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Path Integrals in Quantum Mechanics Statistics Polymer Physics and Financial Markets 3rd Edition by Hagen Kleinert ISBN 9789812381064 9812381066

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Path Integrals in Quantum Mechanics Statistics Polymer Physics and Financial Markets 3rd Edition Hagen Kleinert Digital Instant Download

Author(s): Hagen Kleinert
ISBN(s): 9789812381064, 9812381066
Edition: 3rd
File Details: PDF, 71.63 MB
Year: 2004
Language: english
SKU: EB-1531664 Category: Tag:

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ISBN 10: 9812381066
ISBN 13: 9789812381064
Author: Hagen Kleinert

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 defining uniquely integrals over products of distributions. The powerful Feynman–Kleinert variational approach is explained and developed systematically into a variational perturbation theory which, in contrast to ordinary perturbation theory, produces convergent expansions. The convergence is uniform from weak to strong couplings, opening a way to precise approximate evaluations of analytically unsolvable path integrals. Tunneling processes are treated in detail. The results are used to determine the lifetime of supercurrents, the stability of metastable thermodynamic phases, and the large-order behavior of perturbation expansions. A new variational treatment extends the range of validity of previous tunneling theories from large to small barriers. A corresponding extension of large-order perturbation theory also applies now to small orders. Special attention is devoted to path integrals with topological restrictions. These are relevant to the understanding of the statistical properties of elementary particles and the entanglement phenomena in polymer physics and biophysics. The Chern–Simons theory of particles with fractional statistics (anyons) is introduced and applied to explain the fractional quantum Hall effect. The relevance of path integrals to financial markets is discussed, and improvements of the famous Black–Scholes formula for option prices are given which account for the fact that large market fluctuations occur much more frequently than in the commonly used Gaussian distributions. The author’s other book on ‘Critical Properties of f4 Theories’ gives a thorough introduction to the field of critical phenomena and develops new powerful resummation techniques for the extraction of physical results from the divergent perturbation expansions.

Path Integrals in Quantum Mechanics Statistics Polymer Physics and Financial Markets 3rd Edition Table of contents:

Part 1: Foundations of Path Integrals

  1. Introduction to Path Integrals

    • Historical Background and Feynman’s Formulation
    • Basic Concepts of Functional Integration
    • Classical and Quantum Paths

  2. Path Integrals in Quantum Mechanics

    • The Path Integral Representation of the Propagator
    • Derivation from Schrödinger’s Equation
    • Applications to Simple Quantum Systems

  3. Mathematical Framework of Functional Integration

    • Wiener and Feynman Path Integrals
    • Discretization and Continuum Limits
    • Measure Theory and Convergence Issues

Part 2: Applications in Quantum Mechanics and Field Theory

  1. Quantum Harmonic Oscillator and Coherent States

    • Path Integral Solution of the Harmonic Oscillator
    • Semi-Classical Approximations and the WKB Method
    • Coherent State Path Integrals

  2. Gauge Theories and Quantum Fields

    • Path Integrals in Quantum Electrodynamics (QED)
    • Functional Methods in Quantum Field Theory
    • Fermionic Path Integrals and Grassmann Variables

  3. Instantons and Non-Perturbative Effects

    • Tunneling Phenomena in Quantum Systems
    • Instantons in Quantum Mechanics and Field Theory
    • Topological Effects and Solitons

Part 3: Statistical Physics and Polymer Physics

  1. Path Integrals in Statistical Mechanics

    • Partition Functions and Statistical Averages
    • Connection Between Quantum and Statistical Path Integrals
    • Transfer Matrix Formalism

  2. Polymer Physics and Random Walks

    • Path Integrals in Polymer Chain Models
    • Applications to Brownian Motion and Diffusion Processes
    • Scaling Laws and Renormalization in Polymer Physics

  3. Stochastic Processes and Langevin Dynamics

    • The Fokker-Planck Equation and Path Integrals
    • Applications to Stochastic Differential Equations
    • Nonequilibrium Statistical Mechanics

Part 4: Financial Markets and Stochastic Analysis

  1. Path Integrals in Finance
  • Brownian Motion and the Black-Scholes Model
  • Functional Integration in Option Pricing
  • Risk-Neutral Measures and Martingale Methods
  1. Stochastic Control and Optimal Trading Strategies
  • Path Integral Formulation of Stochastic Control
  • Application to Portfolio Optimization
  • Non-Gaussian and Heavy-Tailed Distributions in Finance
  1. Anomalous Diffusion and Market Dynamics
  • Fractal Models and Levy Paths in Financial Markets
  • Path Integral Methods for Market Fluctuations
  • Applications to High-Frequency Trading and Algorithmic Finance

Part 5: Advanced Topics and Future Directions

  1. Supersymmetry and Path Integrals
  • SUSY in Quantum Mechanics
  • Applications to Stochastic and Disordered Systems
  • Supersymmetric Quantum Field Theories
  1. Functional Renormalization Group Methods
  • Wilsonian Renormalization and Effective Actions
  • Applications in Condensed Matter Physics
  • Non-Perturbative Techniques in QFT
  1. Recent Developments and Open Problems
  • Path Integrals Beyond Gaussian Approximations
  • Quantum Computing and Functional Integration
  • Future Perspectives in Physics, Biology, and Economics

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Tags: Hagen Kleinert, Path Integrals, Quantum Mechanics Statistics, Polymer Physics, Financial Markets