Dynamic correlations in an electron gas. II. Kinetic-equation approach

Starting from the exact hierarchy of quantum kinetic equations for the Wigner distribution functions we develop a theory of dynamical correlations in an electron gas. By making a random-phase-approximation- (RPA) like truncation for the second equation in the hierarchy we obtain an expression for the proper polarizability of the form $Q=Q^0+Q^c$, where $Q^0$ is the Lindhard (RPA) function and $Q^c$ is an additional term which has the following properties: (i) it includes all the three first-order Feynman diagrams for the proper polarizability, (ii) it incorporates in addition the coupled propagation of two particle-hole pairs, and (iii) it leads to an expression for the density-density response function which satisfies exactly the first- and third-frequency-moment sum rules. Detailed calculations of plasmon dispersion, damping, and compressibility have been made. Calculations have also been made for the complex dynamic local field $G(k,\omega )$ for arbitrary values of wave number and frequency. Comparison has been made with the available experimental data for Al ($r_s=2.0\mathrm{}$). A critique of the theory is presented.