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

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 [Formula: see text], 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.