  This paper introduces a new quantum object, the ``coboson'', for composite
particles, like the excitons, which are made of two fermions. Although commonly
dealed with as elementary bosons, these composite bosons -- ``cobosons'' in
short -- differ from them due to their composite nature which makes the
handling of their many-body effects quite different from the existing
treatments valid for elementary bosons. Due to this composite nature, it is not
possible to correctly describe the interaction between cobosons as a potential
$V$. Consequently, the standard Fermi golden rule, written in terms of $V$,
cannot be used to obtain the transition rates between exciton states. Through
an unconventional expression for this Fermi golden rule, which is here given in
terms of the Hamiltonian only, we here give a detailed calculation of the time
evolution of two excitons. We compare the results of this exact approach with
the ones obtained by using an effective bosonic exciton Hamiltonian. We show
that the relation between the inverse lifetime and the sum of transition rates
for elementary bosons differs from the one of composite bosons by a factor of
1/2, whatever the mapping from composite bosons to elementary bosons is. The
present paper thus constitutes a strong mathematical proof that, in spite of a
widely spread belief, we cannot forget the composite nature of these cobosons,
even in the extremely low density limit of just two excitons. This paper also
shows the (unexpected) cancellation, in the Born approximation, of the
two-exciton transition rate for a finite value of the momentum transfer.
