The quantization of classical physical systems whose dynamics is regular or `integrable' is fairly well understood. In fact such systems were thought to be of predominant interest until the discovery of chaotic dynamics back in the early 1970's. The presence of chaos in classical systems is fairly well characterized, and so the natural thing to ask is to what extent this chaotic behavior is preserved when quantizing these systems. The quantization of classical chaotic systems is frequently described as `quantum chaos' although there are some researchers who believe that this designation should be reserved to efforts for characterizing when an actual quantum system could exhibit behavior that is similar to that which occurs in classical chaotic systems. This book exemplifies an approach to quantum chaos that is a mixture of these two outlooks, as it attempts to summarize the tools for studying the `signature' of quantum effects on classical chaos. Ideally, this signature is a collection of criteria that would reduce to the criteria used to characterize classical chaos when Planck's constant approaches zero. The author discusses various approaches to obtaining a "quantum signature" of classical chaos, some of these being techniques drawn from other fields of physics, such as nuclear and condensed matter physics. There are some interesting issues that arise in this book from a mathematical standpoint, such as connections of quantum chaos with the zeros of the Riemann zeta function, but the author refrains from an in-depth discussion because of lack of space.



The book is a fairly comprehensive treatment, and space prohibits a detailed review, but some of the main topics or issues of interest include the following:



Unitarity of the quantum time evolution of state vectors.

In quantum physics, the time evolution of state vectors is represented by an evolution operator that is unitary, which means that for a given Hamiltonian, the distance between two state vectors is preserved under time evolution. This rules out any notion of `sensitive dependence on initial conditions' as is the case in classical chaotic systems. The author though argues for an alternative notion that views quantum dynamics as being dependent on a control parameter. Thus one speaks of the sensitivity to the dynamics that is under the direction of the control parameter. Two states evolving from the same initial state but having slightly different values of the control parameter may have radically different behavior, depending on where the initial state is located (in either the regular or classically chaotic region). Quantum chaos in this view is a kind of measure of "mobility" of the state vectors under slight changes of dynamics (control parameter(s)).



The Ehrenfest time and the correspondence principle

Interestingly, the author seems not to worry too much about the issues with the Ehrenfest time that other researchers do. For regular systems the Ehrenfest time is long enough to not cause worry when comparing classical dynamics with the time evolution of quantum expectation values. For chaotic classical systems though the Ehrenfest time can be much shorter, and so the correspondence principle seems to be problematic for the quantization of classical chaotic systems. The main issue of the Ehrenfest time for the author arises in the validity of the "diagonal" approximation for the short-time form factor in the Gutzwiller semiclassical theory of periodic orbits. In that discussion the Ehrenfest time gives an estimate for the limiting time above which the diagonal approximation fails (due to constructive interference between the periodic orbits). The Ehrenfest time is to be contrasted with the `Heisenberg time', which is the time needed to resolve the discreteness of the quasi-energy spectrum, and which, as the author remarks, gives an optimistic estimate for the range of validity of the diagonal approximation. The Ehrenfest time is proportional to the logarithm of Planck's constant while the Heisenberg time is proportional to the inverse of Planck's constant. Thus the Ehrenfest time is considerably shorter than the Heisenberg time.



Quantum localization

One way of understanding quantum localization is to examine the opposite situation, where the states are "spread out" and the probability amplitudes are the same everywhere up to a change of phase. Such is the situation for example for an electron in a periodic potential, where the states of the electron are the famous `Bloch states.' As is well known, the electron is viewed as a wave that is spread out through the whole solid. Localization then is the case where the wave function of the electron has most of its "support" on a given location, and thus the probability amplitude decays rapidly with increasing distance from this location. Intuitively, one would expect that this would be the case where the perfect periodic lattice is disrupted by the presence of an impurity. This intuition is verified by calculation, with one of the well-known examples being that of `Anderson localization'. The main example studied in this book is that of the `periodically kicked rotator', whose quasi-energy eigenfunctions are localized in the angular momentum representation. But as it turns out, and this is studied in detail in the book, the kicked rotator is related to the Anderson model. This relation is established by considering the eigenvalue problem for the Floquet operator of the kicked rotator. The resulting algebraic equation for the eigenfunctions in the momentum representation has the form of the Schroedinger equation for a particle in a one-dimensional lattice of pseudorandom potentials (in the strict Anderson model these potentials are random). The author is careful to note that a rigorous proof of localization for the kicked rotator has not been accomplished, but that numerical evidence points to an equivalence between the Anderson model and the kicked rotator. And to make the case that not every periodically kicked system will display localization, the author discusses the `kicked top.'



Random matrices

The topic of random matrices has generated a lot of excitement in the mathematical community in recent years due to a possible connection (and resolution) of the Riemann conjecture. In this book, the role of random matrices arises in the discussion of the presence of universality in the local fluctuations in the quasi-energy spectra of classical systems that display global chaos. Random matrices come into play when analyzing the level dynamics of a classical Hamiltonian flow in a manner that is similar to what is done in ordinary equilibrium statistical mechanics.