Theory of dynamical conductivity of interacting electrons

Starting from the current-current correlation function $Q\left(\omega \right)$, and taking into account the fact that the current operator (or the total momentum operator) commutes with the total Hamiltonian of the interacting electrons in the absence of impurities, we generate a series of diagrams for the dynamical conductivity. One essential feature of this approach is that at finite external frequency $\omega$, each of these diagrams vanishes in the limit of zero impurity concentration. We give a formal expression for the conductivity $\sigma \left(\omega \right) \mathrm{at} \omega \tau \gg 1$ ($\tau$ is the electron scattering lifetime), and we evaluate explicity $\sigma \left(\omega \right)$ for $\omega$ and ${\tau }^{-1\mathrm{}}\ll E_F$, the Fermi energy. Our results in the limit $\omega \to 0$ are in complete agreement with those obtained previously.