We study the quantum evolution of many-body Fermi gases in three dimensions, in arbitrarily large domains. We focus on the high-density regime, in the semiclassical scaling. For a class of initial data describing zero-temperature states we prove that, as the density goes to infinity, and for short macroscopic times, the many-body evolution of the reduced one-particle density matrix converges to the solution of the time-dependent Hartree equation. With respect to previous work, the rate of convergence does not depend on the total number of particles, but only on the density: in particular, our result allows us to study the quantum dynamics of extensive many-body Fermi gases.
