Bose-Einstein condensation in the Rindler space Shingo Takeuchi The Institute for Fundamental Study “The Tah Poe Academia Institute” Naresuan University Phitsanulok 65000, Thailand 5 shingo(at)nu.ac.th 1 0 2 p Abstract e S Based on the Unruh effect, we calculate the critical acceleration of the Bose-Einstein 1 condensationinafreecomplexscalarfieldatfinitedensityintheRindlerspace. Ourmodel 2 corresponds to an ideal gas performing constantly accelerating motion in a Minkowski ] space-time at zero-temperature, where the gas is composed of the complex scalar particles h t and it can be thought to be in a thermal-bath with the Unruh temperature. In the - p accelerating frame, the model will be in the Bose-Einstein condensation state at low e h acceleration, on the other hand there will be no condensation at high acceleration by the [ thermal excitation brought into by the Unruh effect. Our critical acceleration is the one 5 at which the Bose-Einstein condensation begins to appear in the accelerating frame when v we decrease the acceleration gradually. To carry out the calculation, we assume that the 1 7 critical acceleration is much larger than the mass of the particle. 4 7 0 . 1 0 5 1 : v i X r a 1 Introduction TheBose-Einstein condensationisgettingalotofattentionrecentlyasaquantumfluidfor the test of the analogy between sound waves in quantum fluids and scalar field fluctuation in curved space-times [1]. Thanks to this analogy, we can expect new progress and insight whicharedifficultonlyinthegravitationalanalysisinthequantumgravityandcosmology. Further, the experiments of the quantum gravity and cosmology are difficult due to the required energy level and the scale of phenomena. However, the experiments of the condensed matters will be free from such problems to some extent. It is thought for these reasons that pseudo experiments of the gravity are possible in the quantum fluids through the analogy. In order to understand the quantum phenomena in the gravity and cosmology such as the Hawking radiation [2] and particle creation [3, 4], the Unruh effect [5, 6, 7][8] is important in terms of the role that the event-horizons play. The Unruh effect is a prediction that one moving in the Minkowski space-time with a linear constant acceleration experiences the space-time as a thermal-bath with the Unruh temperature, T = ~a/(2πck ) 4 10 23a/(cm/s2)[K], where a is the acceleration. U B − ≈ × Now, various experimental attempts in the condensed matters to observe the grav- itational phenomena are being invented (see Ref.[9] for example). Particularly as for the experiments to detect the Unruh effect, there are attempts in Bose-Einstein conden- sates [11], graphenes [12] and Berry phases [13]. For other attempts see Ref.[14], for example, and related references. We also address the issue of the Unruh effect in the Bose-Einstein condensation. Whether the Bose-Einstein condensation occurs or not is determined by temperature. We assume in this paper that the Unruh temperature exists in the constantly accelerat- ing system according to the Unruh effect mentioned above. At this time we can think that the Unruh effect affects the Bose-Einstein condensation. Although an enormous number of studies have been done on the Unruh effect and the Bose-Einstein condensation so far, these are performed separately and little is known about the Unruh effect in the Bose- Einstein condensation at this moment. Since both the Unruh effect and the Bose-Einstein condensation are very important in the fundamental physics, new understandings could be expected by combining the Unruh effect and the Bose-Einstein condensation. In this paper, fromsuch a background, we calculate the critical acceleration for the Bose-Einstein condensation based on the thermal excitation brought into by the Unruh effect. Let us here explain the Bose-Einstein condensation briefly (for more details, see Ref.[10] for example). The Bose-Einstein condensation state is the situation that all the particle stay in the least energy state uniformly owing to the Bose-Einstein statistics, and it appears as a phase transition that all the particles uniformly drop to the least energy state as the entropically-favored state due to the Bose-Einstein statistics at some time. (In the Bose-Einstein condensation state, since all the particles are in a state such as the least energy state, de Broglie wave of each particle becomes longer and even, and eventually the system itself becomes a de Broglie wave.) The situation that the effect of the Bose-Einstein statistics is dominant can be considered as the low temperature region. Actually it is considered at the low temperature region that the particle’s thermal motion energy is extremely small so that it does not excite the state of the particles from the 1 least energy state. Hence, the Bose-Einstein condensation state emerges in the system composed of bosonic particle at extremely low temperature. We calculate in this paper the critical acceleration at which the Bose-Einstein con- densation begins to appear in the accelerating system by the Unruh effect when lowering the acceleration. We mention our critical acceleration more precisely. We first consider an ideal gas in the Minkowski space-time at zero-temperature, where the gas is composed of particles described by a free complex scalar field at finite density. As the gas is now in zero-temperature, it can be considered to be in the Bose-Einstein condensation state. We then start to accelerate such an ideal gas uniformly. At this time, according to the Unruh effect, since each constantly accelerating particle composing the gas will experi- ence the temperature T = ~a/(2πck ) (a is the acceleration) in the accelerating frame, U B the ideal gas performing uniformly accelerating motion will experience the temperature T as a whole in the accelerating frame. Here, what the gas experiences thermal means U that the space which the gas observes is filled with some medium that are performing thermal fluctuation. In this paper, we consider that there is no interaction between the medium and the particles composing the gas without the thermal excitation. Hence it is considered that, as growing the acceleration gradually from lower acceleration, in the accelerating frame, the dissolution of the Bose-Einstein condensation is observed eventu- ally, and finally the Bose-Einstein condensation disappears entirely at some acceleration. The critical acceleration we calculate is the acceleration at that time. We here mention the emergence mechanism of the Bose-Einstein condensation in our model and its critical moment. It is composed of three steps. We first calculate the effective potential of the field meaning the particles composing the gas, and then obtain the particle density by performing a derivative to it with regard to the particle’s chemical potential. Then we find that, if we decrease the acceleration with fixing the particle density to constant, either the particle’s chemical potential or the absolute value of the zero-modeofthefieldhastogrow. Here, theaccelerationplaystheroleofthetemperature in the accelerating frame as mentioned in the above paragraph, and the zero-mode of the field can be considered to correspond to the least energy state. Hence the absolute value of the zero-mode is considered as the expectation value of the Bose-Einstein condensation state, and whether it is zero or not corresponds to its disappearance or appearance, respectively. We consider to start with the high acceleration situation where there is no Bose-Einstein condensation in the accelerating frame, and decrease the acceleration gradually. At this time, we keep the particle density to constant. Namely, the number of particle is always constant in our model. Then, corresponding to no Bose-Einstein condensation, the absolute value of the zero-mode should be zero while the we decrease acceleration. Thus the chemical potential must grow to keep the particle density to constant (step 1). However we find that there is an upper bound for the value of the chemical potential, which is need to avoid a divergence of the probability amplitude. Hence, when the chemical potential reaches the upper bound (step 2), there is no way but the absolute value of the zero-mode starts to grow (step 3), if we further decrease acceleration with keeping the particle density to constant. Thus, the moment that the chemical potential reaches the upper bound is the critical moment that the Bose-Einstein condensation appears, and this is the Bose-Einstein condensation in our model. 2 The situation where our analysis is actually performed is just before the Bose-Einstein condensation state starts to occur. To be more specific, our calculation to obtain the critical acceleration is performed in the situation that the absolute value of the zero-mode and the chemical potential are put to zero and the upper bound value respectively in the equation of the particle density as the critical moment mentioned above. Then the acceleration obtained from that equation is the critical acceleration. In our analysis, due to some technical difficulty mentioned later, we consider the situation m/a 1 (m and a are the mass of the particle and the critical acceleration, c c ≪ respectively), and take the leading contribution of this in our calculation, where m is the mass of the complex scalar particle considered in this paper and a is the critical c acceleration. As the acceleration is proportional to the temperature in the relation given by the Unruh effect, this can be read as k T /2 mc2 (T is the critical temperature B c c ≫ fixed with a ), which is the relativistic situation that the particle’s thermal motion energy c is much higher than its static energy at the critical moment. The result in this paper is hence applicable to the system of the complex scalar field with the critical temperature for the Bose-Einstein condensation that is much higher than the mass. Since the kind of the field considered in this paper is a complex scalar field composing an ideal gas, the actual particle that can correspond to this paper is some ideal complex scalar particle system whose critical temperature of the Bose-Einstein condensa- tion locates in the region k T/2 mc2. B ≫ As stated above, in this study we perform the calculation for the critical acceleration in the Bose-Einstein condensation. So far several kinds of the critical accelerations for the spontaneous symmetry breaking induced by the Unruh effect have been carried out. The critical acceleration for the chiral symmetry restoration was studied in Nambu- Jona-Lasinio model at zero and finite chemical potentials for quarks in refs.[15] and [16] respectively. In Ref.[17] the critical acceleration for the restoration of the spontaneous symmetry breaking of the Z symmetry in the real scalar field theory was studied. 2 There are also papers concluding that the Unruh effect does not contribute to the restoration of the spontaneous symmetry breaking [18]. The system in these papers is not the finite density, and their discussion is based on the effective potential. On the other hand, our Bose-Einstein condensation occurs from the relation between the chemical po- tential and the acceleration playing the role of the temperature. This is a different point between us and him. Because of this, their conclusion would not be applied to our study readily. We should take this issue to a future work. Lastly, we introduce interesting papers that will have some connection to this study. Ref.[19] concludes that a larger acceleration should enhance a condensate as compared to those in a non-accelerated vacuum. In Ref.[20], although the background space-time is not the Rindler space, whether the Bose-Einstein condensation can occur or not is shown in an ideal boson gas model with a point-like impurity at finite temperature in some uniform gravitational force in each of D = 1,2 and 3. Next, it is shown in Ref.[21] that free massless scalar particles are detected by a constantly accelerating detector with not the Bose-Einstein distribution but the Fermi-Dirac distribution in odd dimensions, and this problem is solved in Ref.[22]. 3 2 The model The model in this paper is a free complex scalar model at the finite density correspond- ing to an ideal gas in the constantly accelerating system with the acceleration a. The Lagrangian density is given as = ~2gµν∂ φ ∂ φ c2m2φ φ (1) µ ∗ ν ∗ L − with φ 1 (φ +iφ ) that means the field of the particle composing a gas, and µ,ν are ≡ √2 1 2 the coordinate in the Rindler space explained below. In the Minkowski space-time as an inertial system, the background space-time of a constantly accelerating system is given by the Rindler space. The Rindler coordinate in our notation is given as follows: (η,ρ,y,z) (η,ρ,x ) with x (y,z). (2) ≡ ⊥ ⊥ ≡ This relates with the Minkowski coordinate (t,x,x ) as ⊥ c at at (t,x) = sinh , ccosh ρ(sinhη, ccoshη), (3) a c c ≡ (cid:0) (cid:1) where the accelerating direction has been thought to be in the x-direction. The Rindler metric in our notation is ds2 = (cρ)2dη2 d(cρ)2 dx2. (4) − − ⊥ In what follows, we use the unit system: c = ~ = k = 1. B The constantly accelerating one in the Minkowski space-time corresponds to the one moving along a line on a constant ρ in the Rindler space. The relation between ρ and a are ρ = 1/a as can be seen from eq.(3). Since constantly accelerating one experiences the system as a thermal-bath with the Unruh temperature T (the acceleration are in the U relation: T = a/2π) by the Unruh effect, one moving along a line on a constant ρ in the U Rindler space gets the temperature T = 1/2πρ. (5) U Hence, the gas in our study is considered to be in such a thermal-bath with Unruh temperature T in the accelerating frame. Here, although our gas is considered to be U in a thermal-bath with some medium performing thermal fluctuation giving the Unruh temperature T in the accelerating frame, we assume that there is no interaction between U our gas and the medium other than the thermal excitation. We can see from eq.(5) that varying ρ means varying the temperature. For this reason, how to interpret the results would be unclear if the four-dimensional space-time integration including the ρ-integration was performed. Let us here turn to how this problem is handled in other papers on the critical ac- celeration given by the Unruh effect. In Refs.[15, 16, 17], the action is given with the four-dimensional space-time integration including the ρ-integration. This point is the problem. However, their calculation is once performed using the Green’s function that 4 has dependence on the ρ-direction, the coordinate ρ is treated as a constant in the cal- culation of the effective potential. As a result, the ρ-integration becomes just a volume factor in the calculation of the effective potential. The difficulty in the treatment of the coordinate ρ is also mentioned at the chapter of conclusions and discussions in Ref.[23] in the context of the study of the Larmor radiation with the correction rooted in the Unruh effect. In our analysis, as well as Refs.[15, 16, 17], we obtain the Green’s function that has dependence on the ρ-direction. However, the space-time integration in our action is given without the ρ-integration. As a result, ρ is a parameter in our model. We can see at this time that the action is needed to be multiplied by a quantity with thedimension oflengthso that theactionbecomes dimensionless. To thispurpose, we put dρ in our action. This means that the process to obtain the path-integral representation given in eq.(6) for the probability amplitude from the operator formalism representation has been performed with fixing ρ. If the analysis in Refs.[15, 16, 17] is performed again by our way, there is no difference in the final result. There are four regions separated by the event-horizons in the Rindler space. The region treated in this paper is the right wedge only. 3 The effective potential 3.1 Performance of the path-integral We start with the probability amplitude: Z = π φexp i d3xdργ πη∂ φ+ πη ∂ φ µq , (6) η η ∗ η ∗ Z D D h Z (cid:16) − H− (cid:17)i (cid:0) (cid:1) where γ detg and d3x dηd2x , and µ and q are the chemical potential and µν ≡ − ≡ ⊥ the densitypof particles, respectively. For the reason mentioned in Section.2, to fix the acceleration and Unruh temperature, the integration in the ρ-direction is not included in the space-time integration in the action. As a result, ρ is a parameter in our analysis, and the Unruh temperature that the gas experiences is given according to the relation in eq.(5). At this time, we can see that the action is needed to be multiplied by a quantity with the dimension of length so that it becomes dimensionless. We put dρto this purpose. However the space-time integration is the one without the ρ-direction. Explanation for πη and is in what follows. H πµ,πµ ∂ , ∂ are the momenta given as ∗ ≡ ∂(∂µLφ∗) ∂(∂Lµφ) (cid:0) (cid:1) (cid:0) (cid:1) πη,πη = gηη∂ φ , gηη∂ φ . (7) ∗ η ∗ η (cid:0) (cid:1) (cid:0) (cid:1) is the Hamiltonian density given as H = πη∂ φ+πη ∂ φ η ∗ η H −L = πη π gij∂ φ ∂ φ+m2φ φ, (8) ∗ η i ∗ j ∗ − 5 where πµ is defined as πµ 1 (πµ + iπµ). Correspondingly, the functional integration ≡ √2 1 2 measure of π changes as π = π π . From eq.(7), we can see η η 1η 2η D D D πη = gηη∂ φ . (9) 1,2 η 1,2 It turns out that the conserved current associated with the U(1) global symmetry in our model is obtained as Jµ = igµν(φ∂ φ φ ∂ φ). (10) ν ∗ ∗ ν − − Using this Jµ, the integral of conserved charge density can be written as d3xdργq = d3xdργJη = d3xdργ πηφ +πηφ . (11) Z Z Z − 2 1 1 1 (cid:0) (cid:1) Substituting eqs.(8) and (11) into eq.(6), we can obtain the following Z: i Z = π π φexp d3xdργ 1η 2η Z D D D h2 Z n 1 1 πηπ 2(∂ φ +µφ )πη πηπ 2(∂ φ +µφ )πη − 2 1 1η − η 1 2 1 − 2 2 2η − η 2 1 2 (cid:0) (cid:1) (cid:0) (cid:1) 1 + (∂ φ )2 +(∂ φ )2 m2(φ2 +φ2) , (12) 2 i 1 i 2 − 1 2 oi (cid:0) (cid:1) where i,j = x . Here, in the above, we perform the following rewritings: ⊥ πηπ 2(∂ φ +µφ )πη = gηη π (∂ φ +µφ ) 2 (∂ φ +µφ )2 , (13a) 1 1η − η 1 2 1 1η − η 1 2 − η 1 2 n o (cid:0) (cid:1) πηπ 2(∂ φ +µφ )πη = gηη π (∂ φ +µφ ) 2 (∂ φ +µφ )2 . (13b) 2 2η − η 2 1 2 2η − η 2 1 − η 2 1 n o (cid:0) (cid:1) Furthermore, we redefine the fields as π (∂ φ +µφ ) π , (14a) 1η η 1 2 1η − → π (∂ φ +µφ ) π . (14b) 2η η 2 1 2η − → As a result, we can write Z as i Z = φexp d3xdργ gηη(∂ φ +µφ )2 +(∂ φ )2 η 1 2 i 1 CZ D h2 Z (cid:16) + gηη(∂ φ +µφ )2 +(∂ φ )2 m2(φ 2 +φ 2) . (15) η 2 1 i 2 1 2 − (cid:17)i with π π exp i d3xdργgηη (π )2+(π )2 . Performing the path-integral 1η 2η 1η 2η C ≡ Z D D h Z i (cid:0) (cid:1) of π and π formally, we think that become some factor. We ignore in what follows. 1η 2η C C As a result, with some straight forward calculation, we can write Z in the following form, i Z = φexp d3xdργ φ Gφ +φ Gφ +2gηηµ(φ ∂ φ φ ∂ φ ) , (16) 1 1 2 2 2 η 1 1 η 2 Z D h− 2 Z (cid:16) − (cid:17)i 6 where G ∂2 +γ 1gij∂ (γ∂ )+m2 gηηµ2. ≡ η − i j − Letusnowrewritetherealandimaginarypartsofthefieldintoaconvenient expression fortheanalysisoftheBose-Einsteincondensation. Aswehavewrittenintheintroduction, the Bose-Einstein condensation can be considered as the situation that all particles get to be the least energy state. The least energy state can be considered as the zero-mode of the field in our model, and the situation that all the particles are in the least energy state can be considered as the condensation of the zero-mode. Hence, a convenient expression for the analysis of the Bose-Einstein condensation in our analysis is the one in which the zero-mode is separated as φ √2α cosθ+φˆ , (17a) 1 1 ≡ φ √2α sinθ+φˆ , (17b) 2 2 ≡ where α plays the role of the expectation value of the condensation of the zero-mode, θ ˆ ˆ is a phase in the zero-mode and φ , and φ are the non-zero modes. Depending on just 1 2 before or after the condensation starts, α behaves as follows: α = 0 : before the condensation α = 0 : after the condensation 6 At this time, φ Gφ and φ Gφ can be written as 1 1 2 2 φ Gφ = 2α2(m2 gηηµ2) cos2θ+φˆ Gφˆ , (18) 1 1 1 1 − φ Gφ = 2α2(m2 gηηµ2) sin2θ +φˆ Gφˆ . (19) 2 2 2 2 − As a result, we can rewrite Z into Z = exp iα2 d3xdργ(m2 gηηµ2) φˆexp h− Z − iZ D (cid:20) i G 2gηηµ∂ φˆ d3xdργ φˆ φˆ − η 1 − 2 Z (cid:0) 1 2 (cid:1)(cid:18)2gηηµ∂η G (cid:19)(cid:18) φˆ2 (cid:19) +√2αcosθ G(φ +φ )+(φ +φ )G . (20) 1 2 1 2 (cid:21) (cid:16) (cid:17) Our analysis will be performed just before the condensation as mentioned later. For this reason, we put α = 0 in what follows. As a result, we can write Z as i G 2gηηµ∂ φˆ Z = φˆexp d3xdργ φˆ φˆ − η 1 . (21) Z D (cid:20)− 2 Z (cid:0) 1 2 (cid:1)(cid:18) 2gηηµ∂η G (cid:19)(cid:18) φˆ2 (cid:19)(cid:21) Then, we can see from the form of G given below eq.(16) that there should be the condition: m2 gηηµ2 0. (22) − ≥ Otherwise the path-integrals of Z diverges at the configuration that all the momenta are zero. Further, the above relation gives the upper limit of the chemical potential for given a mass and an Unruh temperature. 7 Performing the diagonalization as i G 2gηηµ∂ φˆ Z = Z Dφˆexp(cid:20)− 2 Z d3xdργ(cid:0) φˆ1 φˆ2 (cid:1)UU−1(cid:18) 2gηηµ∂η − G η (cid:19)UU−1(cid:18)φˆ12 (cid:19)(cid:21). i G+2gηηµ∂ 0 φˆ = φˆ exp d3xdργ φˆ φˆ η ′1 , Z D′ ′ (cid:20)− 2 Z (cid:0) ′1 ′2 (cid:1)(cid:18) 0 G−2gηηµ∂η (cid:19)(cid:18)φˆ′2 (cid:19)(cid:21) (23) i i where U 1 − is a unitary matrix defined to perform the above diagonaliza- ≡ √2 (cid:18) 1 1 (cid:19) ˆ ˆ φ φ tion, and correspondingly ′1 U 1 1 . At this transformation, the functional ˆ − ˆ (cid:18)φ (cid:19) ≡ (cid:18)φ (cid:19) ′2 2 ˆ ˆ measure is also transformed, which we have described as φ φ. However, since U is ′ ′ ˆ D →ˆD a constant unitary matrix, the difference between φ and φ contribute only to some ′ ′ D D constant factor in the path-integral, and we ignore it in what follows. Performing the path-integral, we can obtain 1/2 Z =Det G+2gηηµ∂ G 2gηηµ∂ − . (24) η η − (cid:16) (cid:17) (cid:0) (cid:1)(cid:0) (cid:1) In the above, with regard to the treatment of dρ, we consider that the integration of this has been performed by assigning a value at one point. Hence, the effective action W defined as Z = expiW can be written as i W = LogDet G+2gηηµ∂ G 2gηηµ∂ η η 2 − (cid:16) (cid:17) (cid:0) (cid:1)(cid:0) (cid:1) i = TrLog G+2gηηµ∂ G 2gηηµ∂ η η 2 − (cid:16) (cid:17) (cid:0) (cid:1)(cid:0) (cid:1) i dk3 = V Log ∂2 +γ 1gij∂ (γ∂ )+M2 +2igηηµ∂ 2 Z (2π)3(cid:16) η − i j η (cid:0) (cid:1) +Log ∂2 +γ 1gij∂ (γ∂ )+M2 2igηηµ∂ , (25) η − i j − η (cid:17) (cid:0) (cid:1) where M2 m2 gηηµ2 and V d3xγ is the volume for the (η,x ) space-time, which ≡ − ≡ ⊥ appears from the rewriting of a funRctional trace into an integration: L 3 2π 3 dk3 Tr V (26) → (cid:18)2π(cid:19) (cid:18) L (cid:19) ≡ Z (2π)3 X k with 2π 3 = dk3 and V L3 = d3xγ, where L means just length in each space L k ≡ for th(cid:0)e η(cid:1), xP-direcRtions, and √ detg R= g detg = γ (a,b = η,x except for ρ, ab ρρ ab and g = 1)⊥. d3k dωd2k wit−h k (k ,pk −), which are momenta corres⊥ponding to the ρρ y z ≡ ⊥ ⊥ ≡ coordinate (η,x ) as in eq.(32). ⊥ 8 Defining δ gηη∂2 +γ 1gij∂ (γ∂ ) 2igηηµ∂ , we further rewrite W as ± ≡ η − i j ± η i dk3 W = V Log δ +M2 +Log δ +M2 2 Z (2π)3(cid:16) + − (cid:17) (cid:0) (cid:1) (cid:0) (cid:1) i dk3 M2 M2 = V d∆2(δ +∆2) 1 +Logδ + d∆2(δ +∆2) 1 +Logδ . 2 Z (2π)3 (cid:18) Z + − + Z − − −(cid:19) 0 0 (27) Here, we can ignore dk3 Logδ for the reason in what follows: First, since we will (2π)3 ± perform the derivative wRith regard to the chemical potential to obtain the particle density at the end, we look at only the part concerning the chemical potential as dk3 Logδ Z (2π)3 ± dk3 dk2 2igηηµ∂ = Log ∂2 +γ 1gij∂ (γ∂ ) + dk Log 1 η Z (2π)3 η − i j Z η(2π⊥)3 (cid:18) ± ∂2 +γ 1gij∂ (γ∂ )(cid:19) (cid:0) (cid:1) η − i j dk2 2igηηµ∂ η dk Log 1 . (28) η ⊥ ∼Z (2π)3 (cid:18) ± ∂2 +γ 1gij∂ (γ∂ )(cid:19) η − i j In the above, we have described the integral dk3 separately as dk dk2. Then, η replacing ∂ withik , fromthefact that dkklogR(1+ c1k ) = 0 (c aRresomR ec⊥onstants), i i c2+k2 1,2 we can see that the µ-dependent part vRanishes in the k -integrals. ⊥ Finally, we can write the effective action W in eq.(27) as i dk3 M2 W = V γ d∆2(D +D ), (29) 2 Z (2π)3 Z + − 0 e e where D is ± e γ−1 δ +∆2 −1 D = D (kη,ρ,k ). (30) ± ≡ ± ± ⊥ (cid:0) (cid:1) e e In the above, we wrote the arguments concerning the momenta as k and k despite η ⊥ that these are given by the differential operators in the actual expression, which may be allowed. 3.2 Calculation of D ± From the definition in eq.(30e), we can have the following equation: δ4(x x) =γ δ +∆2 D (x x) ′ ′ − ± ± − =γ(cid:0)gηη∂2 +(cid:1)γ 1gij∂ (γ∂ ) 2igηηµ∂ +∆2 D (x x) (31) η − i j ± η ± − ′ (cid:0) (cid:1) with D (x x) defined as ′ ± − D (x x′) = d3k D ei ω(η−η′)−k⊥(x⊥−x′⊥) . (32) ± − Z (2π)3 ± (cid:0) (cid:1) e 9