-Backward Path Integral1337

18.20.2 Master Equation for Time Evolution in Photon Bath1340

18.20.3 Line Width1341

18.20.4 Lamb shift1342

18.20.5 Langevin Equations1346

18.21 Fokker-Planck Equation in Spaces with Curvature and Torsion1347

18.22 Stochastic Interpretation of Quantum-Mechanical Amplitudes1348

18.23 Stochastic Equation for Schr？dinger Wave Function1350

18.24 Real Stochastic and Deterministic Equation for Schr？dinger Wave Function1351

18.24.1 Stochastic Differential Equation1352

18.24.2 Equation for Noise Average1352

18.24.3 Harmonic Oscillator1353

18.24.4 General Potential1353

18.24.5 Deterministic Equation1354

Appendix 18A Inequalities for Diagonal Green Functions1355

Appendix 18B General Generating Functional1359

Appendix 18C Wick Decomposition of Operator Products1363

Notes and References1364

19 Relativistic Particle Orbits1368

19.1 Special Features of Relativistic Path Integrals1370

19.1.1 Simplest Gauge Fixing1373

19.1.2 Partition Function of Ensemble of Closed Particle Loops1375

19.1.3 Fixed-Energy Amplitude1376

19.2 Tunneling in Relativistic Physics1377

19.2.1 Decay Rate of Vacuum in Electric Field1377

19.2.2 Birth of Universe1386

19.2.3 Friedmann Model1392

19.2.4 Tunneling of Expanding Universe1396

19.3 Relativistic Coulomb System1397

19.4 Relativistic Particle in Electromagnetic Field1400

19.4.1 Action and Partition Function1401

19.4.2 Perturbation Expansion1401

19.4.3 Lowest-Order Vacuum Polarization1404

19.5 Path Integral for Spin-1/2 Particle1408

19.5.1 Dirac Theory1408

19.5.2 Path Integral1412

19.5.3 Amplitude with Electromagnetic Interaction1414

19.5.4 Effective Action in Electromagnetic Field1417

19.5.5 Perturbation Expansion1418

19.5.6 Vacuum Polarization1419

19.6 Supersymmetry1421

19.6.1 Global Invariance1421

19.6.2 Local Invariance1422

Appendix 19A Proof of Same Quantum Physics of Modified Action1424

Notes and References1426

20 Path Integrals and Financial Markets1428

20.1 Fluctuation Properties of Financial Assets1428

20.1.1 Harmonic Approximation to Fluctuations1430

20.1.2 Lévy Distributions1432

20.1.3 Truncated Lévy Distributions1434

20.1.4 Asymmetric Truncated Lévy Distributions1439

20.1.5 Gamma Distribution1442

20.1.6 Boltzmann Distribution1443

20.1.7 Student or Tsallis Distribution1446

20.1.8 Tsallis Distribution in Momentum Space1448

20.1.9 Relativistic Particle Boltzmann Distribution1449

20.1.10 Meixner Distributions1450

20.1.11 Generalized Hyperbolic Distributions1451

20.1.12 Debye-Waller Factor for Non-Gaussian Fluctuations1454

20.1.13 Path Integral for Non-Gaussian Distribution1454

20.1.14 Time Evolution of Distribution1457

20.1.15 Central Limit Theorem1457

20.1.16 Additivity Property of Noises and Hamiltonians1459

20.1.17 Lévy-Khintchine Formula1460

20.1.18 Semigroup Property of Asset Distributions1461

20.1.19 Time Evolution of Moments of Distribution1463

20.1.20 Boltzmann Distribution1464

20.1.21 Fourier-Transformed Tsallis Distribution1467

20.1.22 Superposition of Gaussian Distributions1468

20.1.23 Fokker-Planck-Type Equation1470

20.1.24 Kramers-Moyal Equation1471

20.2 Itō-like Formula for Non-Gaussian Distributions1473

20.2.1 Continuous Time1473

20.2.2 Discrete Times1476

20.3 Martingales1477

20.3.1 Gaussian Martingales1477

20.3.2 Non-Gaussian Martingale Distributions1479

20.4 Origin of Semi-Heavy Tails1481

20.4.1 Pair of Stochastic Differential Equations1482

20.4.2 Fokker-Planck Equation1482

20.4.3 Solution of Fokker-Planck Equation1485

20.4.4 Pure x-Distribution1487

20.4.5 Long-Time Behavior1488

20.4.6 Tail Behavior for all Times1492

20.4.7 Path Integral Calculation1494

20.4.8 Natural Martingale Distribution1495

20.5 Time Series1496

20.6 Spectral Decomposition of Power Behaviors1497

20.7 Option Pricing1498

20.7.1 Black-Scholes Option Pricing Model1499

20.7.2 Evolution Equations of Portfolios with Options1501

20.7.3 Option Pricing for Gaussian Fluctuations1505

20.7.4 Option Pricing for Boltzmann Distribution1509

20.7.5 Option Pricing for General Non-Gaussian Fluctuations1509

20.7.6 Option Pricing for Fluctuating Variance1512

20.7.7 Perturbation Expansion and Smile1514

Appendix 20A Large-x Behavior of Truncated Lévy Distribution1517

Appendix 20B Gaussian Weight1520

Appendix 20C Comparison with Dow-Jones Data1521

Notes and References1522

Index1529