Ground states of a Klein-Gordon field with Robin boundary conditions in global anti–de Sitter spacetime

We consider a real, massive scalar field both on the n-dimensional anti–de Sitter (AdSn) spacetime and on its universal cover CAdSn. In the second scenario, we extend the recent analysis on PAdSn, the Poincaré patch of AdSn, first determining all admissible boundary conditions of Robin type that can be applied on the conformal boundary. Most notably, contrary to what happens on PAdSn, no bound state mode solution occurs. Subsequently, we address the problem of constructing the two-point function for the ground state satisfying the admissible boundary conditions. All these states are locally of Hadamard form being obtained via a mode expansion which encompasses only the positive frequencies associated to the global timelike Killing field on CAdSn. To conclude we investigate under which conditions any of the two-point correlation functions constructed on the universal cover defines a counterpart on AdSn, still of Hadamard form. Since this spacetime is periodic in time, it turns out that this is possible only for Dirichlet boundary conditions, though for a countable set of masses of the underlying field, or for Neumann boundary conditions, though only for even dimensions and for one given value of the mass.