  An explicit holographic correspondence between $AdS$ bulk and boundary
quantum states is found in the form of a one to one mapping between scalar
field creation/annihilation operators. The mapping requires the introduction of
arbitrary energy scales and exhibits an ultraviolet-infrared duality: a small
regulating mass in the boundary theory corresponds to a large momentum cutoff
in the bulk. In the massless (conformal) limit of the boundary theory the
mapping covers the whole field spectrum of both theories. The mapping strongly
depends on the discretization of the field spectrum of compactified $AdS$ space
in Poincare coordinates.
