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Dittrich W., Reuter M. Classical and Quantum Dynamics: From Classical Paths to Path Integrals

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Dittrich W., Reuter M. Classical and Quantum Dynamics: From Classical Paths to Path Integrals
5th Edition. — Springer International Publishing AG, 2017. — 482 p. — ISBN: 3319582976.
The subject of this monograph is classical and quantum dynamics. We are fully aware that this combination is somewhat unusual, for history has taught us convincingly that these two subjects are founded on totally different concepts; a smooth transition between them has so far never been made and probably never will.
An approach to quantum mechanics in purely classical terms is doomed to failure; this fact was well known to the founders of quantum mechanics. Nevertheless, to this very day people are still trying to rescue as much as possible of the description of classical systems when depicting the atomic world. However, the currently accepted viewpoint is that in describing fundamental properties in quantum mechanics, we are merely borrowing names from classical physics. In writing this book we have made no attempt to contradict this point of view. But in the light of modern topological methods we have tried to bring a little twist to the standard approach that treats classical and quantum physics as disjoint subjects.
Contents
Introduction
The Action Principles in Mechanics
The Action Principle in Classical Electrodynamics
Application of the Action Principles
Jacobi Fields, Conjugate Points
Canonical Transformations
The Hamilton–Jacobi Equation
Action-Angle Variables
The Adiabatic Invariance of the Action Variables
Time-Independent Canonical Perturbation Theory
Canonical Perturbation Theory with Several Degrees of Freedom
Canonical Adiabatic Theory
Removal of Resonances
Superconvergent Perturbation Theory, KAM Theorem (Introduction)
Poincaré Surface of Sections, Mappings
The KAM Theorem
Fundamental Principles of Quantum Mechanics
Functional Derivative Approach
Examples for Calculating Path Integrals
Direct Evaluation of Path Integrals
Linear Oscillator with Time-Dependent Frequency
Propagators for Particles in an External Magnetic Field
Simple Applications of Propagator Functions
The WKB Approximation
Computing the Trace
Partition Function for the Harmonic Oscillator
Introduction to Homotopy Theory
Classical Chern–Simons Mechanics
Semiclassical Quantization
The “Maslov Anomaly” for the Harmonic Oscillator
Maslov Anomaly and the Morse Index Theorem
Berry’s Phase
Classical Geometric Phases: Foucault and Euler
Berry Phase and Parametric Harmonic Oscillator
Topological Phases in Planar Electrodynamics
Path Integral Formulation of Quantum Electrodynamics
Particle in Harmonic E-Field E.t/ D Esin!0t; Schwinger–Fock Proper-TimeMethod
The Usefulness of Lie Brackets: From Classical and Quantum Mechanics to Quantum Electrodynamics
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