  We consider a scalar field theory on AdS in both minimally and non-minimally
coupled cases. We show that there exist constraints which arise in the
quantization of the scalar field theory on AdS which cannot be reproduced
through the usual AdS/CFT prescription. We argue that the usual energy, defined
through the stress-energy tensor, is not the natural one to be considered in
the context of the AdS/CFT correspondence. We analyze a new definition of the
energy which makes use of the Noether current corresponding to time
displacements in global coordinates. We compute the new energy for Dirichlet,
Neumann and mixed boundary conditions on the scalar field and for both the
minimally and non-minimally coupled cases. Then, we perform the quantization of
the scalar field theory on AdS showing that, for `regular' and `irregular'
modes, the new energy is conserved, positive and finite. We show that the
quantization gives rise, in a natural way, to a generalized AdS/CFT
prescription which maps to the boundary all the information contained in the
bulk. In particular, we show that the divergent local terms of the on-shell
action contain information about the Legendre transformed generating
functional, and that the new constraints for which the irregular modes
propagate in the bulk are the same constraints for which such divergent local
terms cancel out. In this situation, the addition of counterterms is not
required. We also show that there exist particular cases for which the
unitarity bound is reached, and the conformal dimension becomes independent of
the effective mass. This phenomenon has no bulk counterpart.
