PATH INTEGRALS. Vb World Scientific. in Quantum Mechanics, Statistics, Polymer Physics, and Financial Mar.4ets. 4th Edition.

Transcription

1 PATH INTEGRALS in Quantum Mechanics, Statistics, Polymer Physics, and Financial Mar.4ets 4th Edition Hagen KLEINERT Freie Universitaet Berlin llllll 111 1,1,041,1, t1PC111 flfl ounun Vb World Scientific NEW JERSEY LONDON SINGAPORE SHANGHAI HONG KONG TAIPEI BANGALORE

2 Contents Preface Preface to Third Edition Preface to Second Edition Preface to First Edition vii ix xi xiii 1 Fundamentals Classical Mechanics Relativistic Mechanics in Curved Spacetime Quantum Mechanics Bragg Reflections and Interference Matter Waves Schrödinger Equation Particle Current Conservation Dirac's Bra-Ket Formalism Basis Transformations Bracket Notation Continuum Limit Generalized Functions Schrödinger Equation in Dirac Notation Momentum States Incompleteness and Poisson's Summation Formula Observables Uncertainty Relation Density Matrix and Wigner Function Generalization to Many Particles Time Evolution Operator Properties of Time Evolution Operator Heisenberg Picture of Quantum Mechanics Interaction Picture and Perturbation Expansion Time Evolution Amplitude Fixed-Energy Amplitude Free-Particle Amplitudes Quantum Mechanics of General Lagrangian Systems 51 xvii

3 xviii 1.14 Particle on the Surface of a Sphere Spinning Top Scattering Scattering Matrix Cross Section Born Approximation Partial Wave Expansion and Eikonal Approximation Scattering Amplitude from Time Evolution Amplitude Lippmann-Schwinger Equation Classical and Quantum Statistics Canonical Ensemble Grand-Canonical Ensemble Density of States and Tracelog 81 Appendix 1A Simple Time Evolution Operator 83 Appendix 1B Convergence of Fresnel Integral 84 Appendix 1C The Asymmetric Top 85 Notes and References 87 2 Path Integrals Elementary Properties and Simple Solutions Path Integral Representation of Time Evolution Amplitudes Sliced Time Evolution Amplitude Zero-Hamiltonian Path Integral 91 2,1.3 Schrödinger Equation for Time Evolution Amplitude Convergence of Sliced Time Evolution Amplitude 92 2,1.5 Time Evolution Amplitude in Momentum Space 94 2,1.6 Quantum Mechanical Partition Function Feynman's Configuration Space Path Integral Exact Solution for Free Particle Direct Solution Fluctuations around Classical Path Fluctuation Factor Finite Slicing Properties of Free-Particle Amplitude Exact Solution for Harmonic Oscillator Fluctuations around Classical Path Fluctuation Factor The /37-Prescription and Maslov-Morse Index 114 2,3.4 Continuum Limit Useful Fluctuation Formulas Oscillator Amplitude on Finite Time Lattice Gelfand-Yaglom Formula Recursive Calculation of Fluctuation Determinant Examples Calculation on Unsliced Time Axis D'Alembert's Construction 123

4 xix Another Simple Formula Generalization to D Dimensions Harmonic Oscillator with Time-Dependent Frequency Coordinate Space Momentum Space Free-Particle and Oscillator Wave Functions General Time-Dependent Harmonic Action Path Integrals and Quantum Statistics Density Matrix Quantum Statistics of Harmonic Oscillator Time-Dependent Harmonic Potential Functional Measure in Fourier Space Classical Limit Calculation Techniques an Sliced Time Axis. Poisson Formula Field-Theoretic Definition of Harmonic Path Integral by Analytic Regularization Zero-Temperature Evaluation of Frequency Sum Finite-Temperature Evaluation of Frequency Sum Tracelog of First-Order Differential Operator Gradient Expansion of One-Dimensional Tracelog Duality Transformation and Low-Temperature Expansion Finite-N Behavior of Thermodynamic Quantities Time Evolution Amplitude of Freely Falling Particle Charged Particle in Magnetic Field Action Gauge Properties Time-Sliced Path Integration Classical Action Translational Invariance Charged Particle in Magnetic Field plus Harmonic Potential Gauge Invariance and Alternative Path Integral Representation Velocity Path Integral Path Integral Representation of Scattering Matrix General Development Improved Formulation Eikonal Approximation to Scattering Amplitude Heisenberg Operator Approach to Time Evolution Amplitude Free Particle Harmonic Oscillator Charged Particle in Magnetic Field 193 Appendix 2A Baker-Campbell-Hausdorff Formula and Magnus Expansion 197 Appendix 2B Direct Calculation of Time-Sliced Oscillator Amplitude 200 Appendix 2C Derivation of Mehler Formula 201

5 Notes and References External Sources, Correlations, and Perturbation Theory External. Sources Green Function of Harmonic Oscillator Wronski Construction Spectral Representation Green Functions of First-Order Differential Equation Time-Independent Frequency Time-Dependent Frequency Summing Spectral Representation of Green Function Wronski Construction for Periodic and Antiperiodic Green Functions Time Evolution Amplitude in Presence of Source Term Time Evolution Amplitude at Fixed Path Average External Source in Quantum-Statistical Path Integral Continuation of Real-Time Result Calculation at Imaginary Time Lattice Green Function Correlation Functions, Generating Functional, and Wick Expansion Real-Time Correlation Functions Correlation Functions of Charged Particle in Magnetic Field Correlation Functions in Canonical Path Integral Harmonic Correlation Functions Relations between Various Amplitudes Harmonic Generating Functionals Particle in Heat Bath Heat Bath of Photons Harmonic Oscillator in Ohmic Heat Bath Harmonic Oscillator in Photon Heat Bath Perturbation Expansion of Anharmonic Systems Rayleigh-Schrödinger and Brillouin-Wigner Perturbation Expansion Level-Shifts and Perturbed Wave Functions from Schrödinger Equation Calculation of Perturbation Series via Feynman Diagrams Perturbative Definition of Interacting Path Integrals Generating Functional of Connected Correlation Functions Connectedness Structure of Correlation Functions Correlation Functions versus Connected Correlation Functions Functional Generation of Vacuum Diagrams Correlation Functions from Vacuum Diagrams Generating Functional for Vertex Functions. Effective Action Ginzburg-Landau Approximation to Generating Functional 301

6 xxi Composite Fields Path Integral Calculation of Effective Action by Loop Expansion General Formalism Mean-Field Approximation Corrections from Quadratic Fluctuations Effective Action to Second Order in h Finite-Temperature Two-Loop Effective Action Background Field Method for Effective Action Nambu-Goldstone Theorem Effective Classical Potential Effective Classical Boltzmann Factor Effective Classical Hamiltonian High- and Low-Temperature Behavior Alternative Candidate for Effective Classical Potential Harmonic Correlation Function without Zero Mode Perturbation Expansion First-Order Perturbative Result Perturbative Approach to Scattering Amplitude Generating Functional Application to Scattering Amplitude First Correction to Eikonal Approximation Rayleigh-Schrödinger Expansion of Scattering Amplitude Functional Determinants from Green Functions 338 Appendix 3A Matrix Elements for General Potential 344 Appendix 3B Energy Shifts for gx 4/4-Interaction 345 Appendix 3C Recursion Relations for Perturbation Coefficients 347 3C.1 One-Dimensional Interaction x C.2 One-Dimensional Interactions x 4 and x C.3 Ground-State Energy with External Current 352 3C.4 Recursion Relation for Effective Potential 353 3C.5 Interaction r4 in D-Dimensional Radial Oscillator 356 3C.6 Interaction r2q in D Dimensions 357 3C.7 Polynomial Interaction in D Dimensions 358 Appendix 3D Feynman Integrals for T Notes and References Semiclassical Time Evolution Amplitude Wentzel-Kramers-Brillouin (WKB) Approximation Saddle Point Approximation Ordinary Integrals Path Integrals Van Vleck-Pauli-Morette Determinant Fundamental Composition Law for Semiclassical Time Evolution Amplitude 381

7 Semiclassical Fixed-Energy Amplitude 383 Semiclassical Amplitude in Momentum Space 385 Semiclassical Quantum-Mechanical Partition Function 387 Multi-Dimensional Systems 393 Quantum Corrections to Classical Density of States One-Dimensional Case Arbitrary Dimensions Bilocal Density of States Gradient Expansion of Tracelog of Hamiltonian Operator Local Density of States an Circle Quantum Corrections to Bohr-Sommerfeld Approximation 408 Thomas-Fermi Model of Neutral Atoms Semiclassical Limit Self-Consistent Field Equation Energy Functional of Thomas-Fermi Atom Calculation of Energies Virial Theorem Exchange Energy Quantum Correction Near Origin Systematic Quantum Corrections to Thomas-Fermi Energies Classical Action of Coulomb System Semiclassical Scattering General Formulation Semiclassical Cross Section of Mott Scattering 439 Appendix 4A Semiclassical Quantization for Pure Power Potentials. 440 Appendix 4B Derivation of Semiclassical Time Evolution Amplitude 442 Notes and References Variational Perturbation Theory Variational Approach to Effective Classical Partition Function Local Harmonie Trial Partition Function Optimal Upper Bound Accuracy of Variational Approximation Weakly Bound Ground State Energy in Finite-Range Potential Weil Possible Direct Generalizations Effective Classical Potential for Anharmonic Oscillator Particle Densities Extension to D Dimensions Application to Coulomb and Yukawa Potentials Hydrogen Atom in Strong Magnetic Field Weak-Field Behavior Effective Classical Hamiltonian Effective Potential and Magnetization Curves Variational Approach to Excitation Energies 483

8 5.14 Systematic Improvement of Feynman-Kleinert Approximation Applications of Variational Perturbation Expansion Anharmonic Oscillator at T = Anharmonic Oscillator for T > Convergence of Variational Perturbation Expansion Variational Perturbation Theory for Strong-Coupling Expansion General Strong-Coupling Expansions Variational Interpolation between Weak and Strong-Coupling Expansions Systematic Improvement of Excited Energies Variational Treatment of Double-Well Potential Higher-Order Effective Classical Potential for Nonpolyriomial Interactions Evaluation of Path Integrals Higher-Order Smearing Formula in D Dimensions Isotropic Second-Order Approximation to Coulomb Problem Anisotropic Second-Order Approximation to Coulomb Problem Zero-Temperature Limit Polarons Partition Function Harmonic Trial System Effective Mass Second-Order Correction Polaron in Magnetic Field, Bipolarons, etc Variational Interpolation for Polaron Energy and Mass Density Matrices Harmonic Oscillator Variational Perturbation Theory for Density Matrices Smearing Formula for Density Matrices First-Order Variational Approximation Smearing Formula in Higher Spatial Dimensions Applications 550 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 Path Integrals with Topological Constraints Point Particle an Circle Infinite Wall Point Particle in Box Strong-Coupling Theory for Particle in Box Partition Function 583

9 xxiv Perturbation Expansion Variational Strong-Coupling Approximations Special Properties of Expansion Exponentially Fast Convergence 588 Notes and References Many Particle Orbits Statistics and Second Quantization Ensembles of Bose and Fermi Particle Orbits Bose-Einstein Condensation Free Bose Gas Effect of Interactions Bose-Einstein Condensation in Harmonic Trap Entropy and Specific Heat Interactions in Harmonic Trap Gas of Free Fermions Statistics Interaction Fractional Statistics Second-Quantized Bose Fields Fluctuating Bose Fields Coherent States Second-Quantized Fermi Fields Fluctuating Fermi Fields Grassmann Variables Fermionic Functional Determinant Coherent States for Fermions Hilbert Space of Quantized Grassmann Variable Single Real Grassmann Variable Quantizing Harmonic Oscillator with Grassmann Variables Spin System with Grassmann Variables External Sources in a*, a -Path Integral Generalization to Pair Terms Spatial Degrees of Freedom Grand-Canonical Ensemble of Particle Orbits from Free Fluctuating Field First versus Second Quantization Interacting Fields Effective Classical Field Theory Bosonization Collective Field Bosonized versus Original Theory 683 Notes and References 685

10 XXV 8 Path Integrals in Polar and Spherical Coordinates Angular Decomposition in Two Dimensions Trouble with Feynman's Path Integral Formula in Radial Coordinates Cäutionary Remarks Time Slicing Corrections Angular Decomposition in Three and More Dimensions Three Dimensions D Dimensions Radial Path Integral for Harmonic Oscillator and Free Particle Particle near the Surface of a Sphere in D Dimensions Angular Barriers near the Surface of a Sphere Angular Barriers in Three Dimensions Angular Barriers in Four Dimensions Motion on a Sphere in D Dimensions Path Integrals on Group Spaces Path Integral of Spinning Top Path Integral of Spinning Particle Berry Phase Spin Precession 740 Notes and References Wave Functions Free Particle in D Dimensions Harmonic Oscillator in D Dimensions Free Particle from w --> 0 -Limit of Oscillator Charged Particle in Uniform Magnetic Field Dirac 8-Function Potential 762 Notes and References Spaces with Curvature and Torsion Einstein's Equivalence Principle Classical Motion of Mass Point in General Metric-Affine Space Equations of Motion Nonholonomic Mapping to Spaces with Torsion New Equivalence Principle Classical Action Principle for Spaces with Curvature and Torsion Path Integral in Metric-Affine Space Nonholonomic Transformation of Action Measure of Path Integration Completing Solution of Path Integral on Surface of Sphere External Potentials and Vector Potentials Perturbative Calculation of Path Integrals in Curved Space. 796

11 xxvi Free and Interacting Parts of Action Zero Temperature Model Study of Coordinate Invariance Diagrammatic Expansion Diagrammatic Expansion in d Time Dimensions Calculating Loop Diagrams Reformulation in Configuration Space Integrals over Products of Two Distributions Integrals over Products of Four Distributions Distributions as Limits of Bessel Function Correlation Function and Derivatives Integrals over Products of Two Distributions Integrals over Products of Four Distributions Simple Rules for Calculating Singular Integrals Perturbative Calculation on Finite Time Intervals Diagrammatic Elements Cumulant Expansion of D-Dimensional Free-Particle Amplitude in Curvilinear Coordinates Propagator in 1 e Time Dimensions Coordinate Independence for Dirichlet Boundary Conditions Time Evolution Amplitude in Curved Space Covariant Results for Arbitrary Coordinates Effective Classical Potential in Curved Space Covariant Fluctuation Expansion Arbitrariness of qät Zero-Mode Properties Covariant Perturbation Expansion Covariant Result from Noncovariant Expansion Particle on Unit Sphere Covariant Effective Action for Quantum Particle with Coordinate- Dependent Mass Formulating the Problem Gradient Expansion 867 Appendix 10A Nonholonomic Gauge Transformations in Electromagnetism A.1 Gradient Representation of Magnetic Field of Current Loops A.2 Generating Magnetic Fields by Multivalued Gauge Transformations A.3 Magnetic Monopoles A.4 Minimal Magnetic Coupling of Particles from Multivalued Gauge Transformations A.5 Gauge Field Representation of Current Loops and Monopoles 876 Appendix 10B Comparison of Multivalued Basis Tetrads with Vierbein Fields 878

12 xxvii Appendix 10C Cancellation of Powers of 5(0) 880 Notes and References Schrödinger Equation in General Metric-Affine Spaces Integral Equation for Time Evolution Amplitude From Recursion Relation to Schrödinger Equation Alternative Evaluation Equivalent Path Integral Representations Potentials and Vector Potentials Unitarity Problem Alternative Attempts DeWitt-Seeley Expansion of Time Evolution Amplitude 902 Appendix 11A Cancellations in Effective Potential 905 Appendix 11B DeWitt's Amplitude 908 Notes and References New Path Integral Formula for Singular Potentials Path Collapse in Feynman's formula for the Coulomb System Stable Path Integral with Singular Potentials Time-Dependent Regularization Relation to Schrödinger Theory. Wave Functions 920 Notes and References Path Integral of Coulomb System Pseudotime Evolution Amplitude Solution for the Two-Dimensional Coulomb System Absence of Time Slicing Corrections for D = Solution for the Three-Dimensional Coulomb System Absence of Time Slicing Corrections for D = Geometric Argument for Absence of Time Slicing Corrections Comparison with Schrödinger Theory Angular Decomposition of Amplitude, and Radial Wave Functions Remarks an Geometry of Four-Dimensional u A-Space Solution in Momentum Space Gauge-Invariant Canonical Path Integral Another Form of Action Absence of Extra R-Term 960 Appendix 13A Dynamical Group of Coulomb States 960 Notes and References Solution of Further Path Integrals by Duru-Kleinert Method One-Dimensional Systems Derivation of the Effective Potential Comparison with Schrödinger Quantum Mechanics 974

13 xxviii 14.4 Applications Radial Harmonic Oscillator and Morse System Radial Coulomb System and Morse System Equivalence of Radial Coulomb System and Radial Oscillator Angular Barrier near Sphere, and Rosen-Morse Potential Angular Barrier near Four-Dimensional Sphere, and General Rosen-Morse Potential Hulth6n Potential and General Rosen-Morse Potential Extended Hulthen Potential and General Rosen-Morse Potential D-Dimensional Systems Path Integral of the Dionium Atom Formal Solution Absence of Time Slicing Corrections Time-Dependent Duru-Kleinert Transformation 1004 Appendix 14A Affine Connection of Dionium Atom Appendix 14B Algebraic Aspects of Dionium States 1008 Notes and References Path Integrals in Polymer Physics Polymers and Ideal Random Chains Moments of End-to-End Distribution Exact End-to-End Distribution in Three Dimensions Short-Distance Expansion for Long Polymer Saddle Point Approximation to Three-Dimensional End-to-End Distribution Path Integral for Continuous Gaussian Distribution Stiff Polymers Sliced Path Integral Relation to Classical Heisenberg Model End-to-End Distribution Moments of End-to-End Distribution Continuum Formulation Path Integral Correlation Functions and Moments Schrödinger Equation and Recursive Solution for Moments Setting up the Schrödinger Equation Recursive Solution of Schrödinger Equation From Moments to End-to-End Distribution for D = Large-Stiffness Approximation to End-to-End Distribution Higher Loop Corrections Excluded-Volume Effects Flory's Argument 1061

14 15.12 Polymer Field Theory Fermi Fields for Self-Avoiding Lines 1069 Appendix 15A Basic Integrals 1070 Appendix 15B Loop Integrals 1070 Appendix 15C Integrals Involving Modified Green Function 1072 Notes and References Polymers and Particle Orbits in Multiply Connected Spaces Simple Model for Entangled Polymers Entangled Fluctuating Particle Orbit: Aharonov-Bohm Effect Aharonov-Bohm Effect and Fractional Statistics Self-Entanglement of Polymer The Gauss Invariant of Two Curves Bound States of Polymers and Ribbons Chern-Simons Theory of Entanglements Entangled Pair of Polymers Polymer Field Theory for Probabilities Calculation of Partition Function Calculation of Numerator in Second Moment First Diagram in Fig Second and Third Diagrams in Fig Fourth Diagram in Fig Second Topological Moment Chern-Simons Theory of Statistical Interaction Second-Quantized Anyon Fields Fractional Quantum Hall Effect Anyonic Superconductivity Non-Abelian Chern-Simons Theory 1142 Appendix 16A Calculation of Feynman Diagrams in Polymer Entanglement 1145 Appendix 16B Kauffman and BLM/Ho polynomials 1146 Appendix 16C Skein Relation between Wilson Loop Integrals 1147 Appendix 16D London Equations 1150 Appendix 16E Hall Effect in Electron Gas 1151 Notes and References Tunneling Double-Well Potential Classical Solutions Kinks and Antikinks Quadratic Fluctuations Zero-Eigenvalue Mode Continuum Part of Fluctuation Factor General Formula for Eigenvalue Ratios Fluctuation Determinant from Classical Solution 1178

15 xxx 17.6 Wave Functions of Double-Well Gas of Kinks and Antikinks and Level Splitting Formula Fluctuation Correction to Level Splitting Tunneling and Decay Large-Order Behavior of Perturbation Expansions Growth Properties of Expansion Coefficients Semiclassical Large-Order Behavior Fluctuation Correction to the Imaginary Part and Large- Order Behavior Variational Approach to Tunneling. Perturbation Coefficients to All Orders Convergence of Variational Perturbation Expansion Decay of Supercurrent in Thin Closed Wire Decay of Metastable Thermodynamic Phases Decay of Metastable Vacuum State in Quantum Field Theory Crossover from Quantum Tunneling to Thermally Driven Decay 1248 Appendix 17A Feynman Integrals for Fluctuation Correction 1250 Notes and References Nonequilibrium Quantum Statistics Linear Response and Time-Dependent Green Functions for T Spectral Representations of Green Functions for T Other Important Green Functions Hermitian Adjoint Operators Harmonic Oscillator Green Functions for T Creation Annihilation Operators Real Field Operators Nonequilibrium Green Functions Perturbation Theory for Nonequilibrium Green Functions Path Integral Coupled to Thermal Reservoir Fokker-Planck Equation Canonical Path Integral for Probability Distribution Solving the Operator Ordering Problem Strong Damping Langevin Equations Stochastic Quantization Stochastic Calculus Supersymmetry Stochastic Quantum Liouville Equation Master Equation for Time Evolution Relation to Quantum Langevin Equation Electromagnetic Dissipation and Decoherence Forward Backward Path Integral Master Equation for Time Evolution in Photon Bath 1325

16 Line Width Lamb shift Langevin Equations Fokker-Planck Equation in Spaces with Curvature and Torsion Stochastic Interpretation of Quantum-Mechanical Amplitudes Stochastic Equation for Schrödinger Wave Function Real Stochastic and Deterministic Equation for Schrödinger Wave Function Stochastic Differential Equation Equation for Noise Average Harmonic Oscillator General Potential Deterministic Equation Heisenberg Picture for Probability Evolution 1342 Appendix 18A Inequalities for Diagonal Green Functions 1346 Appendix 18B General Generating Functional 1349 Appendix 18C Wick Decomposition of Operator Products 1354 Notes and References Relativistic Particle Orbits Special Features of Relativistic Path Integrals Proper Action for Fluctuating Relativistic Particle Orbits Gauge-Invariant Formulation Simplest Gauge Fixing Partition Function of Ensemble of Closed Particle Loops Fixed-Energy Amplitude Tunneling in Relativistic Physics Decay Rate of Vacuum in Electric Field Birth of Universe Friedmann Model Tunneling of Expanding Universe Relativistic Coulomb System Relativistic Particle in Electromagnetic Field Action and Partition Function Perturbation Expansion Lowest-Order Vacuum Polarization Path Integral for Spin-1/2 Particle Dirac Theory Path Integral Amplitude with Electromagnetic Interaction Effective Action in Electromagnetic Field Perturbation Expansion Vacuum Polarization Supersymmetry 1413

17 XXXii Global Invariance Local Invariance 1414 Notes and References Path Integrals and Financial Markets Fluctuation Properties of Financial Assets Harmonic Approximation to Fluctuations Levy Distributions Truncated Levy Distributions Student or Tsallis Distribution Asymmetric Truncated Levy Distributions Boltzmann Distribution Meixner Distributions Generalized Hyperbolic Distributions Levy-Khintchine Formula Debye-Waller Factor for Non-Gaussian Fluctuations Path Integral for Non-Gaussian Distribution Semigroup Property of Asset Distributions Time Evolution of Distribution Fokker-Planck-Type Equation Martingales Gaussian Martingales Non-Gaussian Martingales Origin of Heavy Tails Pair of Stochastic Differential Equations Fokker-Planck Equation Solution of Fokker-Planck Equation Pure x-distribution Long-Time Behavior Tau Behavior for all Times Path Integral Calculation Natural Martingales Spectral Decomposition of Power Behaviors Option Pricing Black-Scholes Option Pricing Model Evolution Equations of Portfolios with Options Option Pricing for Gaussian Fluctuations Option Pricing for Boltzmann Distribution Option Pricing for General Non-Gaussian Fluctuations Option Pricing for Fluctuating Variance Perturbation Expansion and Smile 1485 Appendix 20A Large-x Behavior of Truncated Levy Distribution 1488 Appendix 20B Gaussian Weight 1490 Appendix 20C Comparison with Dow-Jones Data 1491

18 Notes and References 1492 Index 1499