Semiclassical theory for many-body fermionic systems

José M. Arias, María Isabel Gallardo, Manuel Lozano

K. Kilimann, D. Kremp, G. Röpke

C.H. Harzer, H. Muether, Markus Stauf +1 more

Michael J. Quist

Thomas Engl

During the last decades, semiclassical techniques for single particle systems have been successfully developed, to the point that they describe the universal and reproducible quantum fluctuations characteristic of quantum systems with chaotic classical limit. By now they not only give a qualitative and intuitive picture about interference, but also yield quantitatively good results. Motivated by this story of success, the aim of this thesis was to first of all rigorously deduce a semiclassical theory for many-body systems in Fock space from path integrals, thus pushing the work of John Hasbrouck Van Vleck, Richard Feynman, Martin Charles Gutzwiller and Michael Victor Berry to the next level. The central object in this development is the propagator in Fock basis, for which a path integral in complex variables has to be found, from which the corresponding classical theory can be deduced. The main focus of attention there lies the action of the classical trajectories being real thus supporting the explicit, illustrative and manifest study of many-body interference. Then, with the help of this semiclassical propagator, universal many-body interference effects in bosonic and fermionic Fock space are studied several of which are novel predictions due to new qualitative tools developed here. The semiclassical tool for quantum fields, and the picture of many-body interference it provides, is put to test against numerical calculations. For a quantitative comparison, the loop expansion developed by Martin Sieber and Klaus Richter going beyond Michael Berry's diagonal approximation is transfered and adopted to the theory in Fock space. All our predictions then show excellent agreement with quantum mechanical simulations.

Chiara Saffirio

The derivation of effective macroscopic theories approximating microscopic systems of interacting particles is a major question in non-equilibrium statistical mechanics. In these notes we present an approximation of systems made by many fermions interacting via inverse power law potentials in the mean-field and semiclassical regimes, reviewing the material presented at the 11th summer school "Methods and Models of Kinetic Theory" held in Pesaro in June 2022. More precisely, we focus on weakly interacting fermions whose collective effect can be approximated by an averaged potential in convolution form, and review recent mean-field techniques based on second quantization approaches. As a first step we obtain a reduced description given by the time-dependent Hartree-Fock equation. As a second step we look at longer time scales where a semiclassical description starts to be relevant and approximate the many-body dynamics with the Vlasov equation, which describes the evolution of the effective probability density of particles on the one particle phase space.