A Comment on the Infra-Red Problem in the AdS/CFT Correspondence

TL;DR

This paper discusses recent progress in rigorously establishing the AdS/CFT correspondence for quantum fields, addressing the unique infra-red problem and exploring potential renormalization strategies.

Contribution

It provides a rigorous proof framework for AdS/CFT using Euclidean path integrals and clarifies the distinct nature of the infra-red problem in this context.

Findings

Proved triviality for ultra-violet regularized $:\,\phi^4:$$ potential.

Highlighted differences between IR problems in AdS/CFT and traditional quantum field theory.

Discussed potential renormalization approaches and associated challenges.

Abstract

In this note we report on some recent progress in proving the AdS/CFT correspondence for quantum fields using rigorously defined Euclidean path integrals. We also comment on the infra-red problem in the AdS/CFT correspondence and argue that it is different from the usual IR problem in constructive quantum field theory. To illustrate this, a triviality proof based on hypercontractivity estimates is given for the case of an ultra-violet regularized potential of type $:\phi^4:$. We also give a brief discussion on possible renormalization strategies and the specific problems that arise in this context.