  We study the non equilibrium dynamics in the fermionic Hubbard model after a
sudden change of the interaction strength. To this scope, we introduce a time
dependent variational approach in the spirit of the Gutzwiller ansatz. At the
saddle-point approximation, we find at half filling a sharp transition between
two different regimes of small and large coherent oscillations, separated by a
critical line of quenches where the system is found to relax. Any finite doping
washes out the transition, leaving aside just a sharp crossover. In order to
investigate the role of quantum fluctuations, we map the model onto an
auxiliary Quantum Ising Model in a transverse field coupled to free fermionic
quasiparticles. Remarkably, the Gutzwiller approximation turns out to
correspond to the mean field decoupling of this model in the limit of infinite
coordination lattices. The advantage is that we can go beyond mean field and
include gaussian fluctuations around the non equilibrium mean field dynamics.
Unlike at equilibrium, we find that quantum fluctuations become massless and
eventually unstable before the mean field dynamical critical line, which
suggests they could even alter qualitatively the mean field scenario.
