The Casimir stress on a spherical shell in de Sitter spacetime for a massless scalar field is calculated using Krein space quantization. In this method, the auxiliary negative frequency states have been utilized, the modes of which do not interact with the physical states and are not affected by the physical boundary conditions. These unphysical states just play the role of an automatic renormalization tool for the theory.

1. Introduction

The Casimir effect is a small attractive force acting between two parallel uncharged conducting plates and it is regarded as one of the most striking manifestation of vacuum fluctuations in quantum field theory. It is due to the quantum vacuum fluctuation of the field operator between two parallel plates. In other words, the Casimir effect can be viewed as the polarization of the vacuum by boundary conditions or geometry. The presence of reflecting boundaries alters the zero-point modes of a quantized field and results in the shifts in the vacuum expectation values of quantities quadratic in the field, such as the energy density and stresses.

In particular, vacuum forces arise acting on the constraining boundaries. The particular features of these forces depend on the nature of the quantum field, the type of spacetime manifold and its dimensionality, the boundary geometries, and the specific boundary conditions imposed on the field. Since the original work by Casimir in 1948 [1], many theoretical and experimental works have been done on this problem [214].

The time dependence of boundary conditions or geometries, the so-called dynamical Casimir effect, is also a new element which has to be taken into account. In particular, in [15] the Casimir effect has been calculated for a massless scalar field satisfying Dirichlet boundary conditions on the spherical shell in de Sitter space. The Casimir stress is calculated for inside and outside of the shell with different backgrounds corresponding to different cosmological constants.

It should be remarked that the procedures of renormalization and regularization have been applied for eliminating the divergences appeared in physical quantities [1620]. The Casimir effect has been also investigated for two parallel plates in Krein space quantization [21].

The historical background of Krein space quantization goes back to the covariant quantization of minimally coupled scaler field in de Sitter spacetime. It has been shown that the linear quantum gravity in the background field method is perturbatively nonrenormalizable and also there appears an infrared divergence. This infrared divergence does not manifest itself in the quadratic part of the effective action in the one-loop approximation. This means that the pathological behavior of the graviton propagator may be gauge dependent and so should not appear in an effective way as a physical quantity [22]. The infrared divergence which appears in the linear gravity in de Sitter space is the same as the minimally coupled scalar field in de Sitter space [2327]. It is shown that one cannot construct a covariant quantization of the minimally coupled scalar field with only positive norm states [28]. It has been proved that the use of the two sets of solutions (positive and negative norm states) is an unavoidable feature if one wants to preserve causality (locality), covariance, and elimination of the infrared divergence in quantum field theory for the minimally coupled scalar field in de Sitter space [29, 30], that is, Krein space quantization.

The singular behavior of Green’s function at short relative distances (ultraviolet divergence) or in the large relative distances (infrared divergence) leads to main divergences in the quantum field theory. It was conjectured that quantum metric fluctuations might smear out the singularities of Green’s functions on the light cone, but it does not remove other ultraviolet divergences [31]. However, it has been shown that quantization in Krein space removes all ultraviolet divergences of quantum field theory (QFT) except the light cone singularity [32]. By using the Krein space quantization and the quantum metric fluctuations in the linear approximation, it has been showed that the infinities in the Green’s function are disappeared [31, 33].

Quantization in Krein space instead of Hilbert space has some interesting features. For example, in this method, the vacuum energy becomes zero naturally, so the normal ordering would not be necessary [29, 32]. The auxiliary negative norm states, which are used in the Krein space quantization, play the regularization of the theory.

Applying the unphysical negative frequency states and defining the field operator in Krein space, we can calculate the gravitational pressure on a spherical shell yielding the standard result obtained.

2. Scalar Casimir Effect for a Sphere in de Sitter Space

The Casimir force due to fluctuations of a free massless scalar field satisfying Dirichlet boundary conditions on a spherical shell in Minkowski space time has been studied in [34]. Doing the calculations in Krein space, the two-point Green's function 𝐺𝐾(𝑥,𝑡;𝑥,𝑡) is defined as the vacuum expectation value of the time-ordered product of two fields (𝐾 and 𝑇 stand for quantities in Krein space and the time-ordered product, resp.)𝐺𝐾𝑥,𝑡;𝑥,𝑡0||𝑖𝑇Φ𝐾(𝑥,𝑡)Φ𝐾𝑥,𝑡||0Φ,(1)𝐾(𝑥,𝑡)=𝑘𝑎𝑘+𝑏𝑘𝑢𝑝𝑎(𝑘,𝑥)+𝑘+𝑏𝑘𝑢𝑛.(𝑘,𝑥)(2)

The operators 𝑎(𝑘) and 𝑎(𝑘) create and destroy, respectively, the mode 𝑢𝑝(𝑘,𝑥) with positive energy (𝑘0=𝜔𝑘), which may be considered as the operators of creation and annihilation of a particle and the operators 𝑏(𝑘) and 𝑏(𝑘) create and destroy, respectively, the mode 𝑢𝑛(𝑘,𝑥) with negative energy (𝑘0=𝜔𝑘), which may be considered as the operators of creation and annihilation of an “antiparticle” in the inverse time direction. The two sets of modes do not affect each other and in the standard QFT, the negative energy states are eliminated in the quantum field operators, which are the origin of the appearance of divergence. On the contrary, the divergence disappears by taking these states into account.

The two-point Green’s function has to satisfy the Dirichlet boundary conditions on the shell: 𝐺𝐾𝑥,𝑡;𝑥,𝑡|||𝑥|=𝑎=0,(3) where 𝑎 is radius of the spherical shell. The stress-energy tensor in Krein space 𝑇𝐾𝜇𝜈(𝑥,𝑡) is given by𝑇𝐾𝜇𝜈(𝑥,𝑡)𝜕𝜇Φ𝐾(𝑥,𝑡)𝜕𝜈Φ𝐾1(𝑥,𝑡)2𝜂𝜇𝜈𝜕𝜆Φ𝐾(𝑥,𝑡)𝜕𝜆Φ𝐾(𝑥,𝑡).(4)

The radial Casimir force per unit area 𝐹/𝐴 on the sphere, called Casimir stress, is obtained from the radial-radial component of the vacuum expectation value of the stress-energy tensor:𝐹𝐴=0||𝑇𝑟𝑟in𝑇𝑟𝑟out||0||𝑟=𝑎.(5)

Taking into account the relation (1) between the vacuum expectation value of the stress-energy tensor 𝑇𝐾𝜇𝜈(𝑥,𝑡) and the Green's function at equal times 𝐺𝐾(𝑥,𝑡;𝑥,𝑡), we obtain𝐹𝐴=𝑖2𝜕𝜕𝜕𝑟𝜕𝑟𝐺𝐾𝑥,𝑡;𝑥,𝑡in𝜕𝜕𝜕𝑟𝜕𝑟𝐺𝐾𝑥,𝑡;𝑥,𝑡out|||𝑥=𝑥,|𝑥|=𝑎.(6)

One may use of the above flat space calculation in de Sitter space time by taking the de Sitter metric in conformally flat form𝑑𝑠2=Ω(𝜂)𝑑𝜂23𝚤=1(𝑑𝑥𝚤)2,(7) where Ω(𝜂)=𝛼/𝜂 and 𝜂 is the conformal time<𝜂<0.(8)

Assuming a canonical quantization of the scalar field in Krein space, the conformally transformed quantized scalar field in de Sitter spacetime is given byΦ𝐾(𝑥,𝜂)=𝑘𝑎𝑘+𝑏𝑘𝑢𝑘𝑎(𝜂,𝑥)+𝑘+𝑏𝑘𝑢𝑘,(𝜂,𝑥)(9) where 𝑎𝑘 and 𝑎𝑘 are creation and annihilation operators, respectively, and the vacuum states associated with the physical modes 𝑢𝑘 defined by 𝑎𝑘|0=0 are called conformal vacuum. Given the flat space Green's function (1), we obtain𝐺𝐾=𝑖0|||𝑇Φ𝐾(𝑥,𝜂)Φ𝐾𝑥,𝜂|||0=Ω1(𝜂)Ω1𝜂𝐺𝐾,(10) where Φ𝐾(𝑥,𝜂)=Ω1(𝜂)Φ𝐾(𝑥,𝜂) has been used. Therefore, using (5), (6), and (10), we obtain the total stress on the sphere in de Sitter spacetime and using Krein space quantization as𝐹𝐴=𝜂2𝛼2𝐹𝐴,(11) in accordance with the standard result.

3. Spherical Shell with Different Vacua in Krein Space Quantization

We assume different vacua inside and outside, corresponding to different 𝛼in and 𝛼out for the Lorentzian metric (7) and use the following relation for the stress on the shell due to boundary conditions in flat spacetime [8]: 𝐹𝐴=14𝜋𝑎2𝜕𝐸,𝜕𝑎(12) where the Casimir energy 𝐸 is the sum of Casimir energies 𝐸in and 𝐸out for inside and outside of the shell. The corresponding relation in de Sitter spacetime and applying Krein method leads to the following result [15]:𝐹𝐴=14𝜋𝑎2𝜕𝐸=𝜂𝜕𝑎28𝜋𝑎4𝑐1𝛼2in+𝑐2𝛼2out,(13) where we have used the automatically renormalized total zero-point energy in Krein Space𝜂𝐸=2𝑐2𝑎1𝛼2in+𝑐2𝛼2out,(14) in which 𝑐1=0.008873,𝑐2=0.003234.

Now, we obtain the pure effect of vacuum polarization due to the gravitational field without any boundary conditions in Euclidean (outside) region with the following metric:𝑑𝑠2=Ω(𝜂)𝑑𝜂2+3𝚤=1(𝑑𝑥𝚤)2.(15)

To this end, we calculate the automatically renormalized stress tensor in Krein space quantization for the massless scalar field in de Sitter spacetime with Euclidean signature. One may use [2]0||𝑇𝜈𝜇𝑔𝑘𝑙||0||𝐾=̃𝑔𝑔1/20||𝑇𝜈𝜇̃𝑔𝑘𝑙||0||𝐾12880𝜋216(1)𝐻𝜈𝜇(3)𝐻𝜈𝜇,(16) where ̃𝑔𝑘𝑙 is the flat Euclidean metric for which 0|𝑇𝜈𝜇[̃𝑔𝑘𝑙]|0|𝐾=0, and(1)𝐻𝜈𝜇=0,(3)𝐻𝜈𝜇=3𝛼4𝛿𝜈𝜇.(17)

We then obtain0||𝑇𝜈𝜇𝑔𝑘𝑙||0||𝐾=1960𝜋2𝛼4𝛿𝜈𝜇,(18) which is exactly the same result for the Lorentzian case [2]. Therefore, the corresponding effective radial pressures for the Euclidean (outside) and Lorentzian (inside) regions with 𝛼out and 𝛼in, due to pure effect of gravitational vacuum polarization without any boundary condition, are given, respectively, by𝑃𝐸out=0|𝑇𝑟𝑟[𝑔𝑘𝑙||]|0𝐾1=960𝜋2𝛼4out,𝑃𝐿in0||𝑇=𝑟𝑟𝑔𝑘𝑙||0||𝐾1=960𝜋2𝛼4in.(19)

The corresponding gravitational pressure on the spherical shell in Krein space quantization is then given by𝑃𝐺=𝑃𝐿in𝑃𝐸out1=960𝜋21𝛼4in1𝛼4out.(20)

4. Conclusion

The Casimir effect has been studied for spherical bubbles with different vacua inside and outside, corresponding to de Sitter metrics. The metrics inside and outside are taken by Lorentzian and Euclidean, respectively. The case of different vacua in a Lorentzian de Sitter spacetime has already been studied in [15]. In the present work, we have shown that the same results could be attained using Krein space quantization. Presence of the unphysical negative frequency states plays the role of an automatic renormalization tool.


In this appendix, we review the elementary facts about Krein space quantization. As we know, the origin of divergences in standard quantum field theory lies in the singularity of the Green's function. The divergence appears in the imaginary part of the Feynman propagator, and the real part is convergent [35]: 𝐺𝑃𝐹𝑥,𝑥1=𝛿𝜎8𝜋0+𝑚2𝜃𝜎8𝜋0𝐽12𝑚2𝜎0𝑖𝑁12𝑚2𝜎02𝑚2𝜎0𝑖𝑚24𝜋2𝜃𝜎0𝐾12𝑚2𝜎02𝑚2𝜎0.(A.1)

Consideration of negative frequency states removes singularity of the Green’s function with exception of delta function singularity:𝐺𝑇𝑥,𝑥1=𝛿𝜎8𝜋0+𝑚2𝜃𝜎8𝜋0𝐽12𝑚2𝜎02𝑚2𝜎0,𝜎00.(A.2)

However, considering the quantum metric fluctuations removes the latter singularity:𝐺𝑇𝑥,𝑥1=8𝜋𝜋2𝜎21𝜎exp202𝜎21+𝑚2𝜃𝜎8𝜋0𝐽12𝑚2𝜎02𝑚2𝜎0,(A.3) where 𝜎21 is related to the density of gravitons. When 𝜎0=0, due to the metric quantum fluctuation 𝜎210, and we have 𝐺𝑇1(0)=8𝜋𝜋2𝜎21+𝑚2.16𝜋(A.4)

By using the Fourier transformation, we obtain [36]𝐺𝑇=𝐺(𝑝)𝑇𝑚(𝑝)+𝑃𝑃2𝑝2𝑝2𝑚2.(A.5)

However, in the one-loop approximation, the contribution of delta function is negligible and the Green’s function in Krein space quantization appearing in the transition amplitude is𝐺𝑇|||(𝑝)one-loop𝐺𝑇||(𝑝)one-loop𝑚𝑃𝑃2𝑝2𝑝2𝑚2.(A.6)


The author would like to thank M. V. Takook for useful discussions.