Quantum Field Theory, Feynman-, Wheeler Propagators, Dimensional Regularization in Configuration Space and Convolution of Lorentz Invariant Tempered Distributions

The Dimensional Regularization of Bollini and Giambiags (Phys. Lett. {\bf B 40}, 566 (1972), Il Nuovo Cim. {\bf B 12}, 20 (1972). Phys. Rev. {\bf D 53}, 5761 (1996)) can not be defined for all Schwartz Tempered Distributions Explicitly Lorentz Invariant (STDELI) ${\cal S}^{'}_L$. In this paper we overcome here such limitation and show that it can be generalized to all aforementioned STDELI and obtain a product in a ring with zero divisors. For this purpose, we resort to a formula obtained in [Int. J. of Theor. Phys. {\bf 43}, 1019 (2004)] and demonstrate the existence of the convolution (in Minkowskian space) of such distributions. This is done by following a procedure similar to that used so as to define a general convolution between the Ultradistributions of J. Sebastiao e Silva [Math. Ann. {\bf 136}, 38 (1958)], also known as Ultrahyperfunctions, obtained by Bollini et al. [Int. J. of Theor. Phys. {\bf 38}, 2315 (1999), {\bf 43}, 1019 (2004), {\bf 43}, 59 (2004),{\bf 46}, 3030 (2007)]. Using the Inverse Fourier Transform we get the ring with zero divisors ${\cal S}^{'}_{LA}$, defined as ${\cal S}^{'}_{LA}={\cal F}^{-1}\{{\cal S}^{'}_L\}$, where ${\cal F}^{-1}$ denotes the Inverse Fourier Transform. In this manner we effect a dimensional regularization in momentum space (the ring ${\cal S}^{'}_{L}$) via convolution, and a product of distributions in the corresponding configuration space (the ring ${\cal S}^{'}_{LA})$. This generalizes the results obtained by Bollini and Giambiagi for Euclidean space in [Phys. Rev. {\bf D 53}, 5761 (1996)]. We provide several examples of the application of our new results in Quantum Field Theory. In particular, the convolution of $n$ massless Feynman propagators and the convolution of n massless Wheeler propagators in Minkowskian space.

L . In this paper we overcome here such limitation and show that it can be generalized to all aforementioned STDELI and obtain a product in a ring with zero divisors.
For this purpose, we resort to a formula obtained in [Int. J. of Theor. Phys. 43, 1019 (2004)] and demonstrate the existence of the convolution (in Minkowskian space) of such distributions. This is done by following a procedure similar to that used so as to define a general convolution between the Ultradistributions of J. Sebastiao e Silva [Math. Ann. 136, 38 (1958)], also known as Ultrahyperfunctions, obtained by Bollini  We provide several examples of the application of our new results in Quantum Field Theory. In particular, the convolution of n massless Feynman propagators and the convolution of n massless Wheeler propagators in Minkowskian space.
The results obtained in this work have already allowed us to calculate the classical partition function of Newtonian gravity, for the first time ever, in the Gibbs' formulation and in the Tsallis' one: Physica A 503, 793 (2018), 497, 310 (2918).
It is our hope that this convolution will allow one to quantize nonrenormalizable Quantum Fields Theories.

Contents 1 Introduction
The problem of defining the product of two distributions (a product in a ring with divisors of zero) is an old problem of hard functional analysis.
In Quantum Theory of Fields the problem of evaluating the product of distributions with coincident point singularities is related to the asymptotic behaviour of loop integrals of propagators.
From a mathematical point of view, practically all definitions of that product lead to limitations on the set of distributions that can be multiplied by each other to give another distribution of the same type.
In fact, Laurent Schwartz showed that he can not define a product of distributions regarded as an algebra, instead of as a ring with divisors of zero.
In references [6,7,8,9] it was demonstrated that it is possible to define a general convolution between the ultradistributions of J. Sebastiao e Silva (Ultrahyperfunctions), resulting in another Ultradistribution, and, therefore, a general product in a ring with zero divisors. Such a ring is the space of Distributions of Exponential Type, or Ultradistributions of Exponential Type, obtained applying the anti-Fourier transform to the space of Tempered Ultradistributions or Ultradistributions of Exponential Type.
We must clarify at this point that the Ultrahyperfunctions are the generalization and extension to the complex plane of the Schwartz Tempered Distributions and the Distributions of Exponential Type. That is, the Temperate Distributions and those of Exponential Type are a subset of the Ultrahyprefunctions.
The problem we then face is that of formulating the convolution between Ultradistributions. This is a complex issue, difficult to manage, even if it has the advantage of allowing to attempt non-renormalizable Quantum Fields Theories.
Fortunately, we have found that a method similar to that used to define the convolution of Ultradistributions can also be used to define the convolution of Lorentz Invariant distributions using the Dimensional Regularization of Bollini and Giambiagi in momentum space. As a consequence, Ultradistributions are not used in the calculations of this paper, which considerably simplifies it. Taking advantage of such Regularization one can also work in configuration space [3]. Thus, one can obtain a convolution of Lorentz Invariant Tempered Distributions in momentum space and the corresponding product in configuration space.
Dimensional Regularization is one of the most important advances in theoretical physics and is used in several disciplines of it [10]- [63].
With this new method the Dimensional Regularization happens to be a convolution of STDELI in momentum space and a product in a ring with divisors of zero in configuration space.
It is our hope that this convolution can then be used to treat nonrenormalizable Quantum Field Theories. This is our present goal.
More to the point let us emphasize that our work is concerned with deeper issues than those regarding QFT axiomatics in Euclidian space and QFT renormalization. Here, we are generalizing Bollini @ Giambiagi dimensional regularization to all Schwartz Tempered, explicitly Lorentz invariant, distributions (STDELI), something that Bollini @ Giambiagi were unable to achieve. This would permit one to deal with non-renormalizable QFT. Indeed, we do not have to use counterterms in a renormalization process devoted to eliminate infinities. This is exactly what we do not want to do, since a non-renormalizable theory involves an infinite numeber of counterterms. The central purpose of our work is to define a STDELI convolution in order to avoid counterterms. We do not appeal to a simple correlation-functions' convolution (not defined for all STDELI). At the same time, we conserve all extant solutions to the problem of running coupling constants and the renormalization group. The STDELI convolution, once obtained, converts configuration space into a ring with zero-divisors. In it, one has now defined a product between the ring-elements. Thus, any unitary-causal-Lorentz invariant theory quantified in such a manner becomes predictive (assumed that it those respects experimental results on which it was based to begin with). The distinction between renormalizable on not-renormalizable QFTs becomes unnecessary now. With our Bollini @ Giambiagi generalization, that uses Laurent expansions in the dimension, all finite constants of the convolutions become completely determined, eliminating arbitrary choices of finite constants. This is tantamount to eliminating all finite renormalizations of the theory. What is the importance of using that term independent of the dimension in Laurent's expansion? That the result obtained for finite convolutions will coincide with such a term. This translates to configuration space the product-operation in a ring with divisors of zero.
As examples, we calculate some convolutions of distributions used in Quantum Field Theory. In particular, the convolution of n massless Feynman propagators and the convolution of n massless Wheeler propagators. For a full discussion about definition and properties of Wheeler propagators see [65,66] which in turn are based on Wheeler and Feynman works [67,68].
The results obtained in this work have already allowed us to calculate the classical partition function of Newtonian gravity, for the first time ever, in the Gibbs' formulation and in the Tsallis' one [64].

Lorentz Invariant Tempered Distributions
In this subsection we give the definitions that we will use in this paper.
We consider first the case on the ν-dimensional Minkowskian space M ν Let S ′ be the space of Schwartz Tempered Distributions [5,69]. Let be g ∈ S ′ . We say that g ∈ S ′ L if and only if: where the derivative is in the sense of distributions, l is a natural number, and is continuous in M ν . The exponent n is a natural number. We say then that f ∈ T 1L . In the case of Euclidean space R ν , let g ∈ S ′ . We say that g ∈ S ′ R if and only if

The Fourier Transform in Euclidean Space
The Fourier transform of a spherically symmetric function is given, according to Bochner's formula, by [70]: is the Bessel function of order (ν − 2)/2. By the use of the equality where K is the modified Bessel function, (2.2. 1) takes the form: By performing the change of variables x = r 2 , ρ = k 2 , (2.2. 1) and (2.2. 3) can be re-written as:

The Fourier Transform in Minkowskian Space
For the Minkowskian case we have, according to ref. [7] f(ρ, ν) = (2π) The corresponding inversion formula is then given by [7]:

An original example
As an example not previously published of this formula we will calculate the Fourier anti-transform of the Dirac's delta δ(ρ) in four dimensions. For this, we make use of the formula given in [71]: . Then: After a simple calculation we obtain: and finally:f The Convolution in Euclidean Space

The generalization of Dimensional Regularization in Configuration Space to the Euclidean Space
The expression for the convolution of two spherically symmetric functions was deduced in ref. [3] (h(k, ν) = (f * g)(k, ν)): However, Bollini and Giambiagi did not obtain a product in a ring with divisors of zero, which we will do now. Consider here that f and g belong to S ′ R . With the change of variables ρ = k 2 , ρ 1 = k 2 1 , ρ 2 = k 2 2 takes the form: Let V be a vertical band contained in the complex ν-plane W. Integral (3.1. 2) is an analytic function of ν defined in the domain V. Then, according to the method of ref. [6], h(ν, ρ) can be analytically continued to other parts of W. In particular, near the dimension ν 0 we have the Laurent expansion: Here, ν 0 is the dimension of the considered space. In particular, ν 0 = 4 is the dimension that we will consider. We now define the convolution product as the (ν − ν 0 )-independent term of the Laurent's expansion.
Thus, in the ring with zero divisors S ′ RA , we have defined a product of distributions.

Example
As an example of the use of (2.2. 1) and (3.1. 1), we evaluate the convolution of a massless propagator with a propagator corresponding to a scalar particle of mass m. The result of this convolution, using this formula, is given in [1]. It is: Now we use the equality: After a tedious calculation, we obtain the corresponding Laurent expansion of h(k, ν):

The generalization of Dimensional Regularization in Configuration Space to the Minkowskian Space
In this section we repeat the efforts of the preceding section for Minkowskian space.
The generalization of the Bochner's formula to Minkowskian space has been obtained in reference [7]. The corresponding expression for ν = 2n is: and therefore, again, we have for the convolution the result: Thus, in the ring with zero divisors S ′ LA we have defined a product of distributions.

Examples
As an example of the use of (4.1. 1) we will consider the convolution of two Dirac's δ-distributions, δ(ρ). The result is Simplifying terms we obtain: Thus, in four dimensions: Note that this convolution does not make sense in a four-dimensional Euclidean space, since in that case δ(ρ) ≡ 0.
As a second example we calculate the convolution δ(ρ − m 2 ) * δ(ρ − m 2 ). In this case we have

The Minkowskian Space Case
Let us now calculate the convolution of n massless Feynman's propagators (n ≥ 2). For this purpose we take into account that According to reference [69], we have and therefore, Using again reference [69] we have now with which we obtain (5.1. 5) We have then, for the convolution of n massless Feynman propagators, the result (5.1. 6) After a tedious calculation we obtain the corresponding Laurent expansion around ν = 4: The independent ν−4 term is the result of the convolution in four dimensions

The Euclidean Space Case
Let us now calculate the convolution of n massless Feynman propagators (n ≥ 2) in Euclidean space, using again (5.1. 1). According to reference [69], we obtain For n propagators we have then Appealing again to reference [69] , we can evaluate the corresponding Fourier Transform F r n(2−ν) = Let ρ = k 2 . We have then for the convolution of n massless Feynman propagators the result By recourse to Laurent expansion we obtain The result of the convolution in four dimensions is then We emphasize that the results of this section are completely original.

The Convolution of massless Wheeler Propagators 6.1 The Convolution of two massless Wheeler Propagators
The Wheeler massless propagator is given by (note that this propagator can not be defined in Euclidean space) and can be written in the form: Therefore, we have After a long and tedious calculation, using (4.1. 1) we obtain For the first convolution of (6.1. 3), we have from (5.1. 6), with n = 2 This equation can be re-written in the form: When ν = 4, the sum of (6.1. 5) and (6.1. 7) has as a result This result was obtained in the reference [8], formula (6.12) using the convolution of even Tempered Ultradistributions. The coincidence of (6.1. 9) with (6.12) of [8] confirms the validity of the results obtained in section 6 of this paper. We emphasize that the present results are obtained in a manner considerably simpler to that of [8].

The Convolution of n massless Wheeler Propagators
According to reference [69], we have

Discussion
In Quantum Field Theory, when we use perturbative expansions, we are dealing with products of distributions in configuration space or, what is the same, with convolutions of distributions in momentum space.
In four earlier papers [6,7,8,9] we have demonstrated the existence of the convolution of Sebastiao e Silva Ultradistributions. This convolution allows us to treat non renormalizable Quantum Fields Theories, but has the disadvantage of being extremely complex.
Following a procedure similar to those of the previously mentioned papers, we defined the convolution of Lorentz Invariant Temperatd Distributions using the Dimensional Regularization of Bollini and Giambiagi.
Using this convolution we have obtained, for example, the convolution of n massless Feynman propagators both in Minkowskian and Euclidean spaces and the convolution of two massless Wheeler propagators.
It is our hope that this convolution will allow one to treat non-renormalizable Quantum Fields Theories.