The Krein-Gupta-Bleuler Quantization in de Sitter Space-time; Casimir Energy-Momentum Tensor for a Curved Brane

In this paper, vacuum expectation value (VEV) of the energy-momentum tensor for a conformally coupled scalar field in de Sitter space-time is investigated through the Krein-Gupta-Bleuler construction. This construction has already been successfully applied to the de Sitter minimally coupled massless scalar field and massless spin-2 field to obtain a causal and fully covariant quantum field on the de Sitter background. We also consider the effects of boundary conditions. In this respect, Casimir energy-momentum tensor induced by Dirichlet boundary condition on a curved brane is evaluated.


Introduction
In quantum field theory (QFT) as usually formulated, the zero-point fluctuations of the fields contribute to the energy of the vacuum. Although this energy does not seem to be observable directly in any laboratory experiment (only energy differences are measurable), but, when gravity is taken into account, the quantum vacuum energy, or more precisely vacuum expectation value of the energy-momentum tensor, appears as the source of curvature in the semi-classical Einstein's equation, so the absolute meaning of energy becomes influential. In curved space-time, however, the gravitational field will in general be expected to produce particles, thereby there is no unique notion of a vacuum state or of particles [1]. With no clear vacuum concept, one cannot give a precise meaning to expectation value of the energy-momentum tensor of the vacuum state, < 0|T µν |0 > .
On the other hand, according to the semi-classical Einstein's equation However, cosmological observations imply that the total effective vacuum energy, ρ v = ρ + Λ 8πG = Λ ef f 8πG , is [3] ρ v ≤ (10 −12 GeV) 4 ∼ 10 −8 erg/cm 3 , which is much smaller than what we had expected from the calculations. This large discrepancy of 10 120 order constitutes the cosmological constant problem.
In this work, we prove that "Krein" space quantization, a new version of indefinite metric field quantization, not only can cure these defects, but also is capable of recovering the results of the conventional QFT.
Krein space as the generalization of the Hilbert space is defined by K = H + ⊕ H − , where K is called the total space and H − (= H * + ) stands for the anti-Hilbert space. Krein space quantization then is a canonical quantization of Gupta-Bleuler type in which the Fock space is constructed over the total space [4,5,6]. In this quantization method, the set of states is different from the set of physical states; The observables are defined on the total space, while the average values of the observables are calculated on the sub-space of physical states. It is worth to mention that the total space is equipped with an indefinite inner product which results in some (un-physical) states have a negative norm. An interesting feature of the theory is that, instead of having a multiplicity of vacuum, we have several possibilities for the space of physical states and only one field and one vacuum which are independent of Bogoliubov transformations [6]. So, the usual ambiguity about vacuum is not suppressed but displaced. This method, for the first time, was applied to the massless minimally coupled scalar field in de Sitter (dS) space [7,8]. 1 Furthermore, pursuing this path, it was shown that covariant quantization for massless spin-2 field in dS space is accessible [10,11,12,13,14,15,16].
Respecting the intrinsic specifications of Krein quantization method, vacuum expectation value of the energymomentum tensor is thoroughly discussed in this paper: Sec. (2) is devoted to a comprehensive introduction of Krein method, supplemented by new practical comments. Also, unitarity of the theory when interactions are taken into account and compatibility of the method with physical achievements of renormalizing process will be discussed in this section. In Sec. (3), the viewpoint of Krein method on the vacuum energy is perused. We show that the entrance of un-physical states in the theory, which act as intrinsic renormalizing devices, makes it possible to determine absolute meaning for energy. Then we consider the theory in the presence of boundary conditions, such as Casimir effect, and also in non-trivial topologies. Eventually, we study the results of generalizing the method to curved space-times. In Sec. (4), we speak of Krein method faced with the trace anomaly subject. In Sec. (5), back reaction effect and Hawking radiation are discussed. Especially, it is shown that the theory is capable of recovering the result obtained by Hawking for black hole radiation even considering the fact that < 0|T µν |0 > of the free theory is zero.

Basic Set-Up
Let us illustrate Krein quantization procedure by giving a simple example, a Klein-Gordon-field in a twodimensional Minkowski space-time which has two sets of solutions The inner product is defined as and these modes are normalized by the following relations The subscripts P and N are respectively referred to the positive and negative norm modes, or simply physical and un-physical states. The field operator in Krein space is defined by in which The quantum field theory is identified by defining the vacuum states as where |1 k is an un-physical state. A significant difference to the standard QFT, which is based on canonical commutation relation, lies in the requirement of the following commutation relations

Unitarity of the Theory and Compatibility with Renormalizing Process
In the context of Krein method, un-physical states are present. However, unitarity of the theory would be preserved by imposing the following conditions on quantum states and probability amplitude • Un-physical states do not interact with the physical ones or real physical world. Indeed, in the Feynman diagrams, such states only appear in the internal legs and disconnected parts of the diagrams.
• Un-physical states, which appear in the disconnected parts of the S-matrix elements, can be eliminated by renormalizing the probability amplitude as [17] S if = physical states, in|physical states, out 0, in|0, out .
The presence of un-physical states in the internal lines plays a key role in the renormalizing procedure, so that, by taking into account the quantum metric fluctuations, the divergences of the Green's function in QFT do not appear any more [18]. Considering these specifications, by using Krein approach, many topics have been studied and the results are in complete agreement with their (Hilbert space) QFT's counterparts, and experimental data. For instance, in [19] the one-loop effective action of QED is calculated by the Schwinger method in Krein approach. It is shown that the theory is free of any divergence and the effective action coincides with standard solution. The magnetic anomaly and lamb shift are also calculated in the one-loop approximation [17]. The results are comparable to the results of conventional QED. In Refs. [20,21,22,23] the interaction QFT in Minkowski space ( λϕ 4 theory) is considered to the one-loop approximation. It is found that the theory is automatically renormalized in this approximation. In Refs. [24,25] Casimir force in Minkowski space-time has been calculated. Once again it is indicated that the theory is automatically renormalized and the results are the same as those that have been experienced.
It should be noted that, the method can also reproduce the very results that are extracted from the renormalization group. To see the points of Krein method, we concentrate on a simple toy model, λϕ 4 scalar field, which is sufficient to give us the basic qualitative results of the approach. For this example, one can extract the effective potential in Krein view as follows [22] As expected, classically, one can acquire the coupling constant by four times differentiating of the V (0) ef f with respect to ϕ . The result would be obtained to order 0 as shown by (11). Following the same path, in order to include the quantum corrections in the theory, we accomplish four times differentiating of the V ef f with respect to ϕ and then, substitute ϕ with a non-zero arbitrary parameter M . The result will be as [22] λ Like routine methods, the dependence of the coupling constant, λ Krein , on the mass scale, M , is governed by a beta function of the theory, which is defined as so that, by applying this process to λϕ 4 scalar field theory, one can obtain [22] β Krein = 3λ 2 16π 2 , which only depends on coupling constant and has no explicit dependence on M . It is worth mentioning that, although the effective potential and coupling constant calculated by using Krein procedure are different from those that come from conventional procedure, however, both methods yield the same beta function. This process can be applied to the other QFTs to calculate the beta function as it has been carried out for QED [26]. Concentrating on the beta function this much is due to the fact that it has important physical outcomes, such as the trace anomaly subject which would be discussed in Sec. (4).

Applying the Method in Flat Space-times
At this stage, we wish to restrict attention to space-times where we are able to formulate quantum field theory constructions in a manner completely analogous to Minkowski space-time. In a two-dimensional Minkowski space-time, the energy density operator is given by the 00-component of the energy-momentum tensor Regarding Krein technique, in which the field operator is constructed by both physical and un-physical states, the vacuum energy in Minkowski space-time intrinsically vanishes, i.e. 0|T Krein 00 |0 = 0 , |0 is the Minkowski space vacuum. In fact, by utilizing un-physical states as renormalizing tools, energy itself is meaningful and can be defined absolutely. Now we will take the field operator (6) to calculate the vacuum energy in the presence of boundary conditions. As the simplest case, we examine the vacuum energy between two parallel plates subjected to Dirichlet boundary condition, where by considering the field operator (6), only physical states would be affected and un-physical states do not interact with the physical ones or real physical world. Therefore, the field operator can be written as here k d are the eigen-frequencies of the system under consideration, where in this case; φ P (k d , t, 0) = φ P (k d , t, a) = 0 , we have = 1, 2, 3, ... . Substituting the field operator (16) into (15) leads to [24] < 0|T Krein Then the total vacuum energy of the interval (0, a) is obtained as follows Surveying the Eq. (17) reveals that in standard method with the choice of Hilbert space, only the first two terms are apparent, which cause an infinite result. In order to remove this infinity or renormalize the theory, one rescales the energy by subtracting the Minkowski vacuum energy, 2 so the third term is added to the equation intelligently. However, in Krein method, because of a new interpretation of un-physical states, as natural renormalizing tools, this term intrinsically exists. Therefore, an absolute meaning for energy is achieved. It exactly coincides with the conventional method and the measured value [28]. Now we study Krein method in non-trivial topologies. Considering a massless Klein-Gordon-field in a twodimensional flat cylindrical space-time ( R 1 × S 1 ) with periodicity length, L , we have In this case, the effect of the space closure, restricts both physical and un-physical states; According to the fundamental postulates of QFT, the spinless wave function must have a definite value at every point in space [29]. Hence, we must demand that the field modes (3) be single-valued, and so, the field operator (6) converts to in which with k n = 2nπ L , n = 0, ±1, ±2, ... and ω = |k| . The vacuum energy density of such field on R 1 × S 1 would be Therefore the vacuum energy is automatically renormalized and it is equal to zero. As before in standard method, just the first term of (21) exists, which is obviously infinite. This was expected, as the R 1 × S 1 spacetime suffers from the same ultraviolet divergence properties as Minkowski space [27], thus the renormalizing process would be accomplished by subtracting the infinite quantum vacuum energy of Minkowski space-time and result in [30] 0 L | : T 00 : the symbol " : : " refers to the energy rescaling process and |0 L , |0 are the vacuum associated with the discrete modes (20) and the Minkowski space, respectively. There is a quite different result.

Generalizing the Method to Curved Space-times
As discussed earlier, due to the lack of a unique vacuum state in general curved space-times, vacuum expectation value of the energy-momentum tensor cannot be determined uniquely. However, based on the rigorous mathematical structure of Krein space quantization, one can address this problem.
To start, we consider a scalar field equation in general curved space-time as follows ϕ = g µν ∇ µ ∇ ν ϕ , R and ξ are the scalar curvature and the coupling constant between the scalar field and the gravitational field, respectively. The inner product is considered as [1] (ϕ 1 , where dΣ µ = dΣ n µ , with n µ , a future-directed unit vector orthogonal to the space-like hypersurface Σ and dΣ is the volume element in Σ . There exists a complete set of mode solutions {φ k } of Eq. (23) which are orthonormal in the product (24), i.e.
the set of positive and negative norm states are {φ k } and {φ * k } respectively. As pointed, for minimally coupled scalar field in de Sitter space, {φ k } is not a complete set, but generally, {φ k , φ * k } form a complete set of solutions of the field equation [7,15], with respect to which one can expand any arbitrary solution. Then the field operator ϕ in Krein method is defined by The new modes F can be expanded in terms of the old ones The matrices α jk , β jk are Bogolubov coefficients and these relations are known as Bogolubov transformations. These coefficients possess the following properties The field operator ϕ can be written regarding any of the two sets The a k /a † k and b k /b † k ( c j /c † j and d j /d † j ) are annihilation/creation operators of the physical and un-physical states, respectively, in the φ (F)-modes. The F-vacuum state can be considered as c j |0 F = 0, d j |0 F = 0 ∀j . By exploiting Eqs. (27) and (29), the two sets of creation and annihilation operators can be written with regard to each other as and Considering the operator N i = a † i a i , that counts particles in the φ -modes, the expectation value of N i for the number of φ -modes particles in the state |0 F is which is to say that the vacuum of the F-modes contains k |β ki | 2 particles in the φ -modes. So, not only Krein method is capable of explaining the particle creation through a time-dependent gravitational field, but also contrary to the usual QFT, one can determine vacuum expectation value of the energy-momentum tensor uniquely in general curved space-times 3 Furthermore, regarding the semi-classical Einstein's equation (1), the mentioned discrepancy between theoretical predictions and experimental data is automatically eliminated.

Trace Anomaly Subject in Krein Quantization Approach
The trace (conformal) anomaly is a quantum anomaly of the semi-classical gravitation that not only has been connected with but also has influenced many important issues in modern physics. A good approach to a more detailed discussion would be an investigation of a self-interacting scalar field, λϕ 4 theory, with the following conformal invariant classical matter action in curved space-time (respecting the semi-classical field equation (1)) where ( g ≡ |det g µν | ) the subscripts 0 and I are respectively referred to free field and interaction-dependent parts of the theory. Classically, conformal invariant theories in which the matter action is invariant under conformal transformations ( g µν (x) → Ω 2 g µν (x) =ḡ µν (x) ), yield a zero trace of the classical energy-momentum tensor [1]. When quantum corrections are included, however, this classical invariance cannot be preserved, and gives rise to an anomalous trace; The trace of expectation value of the energy-momentum tensor, with respesct to the effective action in Krein view W Krein , would be As already stated, in Krein technique, the free field part of vacuum expectation value of the energy-momentum tensor automatically vanishes, and so, the first term in the right hand side of Eq. (36) will be removed. But the second term, the interaction-dependent part of the theory, remains. Actually, affected by the inevitable presence of the arbitrary mass scale (see Eq. (13) and the related discussions), the conformal transformation of the theory leads to a different value of the effective coupling constant in Krein viewpoint, λ Krein , even if the corresponding classical theory is conformal invariant. Therefore, in Krein's point of view, the trace anomaly is just limited to the interaction-dependent part of the theory which is state-dependent and non-local, i.e.
It is worth to mention that, the interaction-dependent anomalous breaking of conformal invariance, as the common point of view between Krein method and the other standard approaches, 4 in quantum chromodynamics (QCD) in the chiral limit, determines the sizes and masses of hadrons, containing protons and neutrons [31,32].

Discussion
Respecting Krein construction, it can be concluded that the differences between Krein approach and the conventional methods of QFT, which can be expressed by the following statements • In Krien context, instead of having a multiplicity of vacuum, we have several possibilities for the space of physical states, so the usual ambiguity about vacuum is not suppressed but displaced.
• Vacuum expectation value of the energy-momentum tensor of the free field and consequently its trace vanish.
actually give rise to the elimination of the theoretical shortcomings in the definition of vacuum expectation value of T µν and of the mentioned discrepancy between theoretical predictions and cosmological observations. Furthermore, removing the free field part of expectation value of the energy-momentum tensor leads to the loss of geometrical effects, caused by renormalizing procedure in the standard approach, 5 on the semi-classical Einstein's equation. While, in the conventional viewpoint, accepting the influence of the renormalizing process of expectation value of the energy-momentum tensor on geometrical part of Einstein's equation, represents a "back-reaction" effect which needs to be dealt with in some self-consistent ways. If < T µν > affects the metric, then this altered metric will change the assumptions for calculating < T µν > . Technically, it is extremely difficult to calculate < T µν > in a given fixed background and incorporation of the back-reaction effect makes it close to impossible (for a more detailed discussion, one can refer to Ref. [33]). On the other side, it is usually esteemed that the trace anomaly, including both free field and interactiondependent parts, is a crucial ingredient of the Hawking radiation [34,35,36]. Therefore, it seems that in Krein context, contrary to the usual QFT, the free field part of the theory does not contribute to black hole radiation. However, following an alternative approach allows us to restore this contribution; As discussed in Refs. [37,38], the gravitational field of a black hole creates an effective potential barrier that acts as a good conductor. It conducts well at low frequencies, but as the frequencies increase, its conductivity diminishes. Therefore, due to the existence of a potential barrier, it is possible that the calculating procedure of Casimir effect in Krein quantization approach be applied to a black hole. Pursuing this path, see [39], we have shown that Krein quantization approach is capable of retrieving the obtained results for the temperature of the radiation of black holes with the exception that vacuum expectation value of the energy-momentum tensor of the free theory is zero. 4 According to Eq. (37), interaction-dependent trace anomaly is directly related to the beta function of the theory and as stated in Sec. (2), at least, the Krein beta functions of the theories that have been studied till today, coincide with the ones that had been obtained in standard approach [22,26]. 5 In the standard approach, confronting the divergence of the free field part of expectation value of the energy-momentum tensor is inevitable. Pursuing a careful renormalizing procedure, < Tµν > 0 can be divided into divergent and renormalized parts, i.e. < Tµν > 0 = (< Tµν > 0 )ren + (< Tµν > 0 ) div , in which (< Tµν > 0 ) div is purely geometrical, built out of the Riemann curvature tensor, Rµνρτ , and its contractions [1]. So, one can split the right hand side of the semi-classical Einstein's equation (1) into two parts, purely geometrical part and dependent part on quantum states of the matter fields.