We present a quantitative estimate for the derivation of the
Hartree dynamics for boson particles at low temperatures in arbitrary
dimension. This achievement is obtained by a normal mode decomposition
of the field operator evolved under the many body quantum dynamics, and
estimates on Wick symbols of the field operator through $L^2 (\mu)$ -
norm with a Gaussian thermal measure $\mu$. The bounding term that
proves this convergence is explicitly written in terms of the
temperature and the number of particles. The interaction potential is
supposed to be in the Hardy class, thus containing the Coulomb type, and
it is not rescaled with respect to the number of particles. The
dependence on time in the main estimates is shown to be globally linear.
This is a joint work in collaboration with A. Ponno (Dept. of Math.
“Tullio Levi-Civita” - University of Padova) and H.P. Singh (SISSA).