Mean-field and semiclassical limit: Wasserstein versus Schatten

In this talk, I will review results concerning the problems of the combined mean-field and semiclassical limits from the N-body Scrödinger equation to the Hartree--Fock and Vlasov equations. As quantum analogue of the stability estimates for the Vlasov equation, the methods to get these limits involve either a quantum analogue of the Wasserstein distances introduced by Golse and Paul, or the quantum analogue of the Lebesgue norms defined using scaled Schatten norms.
The different methods do not give the same advantages, leading to different initial data, types of potentials, rates of convergence and time of validity of the estimates. They involve other classical concepts that are translated to the language of quantum theory, such as moments and quantum Sobolev spaces.