Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 6
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. . . . . 86210.12.5 Covariant Result from Noncovariant Expansion . . . . . . . 86310.12.6 Particle on Unit Sphere . . . . . . . . . . . . . . . . . . . . 86610.13 Covariant Effective Action for Quantum Particle with CoordinateDependent Mass . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 86810.13.1 Formulating the Problem . . . . . . . . . . . . . . . . . . . 86910.13.2 Gradient Expansion . . . . . . . . . . . . . . . . . . . . . . 872Appendix 10A Nonholonomic Gauge Transformations in Electromagnetism 87310A.1 Gradient Representation of Magnetic Field of Current Loops 87310A.2 Generating Magnetic Fields by Multivalued Gauge Transformations . . . . . . . . . . .
. . . . . . . . . . . . . . . . 87810A.3 Magnetic Monopoles . . . . . . . . . . . . . . . . . . . . . . 87910A.4 Minimal Magnetic Coupling of Particles from MultivaluedGauge Transformations . . . . . . . . . . . . . . . . . . . . 88110A.5 Gauge Field Representation of Current Loops and Monopoles882Appendix 10B Comparison of Multivalued Basis Tetrads with VierbeinFields . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884Appendix 10C Cancellation of Powers of δ(0) . . . . . . . . . . . . . . . . 886Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88811 Schrödinger Equation in General Metric-Affine Spaces11.1 Integral Equation for Time Evolution Amplitude . . . . . .11.1.1 From Recursion Relation to Schrödinger Equation .11.1.2 Alternative Evaluation .
. . . . . . . . . . . . . . .11.2 Equivalent Path Integral Representations . . . . . . . . . .11.3 Potentials and Vector 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 . . . . . . . . . . . . . . . ...................................................892892893896899903904907908912914H. Kleinert, PATH INTEGRALSxxviiNotes and References . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 91412 New Path Integral Formula for Singular Potentials12.1 Path Collapse in Feynman’s formula for the Coulomb System12.2 Stable Path Integral with Singular Potentials . . . . . . . . .12.3 Time-Dependent Regularization . . . . . . . . . . .
. . . . .12.4 Relation to Schrödinger Theory. Wave Functions . . . . . . .Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . ......916. 916. 919. 924. 926. 928Integral of Coulomb SystemPseudotime Evolution Amplitude . . . . . . . . . .
. . . . . . . . .Solution for the Two-Dimensional Coulomb System . . . . . . . .Absence of Time Slicing Corrections for D = 2 . . . . . . . . . . .Solution for the Three-Dimensional Coulomb System . . . . . . . .Absence of Time Slicing Corrections for D = 3 . . . . . . . .
. . .Geometric Argument for Absence of Time Slicing Corrections . . .Comparison with Schrödinger Theory . . . . . . . . . . . . . . . .Angular Decomposition of Amplitude, and Radial Wave FunctionsRemarks on Geometry of Four-Dimensional uµ -Space . . . . . . . .Solution in Momentum Space . . . . . .
. . . . . . . . . . . . . . .13.10.1 Gauge-Invariant Canonical Path Integral . . . . . . . . . .13.10.2 Another Form of Action . . . . . . . . . . . . . . . . . . .13.10.3 Absence of Extra R-Term . . . . . . . . . . . . . . . . . .Appendix 13A Dynamical Group of Coulomb States . . . .
. . . . . . .Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .929. 929. 931. 936. 941. 947. 949. 950. 956. 959. 961. 962. 965. 966. 966. 97013 Path13.113.213.313.413.513.613.713.813.913.10..........14 Solution of Further Path Integrals by Duru-Kleinert Method97214.1 One-Dimensional Systems . . . .
. . . . . . . . . . . . . . . . . . . . 97214.2 Derivation of the Effective Potential . . . . . . . . . . . . . . . . . . 97614.3 Comparison with Schrödinger Quantum Mechanics . . . . . . . . . . 98014.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98114.4.1 Radial Harmonic Oscillator and Morse System . . .
. . . . 98114.4.2 Radial Coulomb System and Morse System . . . . . . . . . 98314.4.3 Equivalence of Radial Coulomb System and Radial Oscillator 98414.4.4 Angular Barrier near Sphere, and Rosen-Morse Potential . 99214.4.5 Angular Barrier near Four-Dimensional Sphere, and General Rosen-Morse Potential . . . . . . . . . . .
. . . . . . . 99514.4.6 Hulthén Potential and General Rosen-Morse Potential . . . 99714.4.7 Extended Hulthén Potential and General Rosen-Morse Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .100014.5 D-Dimensional Systems . . . . . . . . . . . . . . . . . . . .
. . . . .100014.6 Path Integral of the Dionium Atom . . . . . . . . . . . . . . . . . .100214.6.1 Formal Solution . . . . . . . . . . . . . . . . . . . . . . . .100314.6.2 Absence of Time Slicing Corrections . . . . . . . . . . . . .1007xxviii14.7 Time-Dependent Duru-Kleinert TransformationAppendix 14A Affine Connection of Dionium Atom .Appendix 14B Algebraic Aspects of Dionium StatesNotes and References . . . . . .
. . . . . . . . . . . ..............................................1010.1013.1014.101415 Path15.115.215.315.415.5Integrals in Polymer Physics1016Polymers and Ideal Random Chains . . . . . . . . . . . . . . . . . .1016Moments of End-to-End Distribution . . . . . . . . . . . . . . . . .1018Exact End-to-End Distribution in Three Dimensions . . .
. . . . . .1021Short-Distance Expansion for Long Polymer . . . . . . . . . . . . .1023Saddle Point Approximation to Three-Dimensional End-to-End Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .102515.6 Path Integral for Continuous Gaussian Distribution . .
. . . . . . .102615.7 Stiff Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .102915.7.1 Sliced Path Integral . . . . . . . . . . . . . . . . . . . . . .103115.7.2 Relation to Classical Heisenberg Model . . . . . .
. . . . . .103215.7.3 End-to-End Distribution . . . . . . . . . . . . . . . . . . . .103415.7.4 Moments of End-to-End Distribution . . . . . . . . . . . . .103415.8 Continuum Formulation . . . . . . . . . . . . . . . . . . . . . . . . .103515.8.1 Path Integral . .
. . . . . . . . . . . . . . . . . . . . . . . .103515.8.2 Correlation Functions and Moments . . . . . . . . . . . . .103615.9 Schrödinger Equation and Recursive Solution for Moments . . . . .104015.9.1 Setting up the Schrödinger Equation . . . . . . . . . . . . .104015.9.2 Recursive Solution of Schrödinger Equation. . . . . . . . . .104115.9.3 From Moments to End-to-End Distribution for D = 3 . . .104515.9.4 Large-Stiffness Approximation to End-to-End Distribution .104615.9.5 Higher Loop Corrections . .
. . . . . . . . . . . . . . . . .105215.10 Excluded-Volume Effects . . . . . . . . . . . . . . . . . . . . . . . .106015.11 Flory’s Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . .106815.12 Polymer Field Theory . . . . . . . . . . . . . . . . . .
. . . . . . . .106815.13 Fermi Fields for Self-Avoiding Lines . . . . . . . . . . . . . . . . . .1076Appendix 15A Basic Integrals . . . . . . . . . . . . . . . . . . . . . . . .1076Appendix 15B Loop Integrals . . . . . . . . . . . . . . . . . . . . . . . . .1077Appendix 15C Integrals Involving Modified Green Function . . . . . . . .1079Notes and References . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . .108016 Polymers and Particle Orbits in Multiply Connected Spaces16.1 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 . . . . . . . . . . . . ...............1082.1082.1086.1095.1100.1114.1117.1123H. Kleinert, PATH INTEGRALSxxix16.8 Entangled Pair of Polymers . .
. . . . . . . . . . . . . . . . . . . . .112716.8.1 Polymer Field Theory for Probabilities . . . . . . . . . . . .112916.8.2 Calculation of Partition Function . . . . . . . . . . . . . . .113016.8.3 Calculation of Numerator in Second Moment . . .
. . . . .113216.8.4 First Diagram in Fig. 16.23 . . . . . . . . . . . . . . . . . .113316.8.5 Second and Third Diagrams in Fig. 16.23 . . . . . . . . . .113416.8.6 Fourth Diagram in Fig. 16.23 . . . . . . . . . . . . . . . . .113516.8.7 Second Topological Moment . . . . .
. . . . . . . . . . . . .113616.9 Chern-Simons Theory of Statistical Interaction . . . . . . . . . . . .113716.10 Second-Quantized Anyon Fields . . . . . . . . . . . . . . . . . . . .114016.11 Fractional Quantum Hall Effect . . . . . . . . . . . . . .
. . . . . .114316.12 Anyonic Superconductivity . . . . . . . . . . . . . . . . . . . . . . .114616.13 Non-Abelian Chern-Simons Theory . . . . . . . . . . . . . . . . . .1148Appendix 16A Calculation of Feynman Diagrams in Polymer Entanglement1151Appendix 16B Kauffman and BLM/Ho polynomials . . . . . . . . .
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