Coset construction of AdS particle dynamics

We analyze dynamics of the AdS$_{N+1}$ particle realized on the coset SO$(2,N)/$SO$(1,N)$. Hamiltonian reduction provides the physical phase space in terms of the coadjoint orbit obtained by boosting a timelike element of ${\frak so}(2,N)$. We show equivalence of this approach to geometric quantization and to the SO$(N)$ covariant oscillator description, for which the boost generators entail a complicated operator ordering. As an alternative scheme, we introduce dual oscillator variables and derive their algebra at the classical and the quantum level. This simplifies the calculations of the commutators for the boost generators and leads to unitary irreducible representations of ${\frak so}(2,N)$ for all admissible values of the mass parameter. We furthermore discuss an SO$(N)$ covariant supersymmetric extensions of the oscillator quantization, with its realization for superparticles in AdS$_2$ and AdS$_3$ given by recent works.