An Alternative Path to the Boundary - the CFT as the Fourier Space of AdS

TL;DR

This paper proposes interpreting the AdS/CFT duality as a Fourier transform, viewing the CFT as the Fourier space of AdS, which offers new insights into the duality and potential applications in QCD and black hole physics.

Contribution

It introduces a novel interpretation of the AdS/CFT correspondence as a Fourier transform, connecting boundary CFT data with bulk AdS functions using integral geometry techniques.

Findings

Fourier transformation of AdS functions yields boundary functions

Green's functions are identified as Fourier weights

Boundary fields correspond to Fourier components

Abstract

In this paper we shed new light on the AdS/CFT duality by interpreting the CFT as the Fourier space of AdS. 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.