  We study the quantum evolution of many-body Fermi gases in three dimensions,
in arbitrarily large domains. We consider both particles with non-relativistic
and with relativistic dispersion. We focus on the high-density regime, in the
semiclassical scaling, and we consider a class of initial data describing
zero-temperature states. In the non-relativistic case we prove that, as the
density goes to infinity, the many-body evolution of the reduced one-particle
density matrix converges to the solution of the time-dependent Hartree
equation, for short macroscopic times. In the case of relativistic dispersion,
we show convergence of the many-body evolution to the relativistic Hartree
equation for all macroscopic times. 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.
