  We compare the optical absorption of extended systems using the
density-density and current-current linear response functions calculated within
many-body perturbation theory. The two approaches are formally equivalent for a
finite momentum $\mathbf{q}$ of the external perturbation. At
$\mathbf{q}=\mathbf{0}$, however, the equivalence is maintained only if a small
$q$ expansion of the density-density response function is used. Moreover, in
practical calculations this equivalence can be lost if one naively extends the
strategies usually employed in the density-based approach to the current-based
approach. Specifically we discuss the use of a smearing parameter or of the
quasiparticle lifetimes to describe the finite width of the spectral peaks and
the inclusion of electron-hole interaction. In those instances we show that the
incorrect definition of the velocity operator and the violation of the
conductivity sum rule introduce unphysical features in the optical absorption
spectra of three paradigmatic systems: silicon (semiconductor), copper (metal)
and lithium fluoride (insulator). We then demonstrate how to correctly
introduce lifetime effects and electron-hole interactions within the
current-based approach.
