  An effective bosonic Hamiltonian of $1s$ excitons with ``spin'' degrees of
freedom in two dimension is obtained through a projection procedure, starting
from a conventional electron-hole Hamiltonian ${\cal H}_{eh}$. We first
demonstrate that a straightforward transformation of ${\cal H}_{eh}$ into a
Hamiltonian of bosonic excitons does not give the two-body interaction between
an ``up-spin'' exciton and a ``down-spin'' exciton, which are created by the
left- and right-circularly polarized light beams, respectively. We then show
that this interaction is generated through a projection procedure onto the
subspace spanned by $1s$ excitons, as a renormalization effect coming from
higher exciton states. The projection also renormalizes the interaction between
$1s$ excitons with the same spins by a large amount. These renormalization
effects are crucial for the polarization dependence of the optical responses
from semiconductors. The present theory gives the microscopic foundation of the
phenomenology that was successfully applied to the analysis of four-wave mixing
experiments in GaAs quantum wells strongly coupled to the radiation field in a
high-Q micro cavity.
