Scalar field in AdS2 and representations of SL̃(2,R)

Abstract

We study the solutions to the Klein–Gordon equation for the massive scalar field in the universal covering space of a 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 SL(2,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 a non-invariant vacuum state and determine the unitary representation to which the vacuum state belongs.