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 formulating the correspondence ADS/CFT (Anti-de Sitter/Conformal Field Theory). This correspondence was established by Maldacena  in 1998 and is universally regarded as a very useful model for many purposes.
The bibliography on this subject, for scalar fields, is quite extensive. We give here just a small representative in  - . For a more complete bibliography the reader is directed to the report .
One of the ADS/CFT correspondence’s prescriptions (see ) will allow us to evaluate the correlators on the boundary of ADS space. The first boundary-bulk propagator was calculated by Witten a few months after the appearance of , entitled Anti de Sitter space and holography. In this case the boundary is a Euclidean space  .
In this work, instead, we evaluate the boundary-bulk propagators for the case in which the boundary is a Minkowskian space. In such regards, remark that some attempts have been made before in   .
1.1. The Wheeler Propagator
The Feynman’s propagator for a free real scalar field is a time-ordered correlation function of two scalar fields and in the vacuum state
This propagator is a Green function of the Klein-Gordon equation, and is discussed in almost any text-book on quantum mechanics. Not so well-known at all is the Wheeler propagator. In fact, to provide a fairly complete description of it constitutes one of the present goals.
More than half a century ago, J. A. Wheeler and R. P. Feynman published a work  in which they represented electromagnetic interactions by means of a half advanced and half retarded Green functions. The charged medium was supposed to be a perfect absorber, so that no radiation could possibly escape the system.
We are going to call this kind of Green function a “Wheeler function’’ (or propagator). It has been used before by P. A. M. Dirac , when trying to avoid some run-away solutions, in which one finds rapid increases that cannot be controlled. Later on, in 1949, J. A. Wheeler and R. P. Feynman showed that, in spite of the fact that the Green function contains an advanced part, the results do no contradict causality .
A causal, unitary, and Lorentz invariant quantification of tachyons was performed in reference . The corresponding propagator is precisely a Wheeler’s one.
The same happens with complex mass particles that appear in higher order supersymmetric models . For these particles, the propagator is also a Wheeler’s propagator.
We review some precedent work below.
1.2. The Starinets and Son Paper
The main previous attempt to try to calculate boundary-bulk propagators in the Minkowskian boundary for the Anti-de Sitter space [in the ADS/CFT correspondence] was made by Son and Starinets (SS) in 2002 . 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”.
1.3. The Freedman et al. Paper
We must also mention the work of Freedman et al. , 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 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.
1.4. 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 (see below for the notation) in momentum space, and therefore a function of 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.
As usual, we use here regularity conditions 1) at the origin (Dirichlet’s) and 2) of rapid decay at infinity (boundary condition). This applies, for instance, to Equations (2.8), (2.9), and (2.10).
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  and Israelovich M. Guelfand et al. .
Note also that, until the 90’s, the only field propagators that had been calculated were Anti-de Sitter (spatial) ones.
1.5. 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.
2. Euclidean Case
2.1. Massless Scalar Field Propagator
The Klein-Gordon equation in for the scalar field reads, in Poincare coordinates,
where plays the role of . We exclude tachyons form of this treatment. Here is the conformal dimension, the boundary’s dimension, and their coordinates. The Fourier transform in the variables of the field is
Using (2.2), (2.1) takes the form
We analyze now the massless case given by . For it we have the motion equation
or equivalently (for ),
In the variable z, this equation is of the Bessel type (see )
The pertinent solution (that does not diverge when the argument tends to infinity) is
Thus, the solution of (2.5) becomes
One easily verifies that, for infinitesimal z ,
Equation (2.9) is just a Bessel-McDonald distribution (defined by Guelfand ) in Euclidean space. As a consequence,
In other words, the solution is regular at the origin and vanishes at infinity (in the variable z). Accordingly, we have, for the field in the bulk, the solution
This solution must reduce itself to the field on the boundary, so that
From this last equation we can obtain as a function of and then write
From (2.14) we then obtain an expression of the boundary-bulk propagator
To carry out the integration in the variable k we appeal to the expressions for the Fourier transform and its inverse obtained by Bochner . For the Fourier transform we have
and for its inverse
Using these relations we have now
So as to evaluate the last integral we appeal to a result from 
Our deduction follows a different, simpler and complete path than that of . Our approach also has a didactic utility.
which leads to
an expression that, in turn, leads to
2.2. Massive Field Propagator
We now consider the massive case . The equation of motion for this case reads
The solution for this last equation is
Since , we select for in (2.26) the plus sign. We have then
For , this solution is not regular at the origin. To overcome this problem we select
where is infinitesimal. From (2.28) we have then
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
From (2.33) we immediately gather that
2.3. Wrong but Popular Approach for Approximate Massive Field Propagators
It is instructive to discuss here a popular but non-valid approach for the function. The issue here is that, although is infinitesimal, it cannot adopt a 0-value. As k is an unbounded variable, when, we have. Notice first that
Some people make now the approximation
From (2.32) one obtains an approximation for the propagator K that can be called M. Ome has then
Using again the Bochner formula one arrives at
By recourse to (2.19) it follows that
one can write
It is then realized that, by construction,
which allows one to write for 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.
The above recounted approximation, not very well founded, is precisely the one most people use in current literature to obtain the propagator (2.32). From it, people deduce the approximation (2.44).
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.
3. Minkowskian Case
3.1. Massless Field Propagator
Let us now deal with the case in which the boundary of the is the -dimensional Minkowskian space. In the massless case the field-equation is
where. Thus, we can write
or, rewriting this last equation,
The distribution is defined as (see reference )
and can be cast in terms of, the Heaviside step function . 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, becoming
One must take into account that (see below in this section and ).
Equation (3.7) is just a Bessel-McDonald distribution (defined by Guelfand ) in Minkowskian space. We have then
From this last equation we deduce that
The ensuing propagator becomes then
Thus, the corresponding Feynman’s propagator is
Note that the Feynman propagator is a function of, 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 can define, as usual, the retarded propagator
and the advanced propagator
We are going to show now that (see ). Let be a test function belonging to a sub-space of Schwartz’s one  . Its Fourier transform is
where belongs to. Then one can verify that
As a consequence, we obtain
(3.21) is an extremely well-known fact established by Distribution Theory, and can be found in the text-book by Jones . The Feynman propagator is, according to (3.12),
Since is exponentially decreasing or oscillating, we can evaluate the integral that defines by means of a Wick rotation over. Therefore we have the change of variables, , , and. Casting the integral that defines the propagator in terms of these new variables, we obtain
Using Bochner’s formula together with (3.19) we have
Now, making the change to Minkowskian variables and taking into account that the Fourier transform of a distribution that depends on is a distribution that depends on, 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
3.2. Massive Field Propagator
For the massive case, the field-motion equation is
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:
3.3. An Approximation
We now evaluate in approximate fashion the propagator
Effecting again the above Wick’s rotation we obtain
This integral is evaluated as in the previous cases. One has
Changing variables as above we arrive at
Now we return to the inequality
The following relation is valid for
Proceeding in analogous fashion with the Anti-Feynman propagator we obtain the approximation
4. Glaring Mistakes of Son and Starinets’ Calculation  Corrected
By appeal to the unproved conjecture mentioned in Subsection 1.2, Son and Starinets evaluated the retarded propagator for a scalar field in a work regarded as a standard-bear of the ADS/CFT field. They found
We will show below that this result is both wrong and incomplete.
The retarded propagator reads
For one has
Using  we have
This, Feynman’s propagator becomes
while the anti-Feynman one turns out to be
With the two last results OUR version of Starinets and Son retarded propagator becomes
and for the advanced one
With a little algebra the two propagators reappear as
Consider now the penultimate term of the retarded propagator. It is
Considering just the first term () in we can write (up to a sign)
that can be recast as
reads, using Son and Starinets’ metrics
which coincides with (4.1) after calling.
Thus, expression (4.1) is just a single term of the full expression for the retarded propagator of (4.11). This last propagator verifies while (4.1) does not. We conclude then that (4.1) CAN NOT be used as a propagator.
Starinets and Son expression (SS) (4.1) cannot be regarded as a propagator for the massless scalar field. The same happens for the Feynman propagator of Eq. (3.21) in page 9 of . These erroneous results demonstrate that their conjecture is inadequate.
4.1. Son and Starinets Surprising Elimination of a Divergence
To justify the results of their paper, in page 22 of , Son and Starinets encounter an infinite in their equation (A.22). They eliminate it by setting, which is absurd since has a pole in, and, as a consequence, it has a divergence in this value of z. This procedure is mathematically unacceptable. However, it was applauded by many ADS/CFT practitioners. Read and learn!
5. Two Points Correlation Functions in Euclidean Space
5.1. Massless Case
To evaluate the two-point correlation function of a scalar operator, we use the result obtained in . This is
where, , , and then
As, we obtain
Using now the expression for K given in Equation (2.20) we have
Accordingly, we have here arrived to the usual, well-known result.
5.2. Massive Case
For the massive case we obtain, similarly,
As we can write
Thus we arrive at
Now, we use the expression for given in (3.32) and write
Using now the following result, given in ,
Note that we have not renormalized the correlation functions. We will do that using the results of  in a forthcoming paper.
6. Two Points Correlation Functions in Minkowskian Space
6.1. Massless Case
Similarly to the Euclidean case we obtain for the Minkowskian one the result
Thus, we obtain for the Feynman’s propagator
For the Anti-Feynman instance one has
and for Wheeler’s situation,
6.2. Massive Case
Again, following the developments of the Euclidean case, we have, for the Minkowskian instance, the two points Feynman’s correlator:
Thus, we have
Using again (5.10) we finally obtain
For the Anti-Feynman propagator we obtain in analogous fashion
and for Wheeler
Note again that we have not re-normalized the correlation functions. We will do that using the results of  in a forthcoming paper.
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  (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 , 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.
 Barci, D.G., Bollini, C.G. and Rocca, M.C. (1993) Il Nuovo Cimento, 106, 603-609.
Barci, D.G., Bollini, C.G. and Rocca, M.C. (1994) International Journal of Modern Physics A, 9, 3497-3502.