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Walter Dittrich Martin Reuter Classical and Quantum Dynamics From Classical Paths to Path Integrals 6th Edition Classical and Quantum Dynamics Walter Dittrich • Martin Reuter Classical and Quantum Dynamics From Classical Paths to Path Integrals Sixth Edition 123 Walter Dittrich Institute of Theoretical Physics University of TRubingen TRubingen, Germany Martin Reuter Institute of Physics Johannes Gutenberg University Mainz, Germany ISBN 978-3-030-36785-5 ISBN 978-3-030-36786-2 (eBook) 1st edition, 2nd edition, 3rd edition: © Springer-Verlag Berlin Heidelberg 1992, 1994, 2001 4th edition, 5th edition: © Springer International Publishing AG 2016, 2017 © Springer Nature Switzerland AG 2020 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations This Springer imprint is published by the registered company The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface to the Sixth Edition In the sixth edition of this book, we have added some new chapters Chapter 39 continues and brings to a close the explicit calculations of the propagation function of a charged spin- 21 particle in a constant electromagnetic field But now we adopt Fock’s and Schwinger’s proper-time method instead of applying the Van Vleck determinant as done in Chap 38 This, in fact, signals the beginning of Schwinger’s withdrawal from operator quantum field theory and his return to c-numbers As it turned out, Schwinger’s article in Phys Rev of 1951 is one of the most-cited papers in the physics literature We will describe his approach in great detail, considering only those parts of his paper that are relevant to the calculations contained in our following chapters The emphasis lies on c-number field theory rather than on operators and their multiplication at the same space–time point because of their highly complicated high-energy behavior As a final application of Schwinger’s proper-time calculation, we derive in Chap 40 the explicit expression for the oneloop effective Lagrangian in QED This is the most elegant way to reproduce the famous Heisenberg-Euler Lagrangian (1936) and sets the stage for all future effective quantum field theories As an introduction to low-energy pion-rho-nucleon chiral physics, in Chaps 41 and 42 we rely on the geometrical as well as the dynamical approach The former is based on differential geometry as developed by Gauss, Riemann, Einstein, and Weyl, where chiral symmetry is realized in a curved isospin space The latter approach was favored by the late Schwinger Our contribution is a reassessment of Schwinger’s Brandeis lecture notes [63] of 1969; the details we present here have—so far as we know—never been published We devote our final Chap 43 to Riemann’s groundbreaking ideas contained in his Habilitation lecture (1854), where he touches on geometrical and physical spaces and consequences for physics and philosophy v vi Preface to the Sixth Edition Finally, at the close of this enterprise, we wish to express our profound thanks to Ms Ute Heuser of Springer-Verlag, who was kind enough to guide us over the years in preparing the various editions of this book for publication Tübingen, Germany Mainz, Germany January 2020 Walter Dittrich Martin Reuter Preface to the First Edition This volume is the result of the authors’ lectures and seminars given at Tübingen University and elsewhere It represents a summary of our learning process in nonlinear Hamiltonian dynamics and path integral methods in nonrelativistic quantum mechanics While large parts of the book are based on standard material, readers will find numerous worked examples which can rarely be found in the published literature In fact, toward the end they will find themselves in the midst of modern topological methods which so far have not made their way into the textbook literature One of the authors’ (W.D.) interest in the subject was inspired by Prof D Judd (UC Berkeley), whose lectures on nonlinear dynamics familiarized him with Lichtenberg and Lieberman’s monograph, Regular and Stochastic Motion (Springer, 1983) For people working in plasma or accelerator physics, the chapter on nonlinear physics should contain some familiar material Another influential author has been Prof J Schwinger (UCLA); the knowledgeable reader will not be surprised to discover our appreciation of Schwinger’s Action Principle in the introductory chapters However, the major portion of the book is based on Feynman’s path integral approach, which seems to be the proper language for handling topological aspects in quantum physics Our thanks go to Ginny Dittrich for masterly transforming a long and complex manuscript into a readable monograph Tübingen, Germany Hannover, Germany January 1992 Walter Dittrich Martin Reuter vii Contents Introduction The Action Principles in Mechanics 3 The Action Principle in Classical Electrodynamics 17 Application of the Action Principles 23 Jacobi Fields, Conjugate Points 45 Canonical Transformations 59 The Hamilton–Jacobi Equation 75 Action-Angle Variables 93 The Adiabatic Invariance of the Action Variables 119 10 Time-Independent Canonical Perturbation Theory 133 11 Canonical Perturbation Theory with Several Degrees of Freedom 141 12 Canonical Adiabatic Theory 157 13 Removal of Resonances 165 14 Superconvergent Perturbation Theory, KAM Theorem (Introduction) 175 15 Poincaré Surface of Sections, Mappings 185 16 The KAM Theorem 197 17 Fundamental Principles of Quantum Mechanics 205 18 Functional Derivative Approach 211 19 Examples for Calculating Path Integrals 223 20 Direct Evaluation of Path Integrals 247 ix x Contents 21 Linear Oscillator with Time-Dependent Frequency 259 22 Propagators for Particles in an External Magnetic Field 275 23 Simple Applications of Propagator Functions 281 24 The WKB Approximation 299 25 Computing the Trace 311 26 Partition Function for the Harmonic Oscillator 317 27 Introduction to Homotopy Theory 325 28 Classical Chern–Simons Mechanics 331 29 Semiclassical Quantization 345 30 The “Maslov Anomaly” for the Harmonic Oscillator 353 31 Maslov Anomaly and the Morse Index Theorem 363 32 Berry’s Phase 371 33 Classical Geometric Phases: Foucault and Euler 389 34 Berry Phase and Parametric Harmonic Oscillator 409 35 Topological Phases in Planar Electrodynamics 425 36 Path Integral Formulation of Quantum Electrodynamics 435 37 Particle in Harmonic E-Field E.t/ D E sin !0 t; Schwinger–Fock Proper-Time Method 445 38 The Usefulness of Lie Brackets: From Classical and Quantum Mechanics to Quantum Electrodynamics 459 39 Green’s Function of a Spin- 12 Particle in a Constant External Magnetic Field 485 40 One-Loop Effective Lagrangian in QED 499 41 On Riemann’s Ideas on Space and Schwinger’s Treatment of Low-Energy Pion-Nucleon Physics 507 42 The Non-Abelian Vector Gauge Particle 539 43 Riemann’s Result and Consequences for Physics and Philosophy 553 Bibliography 557 Index 561 Chapter Introduction 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 The formulation of both classical and quantum mechanics can be based on the principle of stationary action Schwinger has advanced this principle into a powerful working scheme which encompasses almost every situation in the classical and quantum worlds Our treatment will give a modest impression of the wide range of applicability of Schwinger’s action principle We then proceed to rediscover the importance of such familiar subjects as Jacobi fields, action angle variables, adiabatic invariants, etc in the light of current research on classical Hamiltonian dynamics It is here that we recognize the important role that canonical perturbation theory played before the advent of modern quantum mechanics Meanwhile, classical mechanics has been given fresh impetus through new developments in perturbation theory, offering a new look at old problems in nonlinear mechanics like, e.g., the stability of the solar system Here the KAM theorem proved that weakly disturbed integrable systems will remain on invariant surfaces (tori) for most initial conditions and not leave the tori to end up in chaotic motion © Springer Nature Switzerland AG 2020 W Dittrich, M Reuter, Classical and Quantum Dynamics, 42 The Non-Abelian Vector Gauge Particle 547 According to (42.10) the displacement of not only should induce an isotopic transformation on N but also on To achieve this we need higher powers in the terms; these enter because we are considering interactions Fortunately the calculation to realize this program has already been done in section “How the Pion Gets Its Mass” In the present case we merely need to make the following replacements in (41.99): g2 D m cD D2  f0 m Ã2 ; so that we find ı g2 m2 D ı' ı'/ ; or in (41.103) : ı D ı' C g2 m2  ı' 2 à C ı' : The remaining formula can be copied from the end of section “How the Pion Gets Its Mass”, e.g., ı4 1C g m2 Á2 D ı' uE D ıu D 1 g m2 1C g m2 g p 2m 1C g p 2m Á Á2 Á2 ; E Á2 ; p g ı'u4 ; m u4 D ıu4 D Á2 g 2m p 1C p Á2 g 2m 2 ; Eu2 C u24 D ; p g ı' u : m Let us summarize our findings by employing Weinberg’s famous relation between the mass of the non-Abelian vector gauge particles / and A1 1C /:  à  à p p mA mA 1080 mA D 2m ; D D 1:41 ; D D 1:42 : m th m exp 760 We have been considering a partial symmetry which resulted from the low mass of the pion It has the group structure O.4/, which we saw when we defined the three-dimensional isotopic vector u and u4 : uD 1C g mA Á g mA Á2 ; u4 D g mA 1C g mA Á2 Á2 ; u2 C u24 D ; 548 42 The Non-Abelian Vector Gauge Particle where these objects obey the transformations ı! W ıu D ı! ı' W ıu D 2g ı'u4 mA u; ; ıu4 D ; 2g ı' mA ıu4 D u: So far we introduced the four real quantities u˛ ; ˛ D 1; 2; 3; to define the unit vector in four-dimensional space n X u2˛ D : ˛D1 From here follows rotational invariance under O.4/ D O.3/ O.3/ with parameters Now, given a unit vector in four dimension, we can construct a unitary representation using the 2 real matrix: U D u4 C iu with U Ž D U T D u4 ; iu UU Ž D U Ž U D u24 C uE D ; up to a phase factor ei˛ for U An equivalent representation for U can be written as UD g E E C i p2m i pg E E 2m To write U in a more familiar form we introduce eO D E ; jOej D ; g g p E E D p 2m 2m E D eO ; E eO / D E eO / ; Then U takes the form UD C i E eO / : i E eO / Dp g 2m : 42 The Non-Abelian Vector Gauge Particle 549 As familiar from quantum mechanics this can be written as X  C i E eO Ã0 j.E eO /0 ih.E eO /0 j i E eO UD E eO /D˙1 1Ci j.E eO /0 D C1ih.E eO /0 D C1j i i C j.E eO /0 D 1ih.E eO /0 D 1j 1Ci D D i E eO C i C E eO C i 1Ci D 1C 2 C 1C g Dp 2m iE eO ; : Writing U D u4 C iE uE we can identify u4 D pg 2m 1C pg 2m Á2 Á2 2 Á pg 2m 1C pg 2m ; uE D ; sin  D With cos  D 1C 2 1C we rediscover u24 C uE D cos2  C sin2  D : Finally we have U D cos  C iE eO sin  ; D ei E eO D cos  C i eO D E E and since UŽ D e i E eO DU sin  ; E E Á2 : (42.20) 550 42 The Non-Abelian Vector Gauge Particle we obtain the unitarity condition UU Ž D U Ž U D : To write a matrix transcription of the gauge group structure we have for the infinitesimal isotopic rotation ıu4 D ; ıU D iı! u h D iı! ıu D ı! i ; iu u; h D iı! ;U i (42.21) and U C ıU D C iı! Á U Á iı! : Likewise for ıU D ıu D ı'u4 ; ıu4 D ı' u C i ı' u4 D fi ı' ı' u W ; Ug (42.22) and U C ıU D C i ı' Á U C i ı' Á : (42.23) This last equation is not a unitary transformation in the usual sense Also note that (42.21) has zero trace, meaning that only terms linear in can have this property On the other hand (42.22) has non-zero trace, because the one-term in ı' 1u The general infinitesimal transformation becomes U C ıU D C iı!C Á U iı! Á ; (42.24) ı!˙ D ı! ˙ ı' : Both factors in (42.24) are O.3/ representations in O.4/ D O.3/ O.3/ D SU.2/ SU.2/ : In ı!˙ D ı! ˙ ı' we have under space-reflection ı!˙ ! ı! , meaning that ı! is a scalar and ı D ı' is a pseudo-scalar O / Both SU.2/s violate parity Each gives a maximal parity violation However SU.2/ SU.2/ conserves parity ı!˙ has definite handedness, i.e., chirality This is the reason we called this transformation chiral transformation 42 The Non-Abelian Vector Gauge Particle 551 The results of these rather lengthy calculations show convincingly that there is no need for current algebra Also the model is useless as a phenomenological model of strong interactions It is the nature of a numerical effective Lagrangian to give a direct description of phenomena Also there is a fundamental difference: Current algebra uses concepts and numerical parameters derived from the analysis of weak interactions to predict some strong interaction effects We reverse the logical consequence and restore the primary role of strong interactions We also introduce a new chiral transformation The transformations are non-linear and thereby avoid reference to the non-physical field required for their linear transformations Meanwhile, it became clear that N Lagrangians have to be constructed as functional of fields which have a definite transformation under the chiral group SU.2/ SU.2/ Because a three-dimensional linear representation of that group does not exist, it is clear that the triplet isotopic pion field transforms according to the three-dimensional non-linear realization The Riemann geometry of the curved isotopic space replaces the Euclidean geometry of linear-representation spaces The SU.2/ SU.2/ group structure implies that the chiral symmetry is a pure interaction symmetry with similar consequences as those following from the gauge invariance of the second kind in QED, which is also a dynamical (experimental) symmetry Having said this, we return to Chap 41, where a non-linear realization of O.4/ is represented in a curved instead of a Euclidean space We showed that the chiral group is the invariance group of the metric in a three-dimensional space with constant curvature K D ˛12 This fundamental non-linear realization is associated with the pion field It is also interesting to see how the pion mass is generated in the lower-dimensional real isotopic world from the outside O.4/ world by breaking its symmetry down to O.3/ Pion production and interactions are exhibited simultaneously This is not unlike the perturbation of the O.4/ symmetry of the hydrogen atom with 1r potential Here it is an external constant magnetic field in the third direction that destroys the n12 degeneracy by splitting up the energy levels En When we wrote down (41.101), we exhibited a mathematical object which is transformed very simply But it is not necessarily the pion field It is always possible to redefine the pion field in such a way that the original physical identification in terms of non-interacting particles is not changed Nevertheless, the re-definition may change interactions In the absence of m2 , the pion field is undetermined up to an additive constant, ! C ı', where ı' is a constant isotopic vector Under this transformation, nothing happens in the partial-symmetry world, but something does happen in the real world Does this transformation allow us to recognize a wider group? Is this transformation an anomaly or does it play a wider role for interactions? More often than not, we think that an invariance with respect to, let us say, ı', will tell us something about interactions The reason is that particles and interactions have much in common; the properties of both are an aspect of inner dynamics A partial order of the non-interacting situation, i.e., a dimly perceived partial order of the inner world, leads to a partial symmetry of interaction This disproves the common assumption that local particles and interactions are disjoint 552 42 The Non-Abelian Vector Gauge Particle All these various physical facts are reminiscent of Riemann’s ideas to construct the observable physical space by bringing into the extended manifold (structureless void, vacuum), from an outside source, matter, in form of a yardstick whose constituency is composed of interacting atomic particles In our case, it is the breaking of the O.4/ symmetry down to O.3/ that is responsible for the mass generation of the pion All these results have their origin in the revolutionary ideas that Riemann unfolded in his famous habilitation essay of 1854 at the GeorgiaAugusta University in Göttingen in presence of the princeps mathematicorum, C.F Gauss Chapter 43 Riemann’s Result and Consequences for Physics and Philosophy Riemann commented on his main result as follows: “The common character of those manifolds whose curvature is constant may also be expressed thus: that figures may be viewed in them without stretching For clearly figures could not be arbitrarily shifted and turned around in them if the curvature at each point were not the same in all directions at one point as at another, and consequently the same constructions can be made from it; whence it follows that in aggregates with constant curvature, figures may have any arbitrary position given them The measure-relations of these manifolds depend only on the value of the curvature, and in relation to the analytic expression it may be remarked that if this value is denoted by K, the expression for the line-element may be written as in (41.17).” Riemann continues: “By means of these inquiries into the determination of the measure-relations of an n-fold extension, the conditions may be declared which are necessary and sufficient to determine the metric properties of space, if we assume the independence of line-length from position and expressibility of the lineelement as the square root of the quadratic differential, that is to say, flatness in the smallest parts.” This is the basic principle which dominates Riemann’s entire work: to understand the physical behavior of nature from its infinite smallness Riemann regards partial differential equations as the foundation of physics when he says: “It is well known that scientific physics has only existed since the invention of the differential equation,” or, more explicitly, if more than one variable is involved, “partial differential equation.” For the geometrical explanation of the laws of physics, he began looking not so much at long-distance, but short-distance interactions governing all processes in Nature, just as Faraday and Maxwell had done in the theory of electricity The metric field is principally of the same nature as the electromagnetic field The differential formulation of Maxwell’s equations marks the beginning of electromagnetism as field theory and up until now, any of our present theories originate from a differentially formulated action principle, expressed in terms of field variables So Riemann’s formulation of the field equations for the metric field constitutes the © Springer Nature Switzerland AG 2020 W Dittrich, M Reuter, Classical and Quantum Dynamics, 553 554 43 Riemann’s Result and Consequences for Physics and Philosophy starting point for all the field equations that follow, be they of the classical or quantum mechanical kind Riemann, in anticipation of the coming development in field theory, in a short note he presented in Göttingen on February 10, 1858, entitled, “Contribution to electrodynamics,” made the following statement: “I take the liberty of communicating to the Royal Society an observation that brings closely together the theory of electricity and magnetism and that of light and radiant heat I have found that the electrodynamic actions of galvanic currents can be explained if one assumes that the action of one electrical mass on others is not instantaneous but propagates itself toward them with constant speed (equal, within the limits of observational errors, to that of the speed of light) Under this assumption, the differential equation of the electrical force is the same as that for the propagation of light and radiant heat.” For Riemann, local action and finite speed of propagation belonged together In this respect he felt he was a follower of Newton Riemann was deeply impressed by Newton’s pronouncement about the impossibility of direct action at a distance In his draft on gravity and light, Riemann quoted him verbatim: “Newton says: ‘That gravity should be innate, inherent, and essential to matter, so that one body can act upon another at a distance through a vacuum, without the mediation of anything else, by and through which their action and force may be conveyed from one to another, is to me so great an absurdity, that I believe no man who has in philosophical matters a competent faculty of thinking can ever fall into it.”’ Here, Riemann could find in Newton’s own words a denial of the widespread view that their author thought of gravity as action at a distance - a view that suggests itself strongly and yet naively From all of this, Einstein and Hilbert were able to obtain the field equation of general relativity (41.50), which is the result of the response of the Einstein– Hilbert action, Eq (41.33) plus the T term, with regard to the change of the metric field, ıg The result is the intrinsic coupling between the space–time world represented by the Einstein tensor R and matter, i.e., the energy-momentum tensor In modern terms, we can say that the right-hand side, T , is the source of the metric structure of space Einstein identifies the curvature of space, R , with gravitational forces between masses within it But this is exactly the idea that Riemann anticipated, namely that one can grasp space only if one puts matter, e.g., a yardstick for measuring, into it—or, likewise, a point test charge into the neighborhood of a charged body to study the electromagnetic field of the surrounding space This experiment was performed by Faraday about ten years later than Riemann’s habilitation lecture This, then, is the result of Riemann’s space analysis: Contrary to the belief of the entire community of mathematicians and philosophers of that time, including Kant, that the metric of any space is fixed independently of the physical processes occurring in it and reality can just move into it as into a rental apartment, Riemann claimed and proved that space as such is merely a shapeless, three-dimensional manifold, but it is the space-filling matter which provides the space with its shape and metric relations The decisive point is that the metric determination comes from outside The binding forces between the particles of the matter are the external 43 Riemann’s Result and Consequences for Physics and Philosophy 555 source for the metric configuration represented by the metric field, i.e., the metric relations must be thought of as the binding forces that act on space Hence, the Euclidean character of a metric determination of space is not certain a priori The type of world we live in is not something to be settled by logic; the answer can only be reached by a continued path of experiments The contemporary philosophers who shared Kant’s ideas must have considered Riemann’s conception of space as inacceptable It is universally true, however, up to the present day in all of field theory and otherwise, that if a field (here: metric of space) is coupled in an action formulation (here: 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Particles and Rigid Bodies Cambridge University Press, Cambridge (1989) Wiegel, F.W.: Introduction to Path-Integral Methods in Physics and Polymer Science World Scientific, Singapore (1986) Witten, E.: Phys Lett 117B, 324 (1982) Index Symbols -function, 320 Bohr–Sommerfeld quantization rule, 316 Braid group, 433 A Action-angle variables, 93 Action, for the driven harmonic oscillator, 37 one-dimensional harmonic oscillator, 32, 53 particle in a magnetic field, 34 Action functional, Action principles, of Euler–Maupertuis, Hamiltonian, Jacobi, 12, 45 Langrange, Schwinger, 13 Adiabatic approximation, 422 invariance, 119 invariant, 122 Aharonov–Bohm effect, 381 Aharonov–Casher effect, 427 Angle variable, 94 Anyonization, 433 C Canonical adiabatic theory, 157 ensemble, 296 perturbation theory, 133, 141, 157 transformation, 59 Caustics, 241, 243 Chern–Simons action, 43 mechanics, 331 quantum mechanics, 345 term, 336, 433 Classical limit, 208 Commensurability, 101, 200 Complete elliptic integral, 111 Conjugate points, 45, 241, 367 Conservation laws, 14 Coulomb potential with applied constant field, 86 problem, 25 Cycloid pendulum, 97 Cyclotron motion, 149 B Berry amplitude, 423 connection, 376 curvature, 376 phase, 371, 389 D Degeneracy accidental, 154, 172, 183 intrinsic, 154, 174 Degenerate systems, 101 © Springer Nature Switzerland AG 2020 W Dittrich, M Reuter, Classical and Quantum Dynamics, 561 562 Density operator, 293 Double-slit experiment, 206 E Effective action, 345, 419 with adiabatic approximation, 422 Einstein–Brillouin–Keller (EBK) quantization, 351 Einstein–de Broglie relation, 228 Equivalence relation, 326 External sources, 211 F Fermat’s principle, 12 Feynman propagator, 207, 281 Feynman–Soriau formula, 245, 258 Fixed point elliptic, 169 hyperbolic, 169 Fourier path, 254 Fractional statistics, 433 Functional differential equation, 214 G Gauge field, 334 function, 279, 334 Gaussian wave packet, 288 Gauss law constraints, 337 Generating function, 61 Generation functional, 211 Group property for propagators, 209 Gyrophase invariant, 405 transformation, 405 H Hamilton–Jacobi equation, 8, 77, 141 Hamilton’s characteristic function, 79 principal function, 76 Harmonic oscillator damped, 73, 85, 96 in N-dimensions, 26 parametric, 409 perturbed, 137, 144 with time-dependent frequency, 122, 128, 259 Hermite polynomials, 286, 293 Index Homotopy, 325 class, 326 group, 326 I Integrable system, 186 J Jacobi equation, 52, 55, 268, 365 field, 45, 365  -function, 236 K KAM theorem, 144, 175, 177, 194, 197 Kepler problem, 198 Kirkwood gaps, 183 L Lagrange multipliers, 19 Landau damping, 172 Large gauge transformations, 337 Lewis–Riesenfeld theory, 273, 419 Libration, 94, 109 Liouville’s theorem, 185 Lorentz equation, 18 M Magnetic flux, 381 Map area preserving, 185 canonical, 185 Maslov anomaly, 363 index, 361 Mathematical pendulum perturbative treatment, 139 Matsubara rule, 321 Momentum description, 229 Morse Index theorem, 57, 363 N N-dimensional harmonic oscillator invariant, 26 Newton’s procedure, 176 Nonlinearity parameter, 202 Index O One-loop approximation, 345 P Parabolic coordinate, 86 Particle creation, 419 in magnetic field, 49, 105, 127 Partition function, 317 Path integral, for free particle, 223, 269 general quadratic Lagrangian, 239 harmonic oscillator, 241, 255, 269 particle in a box, 231 particle in a circle, 234 particle in a constant force field, 240, 269 Pendulum cycloid, 97 mathematical, 139 periodically driven, 193 Perturbation series, 144 Perturbation theory, 175 Poincaré–Birkhoff theorem, 203 Poincaré surface of section, 185 Point transformation, 62 Poisson identity, 236 Poisson’s summation formula, 235 Primary resonance, 156 Product of paths, 325 Proper action variable, 104 Proper time, 17 Q Quantum Hall effect, 433 R Reduction to an equilibrium problem, 60 initial values, 70 Resonances primary, 168 removal of, 165 Resonant denominators, 172 Rotated frame states, 384 Rotating coordinate system, 66 Rotating number, 186 Runge–Lenz vector, 25 563 S Schrödinger equation, 250 wave function, 207, 227 Semiclassical approximation, 299 quantization, 345 Separatrix, 113, 171, 195 Small divisors, 144, 160, 197 Small gauge transformations, 336 Spectral flow, 357 Standard mapping, 190 generalized, 190 Stationary phase approximation, 312 Stochastic behavior, 194 Stochasticity parameter, 190 Strings, 425 Superconvergent perturbation theory, 175 Superposition principle, 206 Symplectic 2-form, 331 T Time-ordering operation, 212 Toda molecule, 114 Topological quantization, 236 Transition amplitude, 206 Twist mapping, 187 radial, 188 V Vacuum persistence amplitude, 411 Van-Vleck determinant, 57, 269 Variables fast, 157, 371 slow, 157, 371 Virial theorem, 26 W Winding number, 329 WKB ansatz, 123 approximation, 299 propagator, 269, 305 ... TRubingen, Germany Martin Reuter Institute of Physics Johannes Gutenberg University Mainz, Germany ISBN 97 8-3 -0 3 0-3 678 5-5 ISBN 97 8-3 -0 3 0-3 678 6-2 (eBook) 8-3 -0 3 0-3 678 6-2 1st edition,... conditions and not leave the tori to end up in chaotic motion © Springer Nature Switzerland AG 2020 W Dittrich, M Reuter, Classical and Quantum Dynamics, 8-3 -0 3 0-3 678 6-2 _1... © Springer Nature Switzerland AG 2020 W Dittrich, M Reuter, Classical and Quantum Dynamics, 8-3 -0 3 0-3 678 6-2 _3 17 18 The Action Principle in Classical Electrodynamics
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