Phase Space Quantum Mechanics on the Anti-De Sitter Spacetime and its Poincar\'e Contraction

TL;DR

This paper develops a phase space approach to quantum mechanics in anti-de Sitter spacetime, showing how localized states contract to plane waves in the zero curvature limit, and connects geometric quantization with spacetime contraction.

Contribution

It introduces a phase space formulation of quantum mechanics in anti-de Sitter spacetime and analyzes the contraction to Poincaré symmetry using geometric quantization methods.

Findings

Anti-de Sitter localized states contract to plane waves as curvature vanishes.

The invariant Kähler polarization reduces to Poincaré polarization in the zero curvature limit.

Localization notions disappear in the flat spacetime limit.

Abstract

In this work we propose an alternative description of the quantum mechanics of a massive and spinning free particle in anti-de~Sitter spacetime, using a phase space rather than a spacetime representation. The regularizing character of the curvature appears clearly in connection with a notion of localization in phase space which is shown to disappear in the zero curvature limit. We show in particular how the anti-de~Sitter optimally localized (coherent) states contract to plane waves as the curvature goes to zero. In the first part we give a detailed description of the classical theory {\it \a la Souriau\/}. This serves as a basis for the quantum theory which is constructed in the second part using methods of geometric quantization. The invariant positive K\"ahler polarization that selects the anti-de~Sitter quantum elementary system is shown to have as zero curvature limit the Poincar\'e polarization which is no longer K\"ahler. This phenomenon is then related to the disappearance of the notion of localization in the zero curvature limit.