Quantum Chaos: An Introduction

Quantum chaos is the study of the quantum dynamics of systems that are classically chaotic. Contrary to the impression left by most textbooks, almost all conservative dynamical systems are at least partly chaotic in the range of their behavior. This applies to molecules, nuclei, quantum dots, acoustical systems, and dozens of other systems. The field of quantum chaos is still young, growing fast, and rich in conceptual, computational, and experimental challenges. The mathematical questions having a direct bearing on the field are as deep as you care to go, including the Riemann conjecture, convergence of trace formulas, and the intricacies of chaotic Hamiltonian dynamics.

People joining in the fun of studying quantum chaos seem to come from all over physics and chemistry; in the past, most had already established themselves in related fields before turning to quantum chaos. That is changing dramatically in Europe, where brilliant students attracted to the field's challenges are now moving directly into permanent positions. European institutions have discovered that quantum chaologists are fine mathematicians as well as numerical analysts, almost of necessity, and are remarkably broad in the range of applicability of their expertise. The field is coming of age.

Hans-Jürgen Stöckmann's Quantum Chaos appears as the culmination of remarkable accomplishments by perhaps several dozen key researchers, but most notably Michael Berry and Martin Gutzwiller, over the past 25 years. The maturity of the study of quantum chaos was recognized by the Nobel Foundation as recently as June 2000, when one of its Nobel Symposia was devoted to the subject.

Stöckmann's book is a remarkable work, reflecting the growth and excitement of the field and containing something for almost everyone interested in chaos theory and experiment. Experts will appreciate the historical tidbits, many of which are not well known, and will enjoy Stöckmann's compelling presentation of facets of the field that are outside their own areas of specialization. Beginners will find an engaging and approachable introduction to the field, with chapters on most aspects of quantum chaos and its experimental implications. The subjects are current and reflect a broad view, including experiments, inspired by chaos, whose ultimate bearing on the field may have been marginal but whose impacts on other fields can have been considerable.

The book contains the most readable introductions to random matrix theory and supersymmetry that I have seen. The style and clarity of writing are both excellent. Stöckmann is an experimentalist, but his insights and intuition for theory are quite remarkable and worth reading. References to original work are plentiful and well chosen, and for that reason alone the book is valuable and will fast become a "must-have" for those with an interest in quantum chaos.

On the downside, Stöckmann's book is not flawless. A few painful typos exist. For example, the first time the nearest-neighbor level-spacing distribution is displayed, it is missing a factor of the spacing, "s." More subtle but more serious is Stöckmann's treatment of symmetry in quantum mechanics, which is fast and loose at best and wrong at worst. Teaching from this section would be a rocky experience. In fact, experts in a number of subfields will no doubt recognize flaws that they would want to correct.

The sum of all these flaws is not devastating by any means, but because of them Stöckmann's book just misses the mark on being a good textbook from which nonexpert instructors could teach. This is a shame: Courses on quantum chaos are a great way to introduce classical and quantum mechanics of current theoretical and experimental interest. But such courses are rare, partly because of lack of a good text. It wouldn't take much to fix this aspect of Stöckmann's book and make it such a text, and I look forward to a corrected edition. Experts could use the book as a text now, correcting a few things along the way.

Other books on quantum chaos--first generation texts from the early 1990s--include Martin Gutziller's work, Chaos in Classical and Quantum Mechanics (Springer-Verlag, 1990), centered on his famous trace formula, Linda E. Reichl's The Transition to Chaos in Conservative Classical Systems (Springer-Verlag, 1992), and Fritz Haake's Quantum Signatures of Chaos (Springer-Verlag, 1992). Reichl's work is a useful survey of the literature; Haake's work (due out in a new addition soon) is mainly for theorists and much more rigorous than Stöckmann's book. I am glad to own all four, but the book I show to students first is Stöckmann's.