Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 5
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. . . . . . . .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 Problem5.21.4 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, etc. . . . . . . . . .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 . . . . . . .5.23.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . .Appendix 5A Feynman Integrals for T 6= 0 without Zero Frequency . . .Appendix 5B Proof of Scaling Relation for the Extrema of WN .
. . . .Appendix 5C Second-Order Shift of Polaron Energy . . . . . . . . . . . .Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 Path6.16.26.36.4Integrals with Topological ConstraintsPoint Particle on Circle . . . . . . . . . . . . . . . . .Infinite Wall . . . . . . .
. . . . . . . . . . . . . . . .Point Particle in Box . . . . . . . . . . . . . . . . . .Strong-Coupling Theory for Particle in Box . . . . . .6.4.1Partition Function . . . . . . . . . . . . . . .6.4.2Perturbation Expansion . . . . . . . . . . . .6.4.3Variational Strong-Coupling Approximations6.4.4Special Properties of Expansion . .
. . . . . .6.4.5Exponentially Fast Convergence . . . . . . . .Notes and References . . . . . . . . . . . . . . . . . . . . . .......................................................................516519519521522524525529531533539539540541544544546548550554556565567570570576. 576. 580. 584. 587.
588. 588. 590. 592. 593. 5947 Many Particle Orbits — Statistics and Second Quantization5967.1 Ensembles of Bose and Fermi Particle Orbits . . . . . . . . . . . . . 5977.2 Bose-Einstein Condensation . . . . . . . . . . . . . . . . . . . . . . . 6047.2.1Free Bose Gas . . . .
. . . . . . . . . . . . . . . . . . . . . 604xxiv7.37.47.57.67.77.87.97.107.117.127.137.147.15Notes8 Path8.18.28.38.48.58.68.78.87.2.2Effect of Interactions . . . . . . . . . . . . . . . . . . . . .7.2.3Bose-Einstein Condensation in Harmonic Trap . . . . . .7.2.4Entropy and Specific Heat . . . . .
. . . . . . . . . . . . .7.2.5Interactions 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 . . . . . . . . . . . . . . . . . . . . . . .7.10.1 Grassmann Variables . . . . . . . . . . . . . . . . . . . . .7.10.2 Fermionic Functional Determinant . . . . . . . . . . . . .7.10.3 Coherent States for Fermions . . . . . . . . . . . . . . . .Hilbert Space of Quantized Grassmann Variable . . . . . . . . . .7.11.1 Single Real Grassmann Variable . . . .
. . . . . . . . . .7.11.2 Quantizing Harmonic Oscillator with Grassmann Variables7.11.3 Spin System with Grassmann Variables . . . . . . . . . .External Sources in a∗ , a -Path Integral . . . . . . . . . . . . . . .Generalization to Pair Terms . . . . . . . . . . . . . . . . . . . . .Spatial Degrees of Freedom . . . . . . . . . .
. . . . . . . . . . . .7.14.1 Grand-Canonical Ensemble of Particle Orbits from FreeFluctuating Field . . . . . . . . . . . . . . . . . . . . . . .7.14.2 First versus Second Quantization . . . . . . . . . . . . . .7.14.3 Interacting Fields . . . . . . . . . . .
. . . . . . . . . . . .7.14.4 Effective Classical Field Theory . . . . . . . . . . . . . . .Bosonization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7.15.1 Collective Field . . . . . . . . . . . . . . . . . . . . . . . .7.15.2 Bosonized versus Original Theory . . . .
. . . . . . . . . .and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......................613619627630634639644645648654658658659661665667667670671676678680........680682683684686687689691Integrals in Polar and Spherical Coordinates695Angular Decomposition in Two Dimensions .
. . . . . . . . . . . . . 695Trouble with Feynman’s Path Integral Formula in Radial Coordinates 698Cautionary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 702Time Slicing Corrections . . . . . . . . . . . . . . . . . . . . . .
. . 705Angular Decomposition in Three and More Dimensions . . . . . . . 7098.5.1Three Dimensions . . . . . . . . . . . . . . . . . . . . . . . 7108.5.2D Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 712Radial Path Integral for Harmonic Oscillator and Free Particle . . . . 718Particle near the Surface of a Sphere in D Dimensions . . . .
. . . . 719Angular Barriers near the Surface of a Sphere . . . . . . . . . . . . . 7228.8.1Angular Barriers in Three Dimensions . . . . . . . . . . . . 722H. Kleinert, PATH INTEGRALSxxv8.8.2Angular Barriers in Four DimensionsMotion 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 . . . . . . . . . . . . . . . .and References . . . . . . . . . . . . . . . . .........................................................................................................7277327367397407457457479 Wave Functions9.1 Free Particle in D Dimensions . . .
. . . . .9.2 Harmonic Oscillator in D Dimensions . . . .9.3 Free Particle from ω → 0 -Limit of Oscillator9.4 Charged Particle in Uniform Magnetic Field9.5 Dirac δ-Function Potential . . . . . . . . . .Notes and References . . . . . . . . . . . . . . . . ...............................................................................7497497527587607677698.98.108.118.128.138.14Notes10 Spaces with Curvature and Torsion10.1 Einstein’s Equivalence Principle .
. . . . . . . . . . . . . . . . . .10.2 Classical Motion of Mass Point in General Metric-Affine 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 andTorsion . . . . . . . . . . . . . . . . . . . . . . . . . . . .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 Solution of Path Integral on Surface of Sphere . . . . .10.5 External Potentials and Vector Potentials . . . . . . . . . . . . . .10.6 Perturbative Calculation of Path Integrals in Curved Space . . . .10.6.1 Free and Interacting Parts 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 . . . . . . .770. 771. 772. 772. 775. 781....................781786786791797799801801804806808810811818819820822822824825xxvi10.10 Simple Rules for Calculating Singular Integrals . .
. . . . . . . . . . 82710.11 Perturbative Calculation on Finite Time Intervals . . . . . . . . . . 83210.11.1 Diagrammatic Elements . . . . . . . . . . . . . . . . . . . . 83310.11.2 Cumulant Expansion of D-Dimensional Free-Particle Amplitude in Curvilinear Coordinates . . . . . . . . . . .
. . . 83410.11.3 Propagator in 1 − ε Time Dimensions . . . . . . . . . . . . 83610.11.4 Coordinate Independence for Dirichlet Boundary Conditions 83710.11.5 Time Evolution Amplitude in Curved Space . . . . . . . . . 84310.11.6 Covariant Results for Arbitrary Coordinates . . . . . . . . . 84910.12 Effective Classical Potential in Curved Space .
. . . . . . . . . . . . 85410.12.1 Covariant Fluctuation Expansion . . . . . . . . . . . . . . . 85510.12.2 Arbitrariness of q0µ . . . . . . . . . . . . . . . . . . . . . . . 85810.12.3 Zero-Mode Properties . . . . . . . . . . . . . . . . . . . . . 85910.12.4 Covariant Perturbation Expansion . . . . . . . . .
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