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

by Kleinert, Hagen

• ISBN13: 9789812381071
• ISBN10: 9812381074
• Edition: 3rd
• Format: Paperback
• Copyright: 2004-03-01
• Publisher: World Scientific Pub Co Inc

Summary

This is the third, significantly 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 become possible by two major advances. The first is a new euclidean path integral formula which increases the restricted range of applicability of Feynman's famous formula to include singular attractive 1/r and 1/r2 potentials. The second is a simple quantum equivalence principle governing the transformation of euclidean path integrals to spaces 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 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 def

Table of Contents

Preface	vii
Preface to the Second Edition	ix
Preface to the First Edition	xi
1 Fundamentals	1	(76)
1.1 Classical Mechanics	1	(10)
1.2 Relativistic Mechanics in Curved Spacetime	11	(1)
1.3 Quantum Mechanics	12	(6)
1.4 Dirac's Bra-Ket Formalism	18	(8)
1.5 Observables	26	(5)
1.6 Quantum Mechanics of General Lagrangian Systems	31	(6)
1.7 Particle on the Surface of a Sphere	37	(2)
1.8 Spinning Top	39	(7)
1.9 Time Evolution Operator	46	(3)
1.10 Properties of Time Evolution Operator	49	(7)
1.11 Scattering	56	(9)
1.11.1 Scattering Matrix	56	(1)
1.11.2 Cross Section	57	(2)
1.11.3 Born Approximation	59	(1)
1.11.4 Partial Wave Expansion and Eikonal Approximation	59	(2)
1.11.5 Scattering Amplitude from Time Evolution Amplitude	61	(1)
1.11.6 Lippmann-Schwinger Equation	62	(3)
1.12 Heisenberg Picture of Quantum Mechanics	65	(3)
1.13 Classical and Quantum Statistics	68	(6)
1.13.1 Canonical Ensemble	69	(1)
1.13.2 Grand-Canonical Ensemble	70	(4)
Appendix 1A The Asymmetric Top	74	(2)
Notes and References	76	(1)
2 Path Integrals - Elementary Properties and Simple Solutions	77	(110)
2.1 Path Integral Representation of Time Evolution Amplitudes	77	(10)
2.2 Exact Solution for Free Particle	87	(9)
2.3 Finite Slicing Properties of Free-Particle Amplitude	96	(1)
2.4 Exact Solution for Harmonic Oscillator	97	(5)
2.5 Useful Fluctuation Formulas	102	(2)
2.6 Oscillator Amplitude on Finite Time Lattice	104	(2)
2.7 Gelfand-Yaglom Formula	106	(1)
2.7.1 Recursive Calculation of Fluctuation Determinant	106	(1)
2.7.2 Examples	107	(2)
2.7.3 Calculation on Unsliced Time Axis	109	(1)
2.7.4 D'Alembert's Construction	110	(1)
2.7.5 Another Simple Formula	111	(2)
2.7.6 Generalization to D Dimensions	113	(1)
2.8 Path Integral for Harmonic Oscillator with Arbitrary Time-Dependent Frequency	114	(1)
2.8.1 Coordinate Space	114	(3)
2.8.2 Momentum Space	117	(2)
2.9 Free-Particle and Oscillator Wave Functions	119	(2)
2.10 Path Integrals and Quantum Statistics	121	(2)
2.11 Density Matrix	123	(5)
2.12 Quantum Statistics of Harmonic Oscillator	128	(6)
2.13 Tine-Dependent Harmonic Potential	134	(4)
2.14 Functional Measure in Fourier Space	138	(3)
2.15 Classical Limit	141	(1)
2.16 Calculation Techniques on Sliced Time Axis. Poisson Formula	142	(5)
2.17 Field-Theoretic Definition of Harmonic Path Integral by Analytic Regularization	147	(12)
2.17.1 Zero-Temperature Evaluation of Frequency Sun	148	(3)
2.17.2 Finite-Temperature Evaluation of Frequency Sum	151	(2)
2.17.3 Duality Transformation and Low-Temperature Expansion	153	(6)
2.18 Finite-N Behavior of Thermodynamic Quantities	159	(2)
2.19 Time Evolution Amplitude of Freely Falling Particle	161	(2)
2.20 Charged Particle in Magnetic Field	163	(5)
2.21 Charged Particle in Magnetic. Field and Harmonic Potential	168	(3)
2.22 Gauge Invariance and Alternative Path Integral Representation	171	(1)
2.23 Velocity Path Integral	172	(2)
2.24 Path Integral Representation of Scattering Matrix	174	(10)
2.24.1 General Development	174	(3)
2.24.2 Improved Formulation	177	(1)
2.24.3 Eikonal Approximation to Scattering Amplitude	178	(1)
Appendix 2A Derivation of Baker-Campbell-Hausdorff and Magnus Formulas	179	(3)
Appendix 2B Direct Calculation of Time-Sliced Oscillator Amplitude	182	(2)
Appendix 2C Derivation of Mehler Formula	184	(1)
Notes and References	184	(3)
3 External Sources, Correlations, and Perturbation Theory	187	(1)
3.1 External Sources	187	(3)
3.2 Green Function of Harmonic Oscillator	190	(1)
3.2.1 Wronski Construction	191	(4)
3.2.2 Spectral Representation	195	(1)
3.3 Green Functions of First-Order Differential Equation	196	(1)
3.3.1 Time-Independent Frequency	197	(7)
3.3.2 Time-Dependent Frequency	204	(1)
3.4 Summing Spectral Representation of Green Function	205	(2)
3.5 Wronski Construction for Periodic and Antiperiodic Green Functions	207	(1)
3.6 Evolution Amplitude in Presence of Source Term	208	(5)
3.7 External Source in Quantum-Statistical Path Integral	213	(1)
3.7.1 Continuation of Real-Time Result	213	(4)
3.7.2 Calculation at Imaginary Time	217	(7)
3.8 Lattice Green Function	224	(1)
3.9 Correlation Functions, Generating Functional, and Wick Expansion	225	(3)
3.10 Correlation Functions of Charged Particle in Magnetic Field	228	(1)
3.11 Correlation Functions in Canonical Path Integral	229	(8)
3.11.1 Harmonic Correlation Functions	230	(3)
3.11.2 Relations between Various Amplitudes	233	(1)
3.11.3 Harmonic Generating Functionals	234	(3)
3.12 Particle in Heat Bath	237	(4)
3.13 Particle in Heat Bath of Photons	241	(2)
3.14 Harmonic Oscillator in Heat Bath	243	(3)
3.15 Perturbation Expansion of Anharmonic Systems	246	(3)
3.16 Calculation of Perturbation Series with Feynman Diagrams	249	(4)
3.17 Field-Theoretic Definition of Anharmonic Path Integral	253	(1)
3.18 Generating Functional of Connected Correlation Functions	254	(19)
3.18.1 Connectedness Structure of Correlation Functions	255	(3)
3.18.2 Decomposition of Correlation Functions into Connected Correlation Functions	258	(2)
3.18.3 Functional Generation of Vacuum Diagrams	260	(4)
3.18.4 Correlation Functions from Vacuum Diagrams	264	(2)
3.18.5 Generating Functional for Vertex Functions. Effective Action	266	(5)
3.18.6 Ginzburg-Landau Approximation to Generating Functional	271	(1)
3.18.7 Composite Fields	272	(1)
3.19 Path Integral Calculation of Effective Action by Loop Expansion	273	(14)
3.19.1 General Formalism	273	(5)
3.19.2 Quadratic Fluctuations	278	(3)
3.19.3 Effective Action to Second Order in h	281	(3)
3.19.4 Background Field Method for Effective Action	284	(3)
3.20 Nambu-Goldstone Theorem	287	(2)
3.21 Effective Classical Potential	289	(11)
3.21.1 Effective Classical Boltzmann Factor	291	(2)
3.21.2 High- and Low-Temperature Behavior	293	(2)
3.21.3 Alternative Candidate for Effective Classical Potential	295	(1)
3.21.4 Harmonic Correlation Function without Zero Mode	296	(1)
3.21.5 Perturbation Expansion	296	(2)
3.21.6 First-Order Perturbative Result	298	(2)
3.22 Perturbative Calculation of Scattering Amplitude	300	(3)
3.22.1 Generating Functional	300	(1)
3.22.2 Application to Scattering Amplitude	301	(1)
3.22.3 First Correction to Eikonal Approximation	302	(1)
3.23 Rayleigh-Schrödinger Perturbation Expansion	303	(6)
3.23.1 Energy Levels	303	(5)
3.23.2 Scattering Amplitudes	308	(1)
3.24 Functional Determinants from Green Functions	309	(6)
Appendix 3A Feynman Integrals for T not equal to 0	315	(3)
Appendix 3B Energy Shifts for gx4/4-Interaction	318	(2)
Appendix 3C Matrix Elements for General Potential	320	(2)
Appendix 3D Level-Shifts from Schrödinger Equation	322	(2)
Appendix 3E Recursion Relations for Perturbation Coefficients	324	(5)
3E.1 One-Dimensional Interaction x4	324	(4)
3E.2 Interaction r4 in D-Dimensional Radial Oscillator	328	(1)
3E.3 Interaction r2q in D Dimensions	329	(1)
3E.4 Polynomial Interaction in D Dimensions	329	(1)
Notes and References	329	(3)
4 Semiclassical Time Evolution Amplitude	332	(1)
4.1 The Wentzel Kramers Brillouin (WKB) Approximation	332	(4)
4.2 Saddle Point Approximation	336	(1)
4.2.1 Ordinary Integrals	337	(2)
4.2.2 Path Integrals	339	(6)
4.3 Van Vleck-Pauli-Morette Determinant	345	(4)
4.4 Fundamental Composition Law for Semiclassical Time Evolution Amplitude	349	(2)
4.5 Semiclassical Fixed-Energy Amplitude	351	(2)
4.6 Semiclassical Amplitude in Momentum Space	353	(2)
4.7 Semiclassical Quantum Mechanical Partition Function	355	(5)
4.8 Multi-Dimensional Systems	360	(5)
4.9 Quantum Corrections to Classical Density of States	365	(5)
4.10 Thomas-Fermi Model of Neutral Atoms	370	(17)
4.10.1 Semiclassical Limit	370	(8)
4.10.2 Quantum Correction Near Origin	378	(2)
4.10.3 Exchange Energy	380	(2)
4.10.4 Higher Quantum Corrections to Thomas-Fermi Energies	382	(5)
4.11 Classical Action of Coulomb System	387	(9)
4.12 Semiclassical Scattering	396	(4)
4.12.1 General Formulation	396	(3)
4.12.2 Semiclassical Cross Section of Mott Scattering	399	(1)
Notes and References	400	(3)
5 Variational Perturbation Theory	403	(1)
5.1 Variational Approach to Effective Classical Partition Function	403	(1)
5.2 Local Harmonic Trial Partition Function	404	(5)
5.3 Optimal Upper Bound	409	(1)
5.4 Accuracy of Variational Approximation	410	(2)
5.5 Weakly Bound Ground State Energy in Finite-Range Potential Well	412	(2)
5.6 Possible Direct Generalizations	414	(1)
5.7 Effective Classical Potential for Anharmonic Oscillator	415	(6)
5.8 Particle Densities	421	(3)
5.9 Extension to D Dimensions	424	(2)
5.10 Application to Coulomb and Yukawa Potentials	426	(3)
5.11 Hydrogen Atom in Strong Magnetic Field	429	(7)
5.11.1 Weak-Field Behavior	433	(1)
5.11.2 Effective Classical Potential	433	(3)
5.12 Effective Potential and Magnetization Curves	436	(2)
5.13 Variational Approach to Excitation Energies	438	(5)
5.14 Systematic Improvement of Feynman-Kleinert Approximation	443	(3)
5.15 Applications of Variational Perturbation Expansion	446	(6)
5.15.1 Anharmonic Oscillator at T = 0	446	(2)
5.15.2 Anharmonic Oscillator for T > 0	448	(4)
5.16 Convergence of Variational Perturbation Expansion	452	(7)
5.17 Variational Perturbation Theory for Strong-Coupling Expansion	459	(3)
5.18 General Strong-Coupling Expansions	462	(3)
5.19 Variational Interpolation between Weak and Strong-Coupling Expansions	465	(2)
5.20 Systematic Improvement of Excited Energies	467	(1)
5.21 Variational Treatment of Double-Well Potential	468	(2)
5.22 Higher-Order Effective Classical Potential for Nonpolynomial Interactions	470	(12)
5.22.1 Evaluation of Path Integrals	471	(1)
5.22.2 Higher-Order Smearing Formula in D Dimensions	472	(2)
5.22.3 Isotropic Second-Order Approximation to Coulomb Problem	474	(2)
5.22.4 Anisotropic Second-Order Approximation to Coulomb Problem	476	(2)
5.22.5 Zero-Temperature Limit	478	(4)
5.23 Polarons	482	(15)
5.23.1 Partition Function	484	(2)
5.23.2 Harmonic Trial System	486	(6)
5.23.3 Effective Mass	492	(1)
5.23.4 Second-Order Correction	492	(1)
5.23.5 Polaron in Magnetic Field, Bipolarons. etc	493	(1)
5.23.6 Variational Interpolation for Polaron Energy and Mass	494	(3)
5.24 Density Matrices	497	(54)
5.24.1 Harmonic Oscillator	498	(1)
5.24.2 Variational Perturbation Theory for Density Matrices	499	(2)
5.24.3 Smearing Formula for Density Matrices	501	(2)
5.24.4 First-Order Variational Results	503	(5)
5.24.5 Smearing Formula in Higher Spatial Dimensions	508	(2)
5.24.6 Applications	510	(11)
Appendix 5A Feynman Integrals for T not equal to 0 without Zero Frequency	521	(2)
Appendix 513 Proof of Scaling Relation for the Extrema of WN	523	(3)
Appendix 5C Second-Order Shift of Polaron Energy	526	(1)
Notes and References	527	(5)
6 Path Integrals with Topological Constraints	532	(1)
6.1 Point Particle on Circle	532	(4)
6.2 Infinite Wall	536	(5)
6.3 Point Particle in Box	541	(2)
6.4 Strong-Coupling Theory for Particle in Box	543	(1)
6.4.1 Partition Function	544	(1)
6.4.2 Perturbation Expansion	544	(3)
6.4.3 Variational Strong-Coupling Approximations	547	(2)
6.4.4 Special Properties of Expansion	549	(1)
6.4.5 Exponentially Fast Convergence	550	(1)
Notes and References	551	(2)
7 Many Particle Orbits Statistics and Second Quantization	553	(1)
7.1 Ensembles of Bose and Fermi Particle Orbits	554	(7)
7.2 Bose-Einstein Condensation	561	(1)
7.2.1 Free Bose Gas	561	(9)
7.2.2 Effect of Interactions	570	(5)
7.2.3 Bose-Einstein Condensation in Harmonic Trap	575	(9)
7.2.4 Interactions in Harmonic Trap	584	(4)
7.3 Gas of Free Fermions	588	(5)
7.4 Statistics Interaction	593	(5)
7.5 Fractional Statistics	598	(1)
7.6 Second-Quantized Bose Fields	599	(3)
7.7 Fluctuating Bose Fields	602	(6)
7.8 Coherent States	608	(4)
7.9 Dimensional Regularization of Functional Determinants	612	(3)
7.10 Second-Quantized Fermi Fields	615	(1)
7.11 Fluctuating Fermi Fields	616	(8)
7.11.1 Grassiann Variables	616	(3)
7.11.2 Fermionic Functional Determinant	619	(4)
7.11.3 Coherent States for Fermions	623	(1)
7.12 Hilbert Space of Quantized Grassmann Variable	624	(9)
7.12.1 Single Real Grassmann Variable	625	(3)
7.12.2 Quantizing Harmonic Oscillator with Grassmann Variables	628	(1)
7.12.3 Spin System with Grassmann Variables	629	(4)
7.13 External Sources in a*, α -Path Integral	633	(2)
7.14 Generalization to Pair Terms	635	(2)
7.15 Spatial Degrees of Freedom	637	(5)
7.15.1 Grand-Canonical Ensemble of Particle Orbits from Free Fluctuating Field	637	(1)
7.15.2 First versus Second Quantization	638	(1)
7.15.3 Interacting Fields	639	(1)
7.15.4 Effective Classical Field Theory	640	(2)
Notes and References	642	(4)
8 Path Integrals in Spherical Coordinates	646	(1)
8.1 Angular Decomposition in Two Dimensions	646	(3)
8.2 Trouble with Feynman's Path Integral Formula in Radial Coordinates	649	(4)
8.3 Cautionary Remarks	653	(3)
8.4 Time Slicing Corrections	656	(5)
8.5 Angular Decomposition in Three and More Dimensions	661	(1)
8.5.1 Three Dimensions	661	(3)
8.5.2 D Dimensions	664	(5)
8.6 Radial Path Integral for Harmonic Oscillator and Free Particle	669	(1)
8.7 Particle near the Surface of a Sphere in D Dimensions	670	(3)
8.8 Angular Barriers near the Surface of a Sphere	673	(1)
8.8.1 Angular Barriers in Three Dimensions	673	(5)
8.8.2 Angular Barriers in Four Dimensions	678	(5)
8.9 Motion on a Sphere in D Dimensions	683	(4)
8.10 Path Integrals on Group Spaces	687	(3)
8.11 Path Integral of a Spinning Top	690	(1)
Notes and References	691	(2)
9 Fixed-Energy Amplitude and Wave Functions	693	(1)
9.1 General Relations	693	(3)
9.2 Free Particle in D Dimensions	696	(3)
9.3 Harmonic Oscillator in D Dimensions	699	(6)
9.4 Free Particle from ω -> 0 -Limit of Oscillator	705	(2)
9.5 Charged Particle in Uniform Magnetic Field	707	(7)
Notes and References	714	(1)
10 Spaces with Curvature and Torsion	715	(128)
10.1 Einstein's Equivalence Principle	716	(1)
10.2 Classical Motion of Mass Point in General Metric-Affine Space	717	(1)
10.2.1 Equations of Motion	717	(3)
10.2.2 Nonholonomic Mapping to Spaces with Torsion	720	(5)
10.2.3 New Equivalence Principle	725	(1)
10.2.4 Classical Action Principle for Spaces with Curvature and Torsion	726	(4)
10.3 Path Integral in Metric-Affine Space	730	(1)
10.3.1 Nonholonomic Transformation of Action	731	(4)
10.3.2 Measure of Path Integration	735	(6)
10.4 Completing Solution of Path Integral on Surface of Sphere	741	(1)
10.5 External Potentials and Vector Potentials	742	(3)
10.6 Perturbative Calculation of Path Integrals in Curved Space	745	(5)
10.6.1 Free and Interacting Parts of Action	745	(2)
10.6.2 Zero Temperature	747	(3)
10.7 Model Study of Coordinate Invariance	750	(1)
10.7.1 Diagrammatic Expansion	751	(2)
10.7.2 Diagrammatic Expansion in d Time Dimensions	753	(1)
10.8 Calculating Loop Diagrams	754	(7)
10.8.1 Reformulation in Configuration Space	761	(1)
10.8.2 Integrals over Products of Two Distributions	762	(1)
10.8.3 Integrals over Products of Four Distributions	763	(3)
10.9 Distributions as Limits of Bessel Function	766	(1)
10.9.1 Correlation Function and Derivatives	766	(1)
10.9.2 Integrals over Products of Two Distributions	767	(2)
10.9.3 Integrals over Products of Four Distributions	769	(2)
10.10 Simple Rules for Calculating Singular Integrals	771	(5)
10.11 Perturbative Calculation on Finite Time Intervals	776	(26)
10.11.1 Diagrammatic Elements	777	(1)
10.11.2 Cumulant Expansion of D-Dimensional Free-Particle Amplitude in Curvilinear Coordinates	778	(2)
10.11.3 Propagator in 1-epsilon Time Dimensions	780	(2)
10.11.4 Coordinate Independence for Dirichlet Boundary Conditions	782	(7)
10.11.5 Time Evolution Amplitude in Curved Space	789	(6)
10.11.6 Covariant Results for Arbitrary Coordinates	795	(7)
10.12 Effective Classical Potential in Curved Space	802	(15)
10.12.1 Covariant Fluctuation Expansion	803	(2)
10.12.2 Arbitrariness of qμ0	805	(2)
10.12.3 Zero-Mode Properties	807	(3)
10.12.4 Covariant Perturbation Expansion	810	(1)
10.12.5 Covariant Result from Noncovariant Expansion	811	(3)
10.12.6 Particle on Unit Sphere	814	(3)
10.13 Covariant Effective Action fur Quantum Particle with Coordinate-Dependent Mass	817	(6)
10.13.1 Formulating the Problem	817	(3)
10.13.2 Derivative Expansion	820	(3)
Appendix 10A Nonholonomic Gauge Transformations in Electromagnetism	823	(10)
10A.1 Gradient Representation of Magnetic Field of Current Loop	824	(3)
10A.2 Generating Magnetic Fields by Multivalued Gauge Transformations	827	(1)
10A.3 Magnetic Monopoles	828	(1)
10A.4 Minimal Magnetic Coupling of Particles from Multivalued Gauge Transformations	829	(2)
10A.5 Gauge Field Representation of Current Loops	831	(2)
Appendix 10B Difference between Multivalued Basis Tetrads and Vierbein Fields	833	(2)
Appendix 10C Cancellation of Powers of δ(0)	835	(3)
Notes and References	838	(5)
11 Schrödinger Equation in General Metric-Affine Spaces	843	(24)
11.1 Integral Equation for Time Evolution Amplitude	843	(1)
11.1.1 From the Recursion Relation to Schrödinger's Equation	844	(3)
11.1.2 Alternative Evaluation	847	(3)
11.2 Equivalent Path Integral Representations	850	(4)
11.3 Potentials and Vector Potentials	854	(1)
11.4 Unitarity Problem	855	(2)
11.5 Alternative Attempts	857	(1)
11.6 DeWitt-Seeley Expansion of Time Evolution Amplitude	858	(4)
Appendix 11A Cancellations in Effective Potential	862	(3)
Appendix 11B DeWitt's Amplitude	865	(1)
Notes and References	865	(2)
12 New Path Integral Formula for Singular Potentials	867	(13)
12.1 Path Collapse in Feynman's formula for the Coulomb System	867	(3)
12.2 Stable Path Integral with Singular Potentials	870	(5)
12.3 Time-Dependent Regularization	875	(2)
12.4 Relation with Schrödinger Theory. Wave Functions	877	(2)
Notes and References	879	(1)
13 Path Integral of Coulomb System	880	(45)
13.1 Pseudotime Evolution Amplitude	880	(2)
13.2 Solution for the Two-Dimensional Coulomb System	882	(5)
13.3 Absence of Time Slicing Corrections for D = 2	887	(6)
13.4 Solution for the Three-Dimensional Coulomb System	893	(6)
13.5 Absence of Time Slicing Corrections for D = 3	899	(4)
13.6 Geometric Argument for Absence of Time Slicing Corrections	903	(1)
13.7 Comparison with Schrödinger Theory	904	(5)
13.8 Angular Decomposition of Amplitude, and Radial Wave Functions	909	(4)
13.9 Remarks on Geometry of Four-Dimensional uμ-Space	913	(2)
13.10 Solution in Momentum Space	915	(4)
13.10.1 Gauge-Invariant Canonical Path Integral	915	(3)
13.10.2 Another Form of Action	918	(1)
13.10.3 Absence of Extra R-Term	919	(1)
Appendix 13A Group-Theoretic Aspects of Coulomb States	919	(5)
Notes and References	924	(1)
14 Solution of Further Path Integrals by the Duru-Kleinert Method	925	(44)
14.1 One-Dimensional Systems	925	(3)
14.2 Derivation of the Effective Potential	928	(5)
14.3 Comparison with Schrödinger Quantum Mechanics	933	(1)
14.4 Applications	933	(1)
14.4.1 Radial Harmonic Oscillator and Morse System	934	(2)
14.4.2 Radial Coulomb System and Morse System	936	(1)
14.4.3 Equivalence of Radial Coulomb System and Radial Oscillator	937	(8)
14.4.4 Angular Barrier hear Sphere, and Rosen-Morse Potential	945	(3)
14.4.5 Angular Barrier near Four-Dimensional Sphere, and General Rosen-Morse Potential	948	(3)
14.4.6 Hulthén Potential and General Rosen-Morse Potential	951	(2)
14.4.7 Extended Hulthén Potential and General Rosen-Morse Potential	953	(1)
14.5 D-Dimensional Systems	954	(1)
14.6 Path Integral of the Dionium Atom	955	(1)
14.6.1 Formal Solution	956	(3)
14.6.2 Absence of Time Slicing Corrections	959	(4)
14.7 Time-Dependent Durn-Kleinert Transformation	963	(4)
Appendix 14A Affine Connection of Dionium Atom	967	(1)
Appendix 14B Algebraic Aspects of Dionium States	967	(1)
Notes and References	968	(1)
15 Path Integrals in Polymer Physics	969	(49)
15.1 Polymers and Ideal Random Chains	969	(2)
15.2 Moments of End-to-End Distribution	971	(3)
15.3 Exact End-to-End Distribution in Three Dimensions	974	(2)
15.4 Short-Distance Expansion for a Long Polymer	976	(2)
15.5 Saddle Point Approximation to Three-Dimensional End-to-End Distribution	978	(1)
15.6 Path Integral for Continuous Gaussian Distribution	979	(2)
15.7 Stiff Polymers	981	(2)
15.7.1 Path Integral	983	(1)
15.7.2 Moments of End-to-End Distribution	984	(4)
15.8 Schrödinger Equation and Recursive Solution for Moments	988	(1)
15.8.1 Recursive Solution of Schrödinger Equation	989	(3)
15.8.2 Approximation to End-to-End Distribution	992	(5)
15.8.3 From Moments to End-to-End Distribution for D = 3	997	(2)
15.9 Excluded-Volume Effects	999	(7)
15.10 Flory's Argument	1006	(1)
15.11 Polymer Field Theory	1007	(8)
15.12 Fermi Fields for Self-Avoiding Lines	1015	(1)
Notes and References	1015	(3)
16 Polymers and Particle Orbits in Multiply Connected Spaces	1018	(85)
16.1 Simple Model for Entangled Polymers	1018	(3)
16.2 Entangled Fluctuating Particle Orbit: Aharonov-Bohm Effect	1021	(11)
16.3 Aharonov-Bohm Effect and Fractional Statistics	1032	(5)
16.4 Self-Entanglement of Polymer	1037	(14)
16.5 The Gauss Invariant of Two Curves	1051	(3)
16.6 Bound States of Polymers = Ribbons	1054	(6)
16.7 Chern-Simons Theory of Entanglements	1060	(3)
16.8 Entangled Pair of Polymers	1063	(3)
16.8.1 Polymer Field Theory for Probabilities	1066	(2)
16.8.2 Calculation of Partition Function	1068	(2)
16.8.3 Calculation of Numerator in Second Moment	1070	(1)
16.8.4 First Diagram in Fig. 16.23	1071	(2)
16.8.5 Second and Third Diagrams in Fig. 16.23	1073	(1)
16.8.6 Fourth Diagram in Fig. 16.23	1074	(1)
16.8.7 Second Topological Moment	1075	(1)
16.9 Chern-Simons Theory of Statistical Interaction	1076	(21)
16.10 Second-Quantized Anon Fields	1078	(4)
16.11 Fractional Quantum Hall Effect	1082	(3)
16.12 Anyonic Superconductivity	1085	(2)
16.13 Non-Abelian Chern-Simons Theory	1087	(3)
Appendix 16A Calculation of Feymnan Diagrams for Polymer Entanglement	1090	(1)
Appendix 16B Kauffman and BLM/Ho polynomials	1091	(1)
Appendix 16C Skein Relation between Wilson Loop Integrals	1092	(3)
Appendix 16D London Equations	1095	(2)
Appendix 16F Hall Effect in Electron Gas	1097	(1)
Notes and References	1097	(6)
17 Tunneling	1103	(100)
17.1 Double-Well Potential	1103	(3)
17.2 Classical Solutions - Kinks and Antikinks	1106	(4)
17.3 Quadratic Fluctuations	1110	(6)
17.3.1 Zero-Eigenvalue Mode	1116	(3)
17.3.2 Continuum Pact of Fluctuation Factor	1119	(3)
17.4 General Formula for Eigenvalue Ratios	1122	(2)
17.5 Fluctuation Determinant from Classical Solution	1124	(3)
17.6 Wave Functions of Double-Well	1127	(1)
17.7 Gas of Kinks and Antikinks and Level Splitting Formula	1128	(5)
17.8 Fluctuation Correction to Level Splitting	1133	(5)
17.9 Tunneling and Decay	1138	(9)
17.10 Large-Order Behavior of Perturbation Expansions	1147	(29)
17.10.1 Growth Properties of Expansion Coefficients	1148	(3)
17.10.2 Semiclassical Large-Order Behavior	1151	(5)
17.10.3 Fluctuation Correction to the Imaginary Part and Large-Order Behavior	1156	(3)
17.10.4 Variational Approach to Tunneling. Perturbation Coefficients to All Orders	1159	(8)
17.10.5 Convergence of Variational Perturbation Expansion	1167	(9)
17.11 Decay of Supercurrent in Thin Closed Wire	1176	(11)
17.12 Decay of Metastable Thermodynamic Phases	1187	(7)
17.13 Decay of Metastable Vacuum State in Quantum Field Theory	1194	(2)
17.14 Crossover from Quantum Tunneling to Thermally Driven Decay	1196	(1)
Appendix 17A Feynman Integrals for Fluctuation Correction	1197	(3)
Notes and References	1200	(3)
18 Nonequilibrium Quantum Statistics	1203	(100)
18.1 Linear Response and Time-Dependent Green Functions for T not equal to 0	1203	(3)
18.2 Spectral Representations of T not equal to 0 Green Functions	1206	(3)
18.3 Other Important Green Functions	1209	(3)
18.4 Hermitian Adjoint Operators	1212	(1)
18.5 Harmonic Oscillator Green Functions for T not equal to 0	1213	(1)
18.5.1 Creation Annihilation Operators	1213	(3)
18.5.2 Real Field Operators	1216	(2)
18.6 Nonequilibrium Green Functions	1218	(9)
18.7 Perturbation Theory for Nonequilibrium Green Functions	1227	(3)
18.8 Path Integral Coupled to Thermal Reservoir	1230	(5)
18.9 Fokker-Planck Equation	1235	(1)
18.9.1 Canonical Path Integral for Probability Distribution	1236	(2)
18.9.2 Solving the Operator Ordering Problem	1238	(6)
18.9.3 Strung Damping	1244	(3)
18.10 Langevin Equations	1247	(3)
18.11 Stochastic Calculus	1250	(6)
18.12 Supersymmetry	1256	(2)
18.13 Stochastic Quantum Liouville Equation	1258	(3)
18.14 Relation to Quantum Langevin Equation	1261	(1)
18.15 Electromagnetic Dissipation and Decoherence	1261	(5)
18.15.1 Forward Backward Path Integral	1262	(4)
18.16 Master Equation for Time Evolution	1266	(3)
18.17 Line Width	1269	(1)
18.18 Lamb shift	1270	(4)
18.19 Langevin Equations	1274	(1)
18.20 Fokker-Planck Equation in Spaces with Curvature and Torsion	1275	(2)
18.21 Stochastic Interpretation of Quantum-Mechanical Amplitudes	1277	(2)
18.22 Stochastic Equation for Schrödinger Wave Function	1279	(2)
18.23 Real Stochastic and Deterministic Equation for Schrödinger Wave Function	1281	(4)
18.23.1 Stochastic Differential Equation	1281	(1)
18.23.2 Equation for Noise Average	1282	(1)
18.23.3 Harmonic Oscillator	1283	(1)
18.23.4 General Potential	1283	(1)
18.23.5 Deterministic Equation	1284	(1)
18.24 Heisenberg Picture for Probability Evolution	1285	(4)
Appendix 18A Inequalities for Diagonal Green Functions	1289	(4)
Appendix 18B General Generating Functional	1293	(5)
Appendix 18C Wick Decomposition of Operator Products	1298	(2)
Notes and References	1300	(3)
19 Relativistic Particle Orbits	1303	(39)
19.1 Special Features of Relativistic Path Integrals	1305	(3)
19.2 Proper Action for Fluctuating Relativistic Particle Orbits	1308	(1)
19.2.1 Gauge-Invariant Formulation	1308	(2)
19.2.2 Simplest Gauge Fixing	1310	(1)
19.2.3 Partition Function of Ensemble of Closed Particle Loops	1311	(2)
19.2.4 Fixed-Energy Amplitude	1313	(1)
19.3 Relativistic Coulomb System	1313	(4)
19.4 Relativistic Particle in Electromagnetic Field	1317	(1)
19.4.1 Action and Partition Function	1317	(1)
19.4.2 Perturbation Expansion	1318	(2)
19.4.3 Lowest-Order Vacuum Polarization	1320	(4)
19.5 Path Integral for Spin-1/2 Particle	1324	(1)
19.5.1 Dirac Theory	1324	(4)
19.5.2 Path Integral	1328	(2)
19.5.3 Amplitude with Electromagnetic Interaction	1330	(3)
19.5.4 Effective Action in Electromagnetic Field	1333	(1)
19.5.5 Perturbation Expansion	1334	(1)
19.5.6 Vacuum Polarization	1335	(2)
19.6 Supersymmetry	1337	(1)
19.6.1 Global Invariance	1337	(1)
19.6.2 Local Invariance	1338	(2)
Notes and References	1340	(2)
20 Path Integrals and Financial Markets	1342	(61)
20.1 Fluctuation Properties of Financial Assets	1342	(2)
20.1.1 Harmonic Approximation to Fluctuations	1344	(2)
20.1.2 Lévy Distributions	1346	(1)
20.1.3 Truncated Lévy Distributions	1347	(5)
20.1.4 Asymmetric Truncated Lévy Distributions	1352	(3)
20.1.5 Meixner Distributions	1355	(1)
20.1.6 Other Non-Gaussian Distributions	1356	(4)
20.1.7 Levy-Khintchine Formula	1360	(1)
20.1.8 Debye-Waller Factor for Non-Gaussian Fluctuations	1361	(1)
20.1.9 Path Integral for Non-Gaussian Distribution	1361	(2)
20.1.10 Fokker-Planck-Type Equation	1363	(5)
20.2 Martingales	1368	(3)
20.2.1 Gaussian Martingales	1368	(1)
20.2.2 Non-Gaussian Martingales	1369	(2)
20.3 Origin of Heavy Tails	1371	(15)
20.3.1 Pair of Stochastic Differential Equations	1371	(1)
20.3.2 Fokker-Planck Equation	1372	(3)
20.3.3 Solution of Fokker-Planck Equation	1375	(1)
20.3.4 Pure x-Distribution	1376	(2)
20.3.5 Long-Tine Behavior	1378	(4)
20.3.6 Tail Behavior for all Times	1382	(2)
20.3.7 Path Integral Calculation	1384	(1)
20.3.8 Natural Martingales	1385	(1)
20.4 Option Pricing	1386	(17)
20.4.1 Black-Scholes Option Pricing Model	1387	(2)
20.4.2 Evolution Equations of Portfolios with Options	1389	(2)
20.4.3 Option Pricing for Gaussian Fluctuations	1391	(4)
20.4.4 Option Pricing for Non-Gaussian Fluctuations	1395	(3)
20.4.5 Option Pricing for Fluctuating Variance	1398	(2)
20.4.6 Perturbation Expansion and Smile	1400	(3)
Appendix 20A Large-x Behavior of Truncated Levy Distribution	1403	(4)
Appendix 20B Gaussian Weight	1407	(1)
Appendix 20C Comparison with Dow-Jones Data	1408	(1)
Notes and References	1409	(8)
Index	1417