Generalized Free Fields and the AdS-CFT 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 fulfills the requirements of a conserved local density for the Poincaré 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.

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

“…In the absence of mathematically rigorous approaches to path integrals of string type (see however [1]) or even gravity, it seems to be reasonable to use the well-established theory of constructive quantum field theory (QFT) [5] as a testing lab for some aspects of the more complex original AdS/CFT conjecture. That such simplified versions of the AdS/CFT correspondence are in fact possible was already noted by Witten [17] (see also [8]) and further elaborated by [4]. In [6] we give a mathematically rigorous version of the latter work (in [9] one finds some related ideas), leaving however the infra-red (IR) problem open.…”

“…Here F is the hypergeometric function, u = [4,6]. Taking pointwhise scaling limits for z → 0 in one or two of the arguments, the bulk-to-boundary and boundary-toboundary propagators are obtained…”

“…their stochastic and reflection positivity is far from obvious, leaving the linkage to path integrals and relativistic physics open. It was noticed by Dütsch and Rehren [4] that such kinds of generating functionals can however be re-written in terms of ordinary generating functionals, from which the structural properties can be read of in the usual way. These ideas in [6] have been made fully rigorous in the context of constructive QFT.…”

“…In Section 3 we recall the main results from [4,6], i.e. that the generating functional that is obtained from imposing certain boundary conditions at the conformal boundary (which is the way generating functionals are defined in string theory) can in fact be written as a usual generating functional of some other field theory.…”

Partial wave expansion and Wightman positivity in conformal field theory

“…Note, however, that generalized free fields do possess a stressenergy tensor which is more singular than a Wightman field [DR03]: it has infinite vacuum fluctuations, hence formally c = ∞; in particular, this stress-energy tensor does not contribute to the operator product expansion.…”