Quantization of AdS x S particle in static gauge

TL;DR

This paper quantizes particle dynamics in AdS x S spacetime using static gauge, analyzing the energy operator, its spectrum, and constructing unitary group representations, advancing understanding of quantum properties in curved backgrounds.

Contribution

It introduces a novel quantization approach for particles in AdS x S spacetime in static gauge, including spectral analysis and representation construction.

Findings

Energy square operator includes scalar curvature term

Spectrum of the operator is explicitly calculated

Unitary representations of isometry groups are constructed

Abstract

We quantize the particle dynamics in AdS_{N+1} x S^M spacetime in static gauge, which leads to the coordinate representation with wave functions depending only on spatial coordinates. The energy square operator is quadratic in canonical momenta and contains a scalar curvature term. We analyze the self-adjointness of this operator and calculate its spectrum. We then construct unitary representations of the isometry group SO(2,N) x SO(M+1) and calculate the quantum relation between the Casimir numbers.