Scalar field in $\mathrm{AdS}_2$ and representations of $\widetilde{\mathrm{SL}}(2,\mathbb{R})$
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 , 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.
Keywords
Cite
@article{arxiv.2107.08754,
title = {Scalar field in $\mathrm{AdS}_2$ and representations of $\widetilde{\mathrm{SL}}(2,\mathbb{R})$},
author = {Atsushi Higuchi and Lasse Schmieding and David Serrano Blanco},
journal= {arXiv preprint arXiv:2107.08754},
year = {2023}
}