when explicitly stated otherwise. It can conveniently be described in terms of Cartesian coordinates in the ambient space $\RR^{n+1}$ as the quadric surface $${\mbox{{AdS}$^n\,$}}= \{x\in\RR^{n+1} \mid 
x^2 \doteq x_0^2 - x_1^2 - \dots - x_{n-1}^2 + x_n^2 = R^2 \}$$ with metric $g = \mbox{diag}(1,-1,\dots,-1,1)$ in diagonal form. As the value of the radius $R$ is not relevant for the results of this paper, we shall set it equal to $1$ for convenience. The isometry group is O$(2,n-1)$ whose identity component will be denoted by . is a homogeneous space of the group . It is not globally hyperbolic; indeed, it has closed timelike curves and has a timelike boundary at spatial infinity through which physical data can propagate. Although the covering space of eliminates the closed timelike curves, it still has a timelike boundary at spatial infinity. We shall find some notable differences between the properties of quantum field theories on and those of theories on the covering space.

As some of our basic assumptions are motivated by physical considerations concerning certain families of observables, we must collect some basic facts about observers in AdS. Let $x_O \in {\mbox{{AdS}$^n\,$}}$ be any point and let $\lambda (t)$, $t \in \RR$, 