Derivation of Mean-Field Dynamics for Fermions

My talk is about a topic from non-equilibrium quantum mechanics. The starting point is the Schrödinger equation with pair interaction for fermions. I then introduce and discuss different scaling limits which are of a mean-field type. In these scaling limits, it is expected that the microscopic Schrödinger dynamics can be approximated by the timedependent Hartree-Fock equation. This is an effective, non-linear equation with much fewer degrees of freedom. I present the results of coworkers and myself on the rigorous derivation of the Hartree-Fock equation. Several new methods were recently developed for tackling this kind of problems and I will focus on introducing the one I used myself in recent works. This method is based on controlling a well-chosen functional by estimating certain transition amplitudes in the Fermi gas. The functional is directly related to the trace norm difference of reduced density matrices and the estimates lead to explicit rates of convergence.