Linear density response functions are calculated for symmetric nuclear matter of normal density by time-evolving two-time Greens functions in real time. Of particular interest is the effect of correlations. The system is therefore initially time-evolved with a collision term calculated in a direct Born approximation as well as with full (RPA) ring-summation until fully correlated. An external time-dependent potential is then applied. The ensuing density fluctuations are recorded to calculate the density response. This method was previously used by Kwong and Bonitz for studying plasma oscillations in a correlated electron gas. The energy-weighted sum-rule for the response function is guaranteed by using conserving self-energy insertions as the method then generates the full vertex-functions. These can alternatively be calculated by solving a Bethe -Salpeter equation as done in some previous works. The (first order) mean field is derived from a momentum-dependent (non-local) interaction while $2^{nd}$ order self-energies are calculated using a particle-hole two-body effective (or residual) interaction given by a gaussian it local rm potential. We present numerical results for the response function $S(omega,q_0)$ for $q_0=0.2,0.4$ and $0.8 {rm fm}^{-1}$. Comparison is made with the nucleons being un-correlated i.e. with only the first order mean field included, the HF+RPA approximation. We briefly discuss the relation of our work with the Landau quasi-particle theory as applied to nuclear systems by Sjoberg and followers using methods developped by Babu and Brown, with special emphasis on the induced interaction.
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