Understand
We consider a real, massive scalar field on ${\rm PAdS}_{d+1}$, the Poincar\'e 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 ${\rm PAdS}_{d+1}$ and proposing a new definition of Hadamard states which applies to ${\rm PAdS}_{d+1}$.