Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets

Contents
Preface
Preface to Fourth Edition
Preface to Third Edition
Preface to Second Edition
Preface to First Edition
List of Figures
List of Tables
1 Fundamentals
1.1 Classical Mechanics
1.2 Relati vistic Mechanics in Curved Spacetime
l.3 Quantum Mechanics
1.3.1 Bragg Reflections and Interference
1.3.2 Matter Waves
1.3.3 SchrOdinger Equation
1.3.4 Particle Current Conservation
1.4 Dirac's Bra-Ket Formalism.
1.4.1 Basis Transformations
1.4.2 Bracket Notation
1.4.3 Continuum Limit
1.4.4 Generalized Functions
1.4.5 SchrOdinger Equation in Dirac Notation
1.4.6 Momentum States
1.4.7 Incompleteness and Poisson's Summation Formula
1.5 Obscrvables
1.5.1 Uncertainty Relation.
1.5.2 Density Matrix and Wigner Function.
1.5.3 Generalization to Many Particles
1.6 Time Evolution Operator
l.7 Properties of the Time Evolution Operator
1.8 Heisenberg Picture of Quantum Mechanics
l.9 Interaction Picture and Perturbation Expansion
1.10 Time Evolution Amplitude
1.11 Fixed-Energy Amplitude
1.12 Free-Particle Amplitudes.
1.13 Quantum Mechanics of General Lagrangian Systems.
1.14 Particle on the Surface of a Sphere
1.15 Spinning Top
1.16 Scattering
1. 16.1 Scattering Matrix
1.16.2 Cross Section
1.16.3 Born Approximation
1. 16.4 Partial Wave Expansion and Eikonal Approximation
1. 16.5 Scattering Amplitude from Time Evolution Amplitude
1. 16.6 Lippmann-Schwinger Equation
1.17 Classical and Quantum Statistics
1. 17.1 Canonical Ensemble.
1. 17.2 Grand-Canonical Ensemble
1.18 Density of States and Ttacelog
Appendix l A Simple Time Evolution Operator.
Appendix I B Convergence of the Fresnel Integral
Appendix l C The Asymmetric Top
Notes and References.
2 Path Integrals - Elementary Properties and Simple Solutions
2.1 Path Integral Representation of Time Evolution Amplitudcs
2.1.1 Sliced Time Evolution Amplitude.
2.1.2 Zero-Hamiltonian Path Integral.
2.1.3 SchrOdinger Equation for Time Evolution Amplitude
2.1.4 Convergence of of the Time-Sliced Evolution Amplitude
2.1.5 Time Evolution Amplitude in Momentum Space.
2.1.6 Quantum-Mechanical Partition Function.
2.1.7 Feynman's Configurat ion Space Pat h Integral
2.2 Exact Solution for the Frec Particle
2.2. 1 Direct Solution
2.2.2 Fluctuations around the Classical Path
2.2.3 Fluctuation Factor
2.2.4 Finite Slicing Properties of Free-Particle Amplitude.
2.3 Exact Solution for Harmonic Oscillator
2.3. 1 Fluctuations around the Classical Path.
2.3.2 Fluctuation Factor
2.3.3 The i7]-Prescription and Maslov-Morse Index
2.3.4 Continuum Limit.
2.3.5 Useful Fluctuation Formulas.
2.3.6 Oscillator Amplitude on Finite Time Lattice
2.4 Gelfand-Yaglom Formula.
2.4. 1 Recursive Calculation of Fluctuation Detcrminant.
2.4.2 Examples
2.4.3 Calculation on Unsliced Time Axis
2.4.4 D'Alembert's Construction
2.4 .5 Another Simple Formula.
2.4.6 Generalization t o D Dimensions
2.5 Harmonic Oscillator with Time-Dependent Frequency
2.5.1 Coordinate Space.
2.5.2 Momentum Space
2.6 Free-Particle and Oscillator Wave Functions
2.7 General Time-Dependent Harmonic Action
2.8 Path Integrals and Quantum Statistics
2.9 Density Matrix
2.10 Quantum Statistics of the Harmonic Oscillator
2.11 Time-Dependent Harmonic Potential
2.12 Functional Measure in Fourier Space
2.13 Classical Limit.
2.14 Calculation Techniques on Sliced Time Axis via the Poisson Formula
2.15 Ficld·Theorctic Definition of Harmonic Path Integrals by Analytic Regularization
2.15.1 Zero--Temperature Evaluation of the Frequency Sum.
2. 15.2 Finite-Temperature Evaluation of the Frequency Sum.
2.15.3 Quantum.Mechanical Harmonic Oscillator
2. 15.4 Tracelog of the First·Order Differential Operator
2.15.5 Gradient Expansion of the One-Dimensional Tracclog
2.15.6 Duality Transformation and Low· Temperature Expansion
2.16 Finite-N Behavior of Thermodynamic Quantities
2.17 Time Evolution Amplitude of Freely Falling Particle.
2.18 Charged Particle in Magnetic Field
2.18.1 Action.
2.18.2 Gauge Properties.
2. 18.3 Time-Sliced Path Integration
2.18.4 Classical Action
2. 18.5 Translational Invariancc
2.19 Charged Particle in Magnetic Field plus Harmonic Potentia]
2.20 Gauge Invariance and Alternative Path Integral Representation
2.21 Velocity Path Integral.
2.22 Path Integral Representation of the Scattering Matrix
2.22.1 General Development
2.22.2 Improved Formulation
2.22.3 Eikonal Approximation to the Scattering Amplitude
2.23 Heisenberg Operator Approach to Time Evolution Amplitude.
2.23.1 Free Particle
2.23.2 Harmonic Oscillator
2.23.3 Charged Particle in Magnetic Field
Appendix 2A Baker-Campbell-Hausdorff Formula and Magnus Expansion
Appendix 28 Direct Calculation of the Time-Sliced Oscillator Amplitudc
Appendix 2C Derivation of Mehler Formula.
Notes and References
3 External Sources, Correlations, and Perturbation Theory
3.1 External Sources
3.2 Green Function of Harmonic Oscillator
3.2.1 Wronski Construction
3.2.2 Spectral Representation
3.3 Green Functions of First-Order Differential Equation
3.3.1 Time-Independent Frequency
3.3.2 Time-Dependent Frequency
3.4 Summing Spectral Representation of Green Function
3.5 Wronski Construction for Periodic and Antiperiodic Green F\mctions
3.6 Time Evolution Amplitude in Presence of Source Term
3.7 Time Evolution Amplitude at Fixed Path Average
3.8 External Source in Quantum-Statistical Path Integral
3.8.1 Continuation of Real-Time Result
3.8.2 Calculation at Imaginary Time
3.9 Lattice Green Function
3.10 Correlation Functions, Generating Functional, and Wick Expansion
3.10.1 Real-Time Correlation Functions.
3.11 Correlation Functions of Charged Particle in Magnetic Field
3.12 Correlation Functions in Canonical Path Integral.
3.12.1 Harmonic Correlation Functions
3.12.2 Relations between Various Amplitudes
3.12.3 Harmonic Generating Functionals.
3.13 Particle in Heat Bath
3.14 Heat Bath of Photons.
3.15 Harmonic Oscillator in Ohmic Heat Bath
3.16 Harmonic Oscillator in Photon Heat Bath
3.17 Perturbation Expansion of Anharmonic Systems
3.18 Rayleigh-SchrOdinger and Brillouin-Wigner Perturbation Expansion
3.19 Level-Shifts and Perturbed Wave Functions from SchrOdinger Equation
3.20 CrucuiatioIl of Perturbation Series via Fcynman Diagrams.
3.21 Perturbative Definition of Interacting Path Integrals.
3.22 Generating Functional of Connected Correlation Functions
3.22.1 Connectedness Structure of Correlation Functions.
3.22.2 Correlation Functions versus Connected Correlation Functions
3.22.3 Functional Generation of Vacuum Diagrams.
3.22.4 Correlation Functions from Vacuum Diagra.ms
3.22.5 Generating Functional for Vertex Functions. Effective Action
3.22.6 Ginzburg-Landau Approximation to Generating Functional
3.22.7 Composite Fields.
3.23 Path Integral Calculation of Effective Action by Loop Expansion
3.23.1 General Formalism
3.23.2 Mean-Field Approximation
3.23.3 Corrections from Quadratic Fluctuations.
3.23.4 Effective Action to Second Order in Ii
3.23.5 Finite-Temperature Two-Loop Effective Action
3.23.6 Background Field Method for Effective Action
3.24 Nambu-Goldstone Theorem
3.25 Effective Classical Potential
3.25.1 Effective Classical Boltzmann Factor
3.25.2 Effective Classical Hamiltonian
3.25.3 High- and Low-Temperature Behavior
3.25.4 Alternative Candidate for Effective Classical Potential
3.25.5 Harmonic Correlation FUnction without Zero Mode
3.25.6 Perturbation Expansion
3.25.7 Effective Potential and Magnetization Curves
3.25.8 First-Order Perturbative Result.
3.26 Perturbati vc Approach to Scattering Amplitude
3.26.1 Generating Functional
3.26.2 Application to Scattering Amplitude
3.26.3 First Correction to Eikonal Approximation
3.26.4 Rayleigh-SehrOdinger Expansion of Scattering Amplitude.
3.27 Functional Determinants from Green FUnctions
Appendix 3A Matrix Elements for General Potential.
Appendix 3B Energy Shifts for gx4 /4-Interaction.
Appendix 3C Recursion Relations for Perturbation Coefficients.
3C.l One-Dimensional Interaction X4
3C.2 General One-Dimensional Interaction.
3C.3 Cumulative Treatment of Interactions X4 and x3
3C.4 Ground-State Energy with External Current
3C.5 Recursion Relation for Effective Potential
3C.6 Interaction r4 in D-Dimensional Radial Oscillator.
3C.7 Interaction r2q in D Dimensions.
3C.8 Polynomial Interaction in D Dimensions
Appendix 3D Feynman Integrals for T =f 0
Notes and References
4 Semiclassical Time Evolution Amplitude
4. 1 Wentzel-Kramers-Brillouin (WKB) Approximation.
4.2 Saddle Point Approximation
4.2. 1 Ordinary Integrals
4.2.2 Path Integrals
4.3 Van Vleck-Pauli-Morette Det erminant
4.4 Fundamental Composition Law for Semiclassical Time Evolution Amplitude
4 .5 Semiclassical Fix ed-Energy Amplitude
4.6 Se miclassical Amplitude in Momentum Space
4.7 Semiclassical Quantum-Mechanical Partition Function
4.8 Multi-Dimensional Systems
4.9 Quantum Corrections to Classical Density of States
4.9.1 One-Dimensional Case
4.9.2 Arbitrary Dimensions
4.9.3 Bilocal Density of States
4.9.4 Gradient Expansion of Tracelog of Hamiltonian Operator
4.9.5 Local De ns ity of States on Circle
4.9.6 Quantum Corrections to Bohr-Sommerfeld Approximation
4.10 Thomas-Fermi Model of Neutral Atoms
4.10.1 Semiclassical Limit
4.10.2 Self-Consistent Field Equation
4.10.3 Energy Functiona l of Thomas-Fermi Atom
4.10.4 Calcula tion of Energies
4.1 0.5 Virial Theorem
4.10.6 Exchange E nergy
4.10.7 Quantum Correction Near Origin
4.10.8 Systemat ic Quantum Corrections to Thomas-Fermi Energies
4.11 Classical Action of Coulomb System
4.12 Semiclassical Scattering
4.12.1 General Formulation
4.12.2 Semiclassical Cross Section of Mott Scattering
Appendix 4A Se miclassical Quantization for Pure Power Potentials
Appendix 4B Derivation of Se miclassical Time Evolution Amplitude
Notes and References
5 Variation al Perturbation T heory
5.1 Variational Approach to Effective Classical Partition Function
5.2 Local Harmonic Trial Partition Function
5.3 The Optimal Upper Bound
5.4 Accuracy of Variational Approximation
5.5 Weakly Bound Ground State Energy in Finite-Range Potential Well
5.6 Possible Direct Generalizations
5.7 Effective Classical Potential for Anharmonic Oscillator and Double-Well Potential
a) Case w2 > 0, Anharmonic Oscillator
b) Case w2 < 0: The Double-Well Potential
5.8 Particle Densities
5.9 Extension to D Dimensions
5.10 Application to Coulomb and Yukawa Potentials
5.11 Hydrogen Atom in Strong Magnetic Field
5.11.1 Weak-Field Behavior
5.11.2 Effective Classical Hamiltonian
5.12 Variational Approach to Excitation Energies
5.13 Systematic Improvement of Feynman-Kleinert Approximation. Variational Perturbation Theory
5.14 Applications of Variational Perturbation Expansion
5.14.1 Anharmonic Oscillator at T = 0
5.14.2 Anharmonic Oscillator for T > 0
5.15 Convergence of Variational Perturbation Expansion
5.16 Variational Perturbation Theory for Strong-Coupling Expansion
5.17 General Strong-Coupling Expansions
5.18 Variational Interpolation between Weak and StrongCoupling Expansions
5.19 Systematic Improvement of Excited Energies
5.20 Variational Treatment of Double-Well Potential
5.21 Higher-Order Effective Classical Potential for Nonpolynomial Interactions
5.21.1 Evaluation of Path Integrals
5.21.2 Higher-Order Smearing Formula in D Dimensions
5.21.3 Isotropic Second-Order Approximation to Coulomb Problem
5.21.4 Anisotropic Second-Order Approximation to Coulomb Problem
5.21.5 Zero-Temperature Limit
5.22 Polarons
5.22.1 Partition Function
5.22.2 Harmonic Trial System
5.22.3 Effective Mass
5.22.4 Second-Order Correction
5.22.5 Polaron in Magnetic Field, Bipolarons, Small Polarons, Polaronic Excitons, and More
5.22.6 Variational Interpolation for Polaron Energy and Mass
5.23 Density Matrices
5.23.1 Harmonic Oscillator
5.23.2 Variational Perturbation Theory for Density Matrices
5.23.3 Smearing Formula for Density Matrices
5.23.4 First-Order Variational Approximation
5.23.5 Smearing Formula in Higher Spatial Dimensions
Appendix 5A Feynman Integrals for T=fo 0 without Zero Frequency
Appendix 5B Proof of Scaling Relation for Extrema of W N
Appendix 5C Second-Order Shift of Polaron Energy
Notes and References
6 Path Integrals with Topological Constraints
6. 1 Point Particle on Circle.
6.2 Infinite Wall
6.3 Point Particle in Box
6.4 Strong-Coupling Theory for Particle in Box.
6.4. 1 Partition Function
6.4.2 Perturbation Expansion
6.4.3 Variational Strong-Coupling Approximations
6.4.4 Special Properties of Expansion
6.4.5 ExponentiaJly Fast Convergence
Notes and References
7 Many Particle Orbits - Statistics and Second Quantization
7.1 Ensembles of Bose and Fermi Particle Orbits
7.2 Bose-Einstein Condensation
7.2. 1 Free Bose Gas
7.2.2 Bose Gas in Finite Box
7.2.3 Effect of Interactions
7.2.4 Bose-Einstein Condensation in Harmonic 'frap
7.2.5 Thermodynamic Functions
7.2.6 Critical Temperature
7.2.7 More General Anisotropic Trap
7.2.8 Rotating Bose-Einstein Gas
7.2.9 Finite-Size Corrections
7.2.10 Entropy and Specific Heat
7.2. 11 Interactions in Harmonic Trap
7.3 Gas of Free Fermions
7.4 Statistics Interaction
7.5 Fractional Statistics
7.6 Second-Quantized Bose Fields
7.7 Fluctuating Bose Fields.
7.8 Coherent States
7.9 Second-Quantized Fermi Fields
7.10 Fluctuating Fermi Fields
7.10.1 Grassmann Variables
7.10.2 Fermionic Functional Determinant
7.10.3 Coherent States for Fermions
7.11 Hilbert Space of Quantized Grassmann Variable
7.11.1 Single Real Grassmann Variable
7.11.2 Quantizing Harmonic Oscillator with Grassmann Variables
7.11.3 Spin System with Grassmann Variables
7.12 External Sources in a", a -Path Integral
7.13 Generalization to Pair Terms.
7.14 Spatial Degrees of Freedom.
7.14.1 Grand-Canonical Ensemble of Particle Orbits from Free Fluctuating Field
7.14.2 First versus Second Quantization
7.14.3 Interacting Fields.
7.14.4 Effective Classical Field Theory.
7.15 Bosonization
7.15.1 Collective Field.
7.15.2 Bosonized versus Original Theory
Appendix 7A Treatment of Singularities in Zeta-Function
7 A.I Finite Box
7 A.2 Harmonic Trap
Appendix 7B Experimental versus Theoretical Would-be Critical Temperature
Notes and References
8 Path Integrals in Polar and Spherical Coordinates
8.1 Angular Deromposition in Two Dimensions.
8.2 Trouble with Feynman's Path Integral Formula in Radial Coordinates
8.3 Cautionary Remarks
8.4 Time Slicing Corrections
8.5 Angular Decomposition in Three and More Dimensions
8.5.1 Three Dimensions
8.5.2 D Dimensions
8.6 Radial Path Integral for Harmonic Oscillator and Free Particle
8.7 Particle ncar the Surface of a Sphere in D Dimensions.
8.8 Angular Barriers ncar the Surface of a Spherc
8.8.1 Angular Barriers in Three Dimensions
8.8.2 Angular Barriers in Four Dimensions
8.9 Motion on a Sphere in D Dimensions
8.10 Path Integrals on Group Spaces
8.11 Path Integral of Spinning Top
8.12 Path Integral of Spinning Particle
8.13 Berry Phase.
8.14 Spin Precession
Notes and References
9 Wave Functions
9.1 Free Particle in D Dimensions
9.2 Harmonic Oscillator in D Dimensions
9.3 Free Particle from w --+ 0 -Limit of Oscillator.
9.4 Charged Particle in Uniform Magnetic Field
9.5 Dirac a-Function Potential
Notes and References
10 Spaces with Curvature and Torsion
10.1 Einstein's Equivalence Principle
10.2 Classical Motion of Mass Point in General Mctric·Affi ne Space
10.2. 1 Equations of Motion
10. 2.2 Nonholonomic Mapping to Spaces with Torsion
10.2.3 New Equivalence Principle.
10.2.4 Classical Action Principle for Spaces with Curvature and Torsion
10.3 Path Integral in Metric·Affine Space
10.3.1 Nonholonomic Transformation of Action
10.3.2 Measure of Path Integration.
10.4 Completing the Solution of Path Integral on Surface of Sphere
10.5 External Potent ials and Vector Potentials
10.6 Perturbative Calculat ion of Path Integrals in Curved Space
10.6.1 Free and Interacting Part s of Action
10.6.2 Zero Temperature
10.7 Model Study of Coordinate Invariance
10.7.1 Diagrammatic Expansion
10.7.2 Diagrammatic Expansion in d Time Dimensions.
10.8 Calculating Loop Diagrams
10.8.1 Reformulation in Configuration Space
10.8.2 Integrals over Products of Two Distributions
10.8.3 Integrals over Products of Four Distributions
10.9 Distributions as Limits of Bessel Function
10.9.1 Correlation Function and Derivatives.
10.9.2 Integrals over Products of Two Distributions
10.9.3 Integrals over Products of Four Distributions
10.10 Simple Rules for Calculating Singular Integrals
10.11 Perturbative Calculation on Finite Time Intervals
10.11.1 Diagrammatic Elements
10.11.2 Cumulant Expansion of D-Dimensional Free-Particle Amplitude in Curvilinear Coordinates
10.1 1.3 Propagator in 1– Time Dimensions
10.1 1.4 Coordinate Independence for Dirichlet Boundary Conditions
10.11.5 Time Evolution Amplitude in Curved Space
10.1 1.6 Covariant Results for Arbit rary Coordinates.
10.12 Effective Classical Potential in Curved Space
10.12. 1 Covariant Fluctuation Expansion
10.12.2 Arbitrariness of qb
10.12.3 Zero-Mode Properties.
10.12.4 Covariant Perturbation Expansion
10.12.5 Covariant Result from Noncovariant Expansion
10.12.6 Particle on Unit Sphere
10.13 Covariant Effective Action for Quantum Particle with Coordinate- Dependent Mass
10.13.1 Formulating the Problem
10.13.2 Gradient Expansion
Appendix 10A Nonholonomic Gauge Transformations in Electromagnetism
10A.1 Gradient Representation of Magnetic Field of Current Loops
10A.2 Generating Magnetic Fields by Multivalued Gauge TrIlJ1&- formations
10A.3 Magnetic Monopoles.
10A.4 Minimal Magnetic Coupling of Particles from Multivalued Gauge Transformations
10A.5 Gauge Ficld Representation of Current Loops and Monopoles
Appendix 10B Comparison of Multivalucd Basis Tetrads with Vierbein Fields
Appendix 10C Cancellation of Powers of 5(0)
Notes and References
11 Schrodinger Equation in General Metric-Affine Spaces
11.1 Integral Equation for Time Evolution Amplitude.
11.1.1 From Recursion Relation to Schrodinger Equation.
11. 1.2 Alternative Evaluation
11.2 Equivalent Path Integral Representations
11.3 Potentials and Vcctor Potentials.
11.4 Unitarity Problem.
11.5 Alternative Attempts
11.6 DeWitt--Seeley Expansion of Time Evolution Amplitude.
Appendix 11A Cancellations in Effective Potential.
Appendix 11B DeWitt's Amplitude.
Notes and References
12 New Path Integral Formula for Singular Potentials
12.1 Path Collapse in Feynman's formula for the Coulomb System
12.2 Stable Path Integral with Singular Potentials
12.3 Time-Dependent Regularization
12.4 Relation to Schr&lingcr Theory. Wave Functions.
Notes and References
13 Path Integral of Coulomb System
13.1 Pseudotime Evolution Amplitude
13.2 Solution for the Two-Dimensional Coulomb System
13.3 Absence of Time Slicing Corrections for D = 2
13.4 Solution for the Three-Dimensional Coulomb System
13.5 Absence of Time Slicing Corrections for D = 3
13.6 Geometric Argument for Absence of Time Slicing Corrections
13.7 Comparison with Schrodinger Theory
13.8 Angular Decomposition of Amplitude, and Radial Wave Functions
13.9 Remarks on Geometry of Four-Dimensional ul"-Space
13.10 Runge-Lenz-Pauli Group of Degeneracy.
13.11 Solution in Momentum Space
13.11.1 Another Form of Action
Appendix 13A Dynamical Group of Coulomb States.
Notes and References.
14 Solution of Further Path Integrals by Duru-Kleinert Method
14.1 One-Dimensional Systems
14.2 Derivation of the Effective Potential.
14.3 Comparison with SchrOdinger Quantum Mechanics.
14.4 Applications
14.4. 1 Radial Harmonic Oscillator and Morse System
14.4.2 Radial Coulomb System and Morse System
14.4.3 Equivalence of Radial Coulomb System and Radial Oscilla.- tor
14.4.4 Angular Barrier ncar Sphere, and Rosen-Morse Potential
14.4.5 Angular Barrier near Four-Dimensional Sphere, and General Rosen-Morse Potential
14.4.6 Hulthen Potential and General Rosen-Morse Potential
14.4.7 Extended Hulthen Potential and General Rosen-Morse Potential
14.5 D-Dimensional Systems.
14.6 Path Integral of the Dionium Atom
14.6. 1 Formal Solution
14.6.2 Absence of Time Slicing Corrections
14.7 Time-Dependent Duru-Kleinert 'Transformation
Appendix 14A Affine Connection of Dionium Atom
Appendix 14B Algebraic Aspects of Dionium States.
Notes and References
15 Path Integrals in Polymer Physics
15.1 Polymers and Ideal Random Chains.
15.2 Moments of End-to-End Distribution
15.3 Exact End-ta-End Distribution in Three Dimensions.
15.4 Short- Distance Expansion for Long Polymer
15.5 Saddle Point Approximation to Three-Dimensional End-to-End Distribution
15.6 Path Integral for Continuous Gaussian Distribution
15.7 Stiff Polymers
15.7.1 Sliced Path Integral
15.7.2 Relation to Classical Heisenberg Model.
15.7.3 End-to-End Distribution
15.7.4 Momcnts of End-to-End Distribution
15.8 Continuum Formulation
15.8.1 Path Integral
15.8.2 Correlation Functions and Moments
15.9 SchrOdinger Equation and Recursive Solution for Moments
15.9.1 Setting up the SchrOdinger Equation.
15.9.2 Recursive Solution of SchrOdingcr Equation.
15.9.3 From Moments to End-to-End Distribution for D = 3
15.9.4 Large-Stiffness Approximation to End-to-End Distribution
15.9.5 Higher Loop Corrections
15.10 Excluded-Volume Effects
15.11 Flory's Argument
15.12 Polymer Field Theory.
15.13 Fermi Fields for Self-Avoiding Lines
Appendix 15A Basic Integrals
Appendix 15B Loop Integrals
Appendix 15C Integrals Involving Modified Green Function
Notes and References
16 Polymers and Particle Orbits in Multiply Connected Spaces
16.1 Simple Model for Entangled Polymers.
16.2 Entangled Fluctuating Particle Orbit: Aharonov-Bohm Effect
16.3 Aharonov-Bohm Effect and Fractional Statistics
16.4 Self-Entanglement of Polymer
16.5 The Gauss Invariant of Two Curves
16.6 Bound States of Polymers and Ribbons
16.7 Chern-Simons Theory of Entanglements
16.8 Entangled Pair of Polymers
16.8.1 Polymer Field Theory for Probabilities
16.8.2 Calculation of Partition Function
16.8.3 Calculation of Numerat or in Second Moment
16.8.4 First Diagram in Fig. 16.23
16.8.5 Second and Third Diagrams in Fig. 16.23
16.8.6 Fourt h Diagram in Fig. 16.23
16.8. 7 Second Topological Moment
16.9 Chcrn-Simons Theory of Statistical Interaction
16.10 Second-Quantized Anyon Fields
16.11 Fractional Quantum Hall Effect
16.12 Anyonic Superconductivity
16.13 Non-Abelian Chern-Simons Theory
Appendix 16A Calculation of Feynman Diagrams in Polymer Entanglement
Appendix 168 Kauffman and 8LM/Ho polynomials
Appendix 16C Skein Relation between Wilson Loop Integrals
Appendix 160 London Equations
Appendix 16E Hall Effoct in Eioctean Gas
Notes and References
17 Tunneling
17. 1 Double-Well Potential
17.2 Classical Solutions — Kinks and Antikinks
17.3 Quadratic Fluctuations
17.3.1 Zero-Eigenvalue Mode
17.3.2 Continuum Part of Fluctuation Factor
17.4 General Formula for Eigenvalue Ratios
17.5 Fluctuation Determinant fro m Classical Solution.
17.6 Wave Functions of Double-Well
17.7 Gas of Kinks and Antikinks and Level Splitting Formula
17.8 Fluctuation Correction to Level Splitting
17.9 Thnneling and Decay
17.10 Large-Order Behavior of Perturbation Expansions
17.10. 1 Growth Properties of Expansion Coefficients.
17.10.2 Semiclassical Large-Order Behavior.
17.10.3 Fluctuation Correction to the Imaginary Part and Large- Order Behavior.
17.10.4 Variational Approach to Tunneling. Perturbation Coefficients to All Orders
17.10.5 Convergence of Variational Perturbation Expansion.
17.11 Decay of Super current in Thin Closed Wire.
17.12 Decay of Metastable Thermodynamic Phases.
17.13 Decay of Metastable Vacuum State in Quantum Field Theory
17.14 Crossover from Quantum Tunneling to Thermally Driven Decay
Appendix 17A Feynman Integrals for Fluctuation Correction.
Notes and References.
18 Nonequilibrium Quantum Statistics
18.1 Linear Response and Time-Dependent Green Functions for T = 0
18.2 Spectral Representations of Green Functions for T = 0
18.3 Other Important Green Functions
18.4 Hermitian Adjoint Operators.
18.5 Harmonic Oscillator Green Functions for T = 0
18.5. 1 Creation Annihilation Operators
18.5.2 Real Field Operators.
18.6 Nonequilibrium Green Functions.
18.7 Perturbation Theory for Nonequilibrium Green Functions
18.8 Path Integral Coupled to Thermal Reservoir
18.9 Fokker-Planck Equation
18.9.1 Canonical Path Integral for Probability Distribution
18.9.2 Solving the Operator Ordering Problem
18.9.3 Strong Damping
18.10 Langevin Equations
18.11 Path Integral Solution of Klein-Kramers Equation
18.12 Stochastic Quantization
18.13 Stochastic Calculus
18. 13.1 Kubo's stochastic Liouville equation
18.13.2 From Kubo's to Fokker-Planck Equations
18.13.3 Ito's Lemma
18.14 Solving the Langevin Equation.
18.15 Heisenberg Picture for Probability Evolution
18.16 Supcrsymmetry
18.17 Stochastic Quantum Liouville Equation
18.18 Master Equation for Time Evolution
18. 19 Relation to Quantum Langevin Equation
18.20 Electromagnetic Dissipation and Decoherence
18.20.1 Forward- Backward Path Integral
18.20.2 Master Equation for Time Evolution in Photon Bath
18.20.3 Line Width
18.20.4 Lamb shift
18.20. 5 Langevin Equations
18.21 Fokker-Planck Equation in Spaces with Curvature and Torsion
18.22 Stochastic Interpretation of Quantum-Mechanical Amplitudes
18.23 Stochastic Equation for SchrOdingcr Wave Function
18.24 Real Stochastic and Deterministic Equation for SchrOdinger Wave Function
18.24.1 Stochastic Differential Equation
18.24.2 Equation for Noise Average
18.24.3 Harmonic Oscillator
18.24.4 General Potential.
18.24.5 Deterministic Equation
Appendix 18A Inequalities for Diagonal Green Functions
Appendix 18B General Generating Functional
Appendix 18C Wick Decomposit ion of Operator Products
Notes and References
19 Relativistic Particle Orbits
19.1 Special Features of Relativistic Pat h Integra1s
19. 1.1 Simplest Gauge Fixing.
19.1.2 Partition Function of Ensemble of Closed Particle Loops
19. 1.3 Fixed-Energy Amplitude.
19.2 Thnneling in Relativistic Physics.
19.2. 1 Decay Rate of Vacuum in Electric Field
19.2.2 Birth of Universe
19.2.3 Friedmann Model
19.2.4 TUnneling of Expanding Universe
19.3 Relativistic Coulomb System
19.4 Relativistic Particle in Electromagnetic Field.
19.4. 1 Action and Partition Function
19.4.2 Perturbation Expansion
19.4.3 Lowest-Order Vacuum Polarization
19.5 Path Integral for Spin-l/2 Particle.
19. 5.1 Dirac Theory
19. 5.2 Path Integral
19.5.3 Amplitude with Electromagnetic Interaction.
19.5.4 Effective Action in Electromagnetic Field
19.5.5 Perturbation Expansion
19.5.6 Vacuum Polarization
19.6 Supersymmetry
19.6. 1 Global Invariance.
19.6.2 Local Invariance
Appendix 19A Proof of Same Quantum Physics of Modified Action
Notes and References
20 Path Integrals and Financial Markets
20.1 Fluctuation Properties of Financial Assets
20.1.1 Harmonic Approximation to Fluctuations
20.1.2 Levy Distributions.
20.1.3 Truncated Levy Distributions
20.1.4 Asymmetric Truncated Levy Distributions.
20.1.5 Gamma Distribution
20.1.6 Boltzmann Distribution
20.1.7 Student or Tsallis Distribution
20.1.8 Tsallis Distribution in Momentum Space
20.1.9 Relativistic Particle Boltzmann Distribution.
20.1.10 Meixner Distributions
20.1.11 Generalized Hyperbolic Distributions
20.1.12 Debye-Waller Factor for Non-Gaussian Fluctuations
20.1.13 Path Integral for Non-Gaussian Distribution.
20.1.14 Time Evolutio n of Distribution
20.1.15 Central Limit Theorem
20.1.16 Additivity Property of Noises and Hamiltonians.
20.1.17 Uvy-Khintchine Formula
20.1.18 Semigroup Property of Asset Distributions.
20.1.19 Time Evolutio n of Moments of Distribution
20.1.20 Boltzmann Distribution
20.1.21 Fourier-Transformed Tsallis Distribution.
20.1.22 Superposition of Gaussian Distributions
20.1.23 Fokker-Planck-Type Equation
20.1.24 Kramcrs-Moyal Equation
20.2 ito-like Formula for Non-Gaussian Distributions
20.2.1 Continuous Time.
20.2.2 Discrete Times
20.3 Martingales.
20.3.1 Gaussian Martingales
20.3.2 Non-Gaussian Martingale Distributions
20.4 Origin of Semi-Heavy Tails.
20.4.1 Pair of Stochastic Differential Equations
20.4.2 Fokker-Planck Equation
20.4.3 Solution of Fokker-Planck Equation.
20.4.4 Pure x-Distribution
20.4.5 Long-Time Behavior.
20.4.6 Tail Behavior for all Times
20.4.7 Path Integral Calculation
20.4.8 Natural Martingale Distribution
20.5 Time Series
20.6 Spectral Decomposition of Power Behaviors.
20.7 Option Pricing
20.7.1 Black-Scholes Option Pricing Model
20.7.2 Evolution Equations of Portfolios with Options
20.7.3 Option Pricing for Gaussian Fluctuations
20.7.4 Option Pricing for Boltzmann Distribution.
20.7.5 Option Pricing for General Non-Gaussian Fluctuations
20.7.6 Option Pricing for Fluctuating Variance.
20.7.7 Perturbation Expansion and Smile
Appendix 20A Large-x Behavior of Truncated Levy Distribution
Appendix 20B Gaussian Weight..
Appendix 20C Comparison with Dow-Jones Data
Notes and References.
Index