  The bulk S-Matrix can be given a non-perturbative definition in terms of the
flat space limit of AdS/CFT. We show that the unitarity of the S-Matrix, ie the
optical theorem, can be derived by studying the behavior of the OPE and the
conformal block decomposition in the flat space limit. When applied to
perturbation theory in AdS, this gives a holographic derivation of the cutting
rules for Feynman diagrams.
  To demonstrate these facts we introduce some new techniques for the analysis
of conformal field theories. Chief among these is a method for conglomerating
local primary operators to extract the contribution of an individual primary in
their OPE. This provides a method for isolating the contribution of specific
conformal blocks which we use to prove an important relation between certain
conformal block coefficients and anomalous dimensions. These techniques make
essential use of the simplifications that occur when CFT correlators are
expressed in terms of a Mellin amplitude.
