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

We consider a real, massive scalar field on 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 PAdS_d+1 and proposing a new definition of Hadamard states which applies to PAdS_d+1.