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We consider a real, massive scalar field on ${\rm PAdS}_{d+1}$, thePoincar\'e domain of the $(d+1)$-dimensional anti-de Sitter (AdS) spacetime. Wefirst determine all admissible boundary conditions that can be applied on theconformal boundary, noting that there exist instances where "bound states"solutions are present. Then, we address the problem of constructing thetwo-point function for the ground state satisfying those boundary conditions,finding ultimately an explicit closed form. In addition, we investigate thesingularities of the resulting two-point functions, showing that they areconsistent with the requirement of being of Hadamard form in every globallyhyperbolic subregion of ${\rm PAdS}_{d+1}$ and proposing a new definition ofHadamard states which applies to ${\rm PAdS}_{d+1}$.

Hadamard states for a scalar field in anti-de Sitter spacetime with arbitrary boundary conditions