Energy and the AdS/CFT Correspondence

TL;DR

This work challenges the standard AdS/CFT energy prescription by showing that the usual metrical energy fails to capture all bulk information for scalar fields on AdS, especially regarding irregular bulk modes. It introduces a canonical energy derived from the Noether current for time translations, which depends on boundary terms and boundary conditions, and develops a generalized AdS/CFT prescription in which the boundary source is tied to these boundary conditions; a full on-shell action Legendre transform naturally maps bulk constraints to boundary operators and can render divergent local terms harmless without counterterms. The results demonstrate that the new framework reproduces the bulk constraints for irregular mode propagation and provides a unified mapping between bulk and boundary data across Dirichlet, Neumann, and mixed boundary conditions, including ν integer cases and ν=0, while highlighting a peculiar unitarity-bound case with a mass-independent conformal dimension. These insights strengthen the role of the canonical energy and the generalized prescription as foundational tools in AdS/CFT, with potential implications for holographic renormalization and boundary dynamics.

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.