  We propose a general framework for studying quantum field theory on the
anti-de-Sitter space-time, based on the assumption of positivity of the
spectrum of the possible energy operators. In this framework we show that the
n-point functions are analytic in suitable domains of the complex AdS manifold,
that it is possible to Wick rotate to the Euclidean manifold and come back, and
that it is meaningful to restrict AdS quantum fields to Poincare' branes. We
give also a complete characterization of two-point functions which are the
simplest example of our theory. Finally we prove the existence of the AdS-Unruh
effect for uniformly accelerated observers on trajectories crossing the
boundary of AdS at infinity, while that effect does not exist for all the other
uniformly accelerated trajectories.
