We consider a real, massive scalar field on ${\text{PAdS}}_{d+1}$, the Poincaré domain of the ($d+1$)-dimensional anti–de Sitter (AdS) spacetime. We first determine all admissible boundary conditions that can be applied on the conformal boundary, noting that there exist instances where “bound states” solutions are present. Then, we address the problem of constructing the two-point function for the ground state satisfying those boundary conditions, finding ultimately an explicit closed form. In addition, we investigate the singularities of the resulting two-point functions, showing that they are consistent with the requirement of being of Hadamard form in every globally hyperbolic subregion of ${\text{PAdS}}_{d+1}$ and proposing a new definition of Hadamard states which applies to ${\text{PAdS}}_{d+1}$.

We analyze quantum field theories on spacetimes M with timelike boundary from a model-independent perspective. We construct an adjunction which describes a universal extension to the whole spacetime M of theories defined only on the interior \(\mathrm {int}M\). The unit of this adjunction is a natural isomorphism, which implies that our universal extension satisfies Kay’s F-locality property. Our main result is the following characterization theorem: Every quantum field theory on M that is additive from the interior (i.e., generated by observables localized in the interior) admits a presentation by a quantum field theory on the interior \(\mathrm {int}M\) and an ideal of its universal extension that is trivial on the interior. We shall illustrate our constructions by applying them to the free Klein–Gordon field.