, 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, at which all field variations vanish. The Schwinger-Feynman variational principle dictates 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 ”.

Accordingly, the following equality holds:

δ ϕ A = i [ δ S , ϕ A ] . (2.2)

Thus, for a Poincare transformation we have

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

where the field variation is given by

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

From (2.2) one gathers that

μ ϕ A = i [ P μ , ϕ A ] . (2.5)

Specifically,

0 ϕ A = i [ P 0 , ϕ A ] . (2.6)

This last result will be employed in quantizing EG.

3. The Convolution of Two Lorentz Invariant Tempered Ultradistributions

In [3] we have obtained a conceptually simple but rather lengthy expression for the convolution of two Lorentz invariant tempered ultradistributions:

H λ ( ρ , Λ ) = 1 8 π 2 ρ Γ 1 Γ 2 F ( ρ 1 ) G ( ρ 2 ) ρ 1 λ ρ 2 λ { Θ [ ( ρ ) ] { [ l n ( ρ 1 + Λ ) l n ( ρ 1 Λ ) ] × [ l n ( ρ 2 + Λ ) l n ( ρ 2 Λ ) ] 4 ( ρ 1 + Λ ) ( ρ 2 + Λ ) ( ρ ρ 1 ρ 2 2 Λ ) 2 × l n [ 4 ( ρ 1 + Λ ) ( ρ 2 + Λ ) ( ρ ρ 1 ρ 2 2 Λ ) 2 i ( ρ ρ 1 ρ 2 2 Λ ) 2 ( ρ 1 + Λ ) ( ρ 2 + Λ ) ] + [ l n ( ρ 1 + Λ ) l n ( ρ 1 Λ ) ] [ l n ( ρ 2 + Λ ) l n ( ρ 2 Λ ) ]

× 4 ( ρ 1 Λ ) ( ρ 2 Λ ) ( ρ ρ 1 ρ 2 + 2 Λ ) 2 × l n [ 4 ( ρ 1 Λ ) ( ρ 2 Λ ) ( ρ ρ 1 ρ 2 + 2 Λ ) 2 i ( ρ ρ 1 ρ 2 + 2 Λ ) 2 ( ρ 1 Λ ) ( ρ 2 Λ ) ] + [ l n ( ρ 1 + Λ ) l n ( ρ 1 Λ ) ] [ l n ( ρ 2 + Λ ) l n ( ρ 2 Λ ) ]

× { i π 2 [ 4 ( ρ 1 + Λ ) ( ρ 2 Λ ) ( ρ ρ 1 ρ 2 ) 2 i ( ρ ρ 1 ρ 2 ) ] + 4 ( ρ 1 + Λ ) ( ρ 2 Λ ) ( ρ ρ 1 ρ 2 ) 2

× l n [ 4 ( ρ 1 + Λ ) ( ρ 2 Λ ) ( ρ ρ 1 ρ 2 ) 2 i ( ρ ρ 1 ρ 2 ) 2 i ( ρ 1 + Λ ) ( ρ 2 Λ ) ] } + [ l n ( ρ 1 + Λ ) l n ( ρ 1 Λ ) ] [ l n ( ρ 2 + Λ ) l n ( ρ 2 Λ ) ] × { i π 2 [ 4 ( ρ 1 Λ ) ( ρ 2 + Λ ) ( ρ ρ 1 ρ 2 ) 2 i ( ρ ρ 1 ρ 2 ) ] + 4 ( ρ 1 Λ ) ( ρ 2 + Λ ) ( ρ ρ 1 ρ 2 ) 2 × l n [ 4 ( ρ 1 Λ ) ( ρ 2 + Λ ) ( ρ ρ 1 ρ 2 ) 2 i ( ρ ρ 1 ρ 2 ) 2 i ( ρ 1 Λ ) ( ρ 2 + Λ ) ] } }

Θ [ ( ρ ) ] { [ l n ( ρ 1 + Λ ) l n ( ρ 1 Λ ) ] [ l n ( ρ 2 + Λ ) l n ( ρ 2 Λ ) ] × 4 ( ρ 1 Λ ) ( ρ 2 Λ ) ( ρ ρ 1 ρ 2 + 2 Λ ) 2 × l n [ 4 ( ρ 1 Λ ) ( ρ 2 Λ ) ( ρ ρ 1 ρ 2 + 2 Λ ) 2 i ( ρ ρ 1 ρ 2 + 2 Λ ) 2 ( ρ 1 Λ ) ( ρ 2 Λ ) ] + [ l n ( ρ 1 + Λ ) l n ( ρ 1 Λ ) ] [ l n ( ρ 2 + Λ ) l n ( ρ 2 Λ ) ] × 4 ( ρ 1 + Λ ) ( ρ 2 + Λ ) ( ρ ρ 1 ρ 2 2 Λ ) 2

× l n [ 4 ( ρ 1 + Λ ) ( ρ 2 + Λ ) ( ρ ρ 1 ρ 2 2 Λ ) 2 i ( ρ ρ 1 ρ 2 2 Λ ) 2 ( ρ 1 + Λ ) ( ρ 2 + Λ ) ] + [ l n ( ρ 1 + Λ ) l n ( ρ 1 Λ ) ] [ l n ( ρ 2 + Λ ) l n ( ρ 2 Λ ) ] × { i π 2 [ 4 ( ρ 1 Λ ) ( ρ 2 + Λ ) ( ρ ρ 1 ρ 2 ) 2 i ( ρ ρ 1 ρ 2 ) ] + 4 ( ρ 1 Λ ) ( ρ 2 + Λ ) ( ρ ρ 1 ρ 2 ) 2 × l n [ 4 ( ρ 1 Λ ) ( ρ 2 + Λ ) ( ρ ρ 1 ρ 2 ) 2 i ( ρ ρ 1 ρ 2 ) 2 i ( ρ 1 Λ ) ( ρ 2 + Λ ) ] }

+ [ l n ( ρ 1 + Λ ) l n ( ρ 1 Λ ) ] [ l n ( ρ 2 + Λ ) l n ( ρ 2 Λ ) ] × { i π 2 [ 4 ( ρ 1 + Λ ) ( ρ 2 Λ ) ( ρ ρ 1 ρ 2 ) 2 i ( ρ ρ 1 ρ 2 ) ] + 4 ( ρ 1 + Λ ) ( ρ 2 Λ ) ( ρ ρ 1 ρ 2 ) 2

× l n [ 4 ( ρ 1 + Λ ) ( ρ 2 Λ ) ( ρ ρ 1 ρ 2 ) 2 i ( ρ ρ 1 ρ 2 ) 2 i ( ρ 1 + Λ ) ( ρ 2 Λ ) ] } } i 2 { [ l n ( ρ 1 + Λ ) l n ( ρ 1 Λ ) ] [ l n ( ρ 2 + Λ ) l n ( ρ 2 Λ ) ]

× ( ρ 1 ρ 2 ) [ l n ( i ρ 1 + Λ ρ 2 + Λ ) + l n ( i ρ 1 Λ ρ 2 Λ ) ] + [ l n ( ρ 1 + Λ ) l n ( ρ 1 Λ ) ] [ l n ( ρ 2 + Λ ) l n ( ρ 2 Λ ) ] × ( ρ 1 ρ 2 ) [ l n ( i Λ ρ 1 Λ ρ 2 ) + l n ( i Λ + ρ 1 Λ + ρ 2 ) ] + [ l n ( ρ 1 + Λ ) l n ( ρ 1 Λ ) ] [ l n ( ρ 2 + Λ ) l n ( ρ 2 Λ ) ] × { ( ρ 1 ρ 2 ) [ l n ( Λ + ρ 1 Λ ρ 2 ) + l n ( Λ ρ 1 Λ + ρ 2 ) ]

+ ρ 1 ρ 2 2 [ l n ( ρ 1 ρ 2 + Λ ) l n ( ρ 1 ρ 2 Λ ) l n ( ρ 1 + ρ 2 + Λ ) + l n ( ρ 1 + ρ 2 Λ ) ] + ρ 2 [ l n ( ρ 1 ρ 2 + Λ ) l n ( ρ 1 ρ 2 Λ ) ] + ρ 1 [ l n ( ρ 1 + ρ 2 + Λ ) l n ( ρ 1 + ρ 2 Λ ) ] } [ l n ( ρ 1 + Λ ) l n ( ρ 1 Λ ) ] [ l n ( ρ 2 + Λ ) l n ( ρ 2 Λ ) ] × { ( ρ 1 ρ 2 ) [ l n ( Λ ρ 1 Λ + ρ 2 ) + l n ( Λ + ρ 1 Λ ρ 2 ) ]

+ ρ 1 ρ 2 2 [ l n ( ρ 1 + ρ 2 + Λ ) l n ( ρ 1 + ρ 2 Λ ) l n ( ρ 1 ρ 2 + Λ ) + l n ( ρ 1 ρ 2 Λ ) ] + ρ 1 [ l n ( ρ 1 ρ 2 + Λ ) l n ( ρ 1 ρ 2 Λ ) ] + ρ 2 [ l n ( ρ 1 + ρ 2 + Λ ) l n ( ρ 1 + ρ 2 Λ ) ] } } } d ρ 1 d ρ 2 (3.1)

This defines an ultradistribution in the variables ρ and Λ for

| ( ρ ) | > ( Λ ) > | ( ρ 1 ) | + | ( ρ 2 ) |

Let B be a vertical band contained in the complex λ -plane P . Integral (3.1) is an analytic function of λ defined in the domain B . Moreover, it is bounded by a power of | ρ Λ | . Then, H λ ( ρ , Λ ) can be analytically continued to other parts of P . Thus, we define

H ( ρ ) = H ( 0 ) ( ρ , i 0 + ) (3.2)

H λ ( ρ , i 0 + ) = m H ( n ) ( ρ , i 0 + ) λ n (3.3)

As in the other cases, we define now

{ F G } ( ρ ) = H ( ρ ) (3.4)

as the convolution of two Lorentz invariant tempered ultradistributions.

The Feynman propagators corresponding to a massless particle F and a massive particle G are, respectively, the following ultrahyperfunctions:

F ( ρ ) = Θ [ ( ρ ) ] ρ 1

G ( ρ ) = Θ [ ( ρ ) ] ( ρ + m 2 ) 1 (3.5)

where ρ is the complex variable, such that on the real axis one has ρ = k 1 2 + k 2 2 + k 3 2 k 0 2 . For them, the following equalities are satisfied

ρ λ F ( ρ ) = Θ [ ( ρ ) ] ρ λ 1

ρ λ G ( ρ ) = Θ [ ( ρ ) ] ( ρ + m 2 ) λ 1 (3.6)

where we have used: ( ρ + m 2 ) λ ρ λ , since we have chosen m to be very small. On the real axis, the previously defined propagators are given by:

f ( ρ ) = F ( ρ + i 0 ) F ( ρ i 0 ) = ( ρ i 0 ) 1

g ( ρ ) = G ( ρ + i 0 ) G ( ρ i 0 ) = ( ρ + m 2 i 0 ) 1 (3.7)

These are the usual expressions for Feynman propagators.

Consider first the convolution of two massless propagators. We use (3.6), since here the corresponding ultrahyperfunctions do not have singularities in the complex plane. We obtain from (3.1) a simplified expression for the convolution:

h λ ( ρ ) = π 2 ρ ( ρ 1 i 0 ) λ 1 ( ρ 2 i 0 ) λ 1 [ ( ρ ρ 1 ρ 2 ) 2 4 ρ 1 ρ 2 ] + 1 2 d ρ 1 d ρ 2 (3.8)

This expression is nothing other than the usual convolution:

h λ ( ρ ) = ( ρ i 0 ) λ 1 ( ρ i 0 ) λ 1 (3.9)

In the same way, we obtain for massive propagators:

h λ ( ρ ) = ( ρ + m 2 i 0 ) λ 1 ( ρ m 2 i 0 ) λ 1 (3.10)

These last two expressions are the ones we will use later to evaluate the graviton’s self-energy.

4. The Lagrangian of Einstein’s QFT

Our EG Lagrangian reads [7]

L G = 1 κ 2 R | g | 1 2 η μ v α h μ α β h v β , (4.1)

where η μ ν = d i a g ( 1,1,1, 1 ) , h μ ν = | g | g μ ν The second term in (4.1) fixes the gauge. We effect now the linear approximation

h μ v = η μ v + κ ϕ μ v , (4.2)

where κ 2 is the gravitation’s constant and ϕ μ v the graviton field. We write

L G = L L + L I , (4.3)

where

L L = 1 4 [ λ ϕ μ v λ ϕ μ v 2 α ϕ μ β β ϕ μ α + 2 α ϕ μ α β ϕ μ β ] , (4.4)

and, up to 2nd order, one has [7]:

L I = 1 2 κ ϕ μ v [ 1 2 μ ϕ λ ρ v ϕ λ ρ + λ ϕ μ β β ϕ v λ λ ϕ μ ρ λ ϕ v ρ ] , (4.5)

having made use of the constraint

ϕ μ μ = 0. (4.6)

This constraint is required in order to satisfy gauge invariance [14] For the graviton we have then

ϕ μ v = 0, (4.7)

whose solution is

ϕ μ v = 1 ( 2 π ) 3 2 [ a μ v ( k ) 2 k 0 e i k μ x μ + a μ v + ( k ) 2 k 0 e i k μ x μ ] d 3 k , (4.8)

with k 0 = | k | .

5. Quantization of the Theory

We need some definitions. The energy-momentum tensor reads

T ρ λ = L ρ ϕ μ v λ ϕ μ v δ ρ λ L , (5.1)

and the time-component of the four-momentum is

P 0 = T 0 0 d 3 x . (5.2)

Using (4.4) we have

T 0 0 = 1 4 [ 0 ϕ μ v 0 ϕ μ v + j ϕ μ v j ϕ μ v 2 α ϕ μ 0 0 ϕ μ α 2 α ϕ μ j j ϕ μ α + 2 α ϕ μ α 0 ϕ μ 0 + 2 α ϕ μ α j ϕ μ j ] . (5.3)

Consequently,

P 0 = 1 4 | k | [ a μ v ( k ) a + μ v ( k ) + a + μ v ( k ) a μ v ( k ) ] d 3 k . (5.4)

Appeal to (2.6) leads to

[ P 0 , a μ v ( k ) ] = k 0 a μ v (k)

[ P 0 , a + μ v ( k ) ] = k 0 a + μ v ( k ) . (5.5)

From the last relation in (5.5) one gathers that

| k | a + ρ λ ( k ) = 1 2 | k | [ a μ v ( k ) , a + ρ λ ( k ) ] a + μ v ( k ) d 3 k . (5.6)

The solution of this integral equation is

[ a μ v ( k ) , a + ρ λ ( k ) ] = [ δ μ ρ δ v λ + δ v ρ δ μ λ ] δ ( k k ) . (5.7)

As customary, the physical state | ψ of the theory is defined via the equation

ϕ μ μ | ψ = 0. (5.8)

We use now the usual definition

Δ μ ν ρ λ ( x y ) = 0 | T [ ϕ μ ν ( x ) ϕ ρ λ ( y ) ] | 0 . (5.9)

The graviton’s propagator then turns out to be

Δ μ ν ρ λ ( x y ) = i ( 2 π ) 4 ( δ μ ρ δ v λ + δ v ρ δ μ λ ) e i k μ ( x μ y μ ) k 2 i 0 d 4 k . (5.10)

As a consequence, we can write

P 0 = 1 4 | k | [ a μ v ( k ) a + μ v ( k ) + a + μ v ( k ) a μ v ( k ) ] δ ( k k ) d 3 k d 3 k , (5.11)

or

P 0 = 1 4 | k | [ 2 a + μ v ( k ) a μ v ( k ) + δ ( k k ) ] δ ( k k ) d 3 k d 3 k . (5.12)

Thus, we obtain

P 0 = 1 2 | k | a + μ v ( k ) a μ v ( k ) d 3 k , (5.13)

where we have used the fact that the product of two deltas with the same argument vanishes [1], i.e., δ ( k k ) δ ( k k ) = 0 . This illustrates the fact that using Ultrahyperfunctions is here equivalent to adopting the normal order in the definition of the time-component of the four-momentum

P 0 = 1 4 | k | : [ a μ v ( k ) a + μ v ( k ) + a + μ v ( k ) a μ v ( k ) ] : d 3 k . (5.14)

Now, we must insist on the fact that the physical state should satisfy not only Equation (5.8) but also the relation (see [7])

μ ϕ μ v | ψ = 0. (5.15)

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 only one type of graviton emerges, ϕ 12 , while in Gupta’s approach two kinds of graviton arise. Obviously, this happens for a non-interacting theory, as remarked by Gupta.

Undesired Effects of NOT Using Our Constraint

If we do NOT use the constraint (5.8), we have

P 0 = 1 2 | k | [ a + μ v ( k ) a μ v ( k ) 1 2 a μ + μ ( k ) a v v ( k ) ] d 3 k , (5.16)

and, appealing to the Schwinger-Feynman variational principle we find

| k | a ρ λ + ( k ) = 1 2 | k | { a + μ v ( k ) [ a μ v ( k ) , a ρ λ + ( k ) ] 1 2 a μ + μ ( k ) [ a v v ( k ) , a ρ λ + ( k ) ] } d 3 k , (5.17)

whose solution is

[ a μ v ( k ) , a ρ λ + ( k ) ] = [ η μ ρ η v λ + η v ρ η μ λ η μ v η ρ λ ] δ ( k k ) . (5.18)

The above is the customary graviton’s quantification, that leads to a theory whose S matrix in not unitary [7] [10].

6. The Self Energy of the Graviton

To evaluate the graviton’s self-energy (SF)c we start with the interaction Hamiltonian H I . Note that the Lagrangian contains derivative interaction terms.

H I = L I 0 ϕ μ ν 0 ϕ μ ν L I . (6.1)

A typical term reads

Σ G α 1 α 2 α 3 α 4 ( k ) = k α 1 k α 2 ( ρ i 0 ) λ 1 k α 3 k α 4 ( ρ i 0 ) λ 1 . (6.2)

where ρ = k 1 2 + k 2 2 + k 3 2 k 0 2

The Fourier transform of (6.2) is

F [ k α 1 k α 2 ( ρ i 0 ) λ 1 k α 3 k α 4 ( ρ i 0 ) λ 1 ] = 2 4 ( λ + 1 ) [ Γ ( 2 + λ ) 2 ] 4 Γ ( 1 λ ) 2 η α 1 α 2 η α 3 α 4 ( x + i 0 ) 2 λ 4 + 2 4 ( λ + 1 ) Γ ( 2 + λ ) Γ ( 3 + λ ) 2 Γ ( 1 λ ) 2 ( η α 1 α 2 x α 3 x α 4 + η α 3 α 4 x α 1 x α 2 ) ( x + i 0 ) 2 λ 5 2 4 ( λ + 1 ) Γ ( 3 + λ ) 2 Γ ( 1 λ ) x α 1 x α 2 x α 3 x α 4 ( x + i 0 ) ν 2 (6.3)

where x = x 1 2 + x 2 2 + x 3 2 x 0 2

Anti-transforming the above equation we have

k α 1 k α 2 ( ρ i 0 ) λ 1 k α 3 k α 4 ( ρ i 0 ) λ 1 = i π 2 4 Γ ( 1 λ ) 2 { Γ ( λ + 2 ) [ Γ ( 2 + λ ) Γ ( 2 λ + 4 ) 2 Γ ( 3 + λ ) Γ ( 2 λ + 5 ) ] η α 1 α 2 η α 3 α 4 + Γ ( λ + 3 ) 2 Γ ( 2 λ + 6 ) ( η α 1 α 2 η α 3 α 4 + η α 2 α 3 η α 1 α 4 + η α 2 α 4 η α 1 α 3 ) } Γ ( 2 λ 2 ) ( ρ i 0 ) 2 λ + 2 + i π 2 Γ ( λ + 3 ) 2 Γ ( 1 λ ) 2 { Γ ( 2 + λ ) Γ ( 2 λ + 5 ) Γ ( ν + 1 ) ( η α 1 α 2 k α 3 k α 4 + η α 3 α 4 k α 1 k α 2 )

Γ ( λ + 3 ) Γ ( 2 λ + 6 ) ( η α 1 α 2 k α 3 k α 4 + η α 1 α 3 k α 2 k α 4 + η α 1 α 4 k α 2 k α 3 + η α 3 α 4 k α 1 k α 2 + η α 2 α 3 k α 1 k α 4 + η α 2 α 4 k α 1 k α 3 ) } Γ ( 2 λ 1 ) ( ρ i 0 ) 2 λ + 1 + i π 2 Γ ( λ + 3 ) 2 Γ ( 1 λ ) 2 Γ ( 2 λ + 6 ) k α 1 k α 2 k α 3 k α 4 Γ ( 2 λ ) ( ρ i 0 ) 2 λ (6.4)

Self-Energy Evaluation for λ = 0

We appeal now to a λ -Laurent expansion and retain there the λ = 0 independent term [3]. Thus, we Laurent-expand (6.4) around λ = 0 and find

(6.5)

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 in the above expansion, as it is well-known. If the reader is not familiar with this situation, see for instance [3]. We then reach

(6.6)

We have to deal with 1296 diagrams of this kind.

7. Including Axions into the Picture

Axions are hypothetical elementary particles postulated by the Peccei-Quinn theory in 1977 to tackle the strong CP problem in quantum chromodynamics. If they exist and have low enough mass (within a certain range), they could be of interest as possible components of cold dark matter [15]. We include now a massive scalar field (axions) interacting with the graviton. The Lagrangian becomes

(7.1)

We can now recast the Lagrangian in the fashion

(7.2)

where

(7.3)

so that becomes the Lagrangian for the axion-graviton action

(7.4)

The new term in the interaction Hamiltonian is

(7.5)

8. The Complete Self Energy of the Graviton

The presence of axions generates a new contribution to the graviton’s self energy

(8.1)

So as to compute it we appeal to the usual integral together with the generalized Feynman-parameters. After a Wick rotation we obtain

(8.2)

where

(8.3)

After the variables-change we find

(8.4)

where

(8.5)

After evaluation of the pertinent integrals we arrive at

(8.6)

Self-Energy Evaluation for

We need again a Laurent’s expansion and face

(8.7)

Again, the exact result for our four-dimensional convolution becomes

(8.8)

We have to deal with 9 diagrams of this kind.

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

9. Self Energy of the Axion

Here a typical term of the self-energy is:

(9.1)

In four dimensions one has

(9.2)

with the Feynman parameters used above we obtain

(9.3)

where

(9.4)

We evaluate the integral (9.3) and find

(9.5)

Self-Energy Evaluation for

Once again, we Laurent-expand, this time (9.5) around, encountering

(9.6)

The -independent term gives the exact convolution result we are looking for:

(9.7)

10. Discussion

We have developed above a quantum field theory (QFT) of Eintein’s gravity (EG), that is both unitary and finite, by appealing to the Schwinger-Feyman variational principle. We emphatically avoid the functional integral method. Our results critically depend on the use of a rather novel constraint the we introduced in defining the EG-Lagrangian. Laurent expansions were also an indispensable tool for us. As sgtated, in order to quantify the theory we appealed to the variational principle of Schwinger-Feynman’s. This process leads to just one graviton type. The underlying mathematics used in this effort has been developed by Bollini et al. [1] [2] [3] [4] [5]. This mathematics is powerful enough so as to be able to quantize non-renormalizable field theories [1] [2] [3] [4] [5]. We have evaluated here in finite and exact fashion, for the first time as far as we know, several quantities:

• the graviton’s self-energy in the EG-field. This requires full use of the theory of distributions, appealing to the possibility of creating with them a ring with divisors of zero.

• the above self-energy in the added presence of a massive scalar field (axions, for instance). Two types of diagram ensue: the original ones of the pure EG field plus the ones originated by the addition of a scalar field.

• the axion’s self-energy.

• Our central results revolve around Equation (6.6), Equation (8.8), and Equation (9.7), corresponding to the graviton’s self-energy, without and with the added presence of axions. Also, we give the axion’s self-energy.

As a final remark, we would like to point out that our formula for convolutions is a mathematical definition and not a regularization.

Cite this paper
Plastino, A. , Rocca, M. , (2020) Generalization via Ultrahyperfunctions of a Gupta-Feynman Based Quantum Field Theory of Einstein’s Gravity. Journal of Modern Physics, 11, 378-394. doi: 10.4236/jmp.2020.113024.
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