Coordinate representation of particle dynamics in AdS and in generic static spacetimes

TL;DR

This paper develops a coordinate-based quantum particle dynamics framework in static curved spacetimes, especially AdS, linking it to covariant quantization and highlighting the role of scalar curvature and symmetries.

Contribution

It introduces a coordinate representation for quantum particles in static spacetimes, clarifies the relation to covariant quantization, and characterizes the scalar curvature term's role in AdS spaces.

Findings

The energy operator E^2 includes a scalar curvature term fixed by isometry groups.

The quantization scheme reproduces the AdS energy spectrum and is consistent with covariant methods.

In arbitrary static spacetimes, the two quantization approaches are equivalent under specific scalar curvature conditions.

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 E^2, 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 E^2 and in the Klein-Gordon type equation have the same coefficient equal to (N-1)/(4N).