  We discuss how SU$(1,1)$ coherent states from the discrete series allow for a
natural coarse graining operation. The physical application are quantum
theories based on a set of three extensive observables whose Poisson algebra is
isomorphic to su$(1,1)$. In particular, we show that a Perelomov coherent state
with representation label $N j_0$ and spinor label $z$ encodes the physics of
$N$ independent subsystems with labels $j_0, z$. This property is suggested by
existing results for the expectation values and variances of the observables.
We prove that it holds for all higher moments. Our results in particular apply
to a recent quantum cosmology model that has been derived using SU(1,1) group
theoretic quantisation techniques. For it, it follows that a certain notion of
fiducial cell independence holds exactly at the quantum level when using the
coherent states.
