Quantum Entanglement and Thermodynamics of Boson-Antiboson Pair Creation Modes in the Noncommutative Bianchi I Universe
QQuantum Entanglement and Thermodynamics of Boson-Antiboson PairCreation Modes in the Noncommutative Bianchi I Universe
M. F. Ghiti , , N. Mebarki and H. Aissaoui
1. Ecole Normale Sup´erieure (ENSC), Assia Djebar, Constantine, Algeria2. Laboratoire de Physique Math´ematique et Subatomique (LPMS),Constantine 1 University, Constantine, [email protected]
Within the quantum field theory approach and using the technique of Bogoliubov trans-formations, the von Neumann boson-antiboson pair creation quantum entanglement en-tropy is studied in the context of the noncommutative Bianchi I universe. It is shown thatthe latter have a behavior depending strongly on the choice of the noncommutativity θ parameter, k ⊥ -frequency modes and the structure of the curved spacetime. Moreover,the relationship between the Bose-Einstein condensation and quantum entanglement isalso discussed. Keywords : Noncommutative Bianchi I universe, Pair creation, Boson-Antiboson quantumentanglement entropy, Quantum information, Bose-Einstein CondensationPACS numbers: 03.65.Ud, , 03.65.-w, 03.67.-a, 03.75.Nt, 04.62.+v
1. Introduction
During the last few years, The quantum entanglement (Q.E.) phenomenon has beenof great interest for many people in the field of quantum information in both flatand curved spacetime [1–14]. Moreover, it turns out that the pair creation processis intimately related to the quantum entanglement and strongly depends on thenature of the particles as well as on the structure of the spacetime [5, 9, 11, 15, 16].Furthermore, as a possible scenario to explain dark matter and dark energy aswell as the anisotropies found in the cosmic microwave background (C.M.B.), isthe noncommutative geometry (N.C.G) [17–22]. In fact, many models have beenproposed so far where it has been shown that the noncommutativity θ -parameterplays an important role [23–32]. The goal of this paper is to study the process ofcreating boson-antiboson pairs in the noncommutative (N.C.) Bianchi I universe andshow its relationship with quantum entanglement and Bose-Einstein condensation(B.E.C.) phenomena. The role played by the spacetime noncommutativity is alsoemphasized. In section 2, we present the necessary mathematical formalism. Section3 contains our numerical results and discussions. Finally, in section 4, we draw ourconclusions. a r X i v : . [ g r- q c ] D ec
2. Mathematical Formalism
The N.C. spacetime is characterized by the coordinates operators ˆ x µ ( µ = 0 , x µ , ˆ x ν ] = i θ µν (1)where θ µν are antisymmetric matrix elements which control the noncommutativityof the spacetime. The N.C. matter scalar density of the massless Seiberg-Witten(S.W.) bosonic field ˆ ϕ reads : L = ˆ e ∗ (cid:18) ˆ g µν ∗ (cid:16) ˆ D µ ˆ ϕ (cid:17) † ∗ ˆ D ν ˆ ϕ (cid:19) (2)here, the deformed tetrad is given by :ˆ e = det ∗ (cid:0) ˆ e aµ (cid:1) = 14! (cid:15) µνρσ (cid:15) a b c d ˆ e aµ ∗ ˆ e bν ∗ ˆ e cρ ∗ ˆ e dσ (3)where (cid:15) µνρσ and (cid:15) a b c d are the completely antisymmetric tensors in curved andflat spacetime respectively. The gauge covariant derivative is given by : ˆ D µ ˆ ϕ = (cid:16) ∂ µ − ie ˆ A µ (cid:17) ∗ ˆ ϕ . The corresponding Klein-Gordon equation is shown to take thefollowing form (See Appendix A) : (cid:18) g µν ∂ µ ∂ ν − √− g ∂ µ (cid:0) √− gg µν (cid:1) ∂ ν (cid:19) ˆ ϕ − i √− g θ αβ ∂ µ (cid:0) ∂ α + √− g∂ β g µν ∂ ν ˆ ϕ (cid:1) +18 √− g θ αβ ∂ µ (cid:0) ∂ α ∂ ρ (cid:0) √− gg µν (cid:1) ∂ β ∂ σ ∂ ν ˆ ϕ + ∂ ρ (cid:0) ∂ α √− g∂ β g µν ∂ σ ∂ ν ˆ ϕ (cid:1) + ∂ α ∂ ρ √− g∂ β ∂ σ g µν ∂ ν ˆ ϕ (cid:1) + i √− g θ αβ ∂ µ (cid:0) ∂ α (cid:0) √− gg µν (cid:1) ∂ β ∂ ν ˆ ϕ (cid:1) = 0(4)where g is the determinant of the metric. In what follows, using dimensionlessspacetime coordinates ( t, x, y, z ), we consider a Bianchi I universe where the metrichas the form: ds = − dt + t (cid:0) dx + dy (cid:1) + dz (5)(here the time t is related to the cosmological parameters of the model). By con-vention, we choose the spacetime signature as being ( − , + , + , +) and for simplicity,we take : θ µν = θ
00 0 0 0 − θ (6)(notice that the choice of the θ µν does not affect so much our conclusions). UsingMaple 16 tensor package, the non-vanishing components of the vierbeins up to O (cid:0) θ (cid:1) have the following expressions:ˆ e ˜00 = ˆ e ˜33 = 1ˆ e ˜11 = ˆ e ˜22 = 1 t (cid:16) − θ (cid:17) (7) and therefore, the corresponding deformed Bianchi I metric up to O (cid:0) θ (cid:1) is givenby: ds = − dt + t (cid:18) − θ (cid:19) (cid:0) dx + dy (cid:1) + dz (8)It is worth mentioning, as pointed out in references [5,29,33–35] that since the metricpresents a space-like singularity at t = 0, it is difficult to define the particle statewithin the adiabatic approach. To do so, we first follow a quasiclassical approachof Ref. 34 to identify the positive and negative frequency modes and look for theasymptotic behavior of the solutions at t → t → ∞ . Secondly, we solve theN.C. Klein Gordon equation and compare the solutions with the above quasiclassicallimit. After straightforward but lengthy calculations, Eq. (4) is simplified to (cid:18) iθt ˜ ∂ − θ t ˜ ∂ (cid:19) ˜ ∂ ˆ ϕ + 2 t (cid:18) iθ t ˜ ∂ (cid:19) ˜ ∂ ˆ ϕ + (cid:18) t (cid:18) θ (cid:19) (cid:16) ˜ ∂ + ˜ ∂ (cid:17) + (cid:18) iθt ˜ ∂ − θ t ˜ ∂ (cid:19) ˜ ∂ (cid:19) ˆ ϕ = 0 (9)here, the tilde stands for a curved spacetime index. To simplify this equation, weset: ˆ ϕ = ˆ t − f ( t ) h ( t ) exp[ i ( k x x + k y y + k z z )] (10)where ˆ t = t − θ k y , f ( t ) = exp (cid:18) − θ t k y (cid:19) (11)where h ( t ) is some regular function. After straightforward but tedious calculationsEq. (9) becomes: (cid:18) d dt + k ⊥ + θ k ⊥ t + k z (cid:19) h ( t ) = 0 (12)where k ⊥ is given by: k ⊥ = k x + k y (13)Now, we adopt the following change of variables : h ( t ) = ρ µ θ + exp (cid:16) − ρ (cid:17) y ( ρ ) (14a) ρ = − ik z t (14b) µ θ = i (cid:114) −
14 + k ⊥ + 2164 θ k ⊥ (14c)Notice that after direct calculations, Eq.(12) takes the following form: (cid:18) ρ d dρ + (2 µ θ + 1 − ρ ) ddρ − (cid:18) µ θ + 12 (cid:19)(cid:19) y ( ρ ) = 0 (15) It is important to mention that Eq. (15) has the form of the Kummer’s differentialequation [5, 36] : (cid:18) ρ d dρ + ( b − ρ ) ddρ − a (cid:19) y ( ρ ) = 0 (16)with : a = µ θ + 12 (17a) b = 2 µ θ + 1 (17b)The solution of Eq.(15) can be expressed as a linear combination of the two indepen-dent Kummer functions M (cid:0) µ θ + , µ θ + 1 , ρ (cid:1) and U (cid:0) µ θ + , µ θ + 1 , ρ (cid:1) . Thus,the general solution of Eq. (12) is given by : h ( ρ ) = ρ µ θ + exp (cid:16) − ρ (cid:17) (cid:18) C M (cid:18) µ θ + 12 , µ θ + 1 , ρ (cid:19) + C U (cid:18) µ θ + 12 , µ θ + 1 , ρ (cid:19)(cid:19) (18)( C and C are normalization constants). Now, for a better understanding ofthe asymptotic behavior at ρ → in ” fields) and ρ → ∞ (” out ” fields) ofthe solutions, Eq. (18), it is preferable to express the M (cid:0) µ θ + , µ θ + 1 , ρ (cid:1) and U (cid:0) µ θ + , µ θ + 1 , ρ (cid:1) Kummer functions in terms of Whittaker ones, i.e. [36] : M (cid:18)
12 + µ − λ, µ, z (cid:19) = e z z − ( + µ ) M λ,µ ( z ) (19a) U (cid:18)
12 + µ − λ, µ, z (cid:19) = e z z − ( + µ ) W λ,µ ( z ) (19b)Note that the Whittaker function W λ,µ ( z ) can be expressed in terms of M λ,µ ( z )as follows : W λ,µ ( z ) = Γ ( − µ )Γ (cid:0) − µ − λ (cid:1) M λ,µ ( z ) + Γ (2 µ )Γ (cid:0) + µ − λ (cid:1) M λ, − µ ( z ) (20)where : ( W λ,µ ( z )) ∗ = W − λ,µ ( − z ) (21)and : ( M λ,µ ( z )) ∗ = ( − µ + M λ, − µ ( z ) (22)To construct the positive and negative frequency modes in the ” in ” and ” out ” fields,we have to use the asymptotic behavior of the Whittaker functions. It is easy toshow that at t → ρ →
0) one has : h + in ∼ M λ,µ ( ρ ) ∼ e − ρ ρ µ + (23a) h − in ∼ ( M λ,µ ( ρ )) (cid:63) ∼ ( − µ + M λ, − µ ( ρ ) (23b) Similarly, for t → ∞ ( ρ → ∞ ), the corresponding positive and negative frequencymodes are : h + out ∼ W λ,µ ( ρ ) ∼ e − t ρ λ (24a) h − out ∼ ( W λ,µ ( ρ )) (cid:63) ∼ W − λ,µ ( − ρ ) (24b)Furthermore, the Bogoliubov coefficients associated with the asymptotic solutions” in ” and ” out ” (involving in the past and the future respectively) read : h ± out ( t ) = α h ± in ( t ) + β (cid:0) h ± in ( t ) (cid:1) ∗ (25)where the signs ” + ” and ” − ” stand for positive and negative frequency modesrespectively. We note that (cid:0) h ∓ in ( t ) (cid:1) ∗ = h ± in ( t ). Comparing Eq. (20) with the Eq.(25), we deduce that the Bogoliubov coefficients read : γ = βα = − i Γ(2 µ θ )Γ ( + µ θ − λ ) Γ( − µ θ )Γ ( − µ θ − λ ) exp ( − πµ θ ) (26)These coefficients for scalar particles satisfy the following normalization relation : | α | − | β | = 1 (27)it is very important to rewrite the Eq. (25) as :ˆ ϕ ± out ( t, x ) = α ˆ ϕ ± in ( t, x ) + β (cid:0) ˆ ϕ ± in ( t, x ) (cid:1) ∗ (28)Using the fact that :ˆ ϕ in,out ( t, (cid:126)r ) = (cid:90) d k (2 π ) / (cid:2) a in,out ( k ) exp ( − i k X ) + b + in,out ( − k ) exp ( i k X ) (cid:3) (29)where X = ( (cid:126)r, t ) and k = (cid:16) (cid:126)k, k (cid:17) are respectively the four Minkowski positionand momentum vectors. Here a in,out and b + in,out are the annihilation and creationoperators for boson and antiboson respectively. By comparing Eq. (29) with Eq.(28), we can deduce that : b out ( − k ) = α a in ( k ) + β b + in ( − k ) (30)Due to the form of the Bogoliubov transformations, we can show easily that thetensor product of the boson and antiboson vacuum state | (cid:105) out ⊗ | (cid:105) out = | , (cid:105) out can be written in terms of the ” in ” states as (Schmidt decomposition) [16] : | (cid:105) out ⊗ | (cid:105) out = (cid:88) n =0 C n | n k (cid:105) in ⊗ | n − k (cid:105) in (31)where | n k (cid:105) in and | n − k (cid:105) in represent respectively the particle and the antiparticlemode states with momentum k and − k . To find the relationship between the C n coefficients, we impose first the relation b out ( − k ) | , (cid:105) out = 0 leading to : C n +1 = − γ C n − (32) or : | C n | = | γ | n | C | (33)Then, using the normalization condition for the vacuum state (cid:104) , | , (cid:105) out = 1 orequivalently (cid:88) n =0 | C n | = 1 (34)and the fact that : 1 + | γ | + | γ | + ... = 11 − | γ | (35)we end up with : | C | = 1 − | γ | (36)Regarding the von Neumann quantum entanglement entropy S Q.E and since we aredealing with the bipartite pure state, then S Q.E can be written as : S Q.E = − T r ( ρ log ρ ) = − (cid:88) n =0 λ n log λ n (37)where ρ is the density matrix, ” T r ” stands for trace and λ n are the eigenvalues of ρ which are shown to be equal to | C n | . Using Eq. (33) and Eq. (36) together with : (cid:88) n x n = x (1 − x ) (38)and (cid:80) n x n = − x , we obtain : S Q.E = log (cid:32) | γ | ( | γ | ) / ( | γ | − )1 − | γ | (cid:33) (39)such that : | γ | = (cid:12)(cid:12)(cid:12) βα (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ (cid:0) − µ θ − λ (cid:1) Γ (cid:0) + µ θ − λ (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) exp ( − π | µ θ | ) (40)By using the relation : (cid:12)(cid:12)(cid:12) Γ (cid:18)
12 + i y (cid:19) (cid:12)(cid:12)(cid:12) = π cosh ( π y ) (41)and after straightforward and tedious calculations, we get : (cid:12)(cid:12)(cid:12) βα (cid:12)(cid:12)(cid:12) = exp ( − π | µ θ | ) (42)
3. Numerical Results and Discussions
The numerical results of the Q.E. entropy ” S Q.E. ” in the N.C. Bianchi I universeshow that it is a decreasing function of the k ⊥ − frequency modes for a fixed value ofthe N.C. θ parameter. For example for θ = 0 .
4, when k ⊥ = 1 .
0, the S Q.E. = 0 . k ⊥ = 1 . θ , the S Q.E. = 0 . k ⊥ − frequency modes does not start from thevalue k ⊥ = 0, for example, for θ = 0, k min ⊥ = 0 .
5. As a function of θ , one canshow easily that k min ⊥ = √ θ +64 . The reason lies to the existence of the factor µ θ = (cid:112) (25 θ + 64) k ⊥ −
16 in the expression of the S Q.E. such that : S Q.E = log (cid:32) exp ( − π | µ θ | ) exp( − π | µ θ | ) / (exp( − π | µ θ | ) − − exp ( − π | µ θ | ) (cid:33) (43)which has to be real. Fig. 1. The von Neumann Quantum Entanglement entropy S Q.E as a function of the k ⊥ -frequency modes and the noncommutativity θ parameter Notice that in the fermionic case, as it was pointed out in Ref. 5, we have tworegions, namely the first is the one corresponding to the out-of-equilibrium state inwhich the pair creation density has a non thermal behavior.
Furthermore, if we have an anisotropic spacetime, the created particle-antiparticle pair (with the same energy) cannot reach an equilibrium state in all di-rections, unless their energies exceed a certain critical value (cid:0) k ⊥ ≈ (cid:1) beyond whichthe anisotropic effects become negligible. For k ⊥ < , the particle-antiparticle paircreation velocity (energy) is slower than the expansion velocity in the x and y di-rections and the density of the pair creation is in a non thermal out-of-equilibriumstate. However, in the bosonic case, we have just one region corresponding to anequilibrium state, starting from k min ⊥ which is similar to the second region in thefermionic case [5]. Moreover, the behavior of S Q.E as a function of k ⊥ is expected,because if k ⊥ increases, the velocity of the particle’s creation increases too, andtherefore the information will be spread out (decrease of S Q.E ). Figure 1 illustratesthe behavior of S Q.E as a function of k ⊥ − frequency modes for various values ofthe N.C. θ parameter. Notice that and contrary to the fermionic case, S Q.E is adecreasing function of θ for a fixed value of k ⊥ . As an example : for k ⊥ = 0 . θ = 0 .
1, we see that S Q.E = 0 . θ = 0 . k ⊥ , S Q.E = 0 .
051 (see FIG. 1).
Fig. 2. The number density ˆ n as a function of the k ⊥ -frequency modes and the noncommutativity θ parameter In Ref. 5 in region II, the N.C. θ parameter was interpreted as playing the role ofantigravity (e.g. quintessence, dark energy, etc.). However, in this paper (bosoniccase) as θ increases, S Q.E decreases. This does not mean that θ plays the role ofgravity. In fact, in the bosonic case, the B.E.C. phenomenon which of course de- pends on θ increases more than the gravity generated by θ , so that overall, thebehavior of S Q.E looks like we have gravity.Thus, at the end with B.E.C together with antigravitational effects of N.C parame-ter, S Q.E becomes a decreasing function of θ . Therefore, we can say that θ generatesnot only antigravity but the B.E.C phenomenon as well. To have a better under-standing, let us consider the number density ˆ n (see FIG. 2). Notice that ˆ n and S Q.E have the same behavior with respect to k ⊥ or θ . Fig. 3. The chemical potential µ as a function of the k ⊥ -frequency modes and the noncommuta-tivity θ parameter Straightforward calculations show that the chemical potential µ is proportional tolog (cid:16) ˆ ne πα n (cid:17) , where α = (1 / (cid:112) k ⊥ θ + 64 k ⊥ −
16 and it can be shown to be neg-ative (see FIG. 3). The reason for µ being negative is that in the thermodynamicalequilibrium where µ is proportional to − ∆ S Q.E ∆ˆ n and ∆ S Q.E is an increasing functionof ∆ˆ n , the ratio ∆ S Q.E ∆ˆ n is positive and therefore µ is negative. Now, if µ goes to 0implies that α goes to 0, leading to k ⊥ = √ θ +64 , which means that S Q.E goesto S maxQ.E (maximally bipartite boson-antiboson entangled state), and the fugacity z goes to 1. In other words, we have a critical point and one has a sort of B.E.Cphenomenon.Figure 4 displays S Q.E as a function of θ and k ⊥ . Figure 5 shows the density plot of S Q.E as a function of k ⊥ for various values of θ . Figures 6 and 7 illustrate contoursplot of the number density ˆ n and S Q.E as a function of k ⊥ and θ respectively. Figure8 represents the chemical potential µ as a function of S maxQ.E . S Q.E as a function of the k ⊥ -frequencymodes for various values of the N.C. θ parameter Fig. 5. Density plot of S Q.E as a func-tion of the k ⊥ -frequency modes for var-ious values of the N.C. θ parameterFig. 6. Contours plot of ˆ n as a function of the k ⊥ -frequency modes for various values of theN.C. θ parameter Fig. 7. Contours plot of S Q.E as afunction of the k ⊥ -frequency modes forvarious values of the N.C. θ parameter Notice that as S maxQ.E increases, µ increases too. The reason is that, if µ in-creases the boson-antiboson pair creation rate increases and therefore the B.E.C.phenomenon becomes more important leading to an increase in S Q.E (or S maxQ.E ), ora decrease of θ ( S maxQ.E is a decreasing function of θ ). µ as a function of S maxQ.E
4. Conclusion
Throughout this paper, we have studied the creation of quantum entanglementbetween pairs of massless boson-antiboson particles within the framework of theN.C. Bianchi I universe as well as its relationship to thermodynamics. We havefirst derived the modified N.C. Klein Gordon equation for massless bosons and itssolutions. Also, due to the complexity of the N.C. anisotropic Bianchi I spacetimestructure, the behaviors of S Q.E as a function of k ⊥ − frequency modes are not trivialand different from those obtained in Ref. 16 for the case of isotropic F.R.W. universe(taking a particular solvable case). According to our obtained results, the structureand the deformation of the spacetime, as well as the involved particles (fermions orbosons), affect not only the behavior of S Q.E as a function of k ⊥ but also the positionof S maxQ.E as well. It should be noted that, in Ref. 16, due to the spacetime isotropy,the authors have noticed that for massless bosons one has a maximum value of S Q.E independently of the value of k z (not k ⊥ ). Our results show that even with masslessparticles, one can have a non vanishing quantum entanglement which depends onlyon k ⊥ = (cid:113) k x + k y (because of the Bianchi I spacetime anisotropy and the choice ofthe N.C. θ parameter). Similar conclusions were obtained in our previous work (Ref.5) concerning the fermionic case and compared to Ref. 12. Thus, the arguments inRefs.12, 16 do not hold in general. We have also discussed the behavior of somethermodynamic quantities (like the chemical potential µ ) as a function of k ⊥ − frequency modes and the N.C. θ parameter. We have also shown that the behaviorof S Q.E depends on the kind of the involved particles (boson or fermions) duringthe pair creation process as well as on the structure and deformation of spacetime. Contrary to Ref. 16 and because of the spacetime deformation, the S Q.E for masslessbosons does not always have a maximum but do depend on the k ⊥ . Notice also thatour results depend only on the transverse component k ⊥ for some allowed valuesstarting from k min ⊥ = √ θ +64 (not the whole components of k ). The maximallyentangled state where S Q.E → S maxQ.E depends strongly on the N.C. θ parameter andknowing the position of S maxQ.E one can get information about certain thermodynamicquantities (like the chemical potential µ ) of the N.C. Bianchi I universe and viceversa . More interesting, we have shown, our numerical results confirm it, that forthe minimal value of k ⊥ we have a sort of phase transition (critical point) betweenthe out of equilibrium and equilibrium regions (regions I and II). Furthermore, theN.C. θ parameter plays the role of antigravity and contributes to increase the B.E.C.phenomenon. Finally and unlike our previous result in Ref. 5 (where | θ | ≤ ), herewe cannot get an upper bound of the N.C. θ parameter. More studies are underinvestigation. Acknowledgments
We are very grateful to the Algerian Ministry of Higher Education and ScientificResearch and D.G.R.S.D.T. for financial support. This work is also supported bythe P.R.F.U. project.
Appendix A. N.C. Mathematical Formalism
The N.C. metric is given by (see Refs. 29, 37) :ˆ g µν = 12 (cid:0) ˆ e bµ ∗ ˆ e νb + ˆ e bν ∗ ˆ e µb (cid:1) (A.1)where, the N.C. Vierbeins ˆ e aµ up to O ( θ ) are (see Ref. 5):ˆ e aµ = e aµ − i θ νρ e aµνρ + θ νρ θ λτ e aµνρλτ + O (cid:0) θ (cid:1) (A.2)and e aµνρ = 14 (cid:2) ω acν ∂ ρ e dµ + (cid:0) ∂ ρ ω acµ + R acρµ (cid:1) e dν (cid:3) η cd (A.3) e aµνρλτ = 132 [2 { R τν , R µρ } ab e cλ − ω abλ ( D ρ R cdτµ + ∂ ρ R cdτµ ) e mν η dm −{ ω ν , ( D ρ R τµ + ∂ ρ R τµ ) } ab e cλ − ∂ τ { ω ν , ( ∂ ρ ω µ + R ρµ ) } ab e cλ − ω abλ ∂ τ ( ω cdν ∂ ρ e mµ + ( ∂ ρ ω cdµ + R cdρµ ) e mν ) η dm + 2 ∂ ν ω abλ ∂ ρ ∂ τ e cµ − ∂ ρ ( ∂ τ ω abµ + R abτµ ) ∂ ν e cλ − { ω ν , ( ∂ ρ ω λ + R ρλ ) } ab ∂ τ e cµ − ( ∂ τ ω abµ + R abτµ )( ω cdν ∂ ρ e mλ + ( ∂ ρ ω cdλ + R cdρλ ) e mν η dm )] η bc (A.4)here R abµν is the strength field associated with the commutative spin connections ω abµ and is defined as: R abµν = ∂ µ ω abν − ∂ ν ω abµ + (cid:0) ω acµ ω dbν − ω acν ω dbµ (cid:1) η cd (A.5) ( η ab is the Minkowski metric).The N.C. spin connections ˆ ω ABµ up to O ( θ ) are:ˆ ω ABµ = ω ABµ − i θ νρ ω ABµνρ + θ νρ θ λτ ω ABµνρλτ + .... (A.6)where ω ABµνρ = 14 { ω ν , ∂ ρ ω ν + R ρµ } AB (A.7) ω ABµνρλτ = 132 ( −{ ω λ , ∂ τ { ω ν , ∂ ρ ω µ + R ρµ }} + 2 { ω λ , { R τν , R µρ }}−{ ω λ , { ω ν , D ρ R τµ + ∂ ρ R τµ }} − {{ ω ν , ∂ ρ ω λ + R ρλ } , ( ∂ τ ω µ + R τµ ) } +2[ ∂ ν ω λ , ∂ ρ ( ∂ τ ω µ + R τµ )]) AB (A.8)here { α, β } AB = α AC β BC + β AC α BC (A.9)[ α, β ] AB = α AC β BC − β AC α BC (A.10)and D µ R ABρσ = ∂ µ R ABρσ + ( ω ACµ + R DBρσ + ω BCµ R DAρσ ) η CD (A.11)We propose the following action: S = 12 k (cid:90) d x ( L G + L SC ) (A.12)where L G and L SC stand for the pure gravity and matter scalar densities corre-sponding to the charged scalar particle.We define L G = ˆ e ∗ ˆ R (A.13)and L SC = ˆ e ∗ (cid:18) ˆ g µν ∗ (cid:16) ˆ D µ ˆ ϕ (cid:17) † ∗ ˆ D ν ˆ ϕ (cid:19) (A.14)with : ˆ R = ˆ e µ ∗ a ∗ ˆ e ν ∗ b ∗ ˆ R abµν (A.15)Applying the principle of least action, it is easy to show that the modified fieldequations are given by: ∂ L ∂ ˆ ϕ − ∂ µ ∂ L ∂ ( ∂ µ ˆ ϕ ) + ∂ µ ∂ ν ∂ L ∂ ( ∂ µ ∂ ν ˆ ϕ ) − ∂ µ ∂ ν ∂ σ ∂ L ∂ ( ∂ µ ∂ ν ∂ σ ˆ ϕ ) + O (cid:0) θ (cid:1) = 0 (A.16) References
1. S. Hill and W. K. Wootters, Phys. Rev. Lett , 5022 (1997).2. V. Vedral, M. B. Plenio, M. A. Rippin and P. L. Knight, Phys. Rev. Lett , 2275(1997).3. D. C. Santos, Entanglement: from its mathematical description to its experimentalobservation, PhD Thesis, University of Barcelona (2008).4. M. F. Ghiti, N. Mebarki and M. T. Rouabah, Paraquantum Entangled Coherent andSqueezed States: New Type of Entanglement, in Proc. 8th Int. Conf. On Progress inTheoretical Physics, eds. N. Mebarki, J. Mimouni, N. Belaloui and K. Ait Moussa,(Meleville, New York, 2012), p. 238.5. M. F. Ghiti, N. Mebarki and H. Aissaoui, Int. J. Modern. Phys. A , 1550141 (2015).6. M. F. Ghiti and H. Aissaoui, Sciences and Technology A , 65-69 (2014).7. N. Mebarki, A. Morchedi and H. Aissaoui, Int. Jour. Theor. Phys. , 4124 (2015).8. H. Garcia-Comp´ean and F. Robledo-Padilla, Class. Quantum Grav , 235012 (2013).9. I. Fuentes, R. B. Mann, E. M. Martinez and S. Moradi, Phys. Rev. D , 045030(2010).10. J. Hu and H. Yu, Phys. Rev. D , 104003 (2013).11. L. N. Machado, H. A. S. Costa, I. G. da Paz, M. Sampaio and J. B. Araujo, Phys.Rev. D , 125009 (2018).12. I. Fuentes, R. B. Mann, E. M. Martinez and S. Moradi, Phys. Rev. D , 045030(2010).13. J. L. Ball, I. Fuentes-Schuller and F. P. Schuller, Phys. Lett. A , 550 (2006).14. D. E. Bruschi, A. Dragan, I. Fuentes and J. Louko, Phys. Rev. D , 025026 (2012).15. Y. Li, Q. Mao and Y. Shi, Phys. Rev. A , 032340 (2019)16. Z. Ebadi and B. Mirza, Annals of Physics , 363 (2014).17. Peter. K.F. Kuhfitting, J. Mod. Phys , 323 (2017).18. S. Kawamoto et al , Class. Quant. Grav , 177001 (2017).19. M. M. Ettefaghi. Phys. Rev. D , 065022 (2009).20. N. Mebarki, Dark Energy, Induced Cosmological Constant and Matter - AntimatterAsymmetry From Non Commutative Geometry, in Proc. 3rd Int Meeting. On theFrontiers of Physics , eds. S. P. Chia, K. Ratnavelu and M. R. Muhamad, (Melville,New York, 2009), p 38.21. Z. Rezaei and S. Peyman Zakeri, e-Print 2007.01501[hep-ph] (2020).22. A. Espinoza-Garcia and J. Socorro, 10th Workshop of the gravitation and Mathemat-ical physics Division of the Mexican Physical Society, eds. R. B. B´arcenas et al , , 441 (2003).24. M. Buric, D. Latas, V. Radovanovic and J. Trampetic, Phys. Rev. D , 097701(2007).25. M. Haghighat, M. Ettefaghi and M. Zeinali, Phys. Rev. D , 013007 (2006).26. P. K. Das, N. G. Deshpande and G. Rajasekaran, Phys. Rev. D , 035010 (2008).27. A. Prakash, A. Mitra and P. K. Das, Phys. Rev. D , 055020 (2010).28. R. Horvat and J. Trampetic, Phys. Lett. B , 219 (2012).29. N. Mebarki, L. Khodja and S. Zaim, Electron. J. Theor. Phys. , 181 (2010).30. N. Mebarki, S. Zaim, L. Khodja and H. Aissaoui, Phys. Scri. , 045101 (2008).31. S. Zaim and L. Khodja, Phys. Scr. , 055103 (2010).32. N. Seiberg and E. Witten, J. High Energy Phys. , 032 (1999).33. L. Parker, Phys. Rev. Lett. , 562 (1968).34. V. M. Villalba and W. Greiner, Phys. Rev. D. , 025007 (2001).35. V. M. Villalba and W. Greiner, Mod. Phys. Lett. A , 1883 (2002).
36. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas,Graphs, and Mathematical Tables (Dover Publications, New York, 1964).37. M. Chaichian, A. Tureanu and G. Zet, Phys. Lett. B660