Alternative path to the boundary: The CFT as the Fourier transform in AdS space

In this paper we shed new light on the AdS/CFT duality by interpreting the CFT as the Fourier transform in AdS space. We make use of well-known integral geometry techniques to derive the Fourier transformation of a function defined on the AdS hyperboloid. We show that the Fourier transformation of a function on the hyperboloid is a function defined on the boundary. We find that the Green’s functions from the literature are actually the Fourier weights (i.e. plane wave solutions) of the transformation and that the boundary values of fields appearing in the correspondence are the Fourier components of the transformation. One is thus left to interpret the CFT as the quantized version of a classical theory in AdS and the dual operator as the Fourier coefficients. Group theoretic considerations are discussed in relation to the transformation and its potential use in constructing QCD-like theories. In addition, we consider possible implications involving understanding the dual of AdS black holes.

Figure 1

A schematic showing that two representations of $SO(1,2)$ can be found by taking different quotient spaces of the embedding space. The Fourier transformation is a means of going between the two representations. This generalizes for higher dimensional spaces. Because AdS black holes are quotient spaces of AdS, they could potentially be included in the diagram with an analogous transform.