Preface ....................................................... vii
Preface to Fourth Edition .................................... viii
Preface to Third Edition ....................................... ix
Preface to Second Edition ...................................... xi
Preface to First Edition ..................................... xiii
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 Schrödinger 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 Schrödinger 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 IB Convergence of the Fresnel Integral ....... 84
Appendix 1С 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 Schrödinger 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 Gradient Expansion of the One-Dimensional
Tracelog ....................................... 167
2.15.6 Duality Transformation and Low-Temperature
Expansion ...................................... 168
2.16 Finite-JV Behavior of Thermodynamic Quantities ........ 175
2.17 Time Evolution Amplitude of Freely Palling 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 Вакег-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-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 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 Functional ................. 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-Schrödinger and Brillouin-Wigner
Perturbation Expansion ................................ 276
3.19 Level-Shifts and Perturbed Wave Functions from
Schrödinger 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 h .......... 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-Schrödinger Expansion of Scattering
Amplitude ...................................... 342
3.27 Functional Determinants from Green Functions ..... 344
Appendix 3А 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 Expansions ................................... 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 Interactions ............................ 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 Problem ................................ 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 .......................................... 680
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 Coordinates .................................... 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 Amplitude 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 q0μ .......................... 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
Electromagnetism ...................................... 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 Schrödinger Equation in General Metric-Affine Spaces ....... 894
11.1 Integral Equation for Time Evolution Amplitude ........ 894
11.1.1 From Recursion Relation to Schrödinger
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 Schrödinger 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 Geometrie Argument for Absence of Time Slicing
Corrections ........................................... 951
13.7 Comparison with Schrödinger Theory .................... 952
13.8 Angular Decomposition of Amplitude, and Radial Wave
Functions ............................................. 957
13.9 Remarks on Geometry of Four-Dimensional uM-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 Schrödinger 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 Hulthйn 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 Schrödinger Equation and Recursive Solution for
Moments .............................................. 1043
15.9.1 Setting up the Schrödinger Equation ........... 1043
15.9.2 Recursive Solution of Schrödinger 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 Ito'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 Schrödinger Wave Function ... 1350
18.24 Real Stochastic and Deterministic Equation for
Schrödinger 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 Levy Distributions ............................ 1432
20.1.3 Truncated Levy Distributions .................. 1434
20.1.4 Asymmetric Truncated Levy 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 Lйvy-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 Ito-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 Levy
Distribution ......................................... 1517
Appendix 20B Gaussian Weight ............................. 1520
Appendix 20C Comparison with Dow-Jones Data .............. 1521
Notes and References ...................................... 1522
Index ........................................................ 1529
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