Scalar field in $\mathrm{AdS}_2$ and representations of $\widetilde{\mathrm{SL}}(2,\mathbb{R})$

TL;DR

This paper analyzes scalar field solutions in AdS2, focusing on boundary conditions that ensure well-defined evolution and symmetry invariance, and classifies the resulting unitary representations of the isometry group.

Contribution

It characterizes boundary conditions for scalar fields in AdS2 that preserve symmetries and identifies the associated unitary representations of the isometry group.

Findings

Identified boundary conditions leading to self-adjoint Klein-Gordon operators.

Determined which boundary conditions preserve the universal covering group of SL(2,R).

Classified the unitary representations associated with different boundary conditions.

Abstract

We study the solutions to the Klein-Gordon equation for the massive scalar field in the universal covering space of two-dimensional anti-de Sitter space. For certain values of the mass parameter, we impose a suitable set of boundary conditions which make the spatial component of the Klein-Gordon operator self-adjoint. This makes the time-evolution of the classical field well defined. Then, we use the transformation properties of the scalar field under the isometry group of the theory, namely, the universal covering group of $\mathrm{SL}(2,\mathbb{R})$, in order to determine which self-adjoint boundary conditions are invariant under this group, and which lead to the positive-frequency solutions forming a unitary representation of this group and, hence, to a vacuum state invariant under this group. Then we examine the cases where the boundary condition leads to an invariant theory with non-invariant vacuum state and determine the unitary representation to which the vacuum state belongs.