  As a toy model to search for Hamiltonian formalism of the $AdS/CFT$
correspondence, we examine a Hamiltonian formulation of the $AdS_2/CFT_1$
correspondence emphasizing unitary representation theory of the symmetry. In
the course of a canonical quantization of the bulk scalars, a particular
isomorphism between the unitary irreducible representations in the bulk and
boundary theories is found. This isomorphism defines the correspondence of
field operators. It states that field operators of the bulk theory are field
operators of the boundary theory by taking their boundary values in a due way.
The Euclidean continuation provides an operator formulation on the hyperbolic
coordinates system. The associated Fock vacuum of the bulk theory is located at
the boundary, thereby identified with the boundary CFT vacuum. The
correspondence is interpreted as a simple mapping of the field operators acting
on this unique vacuum. Generalization to higher dimensions is speculated.

High Energy Physics - Theory