Casimir Energy Calculation for Massive Scalar Field on Spherical Surface: An Alternative Approach

# Casimir Energy Calculation for Massive Scalar Field on Spherical Surface: An Alternative Approach

M. A. Valuyan Department of Physics, Semnan Branch, Islamic Azad University, Semnan, Iran
December 4, 2018
###### Abstract

In this study, the Casimir energy for massive scalar field with periodic boundary condition was calculated on spherical surfaces with , and topologies. To obtain the Casimir energy on spherical surface, the contribution of the vacuum energy of Minkowski space is usually subtracted from that of the original system. In large mass limit for surface ; however, some divergences would eventually remain in the obtained result. To remove these remaining divergences, a secondary renormalization program was manually performed. In the present work, a direct approach for calculation of the Casimir energy has been introduced. In this approach, two similar configurations were considered and then the vacuum energies of these configurations were subtracted from each other. This method provides more physical meaning respect to the other common methods. Additionally, in large mass limit for surface , it provides a situation in which the second renormalization program is automatically conducted in the calculation procedure, and there was no need to do that anymore manually. Finally, by plotting the obtained values for the Casimir energy of the topologies and investigating their appropriate limits, the logic agreement between the results of our scheme and those of previous studies were discussed.

## I Introduction

Casimir energy is the difference between zero point energy in presence and absence of non-trivial boundary condition. This effect was first predicted and calculated by H.B.G. Casimir in 1948. He was the first one who explained the attraction between two parallel uncharged perfectly conducting plates in vacuum h.b.g. (). First attempts to observe this phenomenon were made in 1958 by M. J. Sparnaay Sparnaay.M.J. (), and then more accurate measurements verified Casimir’s prediction. As Casimir effect was then considered an interesting effect of vacuum polarization, it has found many applications. This effect has an important role in different fields of physics such as quantum field theory  quantum.field.theory.1 (); quantum.field.theory.2 (); quantum.field.theory.3 (); quantum.field.theory.4 (); quantum.field.theory.5 (), condensed matter physics condensed.matter.1 (); condensed.matter.2 (), atomic-molecular physics atomic.molecular.1 (); atomic.molecular.2 (); atomic.molecular.3 (); atomic.molecular.4 (), gravity, astrophysics astro.physics.1 (); astro.physics.2 () and mathematical physics Generalized.Abel.Plana.Saharian (); Mathematical.physics.1 (); Mathematical.physics.2 (). Due to the definition of the Casimir energy, two divergent terms should be subtracted from each other, which is not a simple task. In this regard, to achieve this purpose, various regularization and renormalization techniques were developed that found their own importance. In fact, Casimir energy calculations have provided a situation where various regularization and renormalization techniques have been developed in mathematical physics. Zeta function regularization techniques Zeta.function.1 (); Zeta.function.2 (); Zeta.function.3 (), Green’s function methods Greens.function. () and multiple-scattering expansions multiple.scattering. () are some of these techniques. Advantages and disadvantages of these methods are also reviewed in previous works review.of.some.techniques. (). Whereas the Casimir energy is a physical quantity that its value depends on the system size (e.g. the distance of parallel plates and radius of sphere). In some previous methods to calculate the Casimir energy, the terms, which do not depend on the system size have been eliminated from the Casimir energy expression wolfram. (). Physically, this elimination could be justified. However, the mathematical trend in this way cannot be maintained. In this paper, we have used a method that is free from these kinds of ambiguities in calculation of the Casimir energy. This method was first used by T.H. Boyer for the calculation of the Casimir energy for an electromagnetic field confined in a conducting sphere boyer. () and it was named Box Subtraction Scheme (BSS) in the later works BSS.1 (); BSS.2 (). Up to now, in order to reduce possible ambiguities appearing in the calculation of the Casimir energy, multiple studies used this method other.BSS.1 (); other.BSS.2 (). In the BSS, the Casimir energy is calculated by introducing two similar configurations and then their zero point energies in proper limits are subtracted from each other. To define this method concretely for our problem, as Fig. (1) shows, we consider two similar spheres. These two spheres are named A and B with radii and , respectively. Then, the zero point energies of massive scalar field on the surface of these two spheres are computed and in following, the obtained vacuum energies are subtracted from each other. Finally, by taking the radius of sphere B to be infinite, the Casimir energy of sphere A will be obtained. Therefore, we have,

 ECas.=limb→∞[EA−EB], (1)

where and are zero point energy densities for and configurations, respectively. In the BSS, the contribution of vacuum energy of Minkowski space is substituted with a configuration (like sphere B) that in proper limit (), it will approach to the properties of Minkowski space. Use of two similar configurations in the BSS provides the possibility that parameters of the second configuration (like B configuration) play a useful role as a regulator in divergence removal. These added parameters allow the calculation of the Casimir energy via this scheme to be more clear in details. This scheme also reduces the need for using the analytic continuation techniques in the calculation process. Therefore, the related complications and ambiguities, due to analytic continuation techniques, would be avoided BSS.1 (); BSS.2 (); other.BSS.1 (); other.BSS.2 (). The other successful experience of using the BSS is in higher orders of radiative corrections to the Casimir energy for different configurations, resulting in converged and consistent answers in all previous works in this category other.BSS.Radiative.Correction.2 (). The BSS has been previously used for the calculation of the Casimir energy on flat space, but in this paper, it is intended to calculate the Casimir energy of a massive scalar field with periodic boundary condition on a curved space (e.g. on sphere with and topologies). In common methods, to obtain the Casimir energy on curved manifolds, contribution of the vacuum energy of Minkowski space was subtracted from the vacuum energy of original manifold new.developements. (); New.Paper.On.Sphere. (); Mamaev.1979. (); Advances.book. (). Since the Minkowski space is the ï¬at space, and the original manifold is curved, this subtraction is a comparing between two different kinds of spaces. The BSS, by providing the subtraction between vacuum energies of similar configurations, presents an opportunity for similar spaces to be compared with each other. In fact, the introduction of similar configurations in the BSS, in addition to creating more clarity in the computing process, also has better physical grounds. The other point in using BSS is manifested when we use it in calculation of the Casimir energy for topology. In all previous works, after subtracting the vacuum energy of Minkowski space from that of original system, to reach a physically consistent answer in large mass limit, an extra renormalization procedure has been performed. This secondary renormalization program to remove remaining infinities, that usually appeared due to the mass of the field, has been manually conducted (e.g. Section (3.4) of Ref. new.developements. ()). In our study, the aforementioned renormalization program is automatically performed in the BSS and there is no need to do that manually. It reflects another advantage of the BSS over curved manifolds.

In the next section, through the BSS, the Casimir energy density for massive scalar field with periodic boundary condition on three spherical surface with , and topology with radius are obtained. In the following, for each surface, the Casimir energy values in specific limits of mass of the scalar field (such as or ) will be investigated. In the last section, all of our discussion concerning the BSS are summarized.

## Ii The Casimir energy

In this section, the Casimir energy for massive scalar field with periodic boundary condition on surface , and are calculated. The details of calculation for each topology is investigated in separated subsections and their extreme limits are also discussed separately. At the first step, to present a simple introduction of BSS, we start with calculation of the Casimir energy on a circle with radius .

### ii.1 On a Circle (S1)

The vacuum energy density for massive scalar field with periodic boundary condition on a circle with radius can be obtained from a problem in which the massive scalar field lives in one dimension between two points with periodic boundary condition by distance  (for more details see Refs. new.developements. (); other.BSS.2 ()). Thus, we can write the vacuum energy density of massive scalar field on a circle with radius as:

 E=14πa∞∑n=−∞ωn=12πa∞∑n=0ωn−m4πa, (2)

where is the wave number and is the mass of the field. Now, by using the definition of BSS given in Eq. (1), another circle with radius  (), as shown in Fig. (1), is defined and the vacuum energies of these two circles should be subtracted from each other. Therefore, we have:

 EA−EB=14πa{2∞∑n=0√m2+n2a2−m}−{a→b}. (3)

All summations in Eq. (3) are divergent. Accordingly, to regularize their infinities, the Abel-Plana Summation Formula (APSF) is employed. This formula helps all summation forms given in Eq. (3) to be transformed to the integral form and the removal process of their infinite parts would be conducted with clarity. The usual form of APSF that we have used is:

 ∞∑n=0f(n)=12f(0)+∫∞0f(z)dz+i∫∞0f(it)−f(−it)e2πt−1dt. (4)

Now, by applying Eq. (4) on the summation of Eq. (3) we have:

 EA−EB={m4πa+12πa∫∞0√m2+x2a2dx+B(a)−m4πa}−{a→b}, (5)

where is the Branch-cut term of APSF. The first integral term on the right hand side of Eq. (5) is divergent. Analogously, the same integral appears for configuration B in the second square bracket of Eq. (5). To remove their infinities in subtracting procedure, we first replace the upper limit of these two integrals with multiple cutoffs and respectively. Then, by calculating integrations, we would have two separate expressions as a function of cut-offs and . Now, by expanding the result in the limit , each integral becomes:

 ∫Λ0√x2a2+m2dxΛ→∞⟶Λ22a+m2a4−m2a4ln(m2a24Λ2)+O(Λ−2) (6)

Appropriate adjusting for cut-offs and supplemented by subtracting procedure, which is provided by BSS, helps the infinite terms appeared in the above expansion to be canceled. The BSS also helps the finite term appeared in the expansion to cancel each other out exactly. All of terms in higher order of do not leave any contribution in the limit . Therefore, the only remaining terms after these cancelations would be the Branch-cut terms. Finding an analytical and closed answer for integration of is very cumbersome. Hence, before computation of the integral, the denominator of the integrand is expanded. Then, by calculating integral we have:

 EA−EB=−m2π∫∞1√z2−1e2πmaz−1dz−{a→b}=−m2π2a∞∑j=1K1(2πmaj)j−{a→b}. (7)

Now, by applying the limit given in Eq. (1), the Casimir energy density on a circle with radius becomes:

 ECas.=−μa2π2a2∞∑j=1K1(2πμaj)j, (8)

where and the extreme limit of the result becomes:

 (9)

In Fig. (2), we have plotted the Casimir energy density given in Eq. (8) as a function of radius . This figure shows a sequence of plots for . This sequence of plots indicates that the Casimir energy for massive cases rapidly converges into the massless limit when . This behavior for the Casimir energy is compatible with the previously reported results and it is also expected according to physical grounds new.developements. (); New.Paper.On.Sphere. ().

### ii.2 On a Sphere (S2)

In this subsection, we present the Casimir energy calculation via BSS for massive scalar field with periodic boundary condition on a surface with topology. At the first step, we remind the reader of the metric for this surface on which the scalar field lives on it. Therefore, we have:

 ds2=dt2−a2(dθ2+sin2θdφ2). (10)

This metric describes dimensional space-time on a sphere with radius and the wave equation for a scalar field with mass on this sphere can be written as:

 (∇k∇k+ξR+m2)ϕ(x)=0, (11)

where is the covariant derivative and is the scalar curvature of space-time and is the conformal coupling constant. The coordinate shows the space-time coordinates on the spherical surface. By substituting the values of and for Eq. (11) and expanding the covariant derivative , we have:

 a2∂2∂t2ϕ(x)−1sinθ∂∂θ(sinθ∂∂θϕ(x))−1sin2θ∂2∂φ2ϕ(x)+(14+m2a2)ϕ(x)=0. (12)

The orthonormal set of solutions to Eq. (12) obeying periodic boundary conditions for both and are represented as:

 {ϕ(+)ℓM(x)=1a√2ωℓeiωℓtYℓM(θ,φ),ℓ=0,1,2,3,...M=0,±1,±2,...,±ℓϕ(−)ℓM(x)=(ϕ(+)ℓM(t,θ,φ))∗, (13)

where are the spherical harmonic function and are wave numbers. In order to obtain zero point energy of field on sphere, the field operator of should be expanded according to the following equation:

 ϕ(x)=∞∑ℓ=0ℓ∑M=−ℓ[ϕ(−)ℓM(x)aℓM+ϕ(+)ℓM(x)a†ℓM], (14)

where and are annihilation and creation operators of the field, respectively. The metric energy-momentum tensor is obtained by varying the Lagrangian corresponding to Eq. (11) with respect to the metric tensor . Its diagonal component is:

 T00=(1−2ξ)∂0ϕ∂0ϕ+(2ξ−12)g00∂kϕ∂kϕ−ξ(ϕ∇0∇0ϕ+∇0∇0ϕϕ) +[(12−2ξ)m2g00−ξG00−2ξ2Rg00]ϕ2, (15)

where is Einstien tensor and is Ricci tensor. By substituting Eq.(14) for Eq. (II.2) and then calculating the expectation value of energy-momentum tensor, the vacuum energy density for massive scalar field on sphere will be obtained as follows:

 E=<0∣T00∣0>=14πa2∞∑ℓ=0(ℓ+12)ωℓ. (16)

Due to the spherical symmetry, each energy level degenerates () folds and high frequency modes formally render these sums as divergent.

To start the calculation of the Casimir energy via the BSS, as Fig. (1) symbolically shows, two configurations should be considered. In this problem, these configurations are two similar spheres with radii and , which named A and B configuration, respectively. Now, by using Eqs. (1,16) we have:

 (17)

where and are zero point energy densities for and configurations, respectively. Both of the subtracted expressions in the above equation are divergent. Therefore, a regularization technique is required. At this step, the common method is using the APSF and the prescribed form of this formula for half integer parameters is (for a general review see Ref. Generalized.Abel.Plana.Saharian ()):

 ∞∑n=0f(n+12)=∫∞0f(z)dz−i∫∞0f(it)−f(−it)e2πt+1dt. (18)

By applying the above form of APSF on Eq. (17), all summations are converted to integration form and become:

 EA−EB={14πa3∫∞0z√m2a2+z2dz+B(a)}−{a→b}, (19)

where is the Branch-cut term of APSF and usually has a finite value. While, the first term in both square brackets are divergent and its infinity should be removed. In order to remove infinities due to these two terms, the cutoff regularization technique is employed. Therefore, at the first step, the upper limit of integrals in Eq.(19) is replaced with and , respectively. Then, by calculating the integrations, we will have an answer as a function of the cutoffs and , respectively. When the cutoffs and go to infinity, the following expansion for each integral is obtained:

 ∫Λ0z√m2a2+z2dzΛ→∞⟶Λ33+12m2a2Λ−m3a33+O(Λ−1). (20)

It can be shown that, by selecting proper values for and in the subtraction process given in Eq.(19), all of the finite and infinite terms of above expansion for integral terms will be canceled and the only remaining terms in Eq.(19) are Branch-cut terms. Thus, we have:

 EA−EB=m32π∫10z√1−z2e2πmaz+1dz−{a→b}. (21)

Unfortunately, an analytical and closed answer for the integration of above equation does not exist. Therefore, by expanding the denominator of the integrand as the following form, we have:

 1e2πmaz+1=∞∑j=1(−1)j+1e−2πmazj. (22)

After substituting Eq. (22) into Eq. (21), this becomes:

 EA−EB = m32π∞∑j=1(−1)j+1∫10z√1−z2e−2πmazjdz−{a→b} (23) = μ2a8πa3∞∑j=1(−1)j+1j[43μaj−I2(2πμaj)+L2(2πμaj)]−{a→b},

where and is the modified Bessel function and is Struve function. At the final step, the limit in Eq. (1) should be computed. Thus, the Casimir energy density expression after this limit becomes:

 ECas.=μ2a8πa3∞∑j=1(−1)j+1j[43μaj−I2(2πμaj)+L2(2πμaj)]. (24)

Our obtained expression in Eq. (24) is compatible with the previously reported result obtained in Refs. new.developements. (); New.Paper.On.Sphere. (). The main difference between these two works is only in applying calculation methods. It seems that selecting two similar configurations and subtracting the contributions of vacuum energy of these configurations have provided a situation in which all the infinities are completely removed. It can be also shown that, the Casimir energy density for a massless scalar field vanishes. To find the large-mass limit of the Casimir energy density, we go back to the original expression for the vacuum energy density of surface, given in Eq. (17), and we select the mass as a regulator. Then, we expand the summand in the limit and we have:

 EA−EBm→∞⟶14πa2∞∑ℓ=0[m(ℓ+12)+12ma2(ℓ+12)3+...]−{a→b}. (25)

To regularize the summations, the APSF are employed. The subtraction of vacuum energies, provided by BSS, helps the infinite parts of APSF to be removed and the only remaining terms would be the Branch-cut one. Thus, we have:

 EA−EBm→∞⟶14πa2{2m∫∞0tdte2πt+1+1ma2∫∞0t3dte2πt+1}−{a→b}. (26)

In limit the first term in the both square brackets of Eq. (26) is still divergent. Adjusting the proper value for the parameter , allows the infinities to be cancelled via BSS due to these two terms. At the last step, by computing the limit the final remaining term for the Casimir energy in limit is obtained as: . In Fig. (3), we have plotted the Casimir energy density as a function of radius . In this figure, a sequence of plots for is displayed. This sequence of plots shows that the Casimir energy for massive cases converges rapidly to the massless limit when . This figure also shows that the Casimir energy values become zero, when the radius of sphere becomes infinite. This behavior for the Casimir energy is compatible with the previously reported results and it is also expected according to physical grounds new.developements. (); New.Paper.On.Sphere. ().

### ii.3 Three Dimensional Spherical Surface(S3)

In order to find the vacuum energy density of massive scalar field on a surface with topology we remind the metric of the closed Friedman model as:

 ds2=a2(η)(dη2−dχ2−sin2χdΩ2), (27)

where is a scale factor with dimension of length and . Additionally, is a conformal time variable and . By replacing and scalar curvature with Eq. (11), the form of equation of motion can be written as:

 ϕ′′(x)+2a′aϕ′(x)−△(3)ϕ(x)+(m2a2+a′′a+1)ϕ(x)=0, (28)

where is the angular part of the Laplacian operator on a sphere and . To reach the vacuum energy density of system, the canonical quantization procedure should be conducted. Hence, by solving the differential equation given in Eq. (28), we have found the orthonormal set of solutions. In the following, the field operator expanded in terms of the obtained orthonormal solutions is substituted in component of stress energy-momentum tensor. Finally, the mean value of the tensor in the initial state gives us the vacuum energy density of the system as:

 E=14π2a4∞∑n=1n2ωn, (29)

where is the wave number. It should be noted that for simplicity, we have followed the calculation for static Einstein model ().

Now, to calculate the Casimir energy density, we have employed the BSS again. Therefore, as Fig. (1) shows, two similar spheres are considered and we define the Casimir energy by Eq. (1). Then, by applying the APSF given in Eq. (4) on the summation of vacuum energy, all summations would be transformed to the integration forms and we obtain:

 EA−EB=14π2a4∞∑n=1n2ωn−{a→b}={14π2a4∫∞0z2√z2+m2a2dz+B(a)}−{a→b} (30)

where is the Branch-cut term. As is apparent, the first term in the square bracket of Eq. (30) is divergent. To remove its infinity, the cut-off regularization scenario would be repeated the same as what occurred for Eqs. (5,6). Therefore, we replace the upper limits of integrals in Eq. (30) with a separate cutoffs and , respectively. After calculating each integral and expanding their answers in limit , we have:

 ∫Λ0z2√z2+m2a2dzΛ→∞⟶Λ44+Λ2m2a24−m4a416ln(4Λ2m2a2)+m4a432+O(Λ−2). (31)

The subtraction of vacuum energies of two configurations enable us to remove all contribution of the integral terms and only the remaining terms would be the Branch-cut one. There is not a direct way for finding of analytical and closed answer for the integral of . Therefore, after expanding the denominator of integrand, we calculate it. At the last step, by computing the limit , the Casimir energy density for massive scalar field on a spherical surface with topology is obtained as:

 ECas.=μ2a8π4a4∞∑j=12πμajK1(2πμaj)+3K2(2πμaj)j2, (32)

where and the main following limits for this result are:

 (33)

In Fig. (4), the Casimir energy density as a function of radius for a sequence of plots of is displayed. This plot shows that the Casimir energy for massive cases rapidly converges into the massless limit when . This behavior for the Casimir energy is compatible with the previously reported results new.developements. (); New.Paper.On.Sphere. (). The very simple elimination of divergences in BSS shows that this method is highly powerful in this task and it can be a good candidate in removal divergences in the known regularization techniques, specially for curved spaces. This method, by comparing two similar kinds of manifolds in its definition, provides sufficient degrees of freedom to adjust the cutoffs for removal process and it conceptually provides more physical grounds.

## Iii Conclusions

In the present paper, the Casimir energy for the massive scalar field with periodic boundary condition on the spherical surface with , and topologies were calculated. In the present calculation, procedure of the Box Subtraction Scheme (BSS) as an alternative and direct method was introduced and its various aspects were discussed. In this scheme, the Casimir energy was obtained by subtracting the vacuum energy of two similar configurations in proper limits. This subtraction in the BSS enables us to remove all divergences clearly. We maintain that subtracting the vacuum energy of a curved manifold with nontrivial topology from the vacuum energy of flat space (e.g. the Minkowski space) is irrelevant and at least has less physical meaning. Indeed, in that subtraction, two different kinds of spaces are compared with each other. In the BSS, we allow two similar kinds of vacuum energies to be compared with each other and we maintain that, this subtraction has a more physical ground than the other common definitions of the Casimir energy. The BSS also provides a directive way in calculation of the Casimir energy on spherical surface with topology in proper limit. The obtained results are consistent with expected physical grounds and also in good agreement with the previously reported results. As applying some regularization techniques like analytical continuation technique, give rise to some ambiguities in the Casimir energy calculation other.BSS.Radiative.Correction.2 (), it is hoped that the BSS as a regularization technique will be successful in reducing these sorts of ambiguities. It is also anticipated that the mentioned scheme will be useful in radiative correction to the Casimir energy in various boundary conditions on curved manifolds.

###### Acknowledgements.
The Author would like to thank the research office of Semnan Branch, Islamic Azad University for the financial support.

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