Colloquium di matematica - prof. Benjamin Schlein dell'Università di Zurigo- Derivation of the Hartree-Fock equation for the quantum evolution of weakly coupled particles

The time evolution of many body quantum systems is governed by the Schroedinger equation. In this talk, we focus on fermions (i.e. quantum particles described by wave functions that are antisymmetric with respect to permutations) and we consider the mean field regime, characterised by a large number of weak collisions among the particles. For initial data close to Slater determinants (the simplest example of an antisymmetric wave function), we show that the solution of the many-body Schroedinger equation remains close to a Slater determinant, evolved through the Hartree-Fock equation, providing explicit bounds on the rate of the convergence. Since the mean field regime is linked with a semiclassical limit, the solution of the Hartree-Fock equation can in turn be approximated by the solution of the classical nonlinear Vlasov equation.