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 (AdS n ) spacetime and on its universal cover CAdS n . In the second scenario, we extend the recent analysis on PAdS n , the Poincaré 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.

PROPAGATION OF SINGULARITIES ON AdS SPACETIMES FOR GENERAL BOUNDARY CONDITIONS AND THE HOLOGRAPHIC HADAMARD CONDITION

“…These boundary conditions appear frequently in the physics literature; see e.g. [DF2,DFM]. Further motivation comes from the study of the Dirichlet-to-Neumann operator, which was recently shown to coincide with a power of the wave operator in the case of a static metric [EdMGV].…”

Boundary-to-bulk maps for AdS causal wedges and RG flow

“…In [1,2] it was used that the frequencies corresponding to Dirichlet come equispaced by 2n, with n an integer number [26], and this implied immediately the spacelike support of K. Such a simplification does not occur with Robin boundary conditions, since the frequencies are given by the transcendental equation above and are not equispaced (see [12] as well as [13]). However by the analytic continuation method we will employ, it will be easy to construct K for Neumann boundary condition γ = π/2 and then for generic Robin boundary conditions, and we will see they have spacelike support.…”

“…earlier results of [12] and the recent work of [13], we consider Robin boundary conditions. Immediately we will see the need to proceed with care, since divergences appear when imposing Neumann boundary conditions (Φ ∼ z ∆ − ), at least when working in position space.…”

“…To this end, we start by considering the fact that AdS is not globally hyperbolic, which implies that boundary conditions at timelike infinity need to be imposed in order to have a well-defined evolution of initial conditions. Different boundary conditions have been explored in the literature [11,12] and are well understood from a bulk point of view (see [13] for a thorough treatment). One main point is that the behavior of the scalar field can have both z ∆ + and z ∆ − decays in the Breitenlohner-Freedman (BF) window 0 ≤ ν ≤ 1 [14], where ∆ ± = d/2 ± ν, and ν = d 2 /4 + m 2 .…”

An evolution equation approach to the Klein–Gordon operator on curved spacetime

“…This includes for example compactifications of antide Sitter spacetime with appropriate conditions on its timelike boundary (cf. [8,9] for a recent discussion of boundary conditions on anti-de Sitter spacetime and spacetimes with a timelike boundary).…”