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

Abstract

We consider a real, massive scalar field both on the $n$-dimensional anti--de Sitter (${\mathrm{AdS}}_{n}$) spacetime and on its universal cover ${\mathrm{CAdS}}_{n}$. In the second scenario, we extend the recent analysis on ${\mathrm{PAdS}}_{n}$, the Poincar\'e patch of ${\mathrm{AdS}}_{n}$, first determining all admissible boundary conditions of Robin type that can be applied on the conformal boundary. Most notably, contrary to what happens on ${\mathrm{PAdS}}_{n}$, 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 ${\mathrm{CAdS}}_{n}$. To conclude we investigate under which conditions any of the two-point correlation functions constructed on the universal cover defines a counterpart on ${\mathrm{AdS}}_{n}$, 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.