Mathematics

This seminar concludes Prof. Pickl's graduate course "Many-body quantum dynamics and non-linear effective evolution equations" and focuses on the generalisation of Pickl's "counting method" to systems in interaction with a second-quantised radiation field. First, we introduce the Nelson model with cut-off, which describes a quantum system of non-relativistic particles coupled to a massive (or massless) scalar Bose field. We study the time evolution in a mean-field limit where the number N of charged particles becomes large while the coupling with the radiation field is re-scaled by N^{-1/2}. At time zero we assume that almost all particles are in the same one-body state (a Bose-Einstein condensate) and we assume also that the Bose excitations of the radiation field are not present (vacuum state) or are close to a coherent state. We then show that at later times and in the limit N → ∞ the charged particles remain in a Bose-Einstein condensate, with the time evolution approximatively described by a Schroedinger + Klein-Gordon system of equations, which models the coupling of a nonrelativistic particle to a classical Klein-Gordon field. Afterwards, we shall explain how the proof can be modified so as to derive the Hartree + Maxwell system of equations from the the many-body Pauli-Fierz Hamiltonian.