The Casimir operator of SO(1,2) and the Pöschl-Teller potential: an AdS approach

Abstract

We present and discuss some features of the anti-de Sitter spacetime, that is jointly with de Sitter and Minkowski is only, the unique maximal isotropic manifold. Among all possible lorentzian manifolds, we restrict our attention to the anti-de Sitter (AdS) spacetime, with metric diag(1,$-1,-1$). We start by presenting the conformal time metric on AdS and we then show how we can obtain the Schrödinger formalism~\cite{sch}. The Lie algebra $\mathfrak{so}$(1,2) is introduced and used to construct spin and ladder operators. After presenting the unitary representations, the AdS(1,2) spacetime is suitably parametrized and a representation of SO(1,2) is obtained, from which the Schrödinger equation with Pöschl-Teller potential is immediately deduced. Finally, we discuss some relations between the relativistic harmonic oscillator and the Klein-Gordon equation, using the AdS(1,2) static frame. Possible applications of the presented formalism are provided.