Nonlinear dynamics (``chaos theory'') and quantum mechanics are two of the scientific triumphs of the 20th century. The former lies at the heart of the modern interdisciplinary approach to science, whereas the latter has revolutionized physics. Both chaos theory and quantum mechanics have achieved a fairly large level of glamour in the eyes of the general public. The study of quantum chaos encompasses the application of dynamical systems theory in the quantum regime. In the present article, we give a brief review of the origin and fundamentals of both quantum mechanics and nonlinear dynamics. We recount the birth of dynamical systems theory and contrast chaotic motion with integrable motion. We similarly recall the transition from classical to quantum mechanics and discuss the origin of the latter. We then consider the interplay between nonlinear dynamics and quantum mechanics via a classification and explanation of the three types of quantum chaos. We include several recent results in this discussion.
Nonlinear dynamics (``chaos theory'') and quantum mechanics are two of the scientific triumphs of the 20th century. The former lies at the heart of the modern interdisciplinary approach to science, whereas the latter has revolutionized physics. Both chaos theory and quantum mechanics have achieved a fairly large level of glamour in the eyes of the general public. The study of quantum chaos encompasses the application of dynamical systems theory in the quantum regime. In the present article, we give a brief review of the origin and fundamentals of both quantum mechanics and nonlinear dynamics. We recount the birth of dynamical systems theory and contrast chaotic motion with integrable motion. We similarly recall the transition from classical to quantum mechanics and discuss the origin of the latter. We then consider the interplay between nonlinear dynamics and quantum mechanics via a classification and explanation of the three types of quantum chaos. We include several recent results in this discussion.
This article gives an introduction to chaotic systems in classical mechanics and their quantum mechanical counterparts.The author assumes very few prerequisites and starts with a basic introduction to the mathematical formulation of classical and quantum mechanics and discusses the relation between classical and quantum systems by means of the mathematical procedure of quantization. Then, the article proceeds with discussing the notion of ``chaos'' in classical mechanics and particularly focuses on the concepts of hyperbolicity, ergodicity and mixing. In the last section, the quantum mechanical properties of classically chaotic systems are discussed. Here, the focus lies in particular on quantum ergodicity and the role of Egorov's theorem. Finally, the distribution of eigenvalues, according to Weyl's law, {\it M.C. Gutzwiller}'s trace formula [``Periodic orbits and classical quantization conditions'', J. Math. Phys., 12, No. 3, 343--358 (1971; \url{doi:10.1063/1.1665596})] as well as the random matrix conjecture of {\it O. Bohigas} et al. [``Characterization of chaotic quantum spectra and universality of level fluctuation laws'', Phys. Rev. Lett. 52, No. 1, 1--4 (1984; \url{doi:10.1103/PhysRevLett.52.1})] is also discussed.A particularly nice feature throughout the whole text is that it includes both mathematically rigorous definitions which can be illustrated for toy models such as the cat map, as well as phyiscally relevant sytems such as billiards. For the latter it is then outlined, where the difficulties in a mathematically rigorous description lie.
From the very beginning classical nonlinear dynamics has enjoyed much popularity even among the noneducated public as is documented by numerous articles in well-renowned magazines, including nonscientific papers. For its nonclassical counterpart, the quantum mechanics of chaotic systems, termed in short ‘quantum chaos’, the situation is completely different. It has always been considered as a more or less mysterious topic, reserved to a small exclusive circle of theoreticians. Whereas the applicability of classical nonlinear dynamics to daily life is comprehensible for a complete outsider, quantum chaos, on the other hand, seems to be of no practical relevance at all. Moreover, in classical nonlinear dynamics the theory is supported by numerous experiments, mainly in hydrodynamics and laser physics, whereas quantum chaos at first sight seems to be the exclusive domain of theoreticians. In the beginning the only experimental contributions came from nuclear physics [Por65]. This preponderance of theory seems to have suppressed any experimental effort for nearly two decades. The situation gradually changed in the middle of the eighties, since when numerous experiments have been performed. An introductory presentation also suited to the experimentalist with no or only little basic knowledge is still missing. It is the intention of this monograph to demonstrate that there is no reason to be afraid of quantum chaos. The underlying ideas are very simple. It is essentially the mathematical apparatus that makes things difficult and often tends to obscure the physical background.
The relationship between chaos and quantum mechanics has been somewhat uneasy--even stormy, in the minds of some people. However, much of the confusion may stem from inappropriate comparisons using formal analyses. In contrast, our starting point here is that a complete dynamical description requires a full understanding of the evolution of measured systems, necessary to explain actual experimental results. This is of course true, both classically and quantum mechanically. Because the evolution of the physical state is now conditioned on measurement results, the dynamics of such systems is intrinsically nonlinear even at the level of distribution functions. Due to this feature, the physically more complete treatment reveals the existence of dynamical regimes--such as chaos--that have no direct counterpart in the linear (unobserved) case. Moreover, this treatment allows for understanding how an effective classical behavior can result from the dynamics of an observed quantum system, both at the level of trajectories as well as distribution functions. Finally, we have the striking prediction that time-series from measured quantum systems can be chaotic far from the classical regime, with Lyapunov exponents differing from their classical values. These predictions can be tested in next-generation experiments.
The present collection of papers is devoted to a rather new and very controversial topic, the so-called “quantum chaos”. Some researchers see nothing essentially new at all in this phenomenon (apart from a number of various examples and models), and they have good reason to believe so. Indeed, the problems in this field all belong to the traditional, “old-fashion”, and rather “simple” quantum mechanics of finite-dimensional systems with a given interaction and no quantized fields. Nevertheless, many, including ourselves, consider quantum chaos to be a new discovery, though in an old field, of a great importance for fundamental physics. To understand this, the phenomenon of quantum chaos should be put into its proper perspective in recent developments in physics. The central point of this perspective is the conception of dynamical chaos (also a rather new topic) in classical mechanics (for a good review see, e.g., Refs.[l-3]). Thus before discussing the current understanding of quantum chaos we need briefly to describe classical dynamical chaos
In this work the decoherence formalism of quantum mechanics is explored and applied to a number of interesting problems in quantum physics. The boundary between quantum and classical physics is examined, and demonstration made that quantum histories corresponding to classical equations of motion become more probable for a broad class of models, including linear and nonlinear models of Brownian motion. The link between noise, dissipation, and decoherence is studied. This work is then applied to systems which classically exhibit dissipative chaotic dynamics. A theory is explicated for treating these systems, and the ideas are applied to a particular model of the forced, damped Duffing oscillator, which is chaotic for certain parameter values. Differences between classical and quantum chaos are examined, particularly differences arising in the structure of fractal strange attractors, and the conceptual difficulties in framing standard notions of chaos in a quantum system. A brief discussion of previous work on quantum chaos is included, and the differences between Hamiltonian and dissipative chaos pointed out; a somewhat different interpretation of quantum chaos from the standard one is suggested. A class of histories for quantum systems, in phase space rather than configuration space, is studied. Different ways of representing projections in phase space are discussed, and expressions for the probability of phase space histories are derived; conditions for such histories to decohere are also estimated in the semiclassical limit.
The ubiquity of chaos in classical mechanics (CM), as opposed to the situation in standard quantum mechanics (QM), might be taken as speaking against QM being the fundamental theory of physical phenomena. Bohmian mechanics (BM), as a formulation of quantum theory, may clarify both the existence of chaos in the quantum domain and the nature of the classical limit. Two interesting possibilities are (i) that CM and classical chaos are included in and underwritten by quantum mechanics (BM) or (ii) that BM and CM simply possess a common region of (noninclusive) overlap. In the latter case, neither CM nor QM alone would be sufficient, even in principle, to account for all of the physical phenomena we encounter. In this talk I shall summarize and discuss the implications of some recent work on chaos and on the classical limit within the framework of BM.