  We present a many-body theory for Frenkel excitons which takes into account
their composite nature exactly. Our approach is based on four commutators
similar to the ones we previously proposed for Wannier excitons. They allow us
to calculate any physical quantity dealing with $N$ excitons in terms of "Pauli
scatterings" for carrier exchange in the absence of carrier interaction and
"interaction scatterings" for carrier interaction in the absence of carrier
exchange. We show that Frenkel excitons have a novel "transfer assisted
exchange scattering", specific to these excitons. It comes from indirect
Coulomb processes between localized atomic states. These indirect processes,
commonly called "electron-hole exchange" in the case of Wannier excitons and
most often neglected, are crucial for Frenkel excitons, as they are the only
ones responsible for the excitation transfer. We also show that in spite of the
fact that Frenkel excitons are made of electrons and holes on the same atomic
site, so that we could naively see them as elementary particles, they
definitely are composite objects, their composite nature appearing through
various properties, not always easy to guess. The present many-body theory for
Frenkel excitons is thus going to appear as highly valuable to securely tackle
their many-body physics, as in the case of nonlinear optical effects in organic
semiconductors.
