Semiclassical theory of many-particle systems

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The vast majority of systems found in nature are fully or partly chaotic. Roughly speaking, this means that the trajectory of a particle will be changed completely by just a small change of the initial conditions. Chaos plays a particularly important role for systems consisting of many particles, such as condensed-matter systems or gases of cold atoms. It can come about in several different ways: Some systems already show chaotic behaviour if only one particle is present. For other systems strong interactions between the particles (in particular the Coulomb interaction of electric charges) give rise to chaos. Yet another form of chaos is possible in cold atom gases. At low temperatures these gases display Bose-Einstein condensation, i.e., all atoms are in the same quantum-mechanical state. This state can be described by a so-called macroscopic wavefunction whose dynamics is given by a nonlinear differential equation (the nonlinear Schroedinger equation). Due to the nonlinearity the behaviour of this wavefunction can depend sensitively on the initial conditions just like the trajectory of a particle in a chaotic system. Paradigmatic examples for chaos are the Bose-Hubbard model (one of the most important models in many-body and cold atom theory) as well as recent experiments with cold atoms in chaotic potentials.Developing mathematical methods to deal with chaos in many-particle systems is an urgent priority. Experiments with cold atoms are now reaching a stage where chaotic dynamics and strong interactions play a crucial role. Moreover there is an interesting prediction that important quantum-mechanical properties such as the conductance of strongly interacting many-particle systems should in fact be universal and not depend on the details of the system.Universal behaviour is also expected for the statistics of the energy levels (the discrete values the energy is allowed to assume in a quantum system). Understanding the precise conditions under which many-particle systems behave in a universal way is important both from a fundamental and from a technological point of view: Universal features can be modelled in an efficient way, whereas non-universal features can be tuned to generate desirable behaviour in technological applications.Recent breakthroughs in quantum chaos (that I was involved in myself) now make it possible to tackle these questions. This will be the aim of the proposed project. An important step will be to systematically study how the full theory of quantum many-particle systems is connected to two approximate descriptions: classical many-particle theory, and an effective quantum mechanical description based on the macroscopic wavefunction. To elucidate this connection I will generalize a central result from quantum chaos (the Gutzwiller trace formula) to express properties of the quantum many-particle system as sums over solutions for the macroscopic wavefunction. With these tools in place it will be possible to clarify in which regimes many-particle systems display universal behaviour, and in which regimes important system specific effects are to be expected.