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

Abstract

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

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

“…The techniques used in [8] are rather flexible; they have been studied from a rigorous viewpoint in [9] and applied also to the analysis of a Klein-Gordon field in BTZ spacetime in [10]. In this paper, first we also apply them to the study of a massive, real scalar field in the global chart of CAdS n , in order to investigate if the results obtained in the Poincaré patch do extend globally.…”

“…In a recent paper by two of us [8], it has been shown that, if one considers only the Poincaré patch of AdS n , it is possible to use a mode decomposition together with techniques proper of Sturm-Liouville problems, in order to prove that one can consider a whole one-parameter family of boundary conditions of Robin type, which include as a special case both the Dirichlet and the Neumann ones. In this work it has been shown that, for each of these boundary conditions, the Klein-Gordon equation can be solved in terms of initial data and unique fundamental solutions do exist.…”

“…While, from a classical perspective, this is not a problem, it has rather drastic consequences at a quantum level. As a matter of fact, since the Poincaré patch possesses a global timelike Killing field, in [8], it has been studied the existence for each boundary condition of Robin type of ground states associated with the Klein-Gordon equation. It turned out that, while they do not exist whenever bound state mode solutions occur, in all other cases they can be constructed explicitly in terms of their associated two-point correlation function.…”

Algebraic Quantum Field Theory on Spacetimes with Timelike Boundary

“…We shall illustrate this fact by using the free Klein-Gordon theory as an example. Thus, our results also provide a bridge between the ideas of [30] and the concrete constructions in [8,[10][11][12].…”

“…Recent works addressing this problem are [19,20,33], the latter extending the analysis of [31]. For specific choices of boundary conditions, there exist successful constructions of algebraic quantum field theories on spacetimes with timelike boundary, see e.g., [8,[10][11][12]. The main message of these works is that the algebraic approach is versatile enough to account also for these models, although some key structures, such as for example the notion of Hadamard states [11,32], should be modified accordingly.…”