arXiv:0812.3062v1 [gr-qc] 16 Dec 2008

Path Integral Quantization of Quantum Gauge General Relativity Ning Wu

∗

Institute of High Energy Physics, P.O.Box 918-1, Beijing 100039, P.R.China

December 16, 2008

PACS Numbers: 11.15.-q, 04.60.-m, 04.20.Cv, 11.10.Gh. Keywords: general relativity, gauge field, quantum gravity, path integral quantization, Feynman rules .

Abstract Path integral quantization of quantum gauge general relativity is discussed in this paper. First, we deduce the generating functional of green function with external fields. Based on this generating functional, the propagators of gravitational gauge field and related ghost field are deduced. Then, we calculate Feynman rules of various interaction vertices of three or four gravitational gauge fields and vertex between ghost field and gravitational gauge field. Results in this paper are the bases of calculating vacuum polarization of gravitational gauge field and vertex correction of gravitational couplings in one loop diagram level. As we have pointed out in previous paper, quantum gauge general relativity is perturbative renormalizable, and a formal proof on its renormalizability is also given in the previous paper. Next step, we will calculate one-loop and two-loop renormalization constant, and to prove that ∗

email address: [email protected]

1

the theory is renormalizable in one-loop and two-loop level by direct calculations.

1

Introduction

Quantum gravity is proposed to unify general relativity and quantum theory. One of the biggest troubles for quantum gravity is the problems of perturbative renormalization. Gauge gravity is studied for a long time, and there are many versions of gauge gravity[1, 2, 3, 4]. It is expected that gauge gravity could solve the problem of renormalization of quantum gravity. Quantum gauge general relativity is proposed to solve this problem[5, 6, 7, 8, 9, 3, 10]. It is a quantum theory of gravity proposed in the framework of quantum gauge field theory. In 2003, Quantum Gauge General Relativity(QGGR) is proposed in the framework of QGTG. Unlike Einstein’s general theory of relativity, the cornerstone of QGGR is the gauge principle, not the principle of equivalence, which will cause far-reaching influence to the theory of gravity. In QGGR, the field equation of gravitational gauge field is just the Einstein’s field equation, so in classical level, we can set up its geometrical formulation[11], and QGGR returns to Einstein’s general relativity in classical level. The field equation of gravitational gauge field in QGGR is the same as Einstein’s field equation in general relativity, so two equations have the same solutions, though mathematical expressions of the two equations are completely different. For classical tests of gravity, QGGR gives out the same theoretical predictions as those of GR[12], and for non-relativistic problems, QGGR can return to Newton’s classical theory of gravity[13]. Based on the coupling between the spin of a particle and gravitoelectromagnetic field, the equation of motion of spin can be obtained in QGGR. In post Newtonian approximations, this equation of motion of spin gives out the same results as those of GR[14]. The equation of motion of a spinning test particle in gravitational field can also obtained[15]. It’s found that this motion deviates from traditional geodesic curve, and the deviation effects is detectable[16], which can be regarded as a new classical tests of gravity theory. QGGR is a perturbatively renormalizable quantum theory, and based on it, quantum effects of gravity[17, 18, 19, 20] and gravitational interactions of some basic quantum fields [21, 22] can be explored. Unification of fundamental interactions including gravity can be fulfilled in a semi-direct product gauge group[23, 24, 25, 26]. If we use the mass generation mechanism which is proposed in literature [27, 28], we can propose a new theory on gravity which contains massive graviton and the introduction of massive graviton does not affect the strict local gravitational gauge symmetry of the action and does not affect the traditional long-range gravitational

2

force[29]. The existence of massive graviton will help us to understand the possible origin of dark matter. In literature [3], a formal proof on the renormalizability of quantum gauge general relativity is given. The proof is not based on the calculation of loop diagrams, but based on generalized BRST symmetry and generalized Ward-Takahashi identities. This case is similar to that of traditional gauge field theory. We know that traditional gauge field theory is a renormalizable quantum theory[30, 31, 32, 33, 34, 35]. In gauge field theory, though there are many divergences in loop diagram calculations, the constraints from gauge symmetry will make all divergences cancel each other. Now, we want ask that the divergence cancellation mechanism in quantum gauge general relativity is really work in one- or two-loop level, as what we expected in the literature [3]? In order to prove that quantum gauge general relativity is perturbatively renormalizable in one-loop and two-loop level, we need first to calculate propagators of gravitational gauge field and ghost field, to determine the Feynman rules of various interaction vertices, and to calculate all divergent one-loop and twoloop Feynman diagrams. As a first step, we discuss quantization of quantum gauge general relativity, and determine Feynman rules of various vertices, which is the main goal of this paper. Next step, we will calculate all divergent one-loop Feynman diagram and discuss the renormalization problem of quantum gauge general relativity in one-loop level. Finally, we discuss the renormalization problem in two-loop level. So this paper is the first one of a serial of papers on the renormalization of quantum gauge general relativity. All these calculations are extremely complicated and time consuming. In order to avoid possible mistakes in analytical deductions, all important results are calculated at least two times, and two calculations are completely independent. Some important results are also checked by using Mathematica. How to use Mathematica to perform these calculations will be discussed in another paper.

2

Quantum Gauge General Relativity

In quantum gauge general relativity, the most fundamental quantity is gravitational gauge field Cµ (x),which is a vector in the corresponding Lie algebra. Cµ (x) can be expanded as Cµ (x) = Cµα (x)Pˆα , (µ, α = 0, 1, 2, 3) (2.1) where Cµα (x) is the component field and Pˆα = −i ∂x∂α is the generator of global gravitational gauge group. The gravitational gauge covariant derivative is given by Dµ = ∂µ − igCµ(x) = Gαµ ∂α , 3

(2.2)

where g is the gravitational coupling constant and matrix G is given by G = (Gαµ ) = (δµα − gCµα).

(2.3)

1 = (G−1µ α ). I − gC

(2.4)

Its inverse matrix is G−1 =

Using matrix G and G−1 , we can define two important composite operators g αβ = η µν Gαµ Gβν ,

(2.5)

−1ν gαβ = ηµν G−1µ α Gβ .

(2.6)

In quantum gauge general relativity, space-time is always flat and space-time metric is always Minkowski metric, so g αβ and gαβ are no longer space-time metric. They are only two composite operators which consist of gravitational gauge field. The field strength of gravitational gauge field is defined by △

Fµν (x) =

1 α [Dµ , Dν ] = Fµν (x) · Pˆα −ig

(2.7)

where α Fµν = Gβµ ∂β Cνα − Gβν ∂β Cµα .

(2.8)

The Lagrangian of the quantum gauge general relativity is selected to be L = (detG−1 )L0 ,

(2.9)

where L0 = −

1 1 µρ −1ν −1σ α β 1 µρ νσ α β −1σ α β η η gαβ Fµν Fρσ − η µρ G−1ν β Gα Fµν Fρσ + η Gα Gβ Fµν Fρσ . (2.10) 16 8 4

Its space-time integration gives out the action of the system Z S = d4 xL.

4

(2.11)

3

Path Integral Quantization of Gravitational Gauge Fields

Gravitational gauge field Cµα has 4 × 4 = 16 degrees of freedom. But, if gravitons are massless, the system has only 2 × 4 = 8 degrees of freedom. There are gauge degrees of freedom in the theory. Because only physical degrees of freedom can be quantized, in order to quantize the system, we have to introduce gauge conditions to eliminate un-physical degrees of freedom. For the sake of convenience, we take temporal gauge conditions C0α = 0,

(α = 0, 1, 2, 3).

In temporal gauge, the generating functional W [J] is given by !  Z  Z Y α 4 µ α W [J] = N [DC] δ(C0 (x)) exp i d x(L + Jα Cµ ),

(3.1)

(3.2)

α,x

where N is the normalization constant, Jαµ is a fixed external source and [DC] is the integration measure, [DC] =

3 Y 3 Y Y

εdCµα (τj )/

µ=0 α=0 j

 √ 2πi~ .

(3.3)

We use this generation functional as our starting point of the path integral quantization of gravitational gauge field. Generally speaking, the action of the system has local gravitational gauge symmetry, but the gauge condition has no local gravitational gauge symmetry. If we make a local gravitational gauge transformations, the action of the system is kept unchanged while gauge condition will be changed. Therefore, through local gravitational gauge transformation, we can change one gauge condition into another gauge condition. The most general gauge condition is f α (C(x)) − ϕα (x) = 0,

(3.4)

where ϕα (x) is an arbitrary space-time function. The Fadeev-Popov determinant ∆f (C) is defined by Z Y −1 ∆f (C) ≡ [Dg] δ (f α (g C(x)) − ϕα (x)) , (3.5) x,α

5

where g is an element of gravitational gauge group, g C is the gravitational gauge field after gauge transformation g and [Dg] is the integration measure on gravitational gauge group Y [Dg] = d4 ǫ(x), (3.6) x

where ǫ(x) is the transformation parameter of Uˆǫ . Both [Dg] and [DC] are not invariant under gravitational gauge transformation. Suppose that, [D(gg ′)] = J1 (g ′ )[Dg],

(3.7)

[D g C] = J2 (g)[DC].

(3.8)

J1 (g) and J2 (g) satisfy the following relations J1 (g) · J1 (g −1 ) = 1,

(3.9)

J2 (g) · J2 (g −1 ) = 1.

(3.10)

It can be proved that, under gravitational gauge transformations, the Fadeev-Popov determinant transforms as ′

g −1 ′ −1 ∆−1 f ( C) = J1 (g )∆f (C).

Insert eq.(3.5) into eq.(3.2), we get hQ i R R α W [J] = N [Dg] [DC] α,y δ(C0 (y)) · ∆f (C) ·

hQ

(3.11)

(3.12)

i

 R 4 β g β µ α ) . C δ(f ( C(z)) − ϕ (z)) · exp i d x(L + J α µ β,z

Make a gravitational gauge transformation, C(x) →

g −1

C(x) →

gg −1

C(x),

(3.13)

then, g

C(x).

After this transformation, the generating functional is changed into hQ i R R g −1 α W [J] = N [Dg] [DC] J1 (g)J2(g −1 ) · C (y)) · ∆f (C) δ( 0 α,y ·

hQ

i

n R o β β 4 µ g −1 α δ(f (C(z)) − ϕ (z)) · exp i d x(L + J · C ) . α µ β,z 6

(3.14)

(3.15)

Suppose that the gauge transformation g0 (C) transforms general gauge condition f β (C) − ϕβ = 0 to temporal gauge condition C0α = 0, and suppose that this transformation g0 (C) is unique. Then two δ-functions in eq.(3.15) require that the integration on gravitational gauge group must be in the neighborhood of g0−1(C). Therefore eq.(3.15) is changed into hQ i R β β W [J] = N [DC] ∆f (C) · δ(f (C(z)) − ϕ (z)) β,z  R ·exp i d4 x(L + Jαµ ·g0 Cµα )

(3.16)

hQ i R g −1 α ·J1 (g0−1 )J2 (g0 ) · [Dg] δ( C (y)) . 0 α,y

The last line in eq.(3.16) will cause no trouble in renormalization, and if we consider the contribution from ghost fields which will be introduced below, it will become a quantity which is independent of gravitational gauge field. So, we put it into normalization constant N and still denote the new normalization constant as N. We also change Jαµ g0Cµα into Jαµ Cµα , this will cause no trouble in renormalization. Then we get R Q W [J] = N [DC] ∆f (C) · [ β,z δ(f β (C(z)) − ϕβ (z))] (3.17) R 4 µ α ·exp{i d x(L + Jα Cµ )}. In fact, we can use this formula as our start-point of path integral quantization of gravitational gauge field, so we need not worried about the influences of the third line in eq.(3.16). Use another functional 

i exp − 2α

Z

4

α

β



d xηαβ ϕ (x)ϕ (x) ,

(3.18)

R times both sides of eq.(3.17) and then make functional integration [Dϕ], we get   Z Z 1 α β µ α 4 ηαβ f f + Jα Cµ ) . (3.19) W [J] = N [DC] ∆f (C) · exp i d x(L − 2α Now, let’s discuss the contribution from ∆f (C) which is related to the ghost fields. ˆǫ is an infinitesimal gravitational gauge transformation. The Suppose that g = U

7

gravitational gauge transformation of gravitational gauge field Cµα (x) is[5, 6, 7, 8, 9, 3, 10] 1 ˆ α (3.20) Cµα (x) →g Cµα (x) = Λα β (Uˆǫ Cµβ (x)) − (U ǫ ∂µ ǫ (y)), g Then we have

1 Cµα (x) = Cµα (x) − Dαµ σ ǫσ , g

(3.21)

Dαµ σ = δσα ∂µ − gδσα Cµβ ∂β + g∂σ Cµα .

(3.22)

g

where In order to deduce eq.(3.21), the following relation is used Λαβ = δβα + ∂β ǫα + o(ǫ2 ).

(3.23)

Dµ can be regarded as the covariant derivative in adjoint representation, for Dµ ǫ = [Dµ

,

ǫ],

(3.24)

(Dµ ǫ)α = Dαµ σ ǫσ .

(3.25)

Using all these relations, we have, Z δf α(C(x)) β 1 α g α d4 y f ( C(x)) = f (C) − Dµ σ (y)ǫσ (y) + o(ǫ2 ). β g δCµ (y) Therefore, according to eq.(3.5) and eq.(3.4), we get  Z α Y  1Z σ 4 δf (C(x)) β −1 Dµ σ (y)ǫ (y) . dy ∆f (C) = [Dǫ] δ − g δCµβ (y) x,α Define

Mασ (x, y) = −g δǫσδ(y) f α (g C(x)) =

Then eq.(3.27) is changed into ∆−1 f (C) =

R

R

[Dǫ]

α d4 z δfδC(C(x)) Dβµ σ (z)δ(z β µ (z)

(3.27)

(3.28) − y).

  R 4 1 α σ d yM (x, y)ǫ (y) δ − σ x,α g

Q

(3.26)

(3.29)

−1

= const. × (detM) . Therefore,

∆f (C) = const. × detM. 8

(3.30)

Put the above constant into normalization constant, then generating functional eq.(3.19) is changed into   Z Z 1 α β µ α 4 ηαβ f f + Jα Cµ ) . (3.31) W [J] = N [DC] detM · exp i d x(L − 2α In order to evaluate the contribution from detM, we introduce ghost fields η α (x) and η¯α (x). Using the following relation  Z  Z 4 4 α β [Dη][D η¯]exp i d xd y η¯α (x)M β (x, y)η (y) = const. × detM (3.32) and put the constant into the normalization constant, we can get  Z  Z 1 4 α β µ α W [J] = N [DC][Dη][D η¯]exp i d x(L − ηαβ f f + η¯Mη + Jα Cµ ) , 2α (3.33) R 4 where d x¯ η Mη is a simplified notation, whose explicit expression is Z Z 4 d x¯ η Mη = d4 xd4 y η¯α (x)Mαβ (x, y)η β (y). (3.34) The appearance of the non-trivial ghost fields is a inevitable result of the non-Able nature of the gravitational gauge group. Set external source Jαµ to zero, we get,  Z  Z 1 4 α β W [0] = N [DC][Dη][D η¯]exp i d x(L − ηαβ f f + η¯Mη) , 2α

(3.35)

Now, let’s take Lorentz covariant gauge condition,

Then

Z

4

f α (C) = ∂ µ Cµα .

(3.36)

Z

(3.37)

d x¯ η Mη = −

d4 x (∂ µ η¯α (x)) Dαµ β (x)η β (x).

And eq.(3.35) is changed into   Z Z 1 α β µ α σ 4 ηαβ f f − (∂ η¯α )Dµ σ η ) . W [0] = N [DC][Dη][D η¯]exp i d x(L − 2α (3.38) 9

For quantum gauge general relativity, the external source of gravitational gauge field should be introduced in a special way. Define the generating functional with external sources as R  R ¯ = N [DC][Dη][D η¯]exp i d4 x(L − 1 ηαβ f α f β W [J, β, β] 2α −(∂

µ

η¯α )Dαµ σ η σ

+

Cµα

∼µβ

ν δ αν (x)J0β

+ η¯α β + β¯α η α )

 R = N [DC][Dη][D η¯]exp i d4 x(L − R

∼µγ

α



(3.39)

1 η f αf β 2α αβ

−(∂ µ η¯α )Dαµ σ η σ + Cµα Jαµ + η¯α β α + β¯α η α ) ,

where δ αρ (x) is defined by ∼µγ

δ αρ

△

(x) =

and

 ∼γ ∼µγ ∼ 1 ∼µ η η (x) (x)+ (x) (x) , δ δα αρ 2 ρ △ ∼µβ

ν Jαµ = δ αν (x)J0β . ∼µ

∼µγ

In the above definition, δ ρ (x), η

∼

µ δ ρ (x) = δρ −

∼µγ

∼ η µγ

where

(3.41)

(x) and η µγ (x) are defined by

∼µ

η

(3.40)

∂ µ ∂ρ ,  + iǫ

∂µ∂γ ,  + iǫ ∂µ ∂γ , −  + iǫ

(3.42)

(x) = η µγ −

(3.43)

(x) = ηµγ

(3.44)

△

 = ∂ 2 = ∂ µ ∂µ = η µν ∂µ ∂ν .

(3.45)

Using these relations, we can prove that ∼µβ

Jαµ = δ αν Jβν .

(3.46)

The effective Lagrangian Lef f is defined by Lef f ≡ L −

1 ηαβ f α f β − (∂ µ η¯α )Dαµ σ η σ . 2α 10

(3.47)

Lef f can be separate into free Lagrangian LF and interaction Lagrangian LI , Lef f = LF + LI ,

(3.48)

where β β β 1 µρ νσ α α α − 18 η µρ F0µβ + 14 η µρ F0µα η η ηαβ F0µν F0ρσ F0ρα F0ρβ LF = − 16 1 ηαβ (∂ µ Cµα )(∂ ν Cνβ ) − 2α

µ

(3.49)

α

− (∂ η¯α )(∂µ η ),

LI = +g(∂ µ η¯α )Cµβ (∂β η α ) − g(∂ µ η¯α )(∂σ Cµα )η σ

(3.50)

+ self interaction terms of Gravitational gauge f ield. From the interaction Lagrangian, we can see that ghost fields do not couple to J(C). This is the reflection of the fact that ghost fields are not physical fields, they are virtual fields. Besides, the gauge fixing term does not couple to J(C) either. Using ¯ can be simplified to effective Lagrangian Lef f , the generating functional W [J, β, β]  Z  Z 4 µ α α α ¯ ¯ W [J, β, β] = N [DC][Dη][D η¯]exp i d x(Lef f + Jα Cµ + η¯α β + βα η ) ,

(3.51)

4

Propagators

Using eq.(3.49), we can deduce propagator of gravitational gauge fields and ghost fields. First, after a partial integration, we change the form of eq. (3.49) into   Z Z 1 α µν 4 4 β 2 α d xLF = d x (4.1) C M (x)Cν + η¯α ∂ η , 2 µ αβ where the operator Mµν αβ (x) is defined by Mµν αβ (x) =

1 µν η ηαβ ∂ ρ ∂ρ 4

− 14 ηαβ (1 − α4 )∂ µ ∂ ν − 41 δβµ ∂ ν ∂α

+ 14 δβµ δαν ∂ ρ ∂ρ − 14 δαν ∂ µ ∂β + 12 δβν ∂ µ ∂α − 14 η µν ∂α ∂β − 12 δαµ δβν ∂ ρ ∂ρ + 21 δαµ ∂ ν ∂β .

11

(4.2)

Denote the propagator of gravitational gauge field as −i∆αβ F µν (x),

(4.3)

and denote the propagator of ghost field as −i∆αF β (x).

(4.4)

They satisfy the following equation, ∼µγ

βγ −Mµν αβ (x)∆F νρ (x − y) = δ αρ (x)δ(x − y),

(4.5)

−∂ 2 ∆αF β (x − y) = δβα δ(x − y),

(4.6)

∼µγ

where δ αρ (x) is defined by (3.40). Make Fourier transformations to momentum space Z ∼ αβ d4 k αβ ikx (−i) , −i∆F µν (x) = ∆ F µν (k) · e (2π)4 Z ∼α d4 k α ikx −i∆F β (x) = (−i) , ∆ F β (k) · e (2π)4 ∼ αβ

(4.7) (4.8)

∼α

where −i ∆F µν (k) and −i ∆F β (k) are corresponding propagators in momentum space. They satisfy the following equations, ∼ βγ

∼µγ

−Mµν αβ (k) ∆F νρ (k) = δ αρ (k), ∼α

k 2 ∆F β (k) = δβα ,

(4.9) (4.10)

where the operator Mµν αβ (k) is defined by △

1 4 1 µ ν 1 µν 2 µ ν Mµν αβ (k) = − 4 η ηαβ k + 4 ηαβ (1 − α )k k + 4 δβ k kα

− 14 δβµ δαν k 2 + 41 δαν k µ kβ − 21 δβν k µ kα

(4.11)

+ 14 η µν kα kβ + 21 δαµ δβν k 2 − 12 δαµ k ν kβ , ∼µγ

and δ αρ (k) is defined by ∼µγ

δ αρ

 ∼γ ∼µγ ∼ 1 ∼µ η η (k) = (k) αρ (k) . δ (k) δ α (k)+ 2 ρ 12

(4.12)

The operator Mµν αβ has the following symmetric property νµ Mµν αβ = Mβα . ∼µ

∼µγ

In the above relation, δ ρ (k), η

∼

(k) and η µγ (k) are defined by

∼µ

µ δ ρ (k) = δρ −

∼µγ

η

∼ η µγ

kµ kγ , k 2 − iǫ kµ kγ . − 2 k − iǫ

(k) = ηµγ

(4.16)

∼µγ

∼

∼

(k), η µγ (k), δρµ , η µγ and ηµγ satisfy the ∼µγ

∼

(k)· η γν (k) = η µγ · η γν (k) =η

∼µ

(4.14) (4.15)

∼µ

η

k µ kρ , k 2 − iǫ

(k) = η µγ −

It can be easily proved that δ ρ (k), η following relations: ∼µγ

(4.13)

∼γ

∼γ

∼µ

(k) · ηγν = δ ν (k),

∼µ

∼µ

µ γ δ γ (k)· δ ν (k) = δγ · δ ν (k) = δ γ (k) · δν = δ ν (k),

∼µ

∼γν

∼γ

∼

δ γ (k)· η

∼γν

(k) = δγµ · η

∼µ

∼µν

∼γ

∼

(k) = δ γ (k) · η γν =η

(4.17) (4.18)

(k),

(4.19)

γ δ µ (k)· η γν (k) = δµ · η γν (k) = δ µ (k) · ηγν =η µν (k),

(4.20)

∼

∼µν

∼

k µ η µν (k) = kµ η

∼ν

∼µ

(k) = k µ δ µ (k) = kµ δ ν (k) = 0.

(4.21)

∼µγ

Using all these relations, we can prove that δ αρ (k) satisfies the following relation ∼µγ

∼ρβ

∼µβ

δ αρ (k)· δ γν (k) = δ αν (k).

(4.22)

For the propagator of gravitational gauge field, we require that it should satisfy the following gauge conditions ∼νγ

αβ αγ δ βρ (x) · ∆F µν (x) = ∆F µρ (x),

∼µγ

αβ γβ δ αρ (x) · ∆F µν (x) = ∆F ρν (x).

(4.23) (4.24)

In momentum space, these two gauge conditions become ∼ αβ

∼νγ

∼ αγ

∆F µν (k)· δ βρ (k) =∆F µρ (k), 13

(4.25)

∼µγ

∼ αβ

∼ γβ

(4.26) δ αρ (k)· ∆F µν (k) =∆F ρν (k). These two gauge conditions are related to the zero mass of graviton. The solutions to the two propagator equations (4.9) and (4.10) and gauge conditions (4.25 ) and (4.26)give out the propagators in momentum space,   ∼β ∼α ∼α ∼β ∼ αβ ∼ ∼αβ −i η µν (k) η (k)+ δ µ (k) δ ν (k)− δ µ (k) δ ν (k) , (4.27) −i ∆F µν (k) = 2 k − iǫ

−i α δ . (4.28) − iǫ β The forms of these propagators are quite beautiful and symmetric. it can be easily proved that ∼α

−i ∆F β (k) =

∼ αβ

∼ αβ

k2

∼ αβ

∼ αβ

k µ ∆F µν (k) = kα ∆F µν (k) =∆F µν (k)k ν =∆F µν (k)kβ = 0.

5

(4.29)

Feynman Rules of Interaction Vertices

The interaction Lagrangian LI is a function of gravitational gauge field Cµα and ghost fields η α and η¯α , LI = LI (C, η, η¯). (5.1) Then eq.(3.51) is changed into, R  R ¯ = N [DC][Dη][D η¯] exp i d4 xLI (C, η, η¯) W [J, β, β]

 R ·exp i d4 x(LF + Jαµ Cµα + η¯α β α + β¯α η α )

(5.2)

o n R 1 δ ¯ ) · W0 [J, β, β], = exp i d4 xLI ( 1i δJδ , 1i δδβ¯ , −i δβ

where  R
R ¯ = N [DC][Dη][D η¯]exp i d4 x LF + J µ C α + η¯α β α + β¯α η α W0 [J, β, β] α µ 1 α µν C Mαβ (x)Cνβ 2 µ

R  R = N [DC][Dη][D η¯]exp i d4 x +Jαµ Cµα + η¯α β α + β¯α η α

+ η¯α ∂ 2 η α




n RR h ν = exp i d4 xd4 y 21 Jαµ (x)∆αβ F µν (x − y)Jβ (y) + β¯α (x)∆αF β (x − y)β β (y) 14



.

(5.3)

In order to obtain the above relation, eq. (3.46) is used. The interaction Feynman rules for interaction vertices can be obtained from the interaction Lagrangian LI . For example, the interaction Lagrangian between gravitational gauge field and ghost field is +g(∂ µ η¯α )Cµβ (∂β η α ) − g(∂ µ η¯α )(∂σ Cµα )η σ .

(5.4)

This vertex belongs to Cµα (k)¯ ηβ (−q)η δ (p) three body interactions, its Feynman rule is igδδβ q µ pα − igδαβ q µ kδ . (5.5) To calculate the interaction lagrangian of three gravitational gauge field, four gravitational gauge field and higher gravitational gauge field are extremely complicated. Here I only explain how to calculate then and list related results. First, we can expand detG−1 , G−1ν and gαβ in terms of gravitational gauge field α detG−1 = 1 + gCαα +

gαβ

 g2  α µ Cµ Cα + Cµµ Cαα + · · · , 2

G−1ν = δαν + gCαν + g 2 Cµν Cαµ + · · · , α   = ηαβ + g ηµβ Cαµ + ηµα Cβµ +g

2



ηµβ Cαµ1 Cαα1

+

ηµα Cαµ1 Cβα1

+

ηµν Cαµ Cβν

(5.6) (5.7) (5.8)



+ ···.

Next, we need to expand the lagrangian L0 in terms of gravitational gauge field. We will make the following expanding 2

n

3

4

L0 =L0 + L0 + L0 + · · · ,

(5.9)

where L contains all n-th order interaction terms of gravitational gauge field. Substitute equations (5.7) and (5.8) into (2.10), we can get 2

µνρσ

L0 = Vαβ

∂ρ Cµα

where µνρσ Vαβ =−





∂σ Cνβ ,

1 µνρσ η¯ . 16 αβ

(5.10) (5.11)

µνρσ In the above relation, η¯αβ is defined by µνρσ ρσ µν ρν µσ η¯αβ = η µν η¯αβ + η ρσ η¯αβ − η µσ η¯αβ − η ρν η¯αβ ,

15

(5.12)

where µν η¯αβ = η µν ηαβ + 2δβµ δαν − 4δαµ δβν .

(5.13)

The interaction term of three gravitational gauge field in the L0 is 3

µνλρσ

L0 = Vαβγ where

µνλρσ Vαβγ =

Cλγ ∂ρ Cµα


∂σ Cνβ ,




g µνλρσ νµλσρ (¯ ηαβγ + η¯βαγ ), 16

µνλρσ µνλσ µνρσ η¯αβγ = δγρ η¯αβ − δβλ η¯αγ .

(5.14) (5.15) (5.16)

The interaction term of four gravitational gauge field in the L0 is 4

µνλκρσ

L0 = Vαβγδ where µνλκρσ Vαβγδ

Cλγ Cκδ ∂ρ Cµα





∂σ Cνβ ,

i g 2 h µνλκρσ νµλκσρ µνκλρσ νµκλσρ = . η¯ + η¯βαγδ + η¯αβδγ + η¯βαδγ 64 αβγδ

µνλκρσ µνλρσ νµσλκ µνκρσ µνλρσ η¯αβγδ = δβκ η¯αδγ + δδρ η¯βγα + δβλ η¯αγδ + δακ η¯δβγ .

(5.17)

(5.18) (5.19)

Substitute above results and (5.6) into (2.9), we get 3

µνλρσ γ α L= V¯αβγ Cλ ∂ρ Cµ

where And


∂σ Cνβ ,

g µνρσ µνλρσ µνλρσ . V¯αβγ = Vαβγ − δγλ η¯αβ 16 4

where




L= V¯αβγδ

µνλκρσ µνλκρσ V¯αβγδ = Vαβγδ −

µνλκρσ

g2 32

Cλγ Cκδ ∂ρ Cµα





∂σ Cνβ ,


µνρσ δγκ δδλ + δγλ δδκ η¯αβ

  2 µνλσ νµλρ µνκσ νµκρ + δδκ δγσ η¯βα + δγλ δδρ η¯αβ + δγλ δδσ η¯βα + g32 δδκ δγρ η¯αβ

2 νµσρ µνρσ νµσρ
µνρσ − g32 δδκ δβλ η¯αγ . + δδκ δαλ η¯βγ + δγλ δβκ η¯αδ + δγλ δακ η¯βδ

16

(5.20) (5.21) (5.22)

(5.23)

Feynman rules for the vertex of three gravitational gauge field Cµα (p1 )Cνβ (p2 )Cλγ (p3 ) is h i µνλρσ νλµρσ λµνρσ −2i V¯αβγ p1ρ p2σ + V¯βγα p2ρ p3σ + V¯γαβ p3ρ p1σ . (5.24)

The Feynman rule for the vertex of four gravitational gauge field Cµα (p1 )Cνβ (p2 )Cλγ (p3 )Cκδ (p4 ) is h µνλκρσ µλνκρσ µκνλρσ −4i V¯αβγδ p1ρ p2σ + V¯αγβδ p1ρ p3σ + V¯αδβγ p1ρ p4σ (5.25)

νλκµρσ +V¯βγδα p2ρ p3σ

6

+

νκµλρσ V¯βδαγ p2ρ p4σ

+

λκµνρσ V¯γδαβ p3ρ p4σ

i

.

Discussions

In this paper, path integral quantization of quantum gauge general relativity is discussed, and Feynman rules of various interaction vertices are calculated. These results are needed in the loop diagram calculation. In the literature [3], we have formally proved that quantum gauge general relativity is a perturbatively renormalizable quantum theory. In that proof, detailed calculations of loop diagrams are not performed. In the next step, we will calculate all divergent one-loop diagrams, discuss renormalization of quantum gauge general relativity in one-loop level, and determine the renormalization constant in one-loop level. These results will be summarize in the further paper. [Acknowledgement] The author would like to thank Prof. J.P. Hsu for useful discussions and kindly suggestions on this work.

References [1] F.W.Hehl, P. Von Der Heyde, G.D.Kerlick, J.M.Nester Rev.Mod.Phys. 48 (1976) 393-416 [2] D.Ivanenko and G.Sardanashvily, Phys.Rep. 94 (1983) 1. [3] Ning WU, ”Renormalizable Quantum Gauge General Relativity” grqc/0309041. [4] J.-P.Hsu, Int. J. Mod. Phys. A21 (2006) 5119.

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[5] Ning WU, ”Gauge Theory of Gravity”, hep-th/0109145. [6] Ning WU, Commun. Theor. Phys. (Beijing, China) 38 (2002): 151-156. [7] Ning WU, ”Quantum Gauge Theory of Gravity”, hep-th/0112062. [8] Ning WU, ”Quantum Gauge Theory of Gravity”, talk given at Meeting of the Devision of Particles and Fields of American Physical Society at the College of William & Mary(DPF2002), May 24-28, 2002, Williamsburg, Virgia, USA; hep-th/0207254; Transparancy can be obtained from: http://dpf2002.velopers.net/talks pdf/33talk.pdf [9] Ning WU, Commun. Theor. Phys. (Beijing, China) 42 (2004): 543-552. [10] Ning Wu, Quantum Gauge Theory of Gravity, In Focus on Quantum Gravity Research, chapter 4, ed. David C. Moore, pp.121-169, (Nova Science Publishers, Inc., New York, 2006) [11] Ning WU, Commun. Theor. Phys. (Beijing, China) 40 (2003): 337-340. [12] Ning WU, Commun. Theor. Phys. (Beijing, China) 47 (2007): 503-511; grqc/0508009. [13] Ning WU, Commun. Theor. Phys. (Beijing, China) 44 (2005): 883-886. [14] Ning WU, Commun. Theor. Phys. (Beijing, China) 48 (2007): 469-472; grqc/0603104. [15] Ning WU, Commun. Theor. Phys. (Beijing, China) 49 (2008): 129-132. [16] Ning WU, Commun. Theor. Phys. (Beijing, China) 49 (2008): 1533-1540. [17] Ning WU, Commun. Theor. Phys. (Beijing, China) 41 (2004): 567-572. [18] Ning WU, Theor. Phys. (Beijing, China) 46 (2006): 639-642; gr-qc/0510010. [19] Ning WU, Commun. Theor. Phys. (Beijing, China) 45 (2006): 452-456. [20] Ning WU, Dahua ZHANG, Commun. Theor. Phys. (Beijing, China) 45 (2006): 858-860. [21] Ning WU, Commun. Theor. Phys. (Beijing, China) 40 (2003): 429-434. [22] Ning WU, Commun. Theor. Phys. (Beijing, China) 41 (2004): 381-384. [23] Ning WU, Commun. Theor. Phys. (Beijing, China) 38 (2002): 322-326.

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