The Feynman’s propagator for a free real scalar field is a time-ordered correlation function of two scalar fields Φ ( x ) and Φ ( y ) in the vacuum state

G F ( x − y ) = 〈 0 | T ^ Φ ( x ) Φ ( y ) | 0 〉 . (1.1)

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 [17] 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 [18], 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 [19].

A causal, unitary, and Lorentz invariant quantification of tachyons was performed in reference [20]. The corresponding propagator is precisely a Wheeler’s one.

The same happens with complex mass particles that appear in higher order supersymmetric models [21]. For these particles, the propagator is also a Wheeler’s propagator.

We review some precedent work below.

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 [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

G R ( k ) = − 2 F ( k , z ) | z B (3.15)”

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”.

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 ϵ 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.

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 ρ + i 0 (see below for the notation) in momentum space, and therefore a function of x 2 − i 0 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 [24] and Israelovich M. Guelfand et al. [25].

Note also that, until the 90’s, the only field propagators that had been calculated were Anti-de Sitter (spatial) ones.

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.

The Klein-Gordon equation in A D S ν + 1 for the scalar field ϕ ( z , x ) reads, in Poincare coordinates,

z 2 ∂ z 2 ϕ ( z , x ) + ( 1 − ν ) z ∂ z ϕ ( z , x ) + z 2 ∇ 2 ϕ ( z , x ) − Δ ( Δ − ν ) ϕ ( z , x ) = 0, (2.1)

where Δ ( Δ − ν ) ≥ 0 plays the role of m 2 . We exclude tachyons form of this treatment. Here Δ is the conformal dimension, ν the boundary’s dimension, and x their coordinates. The Fourier transform in the variables x of the field ϕ ( z , x ) is

ϕ ^ ( z , k ) = ∫   ϕ ( z , x ) e i k ⋅ x d ν x . (2.2)

Using (2.2), (2.1) takes the form

z 2 ∂ z 2 ϕ ^ ( z , k ) + ( 1 − ν ) z ∂ z ϕ ^ ( z , k ) − [ z 2 k 2 + Δ ( Δ − ν ) ] ϕ ^ ( z , k ) = 0. (2.3)

We analyze now the massless case given by Δ = 0 , ν . For it we have the motion equation

z 2 ∂ z 2 ϕ ^ ( z , k ) + ( 1 − ν ) z ∂ z ϕ ^ ( z , k ) − z 2 k 2 ϕ ^ ( z , k ) = 0, (2.4)

or equivalently (for z ≠ 0 ),

∂ z 2 ϕ ^ ( z , k ) + 1 − ν z ∂ z ϕ ^ ( z , k ) − k 2 ϕ ^ ( z , k ) = 0. (2.5)

In the variable z, this equation is of the Bessel type (see [26])

F ″ ( z ) + 1 − 2 α z F ′ ( z ) − [ k 2 + μ 2 − α 2 z 2 ] F ( z ) = 0 (2.6)

The pertinent solution (that does not diverge when the argument tends to infinity) is

F ( z ) = z α K μ ( k z ) . (2.7)

Thus, the solution of (2.5) becomes

ϕ ^ ( z , k ) = z ν 2 K ν 2 ( k z ) . (2.8)

One easily verifies that, for infinitesimal z [26],

K ν 2 ( k z ) = 2 ν 2 − 1 Γ ( ν 2 ) ( k z ) ν 2 + O ( ( k z ) − ν 2 + 2 ) . (2.9)

Equation (2.9) is just a Bessel-McDonald distribution (defined by Guelfand [25]) in Euclidean space. As a consequence,

lim z → 0 z ν 2 K ν 2 ( k z ) = 2 ν 2 − 1 Γ ( ν 2 ) k ν 2 . (2.10)

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

ϕ ( z , x ) = z ν 2 ( 2 π ) ν ∫   a ( k ) K ν 2 ( k z ) e − i k ⋅ x d ν k . (2.11)

This solution must reduce itself to the field ϕ 0 ( x ) on the boundary, so that

ϕ ( 0 , x ) = ϕ 0 ( x ) = 2 ν 2 − 1 Γ ( ν 2 ) ( 2 π ) ν ∫   a ( k ) k − ν 2 e − i k ⋅ x d ν k = 1 ( 2 π ) ν ∫   ϕ ^ 0 ( k ) e − i k ⋅ x d ν k . (2.12)

From this last equation we can obtain a ( k ) as a function of ϕ ^ 0 and then write

ϕ ( z , x ) = z ν 2 2 1 − ν 2 ( 2 π ) ν Γ ( ν 2 ) ∫   k ν 2 K ν 2 ( k z ) ϕ ^ 0 ( k ) e − i k ⋅ x d ν k , (2.13)

or, equivalently,

ϕ ( z , x ) = z ν 2 2 1 − ν 2 ( 2 π ) ν Γ ( ν 2 ) ∬   k ν 2 K ν 2 ( k z ) ϕ 0 ( x ′ ) e − i k ⋅ ( x − x ′ ) d ν k d ν x ′ . (2.14)

From (2.14) we then obtain an expression of the boundary-bulk propagator

K ( z , x − x ′ ) = z ν 2 2 1 − ν 2 ( 2 π ) ν Γ ( ν 2 ) ∫   k ν 2 K ν 2 ( k z ) e − i k ⋅ ( x − x ′ ) d ν k . (2.15)

To carry out the integration in the variable k we appeal to the expressions for the Fourier transform and its inverse obtained by Bochner [27]. For the Fourier transform we have

f ^ ( k ) = ∫ f ( x ) e i k ⋅ x d ν x = ( 2 π ) ν 2 k ν 2 − 1 ∫ 0 ∞     r ν 2 J ν 2 − 1 ( k r ) f ( r ) d r , (2.16)

and for its inverse

f ( r ) = 1 ( 2 π ) ν ∫ f ^ ( k ) e − i k ⋅ x d ν k = 1 ( 2 π ) ν 2 r ν 2 − 1 ∫ 0 ∞   k ν 2 J ν 2 − 1 ( k r ) f ^ ( k ) d k . (2.17)

Using these relations we have now

K ( z , x − x ′ ) = z ν 2 2 1 − ν 2 ( 2 π ) ν Γ ( ν 2 ) ( 2 π ) ν 2 | x − x ′ | ν 2 − 1 ∫ 0 ∞   k ν K ν 2 ( k z ) J ν 2 − 1 ( k | x − x ′ | ) d k . (2.18)

So as to evaluate the last integral we appeal to a result from [26]

∫ 0 ∞   x μ + ν + 1 K ν ( b x ) J μ ( a x ) d x = 2 μ + ν a μ b ν Γ ( μ + ν + 1 ) ( a 2 + b 2 ) μ + ν + 1 , (2.19)

Our deduction follows a different, simpler and complete path than that of [2]. Our approach also has a didactic utility.

K ( z , x − x ′ ) = Γ ( ν ) π ν 2 Γ ( ν 2 ) [ z z 2 + ( x − x ′ ) 2 ] ν , (2.20)

which leads to

ϕ ( z , x ) = ∫   K ( z , x − x ′ ) ϕ 0 ( x ′ ) d ν x ′ , (2.21)

an expression that, in turn, leads to

lim z → 0 K ( z , x − x ′ ) = δ ( x − x ′ ) . (2.22)

We now consider the massive case Δ ≠ 0, ν . The equation of motion for this case reads

z 2 ∂ z 2 ϕ ^ ( z , k ) + ( 1 − ν ) z ∂ z ϕ ^ ( z , k ) − [ z 2 k 2 + Δ ( Δ − ν ) ] ϕ ^ ( z , k ) = 0, (2.23)

or equivalently,

∂ z 2 ϕ ^ ( z , k ) + 1 − ν z ∂ z ϕ ^ ( z , k ) − [ k 2 + Δ ( Δ − ν ) z 2 ] ϕ ^ ( z , k ) = 0. (2.24)

The solution for this last equation is

ϕ ^ ( z , k ) = z ν 2 K μ ( k z ) , (2.25)

with

μ = ± ν 2 4 + Δ ( Δ − ν ) . (2.26)

Since K μ ( z ) = K − μ ( z ) , we select for μ in (2.26) the plus sign. We have then

ϕ ( z , x ) = z ν 2 ( 2 π ) ν ∫   a ( k ) K μ ( k z ) e − i k ⋅ x d ν k . (2.27)

For Δ ≠ 0 , this solution is not regular at the origin. To overcome this problem we select

ϕ ( ϵ , x ) = ϕ ϵ ( x ) = ϵ ν 2 ( 2 π ) ν ∫   a ( k ) K μ ( k ϵ ) e − i k ⋅ x d ν k = 1 ( 2 π ) ν ∫   ϕ ^ ϵ ( k ) e − i k ⋅ x d ν k , (2.28)

where ϵ is infinitesimal. From (2.28) we have then

a ( k ) = ϕ ^ ϵ ( k ) ϵ ν 2 K μ ( k ϵ ) . (2.29)

Replacing the result of (2.29) into (2.27) we obtain

ϕ ( z , x ) = 1 ( 2 π ) ν ( z ϵ ) ν 2 ∫ K μ ( k z ) K μ ( k ϵ ) ϕ ^ ϵ ( k ) e − i k ⋅ x d ν k , (2.30)

or similarly,

ϕ ( z , x ) = 1 ( 2 π ) ν ( z ϵ ) ν 2 ∫ ∫ K μ ( k z ) K μ ( k ϵ ) ϕ ϵ ( x ′ ) e − i k ⋅ ( x − x ′ ) d ν k d ν x ′ . (2.31)

From this last equation we see that the propagator is

K m ( z , x − x ′ ) = 1 ( 2 π ) ν ( z ϵ ) ν 2 ∫ K μ ( k z ) K μ ( k ϵ ) e − i k ⋅ ( x − x ′ ) d ν k . (2.32)

As a consequence we can write

ϕ ( z , x ) = ∫ K m ( z , x − x ′ ) ϕ ϵ ( x ′ ) d ν x ′ . (2.33)

From (2.33) we immediately gather that

K m ( ϵ , x − x ′ ) = δ ( x − x ′ ) . (2.34)

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

Defining

one can write

It is then realized that, by construction,

and define

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.

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 [24])

and can be cast in terms of, 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, becoming

One must take into account that (see below in this section and [25]).

Equation (3.7) is just a Bessel-McDonald distribution (defined by Guelfand [24]) in Minkowskian space. We have then

From this last equation we deduce that

or, equivalently,

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

and

we can define, as usual, the retarded propagator

and the advanced propagator

We are going to show now that (see [28]). Let be a test function belonging to a sub-space of Schwartz’s one [24] [25]. 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 [28]. 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

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:

We now evaluate in approximate fashion the propagator

entailing

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

where

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

To justify the results of their paper, in page 22 of [22], 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!

To evaluate the two-point correlation function of a scalar operator, we use the result obtained in [29]. 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.

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

or, equivalently,

Using now the following result, given in [26],

we obtain

Note that we have not renormalized the correlation functions. We will do that using the results of [23] in a forthcoming paper.

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,

Again, following the developments of the Euclidean case, we have, for the Minkowskian instance, the two points Feynman’s correlator:

Thus, we have

or equivalently,

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 [23] in a forthcoming paper.