Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006, страница 7
Описание файла
PDF-файл из архива "Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006", который расположен в категории "". Всё это находится в предмете "математика" из , которые можно найти в файловом архиве . Не смотря на прямую связь этого архива с , его также можно найти и в других разделах. Архив можно найти в разделе "книги и методические указания", в предмете "математика" в общих файлах.
Просмотр PDF-файла онлайн
Текст 7 страницы из PDF
. . .1152Appendix 16C Skein Relation between Wilson Loop Integrals . . . . . . .1153Appendix 16D London Equations . . . . . . . . . . . . . . . . . . . . . . .1156Appendix 16E Hall Effect in Electron Gas . . . . . . . . . . . . . . . . . .1157Notes and References . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .115717 Tunneling116317.1 Double-Well Potential . . . . . . . . . . . . . . . . . . . . . . . . . .116317.2 Classical Solutions — Kinks and Antikinks . . . . . . . . . . . . . .116617.3 Quadratic Fluctuations . . . . . . .
. . . . . . . . . . . . . . . . . .117017.3.1 Zero-Eigenvalue Mode . . . . . . . . . . . . . . . . . . . . .117617.3.2 Continuum Part of Fluctuation Factor . . . . . . . . . . . .118017.4 General Formula for Eigenvalue Ratios . . . . . . . . . . . . .
. . .118217.5 Fluctuation Determinant from Classical Solution . . . . . . . . . . .118417.6 Wave Functions of Double-Well . . . . . . . . . . . . . . . . . . . . .118817.7 Gas of Kinks and Antikinks and Level Splitting Formula . . . . . . .118917.8 Fluctuation Correction to Level Splitting . . . . . . . . . . . . .
. .119317.9 Tunneling and Decay . . . . . . . . . . . . . . . . . . . . . . . . . .119817.10 Large-Order Behavior of Perturbation Expansions . . . . . . . . . .120717.10.1 Growth Properties of Expansion Coefficients . . . . . . . . .120717.10.2 Semiclassical Large-Order Behavior . . . . . . . . . . . . . .121117.10.3 Fluctuation Correction to the Imaginary Part and LargeOrder Behavior . . . .
. . . . . . . . . . . . . . . . . . . . .121617.10.4 Variational Approach to Tunneling. Perturbation Coefficients to All Orders . . . . . . . . . . . . . . . . . . . . . .121817.10.5 Convergence of Variational Perturbation Expansion . . . . .122617.11 Decay of Supercurrent in Thin Closed Wire . . . . .
. . . . . . . . .123517.12 Decay of Metastable Thermodynamic Phases . . . . . . . . . . . . .124617.13 Decay of Metastable Vacuum State in Quantum Field Theory . . . .125317.14 Crossover from Quantum Tunneling to Thermally Driven Decay . .1255xxxAppendix 17A Feynman Integrals for Fluctuation Correction . . . . . .
.1256Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .125918 Nonequilibrium Quantum Statistics126218.1 Linear Response and Time-Dependent Green Functions for T 6= 0 . .126218.2 Spectral Representations of Green Functions for T 6= 0 . . . . . . .126518.3 Other Important Green Functions . . . . . . . . . . .
. . . . . . . .126818.4 Hermitian Adjoint Operators . . . . . . . . . . . . . . . . . . . . . .127118.5 Harmonic Oscillator Green Functions for T 6= 0 . . . . . . . . . . . .127218.5.1 Creation Annihilation Operators . . . . . . . . . . . . . . .127218.5.2 Real Field Operators . . .
. . . . . . . . . . . . . . . . . . .127518.6 Nonequilibrium Green Functions . . . . . . . . . . . . . . . . . . . .127718.7 Perturbation Theory for Nonequilibrium Green Functions . . . . . .128718.8 Path Integral Coupled to Thermal Reservoir . . . . . . . . . . . . .128918.9 Fokker-Planck Equation . . . . . . . . . . . . . . . . . .
. . . . . . .129518.9.1 Canonical Path Integral for Probability Distribution . . . .129618.9.2 Solving the Operator Ordering Problem . . . . . . . . . . .129818.9.3 Strong Damping . . . . . . . . . . . . . . . . . . . . . . . .130418.10 Langevin Equations . . . . . . . . . . . . .
. . . . . . . . . . . . . .130718.11 Stochastic Quantization . . . . . . . . . . . . . . . . . . . . . . . . .131118.12 Stochastic Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . .131418.12.1 Kubo’s stochastic Liouville equation . . . . . . . . . . . . .131418.12.2 From Kubo’s to Fokker-Planck Equations . . . . . . . . .
.131518.12.3 Itô’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . .131818.13 Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . .132218.14 Stochastic Quantum Liouville Equation . . . . . . . . . . . . .
. . .132518.15 Master Equation for Time Evolution . . . . . . . . . . . . . . . . . .132718.16 Relation to Quantum Langevin Equation . . . . . . . . . . . . . . .133018.17 Electromagnetic Dissipation and Decoherence . . . . . . . . . . . . .133018.17.1 Forward–Backward Path Integral . . . . . . . . . .
. . . . .133118.17.2 Master Equation for Time Evolution in Photon Bath . . .133518.17.3 Line Width . . . . . . . . . . . . . . . . . . . . . . . . . . .133618.17.4 Lamb shift . . . . . . . . . . . . . . . . . . . . . . . . . . .133818.17.5 Langevin Equations . . . . . .
. . . . . . . . . . . . . . . .134118.18 Fokker-Planck Equation in Spaces with Curvature and Torsion . . .134318.19 Stochastic Interpretation of Quantum-Mechanical Amplitudes . . . .134418.20 Stochastic Equation for Schrödinger Wave Function . . . . . . . . .134618.21 Real Stochastic and Deterministic Equation for Schrödinger WaveFunction . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .134818.21.1 Stochastic Differential Equation . . . . . . . . . . . . . . . .134918.21.2 Equation for Noise Average . . . . . . . . . . . . . . . . . .134918.21.3 Harmonic Oscillator . . . . . . . . . . . . . . . .
. . . . . .135018.21.4 General Potential . . . . . . . . . . . . . . . . . . . . . . . .135118.21.5 Deterministic Equation . . . . . . . . . . . . . . . . . . . .1352H. Kleinert, PATH INTEGRALSxxxi18.22 Heisenberg Picture for Probability Evolution . . . .Appendix 18A Inequalities for Diagonal Green FunctionsAppendix 18B General Generating Functional .
. . . . .Appendix 18C Wick Decomposition of Operator ProductsNotes and References . . . . . . . . . . . . . . . . . . . . ................19 Relativistic Particle Orbits19.1 Special Features of Relativistic Path Integrals . . . . . . .19.2 Proper Action for Fluctuating Relativistic Particle Orbits19.2.1 Gauge-Invariant Formulation . .
. . . . . . . . .19.2.2 Simplest Gauge Fixing . . . . . . . . . . . . . . .19.2.3 Partition Function of Ensemble of Closed Particle19.2.4 Fixed-Energy Amplitude . . . . . . . . . . . . . .19.3 Tunneling in Relativistic Physics . . . . . . . . . . . . . .19.3.1 Decay Rate of Vacuum in Electric Field . . . . ......................1352.1356.1359.1364.1365.
. . .. . . .. . . .. . . .Loops. . . .. . . .. . . .........1370.1372.1375.1375.1377.1379.1380.1381.1381...........................1370.1370.1372.1374.1376.1381.1384.1384.1386.1387.1388.1391.1392.1394.1395.1396.1397.1399.1400.1404.1406.1406.1408.1409.1409.1411.1414.141420 Path Integrals and Financial Markets20.1 Fluctuation Properties of Financial Assets . . .
. . . . . . . .20.1.1 Harmonic Approximation to Fluctuations . . . . . .20.1.2 Lévy Distributions . . . . . . . . . . . . . . . . . . .20.1.3 Truncated Lévy Distributions . . . . . . . . . . . . .20.1.4 Asymmetric Truncated Lévy Distributions . . . . . .20.1.5 Gamma Distribution . . . . . . . . . . .
. . . . . . .20.1.6 Boltzmann Distribution . . . . . . . . . . . . . . . .20.1.7 Student or Tsallis Distribution . . . . . . . . . . . .20.1.8 Meixner Distributions . . . . . . . . . . . . . . . . .20.1.9 Generalized Hyperbolic Distributions . . . . . . . . .20.1.10 Debye-Waller Factor for Non-Gaussian Fluctuations20.1.11 Path Integral for Non-Gaussian Distribution . . .
. .20.1.12 Time Evolution of Distribution . . . . . . . . . . . .20.1.13 Central Limiting Theorem . . . . . . . . . . . . . . .20.1.14 Additivity Property of Noises and Hamiltonians . . .20.1.15 Lévy-Khintchine Formula . . . . . . . . . . . . .
. .20.1.16 Semigroup Property of Asset Distributions . . . . . .20.1.17 Time Evolution of Moments of Distribution . . . . .20.1.18 Fokker-Planck-Type Equation . . . . . . . . . . . . .20.2 Itô-like Formula for Non-Gaussian Distributions . . . . . . .20.2.1 Contiuous 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 . .
. . . . ............................................................xxxii20.4.2 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 Fluctuations20.7.6 Option Pricing for Fluctuating Variance . . . . . . . .20.7.7 Perturbation Expansion and Smile . . . . . . . . . . .Appendix 20A Large-x Behavior of Truncated Lévy Distribution . .Appendix 20B Gaussian Weight . . . . . . . . . . . . . . .
. . . . .Appendix 20C Comparison with Dow-Jones Data . . . . . . . . . . .Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . .Index...........................................1415.1418.1419.1421.1424.1426.1428.1428.1430.1431.1432.1433.1437.1439.1441.1444.1446.1449.1451.1452.14531461H. Kleinert, PATH INTEGRALSList of Figures1.11.21.31.4Probability distribution of particle behind a double slit . . .
. .P2πiµnRelevant function Nin Poisson’s summation formulan=−N eIllustration of time-ordering procedure . . . . . . . . . . . . . . .Triangular closed contour for Cauchy integral . . . . . . . . . . .........123036852.12.22.32.4Zigzag paths, along which a point particle fluctuates . .Solution of equation of motion . .
