Energy and the AdS/CFT Correspondence

TL;DR

This paper introduces a new definition of energy in AdS scalar field theory, showing it is conserved, positive, and leads to a generalized AdS/CFT correspondence that captures bulk information without counterterms.

Contribution

It proposes a novel energy concept based on Noether currents, providing a consistent quantization framework and a generalized AdS/CFT prescription that accounts for boundary conditions and constraints.

Findings

New energy definition is conserved, positive, and finite.

The generalized AdS/CFT maps all bulk information to the boundary.

Divergent terms relate to the Legendre transformed generating functional.

Abstract

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.