^{1}

^{*}

^{2}

We revisit, advancing a useful approximation, a recently formulated QFT treatment that successfully overcomes any troubles with infinities for non-renormalizable QFTs [J. Phys. Comm.
2 115029 (2018)]. Such methodology was able to successfully deal, in non-relativistic fashion, with Newton’s gravitation potential [Annals of Physics
412, 168013 (2020)]. Our present approximation to the QFT method of [J. Phys. Comm.
2 115029 (2018)] is based on the Einstein’s Lagrangian (EG) elaborated by Gupta [1], save for a different constraint’s selection. This choice allows one to avoid the lack of unitarity for the S matrix that impaired the proceedings of Gupta and Feynman. Moreover, we are able to simplify the handling of such constraint by eliminating the need to involve ghosts for guarantying unitarity.
*Our approximation consists in setting the graviton field ∅<sup>μν</sup>=γ<sup>μν</sup>∅, where γ<sup>μν</sup> is a constant tensor and ∅ a scalar (graviton) field*. The ensuing approximate approach is non-renormalizable, an inconvenience that we are able to overcome in [J. Phys. Comm. 2 115029 (2018)].

Quantifying Einstein gravity (EG) remains an open problem. We can speak of the Holy Grail for quantum field theory (QFT). Many preceding attempts in this vein failed because 1) they use Rigged Hilber Space (RHS) with undefined metric, 2) non-unitarity troubles, and iii) non-renormalizablity problems.

We will construct here a unitary EG’s QFT based on efforts by Gupta [

Rather surprisingly for some contemporary physicists, for a mathematician, quantifying a non-renormalizable field theory is equivalent to properly defining a product of two distributions (a product in a ring with zero-divisors in configuration space). This is an old problem in functional theory, successfully tackled in [

Remember that, in QFT, evaluating products of distributions with coincident point singularities is connected to the asymptotic behavior of loop integrals of propagators.

In [

The ultrahyperfunctions are just the generalization and extension to the complex plane of the Schwartz tempered distributions and the distributions of exponential type. Thus, the tempered distributions and those of exponential type are a subset of the ultrahyprefunctions.

The present work we do not employ counter-terms to eliminate infinities, since our convolutions remain always finite. One should not wish for counter-terms, since a non-renormalizable theory involves an infinite number of them.

Simultaneously, we keep all extant solutions to the problem of running coupling constants and the renormalization group. Our convolution, once available, transforms configuration space into a ring with zero-divisors. In such ring, we have defined now a product between the ring-elements. Accordingly, any unitary-causal-Lorentz invariant theory quantified in such way becomes predictive. One does no need now to distinguish between renormalizable on non-renormalizable QFT’s.

Our convolution uses Laurent’s expansions in the parameter employed to define it. All finite constants of the convolutions become determined, eliminating arbitrary selections of finite constants. This is equivalent to deleting all finite renormalizations of the theory. The independent term in the Laurent expansion yields the convolution value, which translates to configuration space the product-operation in a ring with divisors of zero.

Our paper is structured as indicated belows:

1) Section 2 deals with preliminary materials.

2) Section 3 treats the QFT Lagrangian for EG and introduces a new approximation that consists in setting the graviton field ϕ μ ν = γ μ ν ϕ , where γ μ ν is a constant tensor and ϕ a scalar (graviton) field.

3) Section 4 quantizes the ensuing theory.

4) Section 5 evaluates the graviton’s self-energy up to second order.

5) Section 6 introduces axions into our scenario and considers the axions gravitons interaction.

6) Section 7 calculates the graviton’s self-energy in the presence of axions.

7) Section 8 evaluates the axion’s self-energy, up to second order.

8) Finally, some conclusions are drawn in Section 9.

The most general quantification approach is Schwinger-Feynman’s variational principle [

Consider now the action for a set of fields written in the fashion

S [ σ ( x ) , σ 0 , ϕ A ( x ) ] = ∫ σ 0 σ ( x ) L [ ϕ A ( ξ ) , ∂ μ ϕ A ( ξ ) , ξ ] d ξ , (2.1)

where σ ( x ) if a space-like surface passing through the point x. σ 0 is that surface (at the remote past), for which all field variations vanish. The Schwinger-Feynman variational principle asserts that

“Any Hermitian infinitesimal variation δ S of the action induces a canonical transformation of the vector space in which the quantum system is defined, and the generator of this transformation is this same operator δ S ”.

Thus, this equality holds:

δ ϕ A = i [ δ S , ϕ A ] . (2.2)

Accordingly, for a Poincare transformation one has

δ S = a μ P μ + 1 2 a μ v M μ v , (2.3)

where the field variation is

δ ϕ a = a μ P ^ μ ϕ A + 1 2 a μ v M ^ μ v ϕ A . (2.4)

From (2) one ascertains that

∂ μ ϕ A = i [ P μ , ϕ A ] . (2.5)

More to the point,

∂ 0 ϕ A = i [ P 0 , ϕ A ] . (2.6)

Eq. (ep2.6) will be used below for quantizing EG.

The EG Lagrangian is [

L G = 1 κ 2 R | g | − 1 2 η μ v ∂ α h μ α ∂ β h v β , (3.1)

where Minkowski’s η μ ν = d i a g ( 1,1,1, − 1 ) while h μ ν = | g | g μ ν The second term in (3.1) establishes the gauge fixing. We reach here a critical stage by proceeding to perform a crucial linear approximation. This will be immediately seen to be an approximation to the graviton field. We write:

h μ v = η μ v + κ ϕ μ v , (3.2)

where κ 2 is the gravitation’s constant and ϕ μ v the graviton field. Our approximation based in [

ϕ μ v = γ μ v ϕ , (3.3)

with ϕ a scalar field and where γ μ v is a constant tensor which satisfies

γ μ μ = 0 (3.4)

This approximate casting of ϕ μ v considerably simplifies the handling of matters without sacrifice of rigor. We write now the Lagrangian as a sum of a non-perturbative component plus an interactions one, i.e.,

L G = L L + L I , (3.5)

where

L L = − 1 4 γ μ v γ μ v ∂ λ ϕ ∂ λ ϕ , (3.6)

and, up to 2nd order, one has [

L I = − 1 2 κ γ μ v ϕ [ 1 2 γ ρ λ γ ρ λ ∂ μ ϕ ∂ v ϕ + γ μ β γ λ v ∂ λ ϕ ∂ β ϕ − γ μ ρ γ v ρ ∂ λ ϕ ∂ λ ϕ ] , (3.7)

having made use of the constraint (3.4) This constraint is required in order to satisfy gauge invariance [

□ ϕ = 0, (3.8)

whose solution is

ϕ ( x ) = 1 ( 2 π ) 3 2 ∫ [ a ( k ) 2 k 0 e i k μ x μ + a + ( k ) 2 k 0 e − i k μ x μ ] d 3 k , (3.9)

with k 0 = | k | . Above, a ( k ) and a + ( k ) stand for Fourier coefficients. Up to this point we were using Einstein’s Lagrangian. Its quantization begins next.

As usual in QFT, see for instance Visconti’ celebrated book [

T ρ λ = ∂ L ∂ ∂ ρ ϕ μ v ∂ λ ϕ μ v − δ ρ λ L , (4.1)

and the time-component of the four-momentum is now the quantum operator

P 0 = ∫ T 0 0 d 3 x . (4.2)

Using (3.4) we have

T 0 0 = − 1 4 γ μ v γ μ v [ ∂ 0 ϕ ∂ 0 ϕ − ∂ j ϕ ∂ j ϕ ] . (4.3)

Consequently,

P 0 = 1 4 γ μ v γ μ v ∫ | k | [ a ( k ) a + ( k ) + a + ( k ) a ( k ) ] d 3 k . (4.4)

Appeal to (2.6) leads now to

[ P 0 , a + ( k ) ] = k 0 a + ( k ) . (4.5)

From the last relation in (4.5) one gathers that

| k | a + ( k ′ ) = γ μ v γ μ v 2 ∫ | k | [ a ( k ) , a + ( k ′ ) ] a + ( k ) d 3 k . (4.6)

The solution of this integral equation is

[ a ( k ) , a + ( k ′ ) ] = 2 γ μ v γ μ v δ ( k − k ′ ) . (4.7)

We use now the the usual definition

Δ ( x − y ) = 〈 0 | T [ ϕ ( x ) ϕ ( y ) ] | 0 〉 . (4.8)

The graviton’s propagator then turns out to be

Δ ( x − y ) = i ( 2 π ) 4 2 γ μ v γ μ v ∫ e i k μ ( x μ − y μ ) k 2 − i 0 d 4 k . (4.9)

As a consequence, we can write

P 0 = γ μ v γ μ v 4 ∫ | k | [ a ( k ) a + ( k ′ ) + a + ( k ′ ) a ( k ) ] δ ( k − k ′ ) d 3 k d 3 k ′ , (4.10)

or

P 0 = γ μ v γ μ v 4 ∫ | k | [ 2 a + ( k ′ ) a ( k ) + 2 γ ρ λ γ ρ λ δ ( k − k ′ ) ] δ ( k − k ′ ) d 3 k d 3 k ′ . (4.11)

Thus, we obtain

P 0 = γ μ v γ μ v 2 ∫ | k | a + ( k ) a ( k ) d 3 k , (4.12)

where we have used the fact that the product of two deltas with the same argument vanishes [

Now, we must insist on the fact that the physical state should satisfy the relation (see [

The ensuing theory is similar to the QED-one obtained via the quantization approach of Gupta-Bleuler. This implies that the theory is unitary for any finite perturbative order. In this theory just one type of graviton arises,

So as to compute the graviton’s self-energy (SF)c we begin with the interaction Hamiltonian

A typical term reads

where

In

where

Anti-transforming the above equation one has

We proceed here to perform a

The exact value of the convolution we are interested in, i.e., the left hand side of (5.5), is given by the independent term above, as everyone knows. Should the reader be unfamiliar with this scenario, we direct him/her to [

We face here 1296 diagrams of this type.

Axions are hypothetical elementary particles conjectured by the 1977 Peccei-Quinn theory so as to tackle the strong CP problem in quantum chromodynamics. Should they exist and have low enough mass (within a certain range), they may be of some interest as putative components of cold dark matter [

It is possible to recast the Lagrangian now as

where

so that

A new term in the interaction Hamiltonian appears

Axions necessarily generate a new contribution to a graviton’s self energy

To evaluate it face the customary

where

Effecting a variables-change

where

After computing the associated integrals we find

We appeal once again to a Laurent’s expansion and have

Once again, the exact result for our four-dimensional convolution is

Accordingly, our desired self-energy total is a combination of

The self-energy reads

In

Using the same Feynman parameters as above we have

where

We compute the integral (8.3) and encounter

We Laurent-expand again, this time (8.5) around

The

We have developed above an approximate quantum field theory (QFT) of Eintein’s gravity (EG) that is both unitary and finite. It critically necessitates of a new constraint-introduction in the EG-Lagrangian. Laurent expansions were a main tool for our endeavors. Our approximation consists in defining the graviton field as

• a graviton’s self-energy in the EG-field,

• the self-energy in the presence of a massive scalar field (axions, for example). Two sorts of diagram emerge: the original ones of the pure EG field plus the ones generated by the addition of a scalar field.

• An axion’s self-energy.

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

Plastino, A. and Rocca, M.C. (2020) Approximate Reformulation a Recent Non-Renormalizable QFT’s Methodology and Einstein’s Gravity. Journal of High Energy Physics, Gravitation and Cosmology, 6, 298-311. https://doi.org/10.4236/jhepgc.2020.62023