  We develop a theory to calculate exciton binding energies of both two- and
three-dimensional spin polarized exciton gases within a mean field approach.
Our method allows the analysis of recent experiments showing the importance of
the polarization and intensity of the excitation light on the exciton
luminescence of GaAs quantum wells. We study the breaking of the spin
degeneracy observed at high exciton density $(5 \ \ 10^{10} cm ^2)$. Energy
level splitting betwen spin +1 and spin -1 is shown to be due to many-body
inter-excitonic exchange while the spin relaxation time is controlled by
intra-exciton exchange.
