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

TL;DR

This paper analyzes the ground states of a massive scalar field with Robin boundary conditions in anti-de Sitter spacetime, finding conditions for well-defined states and their properties, including the absence of bound states and specific boundary condition constraints.

Contribution

It determines all admissible Robin boundary conditions for scalar fields in AdS and its cover, constructs the corresponding ground states, and explores conditions for their physical consistency.

Findings

No bound state modes occur on the universal cover of AdS.

Ground states are of Hadamard form and constructed via mode expansion.

Only Dirichlet or Neumann boundary conditions yield states on AdS with Hadamard form.

Abstract

We consider a real, massive scalar field both on the $n$-dimensional anti--de Sitter (AdS$_n$) spacetime and on its universal cover CAdS$_n$. In the second scenario, we extend the recent analysis on PAdS$_n$, the Poincar\'e patch of 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 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 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 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.