Generalized free fields and the AdS-CFT correspondence

TL;DR

This paper revisits generalized free fields in the AdS-CFT context, constructing a conserved stress-energy density as a singular limit of Wightman fields and deriving an explicit holographic formula that expresses the AdS Klein-Gordon field in terms of boundary generalized free fields. It shows that the AdS representation and the boundary conformal representation coincide, enabling a z-integrated AdS stress-energy to reproduce a singular boundary stress-energy, and clarifies how algebraic and projective notions of holography differ in this setting. The results lay groundwork for perturbation theory around generalized free fields and deepen the understanding of holographic duality beyond Lagrangian frameworks, including implications for Haag duality and time-slice properties. Overall, the work provides concrete mechanisms by which bulk AdS dynamics maps to a boundary generalized free field theory and outlines the algebraic structure that underpins this holographic correspondence.

Abstract

Motivated by structural issues in the AdS-CFT correspondence, the theory of generalized free fields is reconsidered. A stress-energy tensor for the generalized free field is constructed as a limit of Wightman fields. Although this limit is singular, it fulfils the requirements of a conserved local density for the Poincar'e generators. An explicit "holographic" formula relating the Klein-Gordon field on AdS to generalized free fields on Minkowski space-time is provided, and contrasted with the "algebraic" notion of holography. A simple relation between the singular stress-energy tensor and the canonical AdS stress-energy tensor is exhibited.