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

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