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Book review

Author(s) Nakamura, K.
Title Quantum chaos - a new paradigm of nonlinear dynamics
Publisher Cambridge University Press
Year of publication 1983
Reviewed by Anatoly Zhigljavsky

The theory of chaotic dynamics finds more and more areas of application in diverse fields of science. Quantum mechanics has appeared to be one of these areas however the whole subject of quantum chaotic dynamics is not yet widely recognized. The current state of art in this field is basically just a few mathematical models of several quantum phenomena and related interpretations. No apparent experimental proof has been given yet that the nonlinear dynamics is a typical way of behaviour in microscopic and mesoscopic world. At the moment, the subject of quantum chaos can be regarded as a fashionable branch of quantum mechanics, allowing to give a description of various quantum effects, such as spectral width in magnetic resonance and fluctuations in conductance, which is different from the ordinary explanation based exclusively on the randomness of the experimental conditions. In other words, the chaotic dynamic models offer a possibility to interpret some randomness evidenced from the experimental study as caused by the internal deteministic reasons rather than by only external noise.

It seems it will take a lot of time for the subject of quantum chaos to be fully recognized and widely accepted as the best justification of different phenomena in quantum mechanics. One of the difficulties is that the nonlinear models of chaotic quantum dynamics are more complicated and consequently harder to study than the ordinary linear ones. But there is no doubts that the subject is very prospective. Specifically, the author believes that some of his ideas related to the background of the quantum chaos, if confirmed, "would be comparable to Einstein's great challenge in 1905 which revolutionized Galilean thought on space and time".

The field of quantum chaos is very young. The literature on the subject is not extensive, the present monograph is one of the very first books on the field. It considers different models of quantum dynamics, both bounded and open systems are chosen from diverse branches of solid-state physics. Well-known concepts such as diamagnetism, antiferromagnetism, spin waves and electric conductance are reconsidered in terms of quantum chaos. It is demonstrated that there may be no contradiction in quantum mechanics between the linear nature of the time-dependent Schrödinger equation and the nonlinear, chaotic dynamics of motion.

The monograph contains 6 chapters. The first one is introductory. The second considers autonomous systems involving an orbital degree of freedom for electrons. The classical and quantum mechanics of noninteracting electrons in a nonintegrable elliptic billiard with a uniform magnetic field applied perpendicularly are under study. There are also chaotic dynamic models for electrical conductance considered. It is shown that quantum chaos provides a possible mechanism for the random motion of electrons. The conductivity is a fundamental measure of the randomness of the potential and it can be seen as a fundamental measure of chaotic instability of electronic motions. Chapter 3 provides several chaotic dynamics models in spin systems, it computationally demonstrates the fractal structure of the eigenvalue distributions. Chapter 4 is devoted to chaotic dynamics in spin-wave instabilities of dissipative systems. In chapter 5 a dynamic system is introduced to model the chaotic behaviour of both eigenvalues and eigenfunctions of bounded quantum systems with a single parameter. Chaotic description of statistical mechanics phenomena is also provided. This could be considered as an alternative to the approach based on the theory of random matricies. Chapter 6 is mainly devoted to the phylosophy and future prospects of the quantum chaos.

The main part of this book is based on a series of the author's recent publications. As was already mentioned, the subject of quantum chaos is not yet widely recognized and it can hardly be regarded as a theory after publication of this book as well. At the moment, the subject of quantum chaos contains a lot of philosophy and speculations as well as a collection of diverse models studied in most cases computationally. Theoretical study of the main part of the models as well as their experimental justification is what should follow.

In general, the monograph seems to be of quite a significant value for the quantum mechanics and nonlinear physics. It can also be put into the list of valuable books on applied chaos. Importance of the book for the mathematicians and specialists in the theory of chaotic dynamics seems not to be so great, there is neither new technique developed or advanced theoretical tools used. Mostly, the computational investigation of the models was done and only few results concern theoretical concepts, such as Lyapunov exponents, fractal dimension, stability analysis.

The volume is of a good production quality and its price is reasonable. It is definitely a worthwhile addition to the literature on quantum mechanics and applied chaotic dynamics. It should certainly be of great interest for the specialists in these fields and can also be valuable for graduate students in physics and other beginners.

Purchase of the volume seems to be a good investment for the specialists in quantum mechanics and for the libraries of the institutions dealing with quantum mechanics and solid-state physics. It might also be worthwile to buy the book for both young and mature scientists interested in chaotic dynamics and in contemporary physics.