Exact Form of Boundary Operators Dual to Interacting Bulk Scalar Fields in the AdS/CFT Correspondence

Using holographic renormalization coupled with the Caffarelli/Silvestre\cite{caffarelli} extension theorem, we calculate the precise form of the boundary operator dual to a bulk scalar field rather than just its average value. We show that even in the presence of interactions in the bulk, the boundary operator dual to a bulk scalar field is an anti-local operator, namely the fractional Laplacian. The propagator associated with such operators is of the general power-law (fixed by the dimension of the scalar field) type indicative of the absence of particle-like excitations at the Wilson-Fisher fixed point or the phenomenological unparticle construction. Holographic renormalization also allows us to show how radial quantization can be extended to such non-local conformal operators.