^{1}

^{2}

^{1}

^{2}

^{1}

^{2}

^{3}.

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.

The AdS/CFT correspondence [

It has been pointed out qualitatively [

Is it possible to fix the explicit form of

To address the generalized conformal transformations including the energy scale, let us firstly review the realization of conformal algebra on the

To include the energy scale (for different approaches of introducing the finite energy scale, see [

From

Given the generalized conformal transformation including the energy scale (

We notice that

As a consistency check, let us use (

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

In the coordinate space, the corresponding divergence takes the following form:

We notice that the

One can also check that the bulk-bulk propagator can be recovered by computing

In Section

The radial quantization for CFT was detailedly reviewed in [

Now let us consider the CFT radial quantization in the presence of the finite energy scale

Simply by using (

Parallel with (

Since our construction directly comes back to the standard CFT language in the

Formula (

In the previous sections, we suggest a CFT construction of the bulk local operators in pure AdS space. The construction is based on considering CFT at finite energy scale. The basic result is that bulk operator is given by acting upon the original CFT primary with an infinite order differential operator

The next challenge is how to generalize our construction to geometries beyond pure AdS. A naive guess is that the bulk local operator is also effectively given by acting upon the original CFT primary with the infinite order differential operator

Bulk geometries are actually dual to the coherent states

The bulk correlators of the dual field

Finally, it is also possible that multiple states with different

As the early version of this work was drawing a conclusion, [

In the momentum space, the general solutions to the bulk free scalar equation are given by the linear combination of the

The Fourier transformation of the

For

For the

In the

Comparing (

The above derivations are performed under the Euclidean signature. For the Minkowskian signature, the 2-point propagator is not unique due to the existence of the light cone singularity. Depending on which kind of 2-point propagator was considered, the corresponding bulk momentum space formulae are different. These different formulae are related to the different choices of quantum states of the boundary QFT [

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors are grateful to Piljin Yi for enlightening discussions as well as collaboration at an early stage of this work and also thanks are due to Jiang Long, Hong Lü, Jun-Bao Wu, Xiao Xiao, and Hossein Yavartanoo for useful conversations. This work is supported by National Natural Science Foundation of China with Grant no. 11305125 and no. 11447607 and the Double First-Class University Construction Project of Northwest University.