Alternative representation of the Kubo formula for the optical conductivity: A shortcut to transport properties

The Kubo formula for the electrical conductivity is expressed in terms of a weighted sum of Drude-like contributions associated to the exact eigenstates of the interacting system, each characterized by its own frequency-dependent relaxation time. This alternative formulation considerably simplifies the access to the static properties (dc conductivity) and resolves the long-standing difficulty to connect the Boltzmann transport theory and the Kubo formula. In particular, at the lowest order of the perturbation theory, the correct transport scattering lifetime depending on the momentum k, which appears in the Boltzmann theory, instead of the single-electron lifetime appearing in the Green function, can be recovered. This alternative formulation is applied to (i) the elastic scattering in metals, and (ii) the inelastic scattering in the Fr\"ohlich polaron model to obtain the exact result of the mobility in the low-temperature weak-coupling limit.