Massless scalar particle on AdS spacetime: Hamiltonian reduction and quantization

TL;DR

This paper analyzes the dynamics of a massless scalar particle in AdS spacetime using Hamiltonian reduction, constructing the physical phase space through conformal symmetry, and quantizing it via unitary irreducible representations of the conformal group.

Contribution

It introduces a Hamiltonian reduction approach to describe massless scalar particles on AdS, linking boundary dynamics to null geodesics, and constructs the quantum theory using conformal group representations.

Findings

Physical phase space as SO(2,N+1) orbit

Representation at the unitarity bound

Connection between AdS null geodesics and boundary dynamics

Abstract

We investigate the massless scalar particle dynamics on $AdS_{N+1} ~ (N>1)$ by the method of Hamiltonian reduction. Using the dynamical integrals of the conformal symmetry we construct the physical phase space of the system as a $SO(2,N+1)$ orbit in the space of symmetry generators. The symmetry generators themselves are represented in terms of $(N+1)$-dimensional oscillator variables. The physical phase space establishes a correspondence between the $AdS_{N+1}$ null-geodesics and the dynamics at the boundary of $AdS_{N+2}$. The quantum theory is described by a UIR of $SO(2,N+1)$ obtained at the unitarity bound. This representation contains a pair of UIR's of the isometry subgroup SO(2,N) with the Casimir number corresponding to the Weyl invariant mass value. The whole discussion includes the globally well-defined realization of the conformal group via the conformal embedding of $AdS_{N+1}$ in the ESU $\rr\times S^N$.