Advances in High Energy Physics

Volume 2017 (2017), Article ID 8185690, 11 pages

https://doi.org/10.1155/2017/8185690

## Bulk Local Operators, Conformal Descendants, and Radial Quantization

^{1}Institute of Modern Physics, Northwest University, Xi’an 710069, China^{2}Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi’an 710069, China

Correspondence should be addressed to Zhao-Long Wang

Received 5 July 2017; Accepted 13 August 2017; Published 2 October 2017

Academic Editor: Zhang-Yu Nie

Copyright © 2017 Zhao-Long Wang and Yi Yan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We establish a construction of the bulk local operators in AdS by considering CFT at finite energy scale. Without assuming any prior knowledge about the bulk, the solution to the bulk free field equation automatically appears in the field theory arguments. In the radial quantization formalism, we find a properly regularized version of our initial construction. Possible generalizations beyond pure AdS are also discussed.

#### 1. Introduction

The AdS/CFT correspondence [1–4] implies a duality between the quantum gravity in -dimensional anti-de Sitter space and the -dimensional conformal field theory which is defined on the boundary of . The AdS metric in the Poincaré patch is given bywhere the boundary is the -dimensional flat space at . The relations between the boundary data of AdS and the CFT quantities have been well established in [4] by the field-operator correspondence. That is, the correlators of a conformal primary operator in the CFT are reproduced by the asymptotical data of a bulk field near the boundary. However, the explicit CFT construction of the bulk local degree of freedoms inside the AdS space is not well understood yet. The earlier attempts [5–7] suggested to reconstruct by propagating the bulk modes from the bulk to the boundary, and then [8–10] showed that it is equivalent to the smearing operator construction. In this letter, we suggest a different construction based on almost purely CFT arguments. In Section 2.1, we establish the construction by considering CFT at finite energy scale. The possible divergence and the prescription of regulator are discussed in Section 2.2. Then in Section 3, we find that our construction can get improved in the radial quantization formalism. We summarize our main results in Section 4, and a possible way of generalizing the construction beyond pure AdS is also proposed there.

#### 2. CFT Construction of Bulk Local Operators

##### 2.1. Renormalized Primary at Finite Energy Scale

It has been pointed out qualitatively [1–4, 11] that the bulk radial direction is related to the energy scale in the dual field theory. In order to reconstruct , the first candidate is to consider in CFT the renormalized primary operator which is defined at a finite energy scale . On the other hand, the behaviors of a primary operator under the conformal transformation have already been encoded in its conformal family. Thus it is natural to expect, at least in the leading order, that the renormalization of the primary operator at a finite energy scale will lead to a mixing between the primary operator and its descendantsFor simplicity, we will only consider the scalar operator from now on. It is also natural to require that the renormalized primary operator recovers the Lorentz properties and the scaling dimension of the original primary. Then we find it can only be in the following form:

Is it possible to fix the explicit form of by imposing certain renormalization condition? The idea is to give the word “primary” a renormalized meaning. In the usual CFT language, the definition of a primary operator is equivalent to requiring that it transforms as a tensor under conformal transformations. We also notice that the renormalization scale will transform nontrivially under conformal transformations. Thus a direct guess is that the proper renormalization condition should be the following.

*The renormalized primary transforms as a tensor under the generalized conformal transformations including the energy scale.*

To address the generalized conformal transformations including the energy scale, let us firstly review the realization of conformal algebra on the -space. Acting on the coordinates , the conformal generator can be expressed as follows:It implies the standard conformal algebra

To include the energy scale (for different approaches of introducing the finite energy scale, see [12, 13].), a straightforward way is to add as well as -dependent coefficients into the realization (4). For latter convenience, we define and equivalently consider instead. From the fact that the energy scale is Poincaré invariant, we conclude that the forms of and remain intact. The scaling dimension of energy scale is obviously 1; thus we can easily write down the following generalized form of dilatation :For the special conformal generator, the strategy is to take its most general ansatzand then try to find the explicit form which satisfies the conformal algebra.

From , we getIt implies thatFrom , we further getwhere is an arbitrary constant. Finally, we can check that the above results satisfy . In conclusion, we haveIn fact, this is exactly the isometry generator of the AdS space when and it suggests to identify here with the standard AdS radial coordinate. We also notice that it corresponds to when is negative, but we will only concentrate on the AdS case in this paper.

Given the generalized conformal transformation including the energy scale (11), we can try to decide the form of by our renormalization condition on primary. For a scalar in the space, we can expand it by powers of The scalar transformation ruleimplies that the terms appearing in the power expansion should transform as follows:Now the task is to construct by the primary and its scalar descendants . From the conformal transformation rules of the primary we can deduce thatComparing with (14), it implies the unique identificationIn conclusion, up to an overall constant, our arguments show that the renormalized primary at energy scale is given byand corresponds to a bulk scalar field . In the limit, it actually comes back to the usual language of CFT.

We notice that obtained in (17) is nothing but the Fourier transformation of the solutions to the bulk free field equation with behavior at the boundary. This construction is different from the one suggested in [5–10]. The approach there encountered only the part of the bulk modes, and thus it cannot be generalized to the Euclidean AdS case. Instead, our construction encounters all the bulk modes since it is the honest Fourier transformation. Obviously, (17) is applicable for both signatures.

##### 2.2. Two-Point Correlators: The Divergent Regime and the Regulator

As a consistency check, let us use (17) to recover the well-known bulk-boundary propagator. We findIn the regime , the series is convergent and gives rise to the expected form of the bulk-boundary propagatorHowever, in the regime , the series (18) is divergent. In fact, this result is not surprising. The given in (17) is just the modes of the bulk solution, while the Fourier transformation of (19) is a linear combination of the modes and the modes [14] which regulate the divergence of (18). The existence of the constituent in (19) can be easily seen from the limit [15]Although both the and modes diverge exponentially as , the combination is well-behaved in the interior since the two divergences cancel with each other. The explicit computation of the corresponding Fourier transformations is performed in Appendix.

In order to understand the above issue better, let us recall a simple fact in field theory. That is, the correlation function for composite operators always has zeroth-order UV divergence due to its composite natural. For example, consider the composite operator :. The two-point correlator receives zeroth-order UV divergence from the following loop diagram even in the free theory.