Recently the damping of the collective charge (and spin) modes of interacting
fermions in one spatial dimension was studied. It results from the nonlinear
correction to the energy dispersion in the vicinity of the Fermi points. To
investigate the damping one has to replace the random phase approximation (RPA)
bare bubble by a sum of more complicated diagrams. It is shown here that a
better starting point than the bare RPA is to use the (conserving) linearized
time dependent Hartree-Fock equations, i.e. to perform a random phase
approximation (with) exchange
(RPAE) calculation. It is shown that the RPAE equation can be solved
analytically for the special form of the two-body interaction often used in the
Luttinger liquid framework. While (bare) RPA and RPAE agree for the case of a
strictly linear disperson there are qualitative differences for the case of the
usual nonrelativistic quadratic dispersion.Comment: 6 pages, 3 figures, misprints corrected; to appear in PRB7