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We revise the unireps. of $U(2,2)$ describing conformal particles withcontinuous mass spectrum from a many-body perspective, which shows massiveconformal particles as compounds of two correlated massless particles. Thestatistics of the compound (boson/fermion) depends on the helicity $h$ of themassless components (integer/half-integer). Coherent states (CS) ofparticle-hole pairs ("excitons") are also explicitly constructed as theexponential action of exciton (non-canonical) creation operators on the groundstate of unpaired particles. These CS are labeled by points $Z$ ($2\times 2$complex matrices) on the Cartan-Bergman domain $\mathbb D_4=U(2,2)/U(2)^2$, andconstitute a generalized (matrix) version of Perelomov $U(1,1)$ coherent stateslabeled by points $z$ on the unit disk $\mathbb D_1=U(1,1)/U(1)^2$. Firstly wefollow a geometric approach to the construction of CS, orthonormal basis,$U(2,2)$ generators and their matrix elements and symbols in the reproducingkernel Hilbert space $\mathcal H_\lambda(\mathbb D_4)$ of analyticsquare-integrable holomorphic functions on $\mathbb D_4$, which carries aunitary irreducible representation of $U(2,2)$ with index $\lambda\in\mathbb N$(the conformal or scale dimension). Then we introduce a many-bodyrepresentation of the previous construction through an oscillator realizationof the $U(2,2)$ Lie algebra generators in terms of eight boson operators withconstraints. This particle picture allows us for a physical interpretation ofour abstract mathematical construction in the many-body jargon. In particular,the index $\lambda$ is related to the number $2(\lambda-2)$ of unpaired quantaand to the helicity $h=(\lambda-2)/2$ of each massless particle forming themassive compound.

On the oscillator realization of conformal U(2,2) quantum particles and their particle-hole coherent states