Classical and quantum dynamics from classical paths to path integrals.

*(English)*Zbl 0769.70001
Berlin: Springer-Verlag. vii, 341 p. (1992).

This monograph is the result of the authors’ lectures and seminars given mainly at Tübingen University. The subject of this volume is classical and quantum dynamics. At first sight, this “combination” seems unusual – due to the complete different bases of these two fields; but the authors show that in the light of modern topology a little twist can be brought to the standard approach that considers classical and quantum physics as disjoint subjects. So, the whole book relies on this basic conception. In order to support it, there are used standard subjects and materials (the first part of the book) as well as subjects based upon the application of modern topology in quantum mechanics (the second part of the book). It is for the first time that the latter subjects find their way in a textbook; they are mainly based upon Feynman’s path integral approach, and they are very minutely worked.

One of the fundamental ideas of the book is that the formulation of both classical and quantum mechanics can be based on the principle of stationary action (Ch. 1: “The action principles in mechanics”). It is emphasized the great efficiency of Schwinger’s action principle; the working scheme based on this principle encompasses almost every situation in classical and quantum mechanics (Ch. 2: “Application of the action principles”).

In the next chapters (Ch. 3: “Jacobi fields. Conjugate points”; Ch. 4: “Canonical transformations”; Ch. 5: “The Hamilton-Jacobi equation”; Ch. 6: “Action-angle variables”; Ch. 7: “The adiabatic invariance of the action variables”) there are presented important subjects as Jacobi fields, action-angle variables, adiabatic invariants etc. in the light of our-days research on classical Hamiltonian dynamics.

It is underlined the significant role played by the canonical perturbation theory before the birth of quantum mechanics (Ch. 8: “Time- independent canonical perturbation theory”; Ch. 9: “Canonical perturbation theory with several degrees of freedom”; Ch. 10: “Canonical adiabatic theory”; Ch. 11: “Removal of resonances”). The recent developments in perturbation theory (Ch. 12: “Superconvergent perturbation theory, KAM theorem (Introduction)”; Ch. 13: “ Poincaré surface of sections, mappings”; Ch. 14: “The KAM theorem”) offer a new look and solutions to old problems in nonlinear mechanics. As an example, the stability of the solar system is analyzed at long times.

In order to give emphasis to the fundamental quantization principles and methods (Ch. 15: “Fundamental principles of quantum mechanics”), it is discussed the important role played by the adiabatic invariants prior to canonical quantization of complementary dynamical variables. There are discussed the common features of the semiclassical quantization (Ch. 25) and (approximations of) Feynman’s path integral (Ch. 16: “Examples for calculating path integrals”; Ch. 17: “Direct evaluation of path integrals”). The point is that Feynman’s path integral methods are suited for the description of a quantum mechanical system even into its classical realm.

Path integrals and other geometrical methods are used for the detailed treatment of the time-dependent oscillator (Ch. 18), and for studying some particles’ propagators (Ch. 19: “Propagators for particle in an external magnetic field”; Ch. 20: “Simple applications of propagator functions”; Ch. 21: “The WKB approximation”). The fact that geometrical methods play an eminent role in quantum relativistic and nonrelativistic theories is again underlined in Ch. 26 (“ Maslov anomaly for the harmonic oscillator”), Ch. 27 (“Maslov anomaly and the Morse index theorem”), and maybe especially in Ch. 28 (“Berry’s phase”), Ch. 29 (“Classical analogues to Berry’s phase”), Ch. 30 (“Berry phase and parametric harmonic oscillator”) – that refer to the famous Berry’s phase. There are also treated the partition function for a harmonic oscillator (Ch. 22), an introduction to homotopy theory (Ch. 23), and the classical Chern-Simons mechanics (Ch. 24). All these modern, topical subjects show that classical and quantum mechanics are far from being closed domains! The book ends with a reference list – that contains 43 important titles in this field.

To conclude, we can say that this monograph is intended for graduate students as well as for research workers in quantum mechanics field. We have to remark the high scientific level of the book, its strictness, minuteness, and high-class didactic style, as well as its modern, original point of view on the tackled subjects. We consider that this monograph is indeed a reference book in its field, and we strongly recommend it to anyone who is interested in classical and quantum dynamics.

One of the fundamental ideas of the book is that the formulation of both classical and quantum mechanics can be based on the principle of stationary action (Ch. 1: “The action principles in mechanics”). It is emphasized the great efficiency of Schwinger’s action principle; the working scheme based on this principle encompasses almost every situation in classical and quantum mechanics (Ch. 2: “Application of the action principles”).

In the next chapters (Ch. 3: “Jacobi fields. Conjugate points”; Ch. 4: “Canonical transformations”; Ch. 5: “The Hamilton-Jacobi equation”; Ch. 6: “Action-angle variables”; Ch. 7: “The adiabatic invariance of the action variables”) there are presented important subjects as Jacobi fields, action-angle variables, adiabatic invariants etc. in the light of our-days research on classical Hamiltonian dynamics.

It is underlined the significant role played by the canonical perturbation theory before the birth of quantum mechanics (Ch. 8: “Time- independent canonical perturbation theory”; Ch. 9: “Canonical perturbation theory with several degrees of freedom”; Ch. 10: “Canonical adiabatic theory”; Ch. 11: “Removal of resonances”). The recent developments in perturbation theory (Ch. 12: “Superconvergent perturbation theory, KAM theorem (Introduction)”; Ch. 13: “ Poincaré surface of sections, mappings”; Ch. 14: “The KAM theorem”) offer a new look and solutions to old problems in nonlinear mechanics. As an example, the stability of the solar system is analyzed at long times.

In order to give emphasis to the fundamental quantization principles and methods (Ch. 15: “Fundamental principles of quantum mechanics”), it is discussed the important role played by the adiabatic invariants prior to canonical quantization of complementary dynamical variables. There are discussed the common features of the semiclassical quantization (Ch. 25) and (approximations of) Feynman’s path integral (Ch. 16: “Examples for calculating path integrals”; Ch. 17: “Direct evaluation of path integrals”). The point is that Feynman’s path integral methods are suited for the description of a quantum mechanical system even into its classical realm.

Path integrals and other geometrical methods are used for the detailed treatment of the time-dependent oscillator (Ch. 18), and for studying some particles’ propagators (Ch. 19: “Propagators for particle in an external magnetic field”; Ch. 20: “Simple applications of propagator functions”; Ch. 21: “The WKB approximation”). The fact that geometrical methods play an eminent role in quantum relativistic and nonrelativistic theories is again underlined in Ch. 26 (“ Maslov anomaly for the harmonic oscillator”), Ch. 27 (“Maslov anomaly and the Morse index theorem”), and maybe especially in Ch. 28 (“Berry’s phase”), Ch. 29 (“Classical analogues to Berry’s phase”), Ch. 30 (“Berry phase and parametric harmonic oscillator”) – that refer to the famous Berry’s phase. There are also treated the partition function for a harmonic oscillator (Ch. 22), an introduction to homotopy theory (Ch. 23), and the classical Chern-Simons mechanics (Ch. 24). All these modern, topical subjects show that classical and quantum mechanics are far from being closed domains! The book ends with a reference list – that contains 43 important titles in this field.

To conclude, we can say that this monograph is intended for graduate students as well as for research workers in quantum mechanics field. We have to remark the high scientific level of the book, its strictness, minuteness, and high-class didactic style, as well as its modern, original point of view on the tackled subjects. We consider that this monograph is indeed a reference book in its field, and we strongly recommend it to anyone who is interested in classical and quantum dynamics.

Reviewer: I.Bena (Iaşi)

##### MSC:

70-02 | Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems |

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |

81S40 | Path integrals in quantum mechanics |