Coordinate representation of particle dynamics in AdS and generic static spacetimes

Abstract

We discuss the quantum dynamics of a particle in static curved spacetimes in a coordinate representation. The scheme is based on the analysis of the squared energy operator E2, which is quadratic in momenta and contains a scalar curvature term. Our main emphasis is on AdS spaces, where this term is fixed by the isometry group. As a byproduct the isometry generators are constructed and the energy spectrum is reproduced. In the massless case the conformal symmetry is realized as well. We show the equivalence between this quantization and the covariant quantization, based on the Klein–Gordon-type equation in AdS. We further demonstrate that the two quantization methods in an arbitrary (N + 1)-dimensional static spacetime are equivalent to each other if the scalar curvature terms both in the operator E2 and in the Klein–Gordon-type equation have the same coefficient equal to .