Comments On Hamiltonian Formalism Of $AdS/CFT$ correspondence

TL;DR

This paper explores a Hamiltonian approach to the $AdS_2/CFT_1$ correspondence, establishing an isomorphism between bulk and boundary representations and clarifying the operator mapping in this duality.

Contribution

It introduces a Hamiltonian formalism emphasizing unitary representation theory and identifies a specific isomorphism linking bulk and boundary field operators.

Findings

Bulk and boundary unitary irreducible representations are isomorphic.

Field operators in bulk and boundary theories correspond via boundary values.

The bulk Fock vacuum is located at the boundary, matching the boundary CFT vacuum.

Abstract

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.