Space-time approach to quantum chaos

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 (D greater than or equal to 2) which can be exactly solved by using the general lattice definition of path integrals in curvilinear coordinates.