  We revise the unireps. of $U(2,2)$ describing conformal particles with
continuous mass spectrum from a many-body perspective, which shows massive
conformal particles as compounds of two correlated massless particles. The
statistics of the compound (boson/fermion) depends on the helicity $h$ of the
massless components (integer/half-integer). Coherent states (CS) of
particle-hole pairs ("excitons") are also explicitly constructed as the
exponential action of exciton (non-canonical) creation operators on the ground
state of unpaired particles. These CS are labeled by points $Z$ ($2\times 2$
complex matrices) on the Cartan-Bergman domain $\mathbb D_4=U(2,2)/U(2)^2$, and
constitute a generalized (matrix) version of Perelomov $U(1,1)$ coherent states
labeled by points $z$ on the unit disk $\mathbb D_1=U(1,1)/U(1)^2$. Firstly we
follow a geometric approach to the construction of CS, orthonormal basis,
$U(2,2)$ generators and their matrix elements and symbols in the reproducing
kernel Hilbert space $\mathcal H_\lambda(\mathbb D_4)$ of analytic
square-integrable holomorphic functions on $\mathbb D_4$, which carries a
unitary irreducible representation of $U(2,2)$ with index $\lambda\in\mathbb N$
(the conformal or scale dimension). Then we introduce a many-body
representation of the previous construction through an oscillator realization
of the $U(2,2)$ Lie algebra generators in terms of eight boson operators with
constraints. This particle picture allows us for a physical interpretation of
our abstract mathematical construction in the many-body jargon. In particular,
the index $\lambda$ is related to the number $2(\lambda-2)$ of unpaired quanta
and to the helicity $h=(\lambda-2)/2$ of each massless particle forming the
massive compound.
