  We have used the Hartree-Fock Random Phase Approximation (HF-RPA) to study
the interacting electron gas in a quantum wire. The spectra of intersubband
spin-flip excitations reveal a considerable red shift with respect to
single-particle HF energies. That signals on appearance of collective
intersubband spind-density excitations due to the exchange interaction. The
long wavelength dispersions of the intrasubband collective spin-density
excitations are linear, but the sound velocities are renormalised due to the
exchange interaction and screening. The in-phase intrasubband charge-density
excitation has the long wavelength form $q[-\ln(q)]^{1/2}$. We found good
qualitative agreement of our results with experimental observations.
