  Starting from the conventional electron-hole Hamiltonian ${\cal H}_{eh}$, we
derive an effective Hamiltonian $\tilde{\cal H}_{1s}$ for $1s$ excitons with
spin degrees of freedom. The Hamiltonian describes optical processes close to
the exciton resonance for the case of weak excitation. We show that
straightforward bosonization of ${\cal H}_{eh}$ does not give the correct form
of $\tilde{\cal H}_{1s}$, which we obtain by a projection onto the subspace
spanned by the $1s$ excitons. The resulting relaxation and renormalization
terms generate an interaction between excitons with opposite spin. Moreover,
exciton-exciton repulsive interaction is greatly reduced by the
renormalization. The agreement of the present theory with the experiment
supports the validity of the description of a fermionic system by bosonic
fields in two dimensions.
