Classical and Quantum Dynamics -- From Classical Paths to Path Integrals

Description

Graduate students seeking to become familiar with advanced computational strategies in classical and quantum dynamics will find in this book both the fundamentals of a standard course and a detailed treatment of the time-dependent oscillator, Chern-Simons mechanics, the Maslov anomaly and the Berry phase, to name just a few topics. Well-chosen and detailed examples illustrate perturbation theory, canonical transformations and the action principle, and demonstrate the usage of path integrals. The sixth edition has been enlarged to include the Heisenberg-Euler Lagrangian, Schwinger’s source theory treatment of the low-energy π-ρ-N physics and general relativity, where Riemann’s (Einstein’s) ideas on space and time and their philosophical implications are discussed. Prof. Dr. Walter Dittrich was head of the quantum electrodynamics group at Tübingen University. His main research activities centered on gauge theories, in particular, on QED, stimulated in part by collaboration with Julian Schwinger. Walter Dittrich has worked for more than 20 years at centers like MIT, the Institute for Advanced Study at Princeton and the National Accelerator Laboratory at Stanford (SLAC). He has over 30 years of teaching experience and is one of the key scientists in developing the theoretical framework of quantum electrodynamics. Prof. Dr. Martin Reuter is head of the quantum Einstein gravity group at the Institute for High Energy Physics at Mainz University. His research focuses on particle physics, quantum field theory and quantum Einstein gravity. He worked at the synchrotron facility DESY and the large hadron collider at CERN. He has more than 30 years of teaching experience in theoretical physics. Preface to the Sixth Edition......Page 5 Preface to the First Edition......Page 7 Contents......Page 8 1 Introduction......Page 10 2 The Action Principles in Mechanics......Page 12 3 The Action Principle in Classical Electrodynamics......Page 26 4 Application of the Action Principles......Page 32 5 Jacobi Fields, Conjugate Points......Page 54 6 Canonical Transformations......Page 67 7 The Hamilton–Jacobi Equation......Page 83 8 Action-Angle Variables......Page 101 9 The Adiabatic Invariance of the Action Variables......Page 126 10 Time-Independent Canonical Perturbation Theory......Page 139 11 Canonical Perturbation Theory with Several Degrees of Freedom......Page 147 12 Canonical Adiabatic Theory......Page 163 13 Removal of Resonances......Page 170 14 Superconvergent Perturbation Theory, KAM Theorem (Introduction)......Page 180 15 Poincaré Surface of Sections, Mappings......Page 189 16 The KAM Theorem......Page 200 17 Fundamental Principles of Quantum Mechanics......Page 208 18 Functional Derivative Approach......Page 213 19 Examples for Calculating Path Integrals......Page 225 20 Direct Evaluation of Path Integrals......Page 248 21 Linear Oscillator with Time-Dependent Frequency......Page 260 22 Propagators for Particles in an External Magnetic Field......Page 276 23 Simple Applications of Propagator Functions......Page 282 24 The WKB Approximation......Page 300 25 Computing the Trace......Page 312 26 Partition Function for the Harmonic Oscillator......Page 318 27 Introduction to Homotopy Theory......Page 325 28 Classical Chern–Simons Mechanics......Page 331 29 Semiclassical Quantization......Page 344 30 The ``Maslov Anomaly'' for the Harmonic Oscillator......Page 351 31 Maslov Anomaly and the Morse Index Theorem......Page 360 32 Berry's Phase......Page 367 33 Classical Geometric Phases: Foucault and Euler......Page 385 34 Berry Phase and Parametric Harmonic Oscillator......Page 404 35 Topological Phases in Planar Electrodynamics......Page 419 36 Path Integral Formulation of Quantum Electrodynamics......Page 429 37 Particle in Harmonic E-Field E(t)= E sinω0 t; Schwinger–Fock Proper-Time Method......Page 438 38 The Usefulness of Lie Brackets: From Classical and Quantum Mechanics to Quantum Electrodynamics......Page 451 39 Green's Function of a Spin- eepic12 Particle in a Constant External Magnetic Field......Page 476 40 One-Loop Effective Lagrangian in QED......Page 489 Riemann's Unit Sphere S2......Page 497 Riemann's Unit Sphere Sn......Page 499 Derivation of Riemann's Main Formula for ds2......Page 501 Calculation of Riemann's Curvature Tensor, Ricci Scalar, etc.; Maximally Symmetric Metric......Page 503 Realization of Chiral Symmetry in a Curved Isospin Space......Page 513 Schwinger's Source Treatment of the Low-Energy π-N System......Page 514 How the Pion Gets Its Mass......Page 522 42 The Non-Abelian Vector Gauge Particle ρ......Page 529 43 Riemann's Result and Consequences for Physics and Philosophy......Page 543 Bibliography......Page 546 Index......Page 549