Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets (5th Edition)

Description

This is the fifth, expanded edition of the comprehensive textbook published in 1990 on the theory and applications of path integrals. It is the first book to explicitly solve path integrals of a wide variety of nontrivial quantum-mechanical systems, in particular the hydrogen atom. The solutions have been made possible by two major advances. The first is a new euclidean path integral formula which increases the restricted range of applicability of Feynman's time-sliced formula to include singular attractive 1/r- and 1/r2-potentials. The second is a new nonholonomic mapping principle carrying physical laws in flat spacetime to spacetimes with curvature and torsion, which leads to time-sliced path integrals that are manifestly invariant under coordinate transformations.In addition to the time-sliced definition, the author gives a perturbative, coordinate-independent definition of path integrals, which makes them invariant under coordinate transformations. A consistent implementation of this property leads to an extension of the theory of generalized functions by defining uniquely products of distributions.The powerful Feynman-Kleinert variational approach is explained and developed systematically into a variational perturbation theory which, in contrast to ordinary perturbation theory, produces convergent results. The convergence is uniform from weak to strong couplings, opening a way to precise evaluations of analytically unsolvable path integrals in the strong-coupling regime where they describe critical phenomena.Tunneling processes are treated in detail, with applications to the lifetimes of supercurrents, the stability of metastable thermodynamic phases, and the large-order behavior of perturbation expansions. A variational treatment extends the range of validity to small barriers. A corresponding extension of the large-order perturbation theory now also applies to small orders.Special attention is devoted to path integrals with topological restrictions needed to understand the statistical properties of elementary particles and the entanglement phenomena in polymer physics and biophysics. The Chern-Simons theory of particles with fractional statistics (anyons) is introduced and applied to explain the fractional quantum Hall effect.The relevance of path integrals to financial markets is discussed, and improvements of the famous Black-Scholes formula for option prices are developed which account for the fact, recently experienced in the world markets, that large fluctuations occur much more frequently than in Gaussian distributions. 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 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