Path Integral Methods and Their Applications

The following sections are included:

• PREFACE

• CONTENTS

The following sections are included:

• Introduction

• Feynman Path Integral
  • Feynman formulation
  • Relation to other approaches of quantum description
  • Functional calculus

• Random Walk, Brownian Motion and Wiener Integral
  • Random walk and Brownian motion
  • Wiener integral

• Trotter Product Formula and Alternative Derivation of Path Integral

• Path Integral Subtleties

The following sections are included:

• Introduction

• Derivation of the Propagator
  • Evaluation of limiting expression for D_N
  • Evaluation of S_c1

• Specific Cases

• Velocity dependent Potentials

• General Quadratic forms

• Propagator Beyond the First Singularity Of VPD

• Notes and References

The following sections are included:

• Elimination of Degrees of Freedom
  • System coupled to a harmonic oscillator
  • An Electron in a random potential

• Two-time Quadratic Actions

• Extension to More General Non-local Actions

• Examples of Explicit Evaluation
  • Generalized one-time actions
  • Transiationally invariant two-time actions
  • Propagator for the polaron kernel

• Applications of Two-time Quadratic Actions
  • Exactly solvable model of electronic density of states (DOS)
  • Polymer distribution functions
  • Propagation of waves in random media

• Notes and References

The following sections are included:

• Introduction

• Path Integrals in Polar Coordinates
  • Polar coordinates in two-dimensions
  • Polar coordinates in three dimensions
  • Generalization to d-dimensional polar coordinates

• Examples of Explicit Evaluation
  • Free particle propagator
  • Rotor in d-dimensions
  • Central potentials
  • Non-spherically symmetric potentials

• Path Integration in General Curved Spaces
  • Quantization of classical Hamiltonian
  • Derivation of a path integral

• Notes and References

The following sections are included:

• Introduction

• Local Time Transformations in Classical Mechanics

• Concept of the Promotor

• Coordinate-Time Transformations in Path Integral

• Illustrative Examples
  • Coulomb potential
  • Morse potential

• Propagators Related to a Rigid Rotor
  • Infinite square well

• Coulomb Problem Based on KS Transformation

• Notes and References

The following sections are included:

• Examples of Constrained Path Integrals
  • Problems in Polymer Physics
  • Aharonov-Bohm effect

• The Constraint As a Functional

• Evaluation of The Path Integral
  • A simple entanglement problems

• Evaluation of The Propagator

• Propagators corresponding to More Than One Constraints

• Total Winding Index and Stochastic Area

• Statistical Mechanics of Entangled Polymers
  • The properties of entangled polymers
  • The properties of entangled polymers: Entanglement with clusters

• Aharonov-Bohm Effect

• Notes and References

The following sections are included:

• Introduction

• Classical and Quanta1 Invariants
  • Noether invariants
  • Derivation of invariant based on Hamiltonian description
  • Examples of invariants

• Schrödinger Equation and Invariants

• Feynman Propagator

• Invariants Quadratic in Momentum and the Propagator
  • Illustrative examples

• Role Played by the Invariant I

• Global Time Transformation in Feynman Path Integral

• Notes and References

The following sections are included:

• Introduction

• The Cumulant Approximation

• Spectrum of Positionally Disordered Systems
  • Basic formulation
  • Evaluation of G
  • The behaviour of density of states
  • The density of states for Gaussian correlation

• The Polaron Problem
  • Free energy and ground state energy
  • The effective mass

• The Bi-Polaron Problem

• Polymer Distribution Functions

• Notes and References

The following sections are included:

• Introduction

• The Perturbation Series

• One-Dimensional Delta-function Potential

• Inverse Square Potential

• The Coulomb Potential

• General Formulation
  • Inverse-square potential (1-dimensional)
  • Harmonic oscillator (1-dimensional)
  • Coulomb potential
  • The singular potentials

• Notes and References

The following sections are included:

• Introduction

• Asymptotic Analysis

• Semiclassical Approximation

• A Particle in an Inverse Square Potential

• Notes and References

The following sections are included:

• Introduction

• Monte Carlo Method

• Path Integral by Monte Carlo

• Deterministic techniques of path summation

• Notes and References

The following sections are included:

• Introduction

• Definition through Limiting Procedures
  • Trotter product formula
  • Generalized Gaussian integrals

• Definition Through White Noise Calculus
  • White noise calculus
  • Feynman propagator
  • A new expansion for the propagator

• Notes and References