Space–time approach to quantum chaos

Abstract

Dedicated to Professor Jozef T. Devreese on the occasion of his 65th birthday PACS 03.65. Sq, 05.45.Mt In the first part of this paper, it is shown that the energy levels of a quantum system, whose classical limit is chaotic, encode certain space-time properties of the corresponding classical system. To see this, one considers the semiclassical limit as Planck's constant tends to zero. As a generalization of Mark Kac's famous question, it is demonstrated that "one can hear the periodic orbits of a quantum billiard". In the second part, some mathematical aspects of the semiclassical limit are reviewed. In order to deal with expressions that are mathematically easier to control, one does not work with the path integral directly, but instead with a smoothed kernel corresponding to a well-defined Fourier integral operator. Applying then the techniques from microlocal analysis and pseudodifferential operators, one arrives at a semiclassical trace formula which is a generalization of the Gutzwiller trace formula originally derived from the path integral. In the third part of this paper, the Hadamard-Gutzwiller model is discussed whose classical limit is a strongly chaotic (Anosov) system. In order to derive exact orbit sum rules for this model, one requires the path integral on hyperbolic D-space ðD ! 2Þ which can be exactly solved by using the general lattice definition of path integrals in curvilinear coordinates.1 What is semiclassical about quantum mechanics? In the field of quantum chaos, one considers quantum systems whose classical limit is chaotic (see, e.g., [1] for a recent review). A central question then is whether one can find some (possibly unique and universal) fingerprints of classical chaos left on the corresponding quantum systems. In view of the correspondence principle, one expects such signatures of quantum chaos to be seen in the semiclassical limit as Planck's constant h tends to zero. It turns out that the semiclassical theory based on the semiclassical approximation to the Feynman path integral (see, e.g., Chapter 5 in [2]) leads to a deep understanding of quantum spectra and wave functions of complex quantum systems in terms of space-time properties of the corresponding classical dynamics.The classical dynamics as well as the quantum mechanics of classically chaotic systems is quite complicated. Therefore, one often studies simple model systems which nevertheless show the main features of typical chaotic systems. Prototype examples of strongly chaotic systems are Euclidean and non-Euclidean billiards. In the Euclidean case, they are defined by the free motion of a point particle with mass m inside a compact Euclidean domain W & R 2 with elastic reflections at the boundary @W. The corresponding quantum systems are governed by the Schraedinger equation inside Ŵ H Hw n ðx Þ ¼ E n w n ðx Þ ; x 2 W ;

Weyl's Law: Spectral Properties of the Laplacian in Mathematics and Physics

“…Pre-trace formulae of this type are the starting point for the derivation of trace formulae in the general case, for example in quantum mechanics, where the righthand side of (1.40) is expressed by the volume of the classical phase space and the classical actions evaluated along the periodic orbits of the corresponding classical system [63]. An alternative way to write the left-hand side of (1.37) is…”