How to Explicitly Calculate Feynman and Wheeler Propagators in the ADS/CFT Correspondence

We discuss, giving all necessary details, the boundary-bulk propagators. We do it for a scalar field, with and without mass, for both the Feynman and the Wheeler cases. Contrary to standard procedure, we do not need here to appeal to any unfounded conjecture (as done by other authors). Emphasize that we do not try to modify standard ADS/CFT procedures, but use them to evaluate the corresponding Feynman and Wheeler propagators. Our present calculations are original in the sense of being the first ones undertaken explicitly using distributions theory (DT). They are carried out in two instances: 1) when the boundary is a Euclidean space and 2) when it is of Minkowskian nature. In this last case we compute also three propagators: Feynman’s, Anti-Feynman’s, and Wheeler’s (half advanced plus half retarded). For an operator corresponding to a scalar field we explicitly obtain, for the first time ever, the two points’ correlations functions in the three instances above mentioned. To repeat, it is not our intention here to improve on ADS/CFT theory but only to employ it for evaluating the corresponding Wheeler’s propagators.


Introduction
Propagators and correlators are one of the essential tools to work, for example, in Quantum Field Theory (QFT) and String Theory (ST), in particular, in for-

The Wheeler Propagator
The Feynman's propagator for a free real scalar field is a time-ordered correlation function of two scalar fields ( ) Minkowskian boundary for the Anti-de Sitter space [in the ADS/CFT correspondence] was made by Son and Starinets (SS) in 2002 [22]. However, SS needed to formulate a conjecture that we show here to become unnecessary if one uses the full distributions-theory of type S' (of Schwartz). SS literally state (the necessary symbols will be explained later in the text) "We circumvent the difficulties mentioned above by putting forward the following conjecture For this conjecture no rigorous mathematical basis is presented. Instead, we will nor need here any conjecture at all. SS' work was entitled "Minkowski-space correlators in AdS/CFT correspondence: recipe and applications".

The Freedman et al. Paper
We must also mention the work of Freedman et al. [23], in which the authors deal with the case of a Euclidean boundary. Freedman, however, did not treat the case of a Minkowskian boundary, at least in the way that Son and Starinets did. To repeat, we make full use here of distribution theory. This does not entail, of course, a simple i prescription, but a much more elaborate treatment, that has not been performed before in this field. Let us also remark, as this is an important point for us, that in this paper we do not evaluate renormalized correlation functions.

Our Treatment
As stated above, in the present effort we evaluate, without any a la Starinets and Son conjecture, the boundary-bulk propagators corresponding to the following three cases i) Feynman, ii) Anti-Feynman, and iii) Wheeler (half advanced plus half retarded). We do this both for massless and massive scenarios (a scalar field involved). Later we calculate the two points correlators (TPC) for operators corresponding to this scalar field in the three instances previously mentioned. We clarify that in this paper we do not evaluate the renormalized TPC.
We demonstrate as well that the Feynman propagator must be a function of 0 i ρ + (see below for the notation) in momentum space, and therefore a function of 2 0 x i − in configuration space. We show that something similar happens with the Anti-Feynman propagator. For the first time ever, we calculate the Wheeler's propagator (half advanced plus half retarded) as well.
It may be asserted that propagators are always to be interpreted in a distributional sense, but most authors do not employ, in dealing with them, the FULL distribution theory developed by Laurent Schwartz [24] and Israelovich M.
Guelfand et al. [25]. Note also that, until the 90's, the only field propagators that had been calcu-lated were Anti-de Sitter (spatial) ones.

Organization of This Work
The paper is organized as follows: Section 2 deals with the Euclidean case. In it, the three different propagators referred to above cannot be distinguished (neither in the massive nor in the massless instances).
In Section 3, we tackle similar scenarios as those of Section 2, but now in Minkowski's space, where the three propagators can be distinguished.
In Section 4, we compute in Euclidean space the TPC for a scalar operator corresponding to a scalar field via Witten's prescription.
In Section 5, we generalize the calculations of Section 4 to Minkowski's space. We obtain in this fashion the two-point correlations functions corresponding to the three different propagators of our list above.
Finally, some conclusions are drawn in Section 6.

Massless Scalar Field Propagator
The Klein-Gordon equation in We analyze now the massless case given by 0,ν ∆ = . For it we have the motion equation In the variable z, this equation is of the Bessel type (see [26]) The pertinent solution (that does not diverge when the argument tends to infinity) is Our deduction follows a different, simpler and complete path than that of [2]. Our approach also has a didactic utility.
an expression that, in turn, leads to

Massive Field Propagator
We now consider the massive case 0,ν ∆ ≠ . The equation of motion for this case reads (2.23) or equivalently, The solution for this last equation is For 0 ∆ ≠ , this solution is not regular at the origin. To overcome this prob- Replacing the result of (2.29) into (2.27) we obtain From this last equation we see that the propagator is As a consequence we can write

Wrong but Popular Approach for Approximate Massive Field Propagators
It is instructive to discuss here a popular but non-valid approach for the function Some people make now the approximation Using again the Bochner formula one arrives at By recourse to (2.19) it follows that It is then realized that, by construction, which allows one to write for m N the expression Therefore, one has constructively proved that Note that (2.44) is indeed the well known expression for the boundary-bulk propagator for a scalar field in configuration space. However, this expression can only be used as an approximation to the propagator K when 2 ν µ ≅ .
The above recounted approximation, not very well founded, is precisely the Indeed, one of the main goals of our paper is to overcome the problems posed by this approximation. We will try below to do better than current usage, and shall indeed achieve our goal.

Massless Field Propagator
Let us now deal with the case in which the boundary of the 1 ADS ν + is the ν -dimensional Minkowskian space. In the massless case the field-equation is or, rewriting this last equation, and can be cast in terms of ( ) H x , the Heaviside step function [24]. We recast now (3.3) in the form of a Bessel equation The solution of this equation that is 1) regular at the origin and 2) vanishes for ρ → ∞ , becoming ( ) ( ) One must take into account that lim e 0 Equation ( or, equivalently, The ensuing propagator becomes then Thus, the corresponding Feynman's propagator is Note that the Feynman propagator is a function of 0 i ρ + , as it should. For the anti-Feynman propagator we have instead The expression for the Wheeler's propagator (half advanced plus half retarded) is: Using the relations We are going to show now that lim e 0 ikx k →∞ = (see [28]). Let φ be a test function belonging to a sub-space  of Schwartz's one [24] [25]. Its Fourier transform is where φ belongs to  . Then one can verify that As a consequence, we obtain lim e 0. (3.21) is an extremely well-known fact established by Distribution Theory, and can be found in the text-book by Jones [28]. The Feynman propagator is, according to (3.12), Now, making the change to Minkowskian variables and taking into account that the Fourier transform of a distribution that depends on 0 i ρ − is a distribution that depends on 2 0 x i + , we obtain ( ) ( ) which is the expression of the Feynman propagator in terms of the variables of the configuration space. For the anti-Feynman propagator we analogously find

Massive Field Propagator
For the massive case, the field-motion equation is with, again, The pertinent solution is now The field-expression in configuration space is then Once again we choose and from (3.23) we obtain We have then the following relation for the solution so that the propagator is now The corresponding Feynman's propagator becomes For the anti-Feynman propagator we obtain the expression Finally, the definition of Wheeler propagators, half retarded and half advanced, is similar to that of the preceding subsection, this is:

An Approximation
We now evaluate in approximate fashion the propagator ( ) Effecting again the above Wick's rotation we obtain Using [26] we have This, Feynman's propagator becomes and for the advanced one

Massless Case
Similarly to the Euclidean case we obtain for the Minkowskian one the result ( ) ( ) ( ) For the Anti-Feynman instance one has A. Plastino, M. C. Rocca

Massive Case
Again, following the developments of the Euclidean case, we have, for the Minkowskian instance, the two points Feynman's correlator: or equivalently, Using again (5.10) we finally obtain For the Anti-Feynman propagator we obtain in analogous fashion Note again that we have not re-normalized the correlation functions. We will do that using the results of [23] in a forthcoming paper.

Conclusions
In this work we have firstly calculated, without using any conjecture, the boundary-bulk Feynman, Anti-Feynman, and Wheeler propagators (half advanced plus half retarded) for both a massless and a massive scalar field, by recourse to the theory of distributions.
We conclusively showed that a previous 2002 work by Son and Starinets [22] (discussing only the Feynman propagator) is wrong.
As further novelties, in the paper we showed that, for massive scalar fields, the expression for the boundary-bulk propagator in Euclidean momentum space does not correspond to the expression used in configuration space, but it is rather a mere approximation.
Subsequently, using the previous results, we have evaluated the correlation functions of scalar operators corresponding to massless and massive scalar fields.
Unlike the results obtained in [22], with the ones obtained here you can calculate the n-points correlation functions from gravity. This is feasible for a scalar operator when n is an arbitrary natural number. This is perhaps our main present contribution.