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Chaos in Classical and Quantum Mechanics by Martin C. Gutzwiller
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Typically, the Fermi wavelength is much smaller than the size of the mesoscopic devices, which justifies the use of the semiclassical approximation in this context. Disorder averaging is performed using the powerful field-theoretical methods which are discussed in A. Mirlin's lectures. One can get an impression of the fertile exchange between quantum chaos and mesoscopics by comparing Bogomolny's periodic orbit approach and Mirlin's field-theoretical discussion of the spectral form factor.
Conductance in mesoscopic systems is one of the most prominent observables which show the fingerprints of the underlying classical chaotic dynamics.
Jalabert discusses this issue at length, and makes use of the semiclassical theory of chaotic scattering in this context. The quantum dynamics of extended systems is another issue of distinct relevance to mesoscopic physics. The motion of a quantum particle in a periodic, quasi-periodic, or disordered potential is the central problem of Localization Theory, which was historically the first meeting point between Quantum Chaos and Solid State Physics.
Here, the central issue is the spectral type of the quantum Hamiltonian, which can be pure-point or continuous, leading to qualitatively different dynamics; it can also exhibit a metal-insulator transition from one spectral type to the other. The case when the spectrum is singular continuous, that is, multifractal, is of special interest, leading to anomalous diffusion. Geisel's lectures some general results about the connection between the multifractal spectral structure and anomalous diffusion were presented; moreover, the role of classical chaos in determining the quantum spectral type was elucidated.
It was in fact shown that the avoided level crossings, which characterize the spectra of bounded, classically chaotic systems, in the case of extended systems are replaced by avoided band crossings, which can sometimes lead to metal-insulator transitions, and to the appearance of singular continuous spectra. The research in quantum chaos in the past years concentrated mostly on systems with the minimum number of degrees of freedom which allow classically chaotic dynamics.
Only recently, systems with a larger number of freedoms are considered, and this will probably be one of the most important directions in future research. In systems composed of many particles, the identity of the constituents is an important symmetry which affects the quantum treatment through Pauli's principle. Its implementation in the semiclassical trace formula was discussed in A.
In most many-body systems, the particles interact pairwise, so that the Hamiltonian matrix is sparse in the Fock space representation. What will the resulting spectral statistics be? Will it approach the predictions of Random Matrix Theory in detail?