Generalized free fields and the AdS-CFT correspondence. (English) Zbl 1038.81051

Motivated by some features of correspondence between field theory in Anti de Sitter (AdS) space-time and conformal field theory (CFT), the theory of generalized free fields is reconsidered in the paper under review. Generalized free fields have been introduced by Greenberg as a new class of models for local quantum fields. They can be characterized in several equivalent ways, for example by the requirement that their correlation functions factorize into 2-point functions or that the generating functional for the correlation functions is a Gaussian. But in distinction from a canonical free field, a generalized free field is not the solution to an equation of motion, and its 2-point function has the form of a superposition of the 2-point functions of the Klein-Gordon fields with different masses (the superposition being determined by the Källen-Lehmann weight). 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.


81T20 Quantum field theory on curved space or space-time backgrounds
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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