Quantum Gravity Field

Transcription

1 Quantum Gravity Field Malik Matwi Department of Physics, Damascus, Syria, Contents 1 The Lagrange of the quantum gravity field 2 2 The Lagrange of the Plebanski two form field 20 3 The static potential of the weak gravity 26 Abstract We study the dynamics of the gravity field according to the quantum fields theory on arbitrary spacetime x µ. Therefore, we suggest a canonical momentum π as a momentum conjugate to the canonical gravity field ẽ = ee µn µ. We derive both the canonical gravity field and its conjugate momentum from the holonomy U γ, A) of the complex selfdual connection A i a. The canonical momentum π is represented in the Lorentz group. We use it in deriving the path integral of the gravity field according to the quantum fields theory. Then, we discuss the situation of the free gravity field like the electromagnetic field). We find that this situation takes place only in the background spacetime approximation, the situation of low matter densityweak gravity). Then, we derive the Lagrange of the Plebanski two form complex field Σ i, which is represented in selfdual representation Σ i. 1

2 We try to combine the two fieldsgravity and Plebanski) in one field: K i µ. Finally, we derive the static potential of exchanging gravitons in scalar and spinor fields, the Newtonian gravitational potential. key words: Conjugate momentum, path integral, free gravity propagator, Plebanski field Lagrange, Plebanski and gravity fields combination. 1 The Lagrange of the quantum gravity field We search for conditions allow us to consider the gravity as a dynamical field. The problem of the dynamics in the general relativity is that spacetime is itself a dynamical. t interacts with matter, it is an operator dˆx µ. So we have to consider it as a quantum field as the other fields. Both the dynamical curved spacetime x µ and the gravity field e have the same entity, it is the gravity. Therefore, we study only one of them as a gravity field: ê. Then the dynamical spacetime x µ is implicitly included in the gravity field e via ê = ê µdx µ. We will see that it is substantially different in the background spacetime, where the gravity field becomes like the usual quantum fields in flat spacetime. To study the dynamics of the gravity field according to the quantum fields theory, we need to find the canonical conjugate momentum π x)represented in the Lorentz group). We find that it acts canonically on the local-lorentz vectors V x) on a closed 3D surface δm immersed in arbitrary curved spacetime x µ of a manifold M. The closed surface δm is parameterized by three parameters X 1, X 2 and X 3. n a certain gauge, we consider that they carry the spatial indices of the local-lorentz inertial frame. The local-lorentz framex : X 0, X 1, X 2, X 3 ) is considered as a tangent-space on the curved spacetime x µ. We will see that the path integral of the gravity field is independent on this gauge. Therefore, the exterior derivative operator, on the surface δm, leads to a change along the norm of that surface, so it causes the change in the time dx 0 direction. That allows the 3D surface δm to extend, thus we get the 4D local-lorentz frame X, which, in our gauge, parameterize the four dimensions x µ coordinates of the curved spacetime x µ in the manifold M. With that the gravity field propagates from one surface to another by the extension 2

3 of those surfaces. To study the gravity field propagation, we suggest canonical states ẽ and π represented in Lorentz group. We use them in deriving the path integral. We find that there is no propagation on the dynamical spacetime x µ. But in the background spacetime, the gravity field propagates freely like the electromagnetic field. 2] The holonomy of the complex connection A i in the quantum gravity is[1, Uγ, A) = T rp e i γ A, 1.1) where the path ordered P is defined via P e i γ A = 1 s 1 n=0 0 0 ds 1 ds 2... s n 1 0 ds n ia γ s n ))...ia γ s 1 )) : γ µ s) = dxµ ds, where γs) is a closed path in the curved spacetime x µ. n irreducible selfdual representation of the Lorentz group, we write A = A i τ i, where τ i are Pauli matrices. The element Uγ, A) is invariant under local-lorentz transformation V L J x)v J and under arbitrary changing of the spacetime coordinates dx µ Λ µ νx)dx ν, therefore the quantum gravity is studied using it[1, 2]. The complex connection A i is selfdual of the real local-lorentz spin connection ωx)[1, 2]: A i µ x) = P i) J ωj µ x), where P i are the selfdual projectors. Let us rewrite the holonomy Uγ, A) using the real spin connection ωµ J dx µ of the local-lorentz frame: Uγ, ω) = T rp e i γ ω J. We expect that it has the same properties of Uγ, A); satisfies the symmetries of GR. 3

4 The connection ω J as: transforms under local Lorentz transformation Lx) J ω = LωL 1 + LdL 1 or ω µdx µ = Lω µ L 1 dx µ + L µ L 1 dx µ, therefore ω µdx µ = γ γ Lω µ L 1 dx µ + γ L µ L 1 dx µ. The connection L µ L 1 is polar: it reverses sign when x µ x µ, where L x µ ) = L x µ ), so its integral over a closed curve γ vanishes. So ω µdx µ = Lω µ L 1 dx µ. γ γ Therefore, the connection ω transforms covariantly inside the integration over a closed curve. Let us write it as ω µ dx µ = Ω µ dx µ + B µ dx µ with B µ dx µ = 0, where the tensor Ω J µ dx µ transforms covariantly under local Lorentz transformation, while the polar connection B µ does not. Therefore L ω B) L 1 = ω B ; Ω = LΩL 1 which yields γ B = LBL 1 + LdL 1 with ω = LωL 1 + LdL 1, 1.2) for B = B J) and ω = ω J). And yields B = LBL 1 dl)l 1 with ω = LωL 1 dl)l 1, 1.3) for B = B J) and ω = ω J ). With that, the condition γ B µdx µ = γ B µdx µ = 0 is satisfied under local Lorentz transformation, because the connection B µ remains polar under it. 4

5 The loop γ ωj = γ ΩJ transforms covariantly under local Lorentz transformation. For free gravity field, let us impose the relation: Ω J = π K J e K, thus π J K x) transforms covariantly under local Lorentz transformation, so we consider it as a conjugate momentum represented in the local Lorentz group and acts on its vectors. Therefore, we consider it as a dynamical operator. The loop becomes ω J = π J K e K. 1.4) γ γ nserting it in the holonomy Uγ, ω), we get ) Uγ, π, e) = T rp exp i πk J e K µ dx µ. γ is anti- n the free gravity field, we expect that the momentum π JK symmetric. Therefore we can write it as π JK = π L ε LJK. This is our starting point in studying the dynamics of the quantum gravity. By that, the last holonomy becomes ) ) U γ, π, e) = T rp exp i π K J ekµ dx µ = T rp exp i ε LK J πl e Kµ dx µ. Let us expand it as 1 n=0 0 ds 1 s 1 0 ds 2... γ s n 1 0 ds n iε LK J π L e Kµ γ µ) s n ) iε L 1K 1 J J1 π L1 e K1 µ 1 γ µ 1 ) s n 1 )... iε L n 1K n 1 J n 2 π Ln 1 e Kn 1 µ n 1 γ µ n 1 ) s 1 ), where iε LK Jπ L e Kµ γ µ) s n ) = iε LK Jπ L s n ) e Kµ s n ) γ µ s n ), with the tangent γ µ s) = dxµ ds on the closed path γs) in the manifold M. Using the properties ε JKL ε JK 1L 1 = 2 δ K 1 K δl 1 L ) δl 1 K δk 1 L and εjkl ε 1JKL = 6δ 1, 5 γ

6 the integrals of the holonomy U γ, π, e) become over terms like...π s j )e µs i ) γ µ s i )ds i...π J s i )e J ν s k ) γ ν s k )ds k... with i j and i k. This holonomy satisfies the general relativity symmetries; invariance under local Lorentz transformation V L Jx)V J and under arbitrary changing of the coordinates dx µ Λ µ νx)dx ν. Therefore, we use it in the quantum gravity. Let us suggest another term: γ π Ke K µ dx µ. We expect that it also satisfies the general relativity symmetries if it is integrated over a closed 3D surface δm instead of the closed path γs). This is because ed 4 x = 1 4 d3 x µ dx µ = 1 4 eε µνρσdx ν dx ρ dx σ dx µ /3! is invariant element, so we can replace π K e K µ dx µ with π K e Kµ d 3 x µ = π K e Kµ eε µνρσ dx ν dx ρ dx σ /3!. With integrating it over a three dimensions closed surface δm, it becomes invariant under GR transformations because in the free gravity there are no sources for the gravity field. Therefore, the flux of the Lorentz vectors is invariant under arbitrary changing of the closed surface δm. The determinant e of the gravity field e µ is defined in e = g with writing the metric g µν x) on the curved spacetime x µ as g µν x) = η J e µe J ν. Under arbitrary transformations, we have invariant element: gεi1...i n = g ε i 1...i n. Therefore, the element eε µνρσ dx ν dx ρ dx σ /3! = d 3 x µ is a co-vector, as µ. Thus, the integral UδM, π, e) = exp i π e µ eε µνρσ dx ν dx ρ dx σ /3! = exp i δm δm 6 π e µ d 3 x µ

7 satisfies the same conditions of the holonomy U γ, A); invariant under local Lorentz transformation V L J x)v J and under arbitrary changing of the coordinates dx µ Λ µ νx)dx ν. This relates to the fact that the integrals of free vectors over a closed surface δm, in a manifold M, are invariant under arbitrary changing of that surface if there are no sources for those vectors. t is the conservation. The spin connection ω µ and so π K e Kµ, as vectors, satisfy this fact in the free gravity. The equation of motion of the gravity field e is De = de + ω J e J = 0. Replacement ω J π K J e K, we get de π N J e N e J. But the tensor e N e J = e N µ e J ν dx µ dx ν = 1 2 e N µ e J ν e N ν e J µ) dx µ dx ν measures the area in the manifold M. Therefore, the changes of the gravity field around a closed path rotation) relate to the flux of the momentum π through the area determined by that closed path. t is like the magnetic field generated by straight electric current. Thus we have e N e J area, de π N J e N e J flux through this area. For this reason, we suggested that the conjugate momentum π JK is antisymmetric. Now, in the integral exp i π e µ eε µνρσ dx ν dx ρ dx σ /3!, δm let us define the canonical gravity field ẽ via ẽ d 3 X = ẽ dx 1 dx 2 dx 3 e µ eε µνρσ dx ν dx ρ dx σ /3!, 7

8 or ẽ = ee µn µ X i), where n µ X i ) is the norm to the surface δm. Thus, the holonomy UδM, π, e) becomes Ũ δm δm, π, ẽ) = exp i π ẽ d 3 X, δm where the parameters X : = 1, 2, 3 parameterize the closed 3D surface δm in the manifold M. As mentioned before, in certain gauge, we consider that the indices = 1, 2, 3 are the spatial indices of the local-lorentz frame = 0, 1, 2, 3). Therefore, the exterior derivative, on the surface δm, becomes along the time dx 0. The time dx 0 is the direction of the norm on the surface δmx 1, X 2, X 3 ). We will see that the result of the path integral is independent on this gauge. As we suggested before, the integral exp i δm π ẽ d 3 X satisfies the same conditions of the holonomy Uγ, A); The GR symmetries, therefore we consider it as a canonical dynamical element. Comparing it with φ π = exp i d 3 XφX)πX)/, a canonical relation in the scalar field φ theory on flat spacetime, for = 1, we suggest canonical states ẽ and π with ẽ π = exp i ẽ X)π δm X)d 3 X, δm where π is the canonical momentum conjugate to ẽ. Let us write this relation on the surface δm as ẽ ẽ π x n + dx n ) π x n ), δm with δm = n, ẽ x n + dx n ) π x n ) δm = exp iẽ x n +dx n )π x n )d 3 X exp iẽ x n )π x n )d 3 X. 8

9 n general, for two points in adjacent surfaces δm 1 and δm 2, let us rewrite it as ẽ x n + dx n ) π x n ) = exp iẽ x n + dx n )π x n )d 3 X. 1.5) Here the variation ẽ x n + dx n ) ẽ x n ) is exterior derivative along the time dx 0 direction, the direction of the norm to the surface δm 1. t allows the extension of the surface: δmx 1, X 2, X 3 ) MX 0, X 1, X 2, X 3 ). This leads to the propagation of those surfaces. We need to make êd 4ˆx commute with ˆẽ d 3 X. For this purpose, we write êd 4ˆx = êdˆx µ ε µνρσ dˆx ν dˆx ρ dˆx σ /4! = êdˆx µ ε µνρσ 4! ˆx ν X i ˆx ρ X j ˆx σ X k ε ijk 3! d3 X = 1 4êdˆxµˆn µ d 3 X. The indexes i, j and k, in our gauge, are the local-lorentz frame indices for = 1, 2, 3. We can rewrite it, in our gauge, as: ed 4 x = 1 4 edxµ n µ d 3 X = 1 4 e xµ X 0 n µd 3 XdX 0 = 1 4 eeµ 0n µ d 3 XdX 0. Comparing it with the term ẽ d 3 X = e µ eε µνρσ dx ν dx ρ dx σ /3! = ee µ n µ d 3 X, we find that it commutes with it: [êê µˆn µ d 3 X, êê µ 0 ˆn µ d 3 XdX 0] = 0 [ˆẽ 4ˆx] d 3 X, êd = 0, where [ ê µ, ê J ν ] = 0. Thus, the operator êd4ˆx takes eigenvalues when it acts on the states ẽ. The action of the gravity field is[1] Se, ω) = 1 ε JKL e e J R KL ω) + λe e J e K e L). 16πG 9

10 Let us consider only the first term: Se, ω) = c ε JKL e e J R KL ω), where C is constant. The Riemann curvature here is R KL ω) = dω KL + ω K M ω ML. nserting the relation we suggested before: ω) J = Ω J + B J ; Ω J = π K J e K, the action becomes Se, π) = c ε JKL e e J dω + db + Ω Ω + Ω B + B Ω + B B) KL. 1.6) As we assumed before that Ω J transforms covariantly, so the remaining term dω + db + Ω B + B Ω + B B also transforms covariantly, this is because the action transforms covariantly. Let us choose a gauge for the polar vector B µ : dω + db + Ω B + B Ω + B B = 0. This gauge satisfies the conditions 1.2) and 1.3), let us test them: d LΩL 1 ) + d LBL 1 + LdL 1 ) + LΩL 1 LBL 1 + LdL 1 ) + LBL 1 dl)l 1 ) LΩL 1 + LBL 1 dl)l 1 ) LBL 1 + LdL 1 ). Expanding it: dl) ΩL 1 + L dω) L 1 LΩ dl 1 ) + dl) BL 1 + L db) L 1 LB dl 1 + dl) dl 1 + LΩL 1 LBL 1 + LΩL 1 LdL 1 + LBL 1 LΩL 1 dl)l 1 LΩL 1 + LBL 1 LBL 1 dl)l 1 LBL 1 + LBL 1 LdL 1 dl)l 1 LdL 1, 10

11 so dl) ΩL 1 + L dω) L 1 LΩ dl 1 ) + dl) BL 1 + L db) L 1 LB dl 1 + dl) dl 1 + LΩ BL 1 + LΩ dl 1 + LB ΩL 1 dl) ΩL 1 +LB BL 1 dl) BL 1 + LB dl 1 dl) dl 1, it becomes L dω) L 1 + L db) L 1 + LΩ BL 1 + LB ΩL 1 + LB BL 1, therefore, as expected, it transforms covariantly and satisfies the conditions 1.2) and 1.3), therefore we can choose the gauge dω + db + Ω B + B Ω + B B = 0. We need it, because the polar vector B µ is not included in the loop 1.4), so it must not be included in the Lagrange. Finally, the action becomes Se, π) = c ε JKL e e J Ω Ω) KL, or Se, π) = c ε JKL e e J Ω K M Ω ML. nserting our assumption Ω J = π J K e K, this action becomes Se, π) = c ε JKL e e J ) K π K1 M e K 1 π ) ML K2 e K 2. Making the replacement e e J e K 1 e K 2 ε JK 1K 2 e 0 e 1 e 2 e 3, we get Se, π) = c ε JKL πk1 K M ) πk2 ML ) ε JK 1K 2 e 0 e 1 e 2 e 3. 11

12 nserting the relation π JL = π K ε KJL we imposed before, and using ε JKL ε JK 1K 2 = 2 δ K 1 K δk 2 L ) δk 1 L δk 2 K, the action becomes Se, π) = c 2π ε LMK π J ε JLMK e 0 e 1 e 2 e 3, and using ε LMK ε JLMK = 6δ J, it becomes Se, π) = 12c π π e 0 e 1 e 2 e 3 Or S 0 e, π) = 12c π 2 ed 4 x. n the background spacetime, we have e 1 + δe, so this action becomes S 0 δe, π) 12c π 2 d 4 x +... To find its meaning, we compare it with the scalar field Lagrange in the flat spacetime: 1 Ld 4 x = π 0 φ Hφ, π)) d 4 x with Hφ, π)d 4 x = 2 π φ)2 + 1 ) 2 m2 φ 2 d 4 x. For = 1, we conclude that the term 12cπ 2 d 4 x 0 is the energy of the gravity field in the background spacetime. The missed term is the kinetic term, like π 0 φ, we find it using the path integral. We derive the path integral according to the quantum fields theory. As we saw before, in our gauge, the operator êd 4ˆx takes eigenvalues when it acts on the states ẽ. Using eq.1.2), we get the amplitude ẽ x + dx) e is π x) ẽ x + dx) e i12cˆπ 2 êd 4 ˆx π x) = exp i12cπ 2 x) e x + dx) d 4 x + iẽ x + dx) π x)d 3 X ) exp i12cπ 2 x) e x) d 4 x + iẽ x + dx) π x)d 3 X ), 12

13 where we let the momentum π acts on the left. The amplitude of the propagation between two points x and x + dx of adjacent surfaces δm 1 and δm 2 is ẽ x + dx) e ic12ˆπ2 êd 4 ˆx ẽ x) δm 1 δm 2 = = dπ ẽ x + dx) e ic12ˆπ2 êd 4 ˆx π x) δm 1 δm 2 π x) ẽ x) δm 1 dπ exp [ i12cπ 2 x) e x + dx) d 4 x + iẽ x + dx)π x)d 3 X ] exp iẽ x)π x)d 3 X ) dπ exp [ i12cπ 2 x) e x) d 4 x + i ẽ x + dx) ẽ x) ) π x)d 3 X ]. The exterior derivative ẽ x + dx) ẽ x) ) d 3 X = ẽ x) X 0 d3 XdX 0 = dẽ x)d 3 X is along the time dx 0 direction, the direction of the norm to the surface δmx 1, X 2, X 3 ), so it leads to the propagation from one surface to another. Thus, we write the amplitude as ẽ x + dx) e ic12ˆπ2 êd 4 ˆx ẽ x) δm 1 δm 2 = dπ exp [ i12cπ 2 x) e x) d 4 x + iπ x)dẽ x)d 3 X ]. The path integral is the integral of ordered product of those amplitudes on all spacetime pointsover all ordered 3D surfaces), thus we write it as 12cπ W ST = Dẽ Dπ exp i 2 ed 4 x + π dẽ d 3 X ) 12cπ = Dẽ Dπ exp i 2 e 0 e 1 e 2 e 3 + π dẽ d 3 X ). There is no problem with Lorentz non-invariance in ẽ x) d 3 XdX 0, because X 0 the equation of motion we find in the result of the path integral is ẽ x) X 0 π, 13

14 thus we have ẽ x) X 0 π d 3 XdX 0 π π d 3 XdX 0. This is Lorentz invariant. This is like the equation of motion of the scalar field φ; π = 0 φ which solves the same problem. n our gauge, we have π π d 3 XdX 0 π 2 dx 0 dx 1 dx 2 dx 3 = π 2 e 0 µe 1 νe 2 ρe 3 σdx µ dx ν dx ρ dx σ t is invariant element, we find it in the path integral. = π 2 e 0 µe 1 νe 2 ρe 3 σε µνρσ d 4 x = π 2 ed 4 x. The path integral 12cπ W ST = Dẽ Dπ exp i 2 e 0 e 1 e 2 e 3 + π dẽ d 3 X ) vanishes unless δsπ, e) δπ = δ δπ 12cπ 2 e 0 e 1 e 2 e 3 + π dẽ d 3 X ) = 24cπ e 0 e 1 e 2 e 3 +dẽ d 3 X = 0. Therefore we get the pathequation of motion): or So, ˆπ = 1 ê0 ê 1 ê 2 ê 3) 1 dˆẽ d 3 X, 1.7) 24c π π J = 1 24c) 2 e 0 e 1 e 2 e 3) 2 dẽ d 3 Xdẽ J d 3 X. 1.8) 12cπ 2 e 0 e 1 e 2 e 3 + π dẽ d 3 X = 1 e 0 e 1 e 2 e 3) 1 dẽ d 3 X ) dẽ d 3 X ) 48c + 1 e 0 e 1 e 2 e 3) 1 dẽ d 3 X ) dẽ d 3 X ). 24c nserting it in the path integral, thus we get W ST = Dẽ exp i e 0 e 1 e 2 e 3) 1 dẽ d 3 X ) dẽ d 3 X ). 48c 14

15 The canonical field ẽ is defined in ẽ K d 3 X = e Kµ eε µνρσ dx ν dx ρ dx σ /3!. Applying the exterior derivative, we get ) ) dˆẽ K d 3 X = ˆDµ1ê Kµ êε µνρσ dˆx µ 1 dˆx ν dˆx ρ dˆx σ /3!, where D is the co-variant derivative defined in DV = dv + ω J V J. Thus, the term e 0 e 1 e 2 e 3) 1 dẽ d 3 X ) dẽ d 3 X ) = dẽ d 3 X) dẽ d 3 X ) e 0 e 1 e 2 e 3, in the path integral, becomes ) ) ˆDµ1ê µ êε µνρσ dˆx µ 1 dˆx ν dˆx ρ dˆx σ ˆDµ2 ê µ êε µ ν ρ σ dˆxµ 2 dˆx ν dˆx ρ dˆx σ 3!3!ê 0 µ 3 ê 1 ν 3 ê 2 ρ 3 ê 3 σ 3 dˆx µ 3 dˆx ν 3 dˆx ρ 3 dˆx σ 3. Let us define the inverse: e 0 µ e 1 νe 2 ρe 3 σdx µ dx ν dx ρ dx σ) 1 = E µ 0 E ν 1 E ρ We can rewrite: 2 E3 σ e 0 µe 1 νe 2 ρe 3 σdx µ dx ν dx ρ dx σ = 1 4 ed3 x µ dx µ.. x σ x ρ x ν x µ Actually, we have to rewrite the tensors ε µνρσ and ε µνρσ as e 1 ε µνρσ and eε µνρσ, but here we neglect this because we get the same results). Also, we can rewrite: E µ 0 E ν 1 E ρ 2 E σ 3 σ ρ ν µ = E ν 3ν, with inner product like E ν 3ν) ) 1 4 ed3 x µ dx µ = 1 4 Ee ν 3ν d 3 x µ dx µ = 1 4 Ee δ ν µ) ν dx µ = Ee = 1. 15

16 n general, we can write it as E ν 3ν) ed 3 x µ dx µ) = Ee ν 3ν d 3 x µ dx µ = Eeδ ν µ νdx µ = δ µ µ. n the path integral term, let us make the replacements: D µ1 e µ ) eε µνρσdx µ 1 dx ν dx ρ dx σ /3! D µ1 e µ ) edxµ 1 d 3 x µ = D µ1 e µ ) ed3 x µ dx µ 1, and D µ2 e µ ) eε µ ν ρ σ dxµ 2 dx ν dx ρ dx σ /3! D µ2 e µ ) ed 3 x µ dx µ 2. Let us assume the following replacing: d 3 x µ dx µ = dx µ d 3 x µ d 3 x µ dx µ 1 = dx µ d 3 x µ 1. There is no problem with this trick because in any 4D spacetime we have the contraction d 3 x µ dx ν ) = δ ν µd 4 x. Therefore, we make the replacement: D µ1 e µ ) ed3 x µ dx µ 1 D µ1 e µ ) edx µ d 3 x µ 1. By that, the term ) ) ˆDµ1ê µ êε µνρσ dˆx µ 1 dˆx ν dˆx ρ dˆx σ ˆDµ2 ê µ êε µ ν ρ σ dˆxµ 2 dˆx ν dˆx ρ dˆx σ 3!3!ê 0 µ 3 ê 1 ν 3 ê 2 ρ 3 ê 3 σ 3 dˆx µ 3 dˆx ν 3 dˆx ρ 3 dˆx σ 3, in the path integral, becomes E ν 3ν) D µ1 e µ ) edx µ d 3 x ) µ 1 D µ2 e µ ) ) ed 3 x µ dx µ 2 = D µ 1 e µ ) D µ2 e µ ) e ν 3ν) d 3 x µ1 dx µ) d 3 x µ dx ) µ 2, where we used dx µ d 3 x µ 1 = d 3 x µ 1 dx µ then d 3 x µ1 dx µ. Thus we can write dẽ d 3 X) dẽ d 3 X ) e 0 e 1 e 2 e 3 D µ 1 e µ ) D µ2 e µ ) e ν 3ν) d 3 x µ1 dx µ) d 3 x µ dx µ 2 ). 16

17 Let us choose the contraction: ν 3ν) d 3 x µ1 dx µ) d 3 x µ dx µ 2 ) = ν 3ν d 3 x µ1 dx µ) d 3 x µ dx µ 2 ) = δµ ν 1 ν dx µ ) d 3 x µ dx ) µ 2 = δµ ν 1 dx µ ν ) dx µ 2 ) d 3 x µ = δ ν µ 1 dx µ ν dx µ 2 d 3 x µ = δ ν µ 1 δ µ 2 ν dx µ d 3 x µ. Thus we can write the term in the path integral as dẽ d 3 X) dẽ d 3 X ) D µ 1 e e 0 e 1 e 2 e 3 µ ) D µ2 e µ ) eδµ ν 1 δ µ 2 ν dx µ d 3 x µ = D ν e µ ) D ν e µ ) edx µ d 3 x µ = D ν e µ ) D ν e µ ) ed 3 x µ dx µ = D ν e µ ) D ν e µ ) eδ µ µ d 4 x = D ν e µ ) D ν e µ) ed 4 x. We can also choose another contraction: D µ 1 e µ ) D µ2 e µ ) e ν 3ν) d 3 x µ1 dx µ) d 3 x µ dx ) µ 2 Thus, we get D µ 1 e µ ) D µ2 e µ ) e ν 3ν d 3 x µ1 dx µ) d 3 x µ dx ) µ 2 = D µ 1 e µ ) D µ2 e µ ) e δ νµ1 ν dx µ) d 3 x µ dx ) µ 2 = δµ ν 1 δ ν µ D µ 1 e µ ) D µ2 e µ ) e d 3 x µ dx ) µ 2. dẽ d 3 X) dẽ d 3 X ) e 0 e 1 e 2 e 3 D µ e µ ) D µ e µ ) ed 4 x. By the two possible contractions, we can write the final result as e 0 e 1 e 2 e 3) 1 dẽ d 3 X ) dẽ d 3 X ) = 1 2 Dµ e ν D µ e ν D µ e ν D ν e µ) ed 4 x. This Lagrange is like the Lagrange of the electromagnetic field, but with opposite sign. t is also independent on the gauge we chose for the surface δm: t is invariant under local Lorentz transformation V L Jx)V J and 17

18 under any coordinate transformation V µ xµ x ν V ν. The path integral of the gravity field W ST = Dẽ exp i e 0 e 1 e 2 e 3) 1 dẽ d 3 X ) dẽ d 3 X ) 48c becomes W ST = De exp i 1 Dµ e ν D µ e ν + D µ e ν D ν e µ) ed 4 x, 48c 2 with the free gravity field Lagrange: Ld 4 x = 1 1 Dµ e ν D µ e ν + D µ e ν D ν e µ) ed 4 x. 1.9) 48c 2 We determine the constant c in the Newtonian gravitational potential c 0. n the background spacetime, weak gravity; D µ µ and e 1 + δe, we get or L 1 1 µ e ν µ e ν + µ e ν ν e µ), 48c 2 L 0 = c 2 η Je µ g µν 2 µ ν) e J ν. Without background spacetime approximation, in strong gravity field, we have a problem with the determinant e in the path integral W ST = De exp i 1 Dµ e ν D µ e ν + D µ e ν D ν e µ) e 0 µ 48c 2 1 e 1 ν 1 e 2 ρe 3 σε µ 1ν 1 ρσ d 4 x, with η 0123 = 1, we rewrite it as De exp i 1 Dµ e ν D µ e ν + D µ e ν D ν e µ) η 1 JKL) e 1 µ 48c 2 1 e J ν 1 e K ρ e L σε µ 1ν 1 ρσ d 4 x/4!. 18

19 The path integral is independent on arbitrary changing of the coordinates x µ : consider e K ρ e K ρ + δe K ρ, then we have δse) δe K ρ = 0, it yields Dirac delta: δ D µ e ν D µ e ν + D µ e ν D ν e µ), so D µ e ν D µ e ν + D µ e ν D ν e µ = 0. Thus, we get π 2 = 0, S π, e) = 12c π 2 ed 4 x = 0 then H π, e) = 0. This path integral is trivial; there is no propagation because there is no gravity energy: H π, e) = 0, as the Wheeler-DeWitt equation Ĥψ = 0. The reason is that the gravity field e µ has the entity of spacetime. t is impossible for spacetime to be a dynamical on itself, to propagate over itself. But if we write e µx) δ µ + h µx), the path integral exists. Then the propagation is possible. Thus, the gravity field propagates freely only on the background spacetime. This is the situation of the weak gravity low energy densities). The path integral of the weak gravity field, in the background spacetime, becomes W = De 1 1 expi 48c 2 e µ ηj g µν 2 η J µ ν) e J ν d 4 x. 1.10) Thus, the gravity field propagator, g = η and k µ e µ = 0, is d µν 4 J x k η J g µν e ikx2 x1) 2 x 1 ) = 48c, 2π) 4 k 2 iε or d µν 4 k g ρσ g µν e ikx2 x1) ρσx 2 x 1 ) = 48c. 1.11) 2π) 4 k 2 iε We will use this propagation in the gravity interaction with the scalar and spinor fields. 19

20 2 The Lagrange of the Plebanski two form field The Plebanski two form complex field Σ i, in selfdual representation Σ i, is defined in Σ i = P i J ΣJ, where Σ J = e e J is real anti-symmetric two form and P i is the selfdual projector given in[3, 4]: P i ) jk = 1 2 εi jk, P i) 0j = i 2 δi j : i = for = 1, 2, 3. We start with the Lagrange of the gravity field 1.6): Se, ω) = c ε JKL e e J dω + db + Ω Ω + Ω B + B Ω + B B) KL. As we did before, we try to find the Lagrange Le, Σ, B), which transforms covariantly under Local Lorentz transforms. For that, we separate the action Se, ω) to Se, Ω)+Se, Ω, B). Because the actions Se, ω) and Se, Ω) transform covariantly the action Se, Ω, B) also transforms covariantly. Therefore, we can choose Se, Ω, B) = 0. Let us write the action as Se, ω) = c ε JKL e e J dω + Ω Ω) KL + Se, Ω, B). We start with Se, Ω) = c ε JKL e e J dω + Ω Ω) KL. Using the assumption Ω J = π J M e M, the action becomes Se, π) = [εjkl c e e J d π KL M e M) + ε JKL e e J ) K π K1 M e K 1 π ) ML K2 e ] K 2. We assume that the integral of ε JKL d e e J π KL M e M)) = ε JKL d Σ J π KL M e M)) is zero at the infinities. Using dσ J π ) KL M e M +e e J d π KL M e M) = π ) KL M e M dσ J +e e J d π KL M e M), 20

21 the Action becomes [εjkl Se, π) = c ) KL πm e M dσ J + ε JKL Σ J ) K π ) ML K1 M πk2 e K 1 e ] K 2, or [εjkl Se, π) = c ) ) KL πm e M dσ J K + ε ) ML JKL πk1 M πk2 Σ J Σ K 1K 2 ]. nserting our assumption: π JK = π L ε LJK, we get ε ) KL JKL πm e M = ε JKL π MKL e M = ε JKL π N ε NMKL e M = 2 π e J π J e ). Making the replacement: Σ J Σ K 1K 2 ε JK 1K 2 Σ 01 Σ 23, we get ) K ε ) ML JKL πk1 M πk2 Σ J Σ K 1K 2 ) K = ε ) ML JKL πk1 M πk2 ε JK 1K 2 Σ 01 Σ 23 = 2 ) K π ) ML L M πk Σ 01 Σ 23 = 2 π LKM ) π KML) Σ 01 Σ 23 = 2 π KML ) π KML) Σ 01 Σ 23 = 2π ε KML π J ε JKML Σ 01 Σ 23 = 12π 2 Σ 01 Σ 23. Therefore, the Action becomes [ 2 Se, π, Σ) = c π e J π J e ) dσ J 12π π Σ 01 Σ 23]. Because the real Plebanski two form Σ J = e e J is anti-symmetric, we can rewrite: Se, π, Σ) = c [ 4π e J dσ J 12π π Σ 01 Σ 23], then using ε 0123 = 1, we can rewrite it as Se, π, Σ) = c [ 4π e J dσ J + 12π π ε JKL Σ J Σ KL /4! ], 21

22 or Se, π, Σ) = c [ 4π e J dσ J + 12 ] π2 ε JKL Σ J Σ KL. The integral over the momentum π vanishes unless the equation of motion) δse, π, Σ) = δ [ 4π e J dσ J + 12 ] δπ δπ π2 ε JKL Σ J Σ KL = 0. But, it is not easy to separate Σ from e. t is like the gravity field, it is separable only in weak gravitybackground spacetime). Therefore, we solve it in the background spacetime. On arbitrary spacetime, we get the integral: 4π e J dσ J + 12 π2 ε JKL Σ J Σ KL ) 4π e µj ν Σ J ρσε µνρσ π2 ε JKL Σ J µνσ KL ρσ ε µνρσ )d 4 x. The background spacetime approximation is e µx) δ µ + h µx), e 1 + δe, thus we get Σ J µν = 1 e 2 µ e J ν e νeµ) J 1 δ 2 µ δν J δνδµ) J 1 + h 2 µ δν J h νδµ) J 1 ) + δ 2 µ h J ν δνh J µ. nserting it in the action S e, Σ) = c 4π e µj ν Σ J ρσε µνρσ π2 ε JKL Σ J µνσ KL ρσ ε )d µνρσ 4 x, it becomes S e, Σ) S h, δσ) = c 4π ν Σ J ρσε νρσ J + 1 ) 2 π2 24) +... d 4 x. Therefore, the conditionequation of motion) δ [ 4π e J dσ J + 12 ] δπ π2 ε JKL Σ J Σ KL = 0 22

23 approximates to δ 4π ν Σ Jρσε Jνρσ + 1 ) δπ 2 π2 24) d 4 x = 0. ts solution is π = 1 6 νσ Jρσε Jνρσ = 1 6 ν Σ Jρσ ε Jνρσ. Thus, the action in the background spacetime is approximated to [ ] 2 SΣ) c 3 ν 1 Σ J 1ρ 1 σ 1 ε J1 ν 1 ρ 1 σ 1 ν Σ Jρσ ε Jνρσ +... d 4 x. Defining inner product via Σ J 1ρ 1 σ 1 Σ Jρσ = Σ 2 δ J 1 J δρ 1 ρ δ σ 1 σ, we get 4 µ SΣ) c Σ νρ J µ Σ J νρ +... ) d 4 x with µ Σ µρ J = 0. This is the action of the real Plebanski two form in weak gravity field background spacetime). t is like the electromagnetic field. The corresponding Lagrange is L 0 Σ) 4c µ Σ νρ J ) ) µ Σ J νρ with µ Σ µρ J = 0. n curved spacetime, we rewrite it as L 0 Σ)d 4 x 4c µ Σ νρ J ) µ Σ J νρ) ed 4 x. 2.1) t does not transform covariantly, because of the partial derivative. But the total Lagrange Le, ω) transforms covariantly, thus Le, ω) = Le, Σ) + Le, Σ, B) transforms covariantly. Let us rewrite: Le, Σ) + Le, Σ, B) = Le, Σ) + L L + Le, Σ, B), with Le, Σ) + L transforms covariantly, and L + Le, Σ, B) = 0, which determines B. We get Le, Σ) + L LΣ)d 4 x = 4c D µ Σ νρ J ) D µ Σ J νρ) ed 4 x, which transforms covariantly. 23

24 We get the Lagrange of the complex Plebanski two form field Σ i = P i J ΣJ using the selfdual projector P i, which projects the real Plebanski two form Σ J to two states: selfdual Σ i and anti-selfdual Σi. Thus, the term D µ Σ νρ J Dµ Σ J νρ, in the Lagrange, becomes D µ Σ νρ i D µ Σ i νρ + D Σνρ µ i D µ Σi νρ, where the hermitian conjugate Σ i = P J i ΣJ. νρ Σ i Σ i νρ is represented in the anti-selfdual We search for conditions allow us to rewrite the complex Plebanski two form field Σ i as a real field. As done for the left and right spinor fields: n the left spinor field representation, the acting of the right spinor field is zero. While, in the right spinor field representation, the acting of left spinor field is zero[5]. Therefore, in the selfdual representation Σ i, we assume that the acting of the complex Plebanski field, represented in the anti-selfdual, is zero, thus Σ j Σ i = 0. Like that is in the anti-selfdual representation Σi, thus Σ j Σi = 0. Therefore, in the selfdual representation Σ i, we get Σ i = 1 2 εijk Σ jk iσ 0i = εijk Σ jk = iσ 0i. Thus, the Plebanski two form field, in the selfdual representation, becomes Σ i = 1 2 εijk Σ jk + iσ 0i = ε ijk Σ jk, which is real as required for satisfying the reality condition. t is equivalent to the replacement x 0 ix 0. Same result we get in the anti-selfdual representation Σi : Σ i = 0 Σ i = ε ijk Σ jk, it is equivalent to the replacement x 0 ix 0. That allows the splitting: SO3, 1) SU2) SU2). Σ i νρ) be- n the two representations, the term 1 2 comes 1 2 ε ij k k Σj νρ ε ijk Σ νρ jk = Σνρ jk Σjk νρ. Σ νρ J ΣJ νρ ) = 1 2 Σ νρ i Σ i νρ + Σ νρ i 24

25 Therefore, the Lagrange of the Plebanski two form field, in selfdual representation, becomes: L 0 Σ)d 4 x = 4c D µ Σ νρ i ) D µ Σ i νρ) ed 4 x. 2.2) Let us combine the gravity and the real Plebanski two form in one field Kµ i as Kµ i = 1 e i µ + i ) 2 4 εijk ε 0µρσ Σ ρσ jk, and its hermitian conjugate is K µ i = 1 e i µ i ) 2 4 εijk ε 0µρσ Σ ρσ jk, the local-lorentz frame indices are for = i = 1, 2, 3. Thus we get K i µ + K i µ = e i µ and K i µ K i µ = i 4 εijk ε 0µρσ Σ ρσ jk = i 2 ε 0µρσΣ iρσ, then we get ) ) ν Kµ i + ν Ki µ ν K µ i + ν Kµ i = ν e i µ ν e µ i, and ) ) ν Kµ i ν Ki µ ν K µ 1 i ν Kµ i = 4 ε µρ σ 0µρσε 0 ν Σ iρσ ν Σ iρ σ = 1 4 ε 0µρσε 0µρ σ ν Σ iρσ ν Σ iρ σ = ν Σ iρσ ν Σ iρσ. Therefore, we have ν e µ i ν e i µ + ν Σ iρσ ν Σ iρσ 4 µ Ki µ K i. Using them in the gravity Lagrange 1.9) and in the Plebanski Lagrange 2.2): L e) d 4 x = c 2 µe ν ) ) µ e ν ed 4 x, 25

26 and L Σ) d 4 x = 8c 1 2 ν Σ iρσ ν Σ iρσ ) ed 4 x, for the gauge e µ = 0 and 8c = 1/48c), we get L e) d 4 x + L Σ) d 4 x 1 1 ) ν Ki 12c 2 µ ν K µ i ed 4 x. 2.3) To make it invariance, we rewrite: L K) d 4 x = 1 1 ) D ν Ki 12c 2 µ D ν K µ i D ν Ki µ D µ K iν ed 4 x. t satisfies the reality condition as required, like the Lagrange of the electromagnetic field. 3 The static potential of the weak gravity We derive the static potential of the scalar and spinor fields interactions with the weak gravity field in the static limit; the Newtonian gravitational potential. We find that this potential has the same structure for the both fields, it depends on the fields energy. By that, we determine the constant c 0. The action of the scalar field in arbitrary curved spacetime is[1] Se, φ) = d 4 xe η J e µ eν JD µ φ + D ν φ V φ) ). n the weak gravity, the background spacetime approximation is given by e µ x) δµ + hµ x), e 1 + δe. Thus, the action is approximated to Se, φ) = d 4 x µ φ + µ φ + h µν x) µ φ + ν φ + h νµ x) µ φ + ν φ V φ) +... ). 26

27 The gravity field is symmetric, so we get Se, φ) = d 4 x µ φ + µ φ + 2h µν x) µ φ + ν φ V φ) +... ). The energy-momentum tensor of the scalar field is[5] hence T µν = µ φ + ν φ + g µν L, µ φ + ν φ = T µν g µν L. nserting it in the Lagrange, it becomes or L = µ φ + µ φ + 2h µν x) T µν g µν L) V φ) +... L = µ φ + µ φ + 2h µν T µν V φ) 2h µν g µν L +... Therefore, in the interaction term, we make the replacement: µ φ + ν φ T µν and V V + 2h µν g µν L. Because the gravity field is weak background spacetime), so 2h µν g µν L is neglected compared with L. We find the potential V r) of the exchanged virtual gravitons by two particles k 1 and k 2, using M k 1 + k 2 k 1 + k 2) matrix element like Born approximation to the scattering amplitude in non-relativistic quantum mechanics [6]). For one diagram of Feynman diagrams, we have with im k 1 + k 2 k 1 + k 2) = i ik 2) µ ik 2 ) ν q = k 1 k 1 = k 2 k 2. µνρσ q) i i ik 1) ρ ik 1 ) σ, 27

28 The propagator µνρσ x 2 x 1 ) is the gravitons propagator 1.11), we get it from the Lagrange of the free gravity field in background spacetime) we had before: L 0 = c 2 η Je µ g µν 2 µ ν) e J ν c 2 η Jh µ g µν 2 µ ν) h J ν. With the gauge µ e µ = 0, we get J µν y x) = d 4 q 2π) J 4 µν Therefore, the M matrix element becomes q 2 ) e iqy x) : J µν q 2 ) = 48c g µνη J q 2 iε. im k 1 + k 2 k 1 + k 2) = i48c ik 2) µ ik 2 ) ρ g µν g ρσ where g = η and q = k 1 k 1 = k 2 k 2. Comparing it with[6] we get im k 1 + k 2 k 1 + k 2) = i V q) δ 4 k out k in ), V q 2) = 48c ik 2) µ ik 2 ) ρ g µν g ρσ q 2 ik 1) σ ik 1 ) ν. Then, comparing this relation with the replacement: µ φ + ν φ T µν, and evaluating the inverse Fourier transform, thus we get q 2 ik 1) σ ik 1 ) ν, V y x) = 48cT µρ y) g µν g ρσ 1 T νσ x) 4π y x = 48cT µν y) T µν x), 4π y x where T µν is transferred energy-momentum tensor, it is anti-symmetric, so the summation over the indices µ and ν is repeated twice. Therefore, we divide the right side by 2: V y x) = 48c 2 T µν y) T µν x). 4π y x 28

29 n the static limit, for one particle, we approximate T 00 to m: m is the mass of interacted particles. Thus, we get the Newtonian gravitational potential: V y x) = 48c 2 m 2 m2 4π y x = G 48c = 8πG. y x Therefore, the weak gravity Lagrange becomes L 0 = 1 1 4πG 4 η Je µ g µν 2 µ ν) e J ν. We do the same thing for the spinor fields interaction with the gravity. The action is[1] Se, ψ) = d 4 xe ie µ ψγ D µ ψ m ψψ ), where the covariant derivative D µ is D µ = µ + ω µ ) J LJ +A a µt a. n the background spacetime, it becomes Se, ψ) = d 4 x i ψγ µ D µ ψ + ih µ ψγ D µ ψ m ψψ +... ). Let us consider only the terms: d 4 x i ψγ µ µ ψ + ih µ ν ψγ ν µ ψ m ψψ ) : g = η. The energy-momentum tensor of the spinor field is[5] T µν = i ψγ µ ν ψ + g µν L. Therefore, in the interaction term, we have the replacement i ψγ µ ν ψ T µν and L L + h µν g µν L. The term h µν g µν L is neglected compared with the Lagrange L. We find M matrix element of the exchanged virtual gravitons p 1 + p 2 p 1 + p 2, for one diagram of Feynman diagrams[6]: im p 1 + p 2 p 1 + p 2) = i48cū p 1) γ µ ip 1 ) ν u p 1 ) g µσg νρ 29 q 2 ū p 2) γ σ ip 2 ) ρ u p 2 ),

30 with we get q = p 1 p 1 = p 2 p 2 and g = η, V q 2) = 48cū p 1) γ µ ip 1 ) ν u p 1 ) g µσg νρ Comparing this relation with the replacement: i ψγ µ ν ψ T µν, and evaluating the inverse Fourier transform, we get q 2 ū p 2) γ σ ip 2 ) ρ u p 2 ). V y x) = 48c T µρ y)) g µν g ρσ 1 T νσ x)) 4π y x = 48cT µν y) T µν x), 4π y x where T µν is transferred energy-momentum tensor. Dividing the right side by 2: V y x) = 48c 2 T µν y) T µν x). 4π y x n the static limit, for one particle, we approximate T 00 to m: m is the mass of interacted particles. Thus, we get the Newtonian gravitational potential: V y x) = 48c 2 m 2 m2 4π y x = G 48c = 8πG. y x t is the same potential we found in the scalar field interaction with the gravity field. References [1] Carlo Rovelli, Quantum Gravity; Cambridge University Pres, Cambridge, United Kingdom, [2] Carlo Rovelli and Francesca Vidotto, Covariant Loop Quantum Gravity, An elementary introduction to Quantum Gravity and Spinfoam Theory; Cambridge University Press, Cambridge, United Kingdom,

31 [3] Steffen Gielen and Daniele Oriti, Classical general relativity as BF- Plebanski theory; Classical and Quantum Gravity, OP Publishing, United Kingdom, [4] J. E. Rosales-Quintero, Antiself-dual gravity and supergravity from a pure connection formulation; nternational Journal of Modern Physics A, World Scientific Publishing Co, [5] Mark Srednicki, Quantum Field Theory; Cambridge University Press, Cambridge, United Kingdom, [6] Michael E. Peskin and Daniel V. Schroeder, An ntroduction to Quantum Field Theory; Westview Press, Colorado, USA, [7] Carlo Rovelli, Ashtekar formulation of general relativity and loop space nonperturbative quantum gravity: a report; OP Publishing, United Kingdom, [8] Bryce S. DeWitt, Quantum Theory of Gravity. 1. The Canonical Theory; Phys. Rev, APS, USA, [9] Christopher Pope, Geometry and Group Theory; Texas A&M University Press, Texas, USA,