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Vacuum Polarization of a Scalar Field in a Rectangular Waveguide

An analysis of the one-loop vacuum fluctuations associated with a scalar field confined in the interior of a infinite waveguide of rectangular cross section is presented. We first consider the massless scalar field defined in a four-dimensional Euclidean space. To identify the infinities of the vacuum fluctuations we use a combination of dimensinal and zeta function analytic regularization procedures. The infinities which occur in the one-loop vacuum fluctuations fall into two distinct classes: ultraviolet divergences that are renormalized by the introduction of bulk counterterms and also surface and edges divergences that demand countertems concentrated on the boundaries. We present the detailed form of the surface and edge divergences. Finally we discuss how to generalize our calculations for a confined massive scalar field defined in a higher dimensional Euclidean space.


Introduction
In 1948 Casimir [1] predicted that uncharged, parallel, perfectly conducting plates should attract with a force per unit area, F (L) ∝ 1 L 4 , where L is the distance between the plates. This force can be interpreted as the manifestation of the zero-point energy of the electromagnetic field in the presence of the plates. Complete reviews of this effect can be found in refs. [2] [3] [4].
The aim of this paper is to generalize the results obtained by Fulling [5] and others, for the renormalized vacuum fluctuations of a massless scalar field calculated between two parallel flat mirrors. Our purpose is to analyse the vacuum fluctuations near surfaces where the field satisfies some classical boundary condition, the boundary being defined by two of the coordinates which we assume are closed intervals. A special case of this situation is the infinite waveguide of rectangular cross section. We are thus interested in obtaining the renormalized one-loop vacuum fluctuations associated with a scalar field defined in the interior of a infinite waveguide of rectangular cross section. We first consider a massless scalar field in a four-dimensional Euclidean space, and then extend our calculations to a massive scalar field defined in a higher dimensional Euclidean space.
As stated by Deutsch and Candelas [6] and also Candelas [7] the calculation of the renormalized one-loop vacuum fluctuations contains all the important steps that we need to calculate the renormalized stress-energy tensor associated with the scalar field.
It is well known that there are two quantities which might be expected to correspond to the total renormalized energy of quantum fields [6]. The first is the mode sum energy, defined as where 1 2 ω is the zero-point energy for each mode, N(ω)dω is the number of modes with frequencies between ω and ω + dω in the presence of boundaries, and N 0 (ω)dω is the corresponding quantity evaluated in empty space. In this case the divergences that appear in the regularized energy are given by Weyl's theorem and its generalization, that relate the asymptoptic distribution of eigenvalues of some elliptic differential operator with geometric parameters of the surface where the fields satisfy some boundary condition [8] [9]. The second is the volume integral of the renormalized energy density E vol ren obtained by the Green's functions method [10] [11] [12] [13].
For special configurations where the modes and the eigenfrequencies of the electromagnetic field in the presence of the boundaries can be found, it is possible to find the Casimir energy and the Casimir force. For the case of cylindrical geometry, Balian and Duplantier and also others calculated the Casimir energy of the electromagnetic field [14] [15] [16] [17]. For the important case of scalar or fermionic fields in the presence of a spherical shell, Bender and Hays [13], studying the global problem, found the renormalized zero-point energy of these fields assuming that the fields are confined in a spherical region of the space. Many years before Bender and Hays, Boyer [18] and Davies [19] also studying the global problem, obtained the Casimir energy of an electromagnetic field in the presence of a perfectly conducting spherical shell. A systematic study of the spherical shell configuration was made by Milton [20]. He calculated the zero-point energy for gluons and fermions, assuming that the fields are confined in the interior of the shell. He later calculated the Casimir energy of massless fermions in the presence of a spherical shell, in this case taking into account the external modes of the fermions [21]. More recently, working in a generic flat d-dimensional spacetime Bender and Milton [22] obtained the Casimir energy associated with a massless scalar field taking into account the interior and exterior contributions of the modes in the presence of the hypersphere. Still studying the spherical configuration, Romeo [23] investigated the Casimir energy of a massless scalar field and for QED assuming that the field is confined in the interior of the spherical shell. Finally Bordag et.al. [24] have calculated the Casimir energy associated with a massive scalar field in the presence of a spherical shell assuming that the interior and the exterior modes give contributions to the energy.
Although the main interest in the literature is the global approach, where the Casimir energy can be found and from which one can derive the force on the boundaries, the necessity of studying the local problem has often been suggested. First, the local properties of the vacuum fluctuations can in principle be observed by measuring the energy level shift of atoms interacting with an electromagnetic field [25] [26]. Second, the distortion of the vacuum fluctuations due to the presence of classical boundaries can also be measured by studying the spontaneous and induced emission of excited atoms in the presence of classical boundaries [27] [28]. For a update discussion of QED vacuum effects, see for example [29]. Finally, it is clear that local results contain more information than the global ones. A few years ago, Actor [30] and also Actor and Bender considered this kind of problem [31]. These authors studied the use of the generalized zeta function method [32] [33] [34] [35] to find the one-loop effective action associated with a scalar field defined in the interior of a infinite waveguide of rectangular cross section. The local problem has also been considered by other authors. For the case of parallel plates geometric configuration Brown and Maclay [12], using the Green's function method obtained the renormalized vacuum expectation value of the stress-energy tensor associated with an electromagnetic field. Deutsch and Candelas [6] evaluated the renormalized stress-energy tensors associated with a conformally coupled scalar field and also with an electromagnetic field in the wedge-shaped region formed by two plane boundaries.
Recently, Brevik et al [36] repeated these calculations using Schwinger's source theory. Their results agree with those of Deutsch and Candelas. In the spherical geometry, the local problem was also investigated by Olaussen and Ravndal [37], who studied the vacuum fluctuations of an electromagnetic field within a perfectly conducting spherical cavity. They found that the vacuum expectation value of the squared electric and magnetic fields diverge as one approaches the boundary. This result has also been obtained by Deutsch and Candelas [6], DeWitt [38], and Kennedy et al [39]. Olaussen and Ravndal [40] and also Milton [41] generalized this result to the non-abelian gluon fields in the MIT bag model. It has often been suggested that these surface divergences are related to the uncertainly relation between the field and the canonical conjugate momentum associated with the field [37] [42] [43]. In other local calculations, Ford [44] and Ford and Svaiter [45] discuss the possibility of amplification of the vacuum fluctuations. These authors studied the renormalized vacuum fluctuations associated with a scalar and electromagnetic field near the focus of a parabolic mirror. Using the geometric optics approximation they found that the parabolic mirror geometry can produce large vacuum fluctuations near the focus, much as what happens in the classical focusing effect by the parabolic mirror geometry.
In our study of the one-loop vacuum fluctuations associated with a scalar field we develop a method adequate to deal with rectangular geometries. In this paper, using analytic regularization procedures, we first calculate the regularized vacuum fluctuations associated with a massless scalar field, confined in the interior of a infinite waveguide with rectangular cross section, in a fourdimensional Euclidean space. We first rederive the well-known result that there are surface and edge divergences that require the introduction of surface [42] [46] [47] and edge counterterms in the renormalization procedure. Then, we show how it is possible to generalize our results to a massive scalar field in a higher dimensional Euclidean space. Preliminary calculations of the renormalized vacuum expectation value of the stress-energy tensor in the rectangular waveguide were performed by Dowker and Banach [48]. Also E mode ren in rectangular geometries has been calculated in refs. [49] [50]. A seminal paper studying these geometries was made by Ambjorn and Wolfram [51]. More recently Milton and Ng studied the Casimir effect in (2 + 1) Maxwell-Chern-Simons electrodynamics in a rectangular domain [52]. Also Hacyan et al [53] and Maclay [54] respectively studied the vacuum fluctuations of the electromagnetic field and the Casimir force in the interior of a rectangular waveguide.
When studying interacting field theories for translationaly invariant systems, one usually goes from coordinate to momentum space representation, a more convenient framework to analyse the divergences of the theory. In this representation translational invariance is expressed by momentum conservation conditions. Because the system of interest for this work possesses translational invariance along two directions, it is more convenient to use a mixed coordinate-momentum representation for the Green's functions [55]. For a recent treatment of systems without translational invariance, see for example [56] and also [57]. As we discussed, the fundamental problem in the infinite waveguide of rectangular cross section is the lack of translational invariance, which manifests itself by the fact that the Green's functions associated with the scalar field are expressed in terms of infinite double summations. Although the one-loop vacuum fluctuation is written in terms of expressions involving double summations, a simple trigonometric identity allows us to obtain expressions with only one summation. The advantage is that all the calculations can then be done analytically. We show that the form of surface and edge counterterms that we have to introduce to renormalize the one-loop vacuum fluctuations, can be explicitly calculated and we discuss some different physical arguments that support the introduction of surface countertems that remove these divergences.
Throughout the paper we use the term bulk counterterms to those conterterms that are not related to the existence of the boundaries. For instance, in the case of periodic boundary conditions we have these counterterms. As well as a second group that we call surface and edge counterterms that are related to the existence of the boundaries.
The organization of the paper is the following: In the section II we study the one-loop vacuum fluctuations of a massive scalar field confined within a rectangular waveguide. In section III we use an analytic regularization method to identify the infinities that appear in the expression of the vacuum fluctuations of the massless field near the boundaries. In section IV, we show how it is possible to generalize our calculations to a massive scalar field defined in a higher dimensional Euclidean space. Conclusions are given in section V. Finally in the appendix we present the formal relation between the one-loop vacuum fluctuations and the local generalized zeta function. In this paper we useh = c = 1.
2 Vacuum fluctuations of a scalar field confined in a rect-

angular waveguide
In this section and in the next one we will investigate the one-loop vacuum fluctuations associated with a scalar field defined in the interior of a infinite waveguide with rectangular cross section.
First, we assume that the scalar field is defined in a four-dimensional Euclidean space, where the last two coordinates are unbounded, while the first two, which we call x 1 and x 2 , lie in the interval [0, a] and [0, b], respectively. We assume Dirichlet boundary conditions on the boundaries. The free field is defined in the region with Dirichlet boundary conditions at x 1 = 0 and x 1 = a, and also x 2 = 0 and x 2 = b. As stressed previously, the lack of translational invariance introduces surfaces and edges divergences.
One way to reduce the degree of these divergences is to smooth out the surface of the plates, for example by using soft, hard or semi-hard boundary conditions [58] [59] [60]. The one-loop graphs will depend on the ad-hoc model assumptions, and consequently we prefer to maintain the hard walls. Instead of smoothing the plates surfaces, another way to avoid surface divergences, discussed by Kennedy et al [39], is to treat the boundary as a quantum mechanical object. This approach was used recently, by Ford and Svaiter in the case of parallel flat plates, to solve a long standing paradox concerning the renormalized energy of minimally and conformally coupled scalar fields [43]. As we stressed before, we prefer, at least at the moment to keep only a hard classical boundary conditions.
We would like to stress that we are studying first the massless four-dimensional case since this simpler case will give us an indication of the behavior that may be expected in the most general case of a massive scalar field in a higher dimensional Euclidean space. A logical going is to use analytic regularization procedures to identify the divergent contributions that appear in the one-loop vacuum fluctuations. Let us start using dimensional regularization, working at the begining in a d-dimensional Euclidean space. Assuming Dirichlet boundary conditions on the walls we calculate first the two-point Schwinger function at coincident points in the interior of the waveguide. Then we use analytic regularization procedures to identify the form of the surface and edge divergences. In the one-loop vacuum fluctuations, ϕ 2 (x 1 , x 2 , a, b) , we will change the notation of the previous section to x 1 = x, x 2 = y (note that in our notation ϕ 2 (x, y) means the one-loop vacuum fluctuations in one point with cartesian coordinates (x, y) of the rectangular cross section of the waveguide). Let the waveguide be oriented along the z axis with walls at x = 0 and a and y = 0 and b. In this mixed representation, since we are assuming Dirichlet boundary conditions, the expression for the one-loop vacuum fluctuations, T DD (x, y, a, b, d) = ϕ 2 (x, y) can be written as: There are two points that we would like to stress. First is the fact that to perform analytic regularizations, we have to introduce a parameter µ with dimension of mass in order to have dimensionless quantities raised to a complex power. For sake of simplicity, we omit the µ factor in the following. Second is the fact that the generalization for the case of Neumann boundary conditions is straightforward, although in this case infrared divergences associated with the n = 0 mode will appear in the case of massless scalar field. To circumvent this situation we must have a finite Euclidean volume to regularize the model in the infrared.
Using trigonometric identities, it is convenient to write the one-loop vacuum fluctuation in the following way : where each expression of the above equation are given by the following. For T (a, b, d) we have The second term in Eq.(4), T (x, a, b, d) is given by The expression for T (y, a, b, d) has the same functional form of the above equation only changing x by y and a by b. Consequently we have : and finally : Thus the contribution given by Eq.(8) contains the same contributions given by Eqs. (5), (6), (7), as well as a contribution that contains edge divergences that we will define as N(x, y, a, b, d). It is given by Let us study each contribution separately. Using dimensional regularization on Eq.(5) it is possible to write T (a, b, d) in terms of the Epstein zeta function. Thus we have The contribution given by T . This expression is given by Although Eq. (11) is written in terms of two sums, one of the sums can be easily done using a trigonometric expression given by [64] [65] : Using Eq. (12), it is possible to write Eq.(11) as where : and . (15) It is clear that to calculate the analytic structure for the case of the position dependent divergent part T (y, a, b, d) we can use the same method that we use for T (x, a, b, d). Consequently the where : and . (18) Using dimensional regularization in both expressions I 1 (a, b, d) and R 1 (a, b, d), we obtain a special Epstein-Hurwitz zeta function. The analytic extension of this function for general d in the massive and massless case can be found in the literature. For the massive case see, for example [66]. For the massless case, the duplication formula for the Gamma function allows us to write where is an entire function of d. Since ζ(z) can be analytically continued from an open connected set of points in the complex z plane into the entire domain of z, it is easy to find the analytic structure counterterms. To find the analytic structure of I 2 (y, a, b, d) and R 2 (x, a, b, d), let us concentrate on Integrating over the solid angle in Eq. (18), i.e using the fact that d d p = p d−1 dp dΩ d we have : where the function h(d) is given by : Performing a change of variables v = p 2 + m 2 + ( nπ a ) 2 1 2 , and now going back to the fourdimensional case it is possible to write I 2 (y, a, b, d)| d=4 ≡ I 2 (y, a, b) as : where the lower limit of the above integral is given by It is important to keep in mind that the situation is completely different for d = 4, since after it will appear in the integrand of Eq. (21). As a consequence of this fact, it is more difficult to perform algebraic manipulations that allow us to analytically regularize R 2 (x, a, b, d) and I 2 (y, a, b, d) and also other expressions. Nevertheless, utilizing the same techniques used in refs. [67] [68] [69], it is possible to perform all the calculations for a higher dimensional Euclidean space. This generalization will be presented in section IV. Going back to Eq.(23) and using trigonometric identities, we have : where : and finally An exact expression in the massless case can be obtained from I 21 (y, a). Summing the geometric series I 21 (y, a)| m=0 can be written as We will use the Bernoulli polynomials and numbers in the next sections. Consequently, let us introduce this Laurent expansion to study the behavior of I 21 (y, a)| m=0 in the neighborhood of y = 0. To find the principal part of the Laurent series of I 21 (y, a)| m=0 around y = 0 let us use the Bernoulli polynomials, which are defined by the generating function and the Bernoulli numbers Using this Laurent expansion one finds We note that we still have to calculate N(x, y, a, b). Using again the trigonometric identity given by Eq. (12), it is possible to write N(x, y, a, b, d) = N 1 (x, a, b, d) + N 2 (x, y, a, b, d) where we have : and Let us study the expression given by N 1 (x, a, b, d). It is possible to write N 1 (x, a, b, d) = N 11 (a, b, d)+ N 12 (x, a, b, d), where N 11 (a, b, d) and N 12 (x, a, b, d) are given by and The expression given by Eq.(34) can be easily calculated using dimensional regularization and can be written as This contribution for the vacuum fluctuations also demands bulk conterterm. We have to deal with the expression of N 12 (x, y, a, b, d). Integrating over the solid angle, changing the variables and using the fact that we are in d = 4, i.e. defining N 12 (x, a, Again, using trigonometric identities, we have : We note that Eq.(38) has surface divergences when x → 0 and x → a. The structure of the divergences of N 12 (x, a, b) will be studied further.
As a next step in the discussion, let us investigate the part of the one-loop vacuum fluctuations that contains edge divergences that is given by N 2 (x, y, a, b). Again, integrating over the solid angle, changing the variables and using the fact that we are in d = 4, the expression for N 2 (x, y, a, b)| d=4 ≡ N 2 (x, y, a, b) is given by where the lower limit of the above integral is given by α, defined in Eq. (24). Using trigonometric identities it is possible to write N 2 (x, y, a, b) as where : and From now we will put m = 0. Using the Poisson's kernel given by Again using the generating function of the Bernoulli numbers it is possible to find the principal part of the Laurent expansion of N 21 (x, y, a, b)| x=0 around y = 0, and also N 21 (x, y, a, b)| x=a around y = 0. The Laurent expansion around y = 0 allow us to write N 21 (x, y, a, b)| x=0 as We note that Eq.(41) has two edge divergences, one at x = y = 0 and the other at x = a, y = 0.
To investigate the divergences of Eq.(42) let us rewrite N 22 (x, y, a, b) as where In the next section, we will show that N + 22 (x, y, a, b) has surface divergences when y → b for all x, and also two edge divergences, one at x = 0, y = b and the other at x = a, y = b. We note also that N − 22 (x, y, a, b) is finite everywhere. Using a combination of dimensional and zeta function regularization, in the next section we will present the general method to regularize the one-loop vacuum field fluctuations in the rectangular waveguide.
3 Analysis of the surface divergences in the one-loop vacuum fluctuations.
The purpose of this section is to present a general method to analytic regularize the one-loop vacuum fluctuations associated with the confined scalar field. We first present the structure of the surface and edge divergences of the vacuum fluctuations associated with a massless scalar field in a four dimensional Euclidean space. As we discussed in the previous section, it is possible to write the one-loop vacuum fluctuations as The first expression that we have to deal with is T (a, b, d). As we discussed in the previous section, the analytic structure of T (a, b, d) was carefully analysed by Kirsten [63] and it is not necessary to repeat the calculation again. The second term of the above equation and third one can be written respectively in terms of R 1 (a, b, d), R 2 (x, a, b, d), I 1 (a, b, d) and finally I 2 (y, a, b, d).
The polar structure of R 1 (a, b, d) and I 1 (a, b, d) can be found in the literature, and we will not repeat the analysis that was done in these papers. The next quantity that we have to regularize is R 2 (x, a, b, d) and also I 2 (y, a, b, d). Since both cases are equivalent, let us study only the expression given by R 2 (x, a, b, d). It is instructive to study first the simpler case b ≫ a which will give us an indication of the behavior that may be encounter. Let us now proceed with the calculations in a general d-dimensional Euclidean space. Defining R 2 (x, a, b, d)| b≫a = r 2 (x, a, d), we have : We will again use the fact that d d−1 p = p d−2 dp dΩ d−1 and dΩ d−1 = 2π 2 ) . Let us choose now m = 0. Defining h 2 (d) by : it is possible to write r 2 (x, a, d)| m=0 as In the first integral for large k, (coth ka − 1) has the behavior: (coth ka − 1) ∼ e −2ka . Moreover, the second integral in the above equation is ultraviolet finite for x = 0. Let us define t = ka and q = kx in the first and second integrals above respectively. Then Eq.(52) becomes : The second term in the above equation gives us the well known result that for a massless scalar field ϕ 2 (x) diverges as 1 x 2 (in a four-dimensional space) as we approach the plate [5]. In order to analyze the behavior of r 2 (x, a, d)| m=0 around x = a, let us use the following integral representation of the gamma function, and also the following integral representation of the product of the Gamma function times the Hurwitz zeta function where ζ(s, u) is the Hurwitz zeta function defined by [64] ζ(s, u) = ∞ n=0 1 (n + u) s , Re(s) > 1, u = 0, −1, −2...
Then, using Eqs.(54), (55) and (56) in Eq.(53) we have that : Using the definition of the Hurwitz zeta function, it is evident that : We see that the regularized r 2 (x, a, d)| m=0 has two poles of order (d − 2) in x = 0 and in x = a.
Note that the residues of the poles in x = 0 and in x = a are a-independent. The same analysis can be done for I 2 (y, a, b, d) assuming a ≫ b. In the next section the general case will be studied.
Let us finally analyze N 21 (x, y, a) and N 22 (x, y, a, b), given respectively by and To find the analytic structure of N 21 (x, y, a) we can expand the general term in the sum in power series, commute the two summations, and use analytic continuation in the zeta function that will appear. The process will in general produce an extra term, which is generated by commuting the convergent exponential summation m with the new divergent summation k (for details see for e.g. refs. [70] [71] [72]). In our case this term vanishes due to the power of n. Let us express the sum that appears in Eq. (59) in terms of the complex variable z = ix − y: Expanding around z = 0, (using the Bernoulli expansion) will produce : We see that the edge divergence appears in the term 1/z. Taking the real part : Re where f 1 (z) is an entire function of z and is given by : Expanding around z = ia will produce : Taking the real part of 1/(z − ia), we have : where f 2 (z) is also an entire function of z, and is given by: To find the analytic structure of N 22 (x, y, a, b), it is enough to analyse the quantity N + 22 (x, y, a, b), which is given by the Eq.(47) (the other quantity N −  22 (x, y, a, b) is finite everywhere). To calculate the integral in Eq.(47), we can express coth(bv) in exponential functions and expand the integrand in power series. The integral can be easily evaluated, and the result is given by This result can be written in terms of the Lerch's trancendent function Φ(z, c, v) which is defined as follows [73] : which is valid for |z| < 1. By analytic continuation it can be extended to the whole complex plane.
This function has singularities at z = 1 and c = 0 or c = 1 and when v is a non-positive integer and Re(c) is also non-positive. We note that only the term k = 0 has a surface divergence at y = b. The remaining part of the series in k is finite everywhere and we call it F (x, y, a, b). Using the same procedure used to analyse N 21 (x, y, a), we can define z = (b − y) + ix and expand the term k = 0 around z = 0 or around z = ia. Therefore we have where the first expansion B(x, y, a, b) is given by where h 1 (z) is an entire function given by We note that in this case B(x, y, a, b) has a edge divergence for x = 0, y = b. For the second expansion, B(x, y, a, b) is given by where h 2 (z) is an entire function given by We note that in this case B(x, y, a, b) has a edge divergence for x = a, y = b. A general picture that emerges from the above discussion is the following: we have found that in order to eliminate the ultraviolet divergences of the one-loop vacuum fluctuations we have to introduce not only the usual bulk counterterms, but also counterterms concentrated on the boundaries. In the next section we will show how it is possible to generalize our local analysis for a massive field defined in a higher dimensional Euclidean space.
4 The one-loop vacuum fluctuation in the waveguide in a higher dimensional Euclidean space.
In this section we will discuss how to generalize our calculations to a higher dimensional Euclidean space. We are interested in investigating the one-loop vacuum fluctuations associated with a massive scalar field defined in a generic d-dimensional Euclidean space, where d − 2 coordinates are unbounded while the first two lie in some finite interval. As discussed in the previous sections, we found that the vacuum fluctuations can be expressed as : N(x, y, a, b, d).
In the previous sections, we studied the expression given by T (a, b, d), R 1 (a, b, d) and also I 1 (a, b, d) in the massless case, which demands bulk counterterms. In the following, we will present a detailed analysis of the contributions of the vacuum fluctuations that demand counterterms concentrated on the boundaries. We will show that some of the terms of the one-loop vacuum fluctuations, can be expressed in terms of Bessel functions and also Hurwitz zeta functions.
From the previous analysis for the massless field in a four-dimensional Euclidean space, we found that to perform our calculations in a generic d-dimensional space, for the massive field, it is natural to write I 2 (y, a, b, d) in the following way : where the functions I 21 (y, a, d) and I 22 (y, a, b, d) are given by : and The lower limit of both integrals is given by α defined in Eq.(24) and the coefficient h(d) is an entire function defined in Eq. (22). Using the following integral representation of the modified Bessel functions of third kind, or the Macdonald's functions K ν (x) : which is valid for for u > 0, Re µ > 0 and Re ν > 0, I 21 (y, a, b, d) can be written in terms of these functions. A simple substitution gives Equation (80) can not be evaluated exactly, even for the massless case, but for small arguments of : where the C k p are the generalizations of the binomial coeficients, given by: − 1)..(p − k + 1)) and p = d−4 2 . Note that the generalization of the binomial series is valid even for any complex exponent p. In other words, for v > nπ a we have an everywhere convergent power series in p, hence a continuous function on p in the complex d plane. In the following, it is convenient to define the quantities: . After a change of variables, using Eq.(81) in Eq.(78), I 22 (y; a, b, d)| m=0 becomes appear. Let us split the I 22 (y, a, b, d)| m=0 in the following way : where and Although the integral that appears in Eq.(85) can not be expressed in tems of known functions, the leading divergences of I 22 (y, a, b, d)| m=0 are contained in the Eq. (84). Consequently, let us analyze the divergences that appear in Eq.(84). Using the fact that ∞ we reexpress Eq.(84) as where and Expanding cosh( 2uy b ) in powers series, the integral that appears in I b 22 (y, a, b, d) is a Debye integral.
Although this can be evaluated, the study of I a 22 (y, a, b, d) is sufficient to obtain the divergences of I < 22 (y, a, b, d). Again, we note that the integral that appear in Eq.(87) can be written in tems of products of zeta, Gamma function and Hurwitz zeta function. Thus we have :  N(x, y, a, b, d) in the following way : where the N 11 (a, b, d) term contains contributions to the vacuum fluctuations that demands bulk counterterms in order to renormalize the one-loop vacuum fluctuations. The next contribution that we have to study is N 12 (x, a, b, d). Starting from Eq. (35) integrating over the solid angle and changing variables, it is also convenient to split N 12 (x, a, b, d) in the following way : where each of the terms of Eq.(91) are given respectively by and We can again use the integral representation of the modified Bessel functions of the third kind given by Eq.(79), to write Eq.(92) as To calculate N b 12 (x, a, b, d), let us study first the massless case. In this case, first changing variables and using Eq.(55) we get Next we consider the massive case. Note that we have exactly the same situation discussed before.
After a change of variables, N b 12 (x, a, b, d) also contains a power of a binomial. Let us discuss again the most interesting situation, the odd dimensional case. First, let us the same power series expansion that we used before for the quantity given by Substituting Eq.(96) in Eq.(93) and again changing variable, we have A natural way to obtain N b 12 (x, a, b, d) expressed in terms of the Hurwitz zeta function is the following. Let us split N b 12 (x, a, b, d) in the following way : where and We have the same situation that we studied before. Let us investigate first the N b< 12 (x, a, b, d).
it is possible to write Eq.(99) in the following way: where the singularities of N b< 12 (x, a, b, d) appear at x → a. We have to analyze the N b> 12 (x, a, b, d). It is clear that in a even dimensional Euclidean space for x < a the integral that appear in Eq. (100) is convergent. The odd dimensional case also can be studied. In this case we have to expand cosh( 2uy a ) in power series and it is clear that in the new integral that appears in Eq.(100) demands a generalization of the Debye integrals. The calculations that we presented can also be used to write N 2 (x, y, a, b, d) in the same form that we obtained for the other expressions. Unfortunatelly one cannot perform the summation in n. For completeness, we will only write the remaining expressions. Let us start from : where each term of the above equation is given by and Note that the lower limit of both integrals are given by α defined in Eq. (24) and h(d) was defined in Eq. (22). Let us present the N 21 (x, y, a, b, d) in the massless case. Using the same integral representation of the Bessel functions that we used before gives The massive case can also be presented but in both cases it is not possible to perform the summation in n. Finally, let us present the expression for N 22 (x, y, a, b, d) in the massless case and for the odd dimensional case : The same procedure that we used in I 22 (y, a, b, d), can be repeated again. Unfortunatelly one can not perform the summation in n that appears in Eq.(106). Nevertheless the same analysis that we did in the end of section III can be performed to study the edge divergences that appear in the one-loop vacuum fluctuations associated with the scalar field confined in the interior of a waveguide in a higher dimesional Euclidean space.

Discussions and conclusions
In this paper, we first obtained the regularized one-loop vacuum fluctuations associated with a massless scalar field defined in the interior of an infinity waveguide of rectangular cross section in a four-dimensional Euclidean space. Then, we discussed how it is possible to generalize our results for a massive field defined in a higher dimensional Euclidean space. We first rederive a well-know result that surface and edge divergences appear in the one-loop vacuum fluctuations as consequence of the uncertaintly principle. There are at least two different possible solutions that can eliminate these divergences. The first is to take into account that real materials have imperfect conductivity at high frequencies. As was stressed by many authors, the infinities that appear in renormalized values of local observables for the ideal conductor (or perfect mirror) represent a breakdown of the perfect-conductor approximation. A wavelength cutoff corresponding to the finite plasma frequency must be included. The second would be given by a quantum mechanical treatment of the boundary conditions [43]. It was shown [43] that position fluctuations of a reflecting boundary also remove divergences in the renormalized values of local observables at least in the flat plate configuration. A logical going is to use an analytic regularization procedure to identify these divergent terms. As we discussed counterterms concentrated on the boundaries produce a finite one-loop vacuum fluctuations in the interior of the waveguide. A natural extension of this paper is to go beyond the one-loop approximation, investigating interacting field models in the presence of a waveguide in a d-dimensional Euclidean space. This topic is under investigation by the authors.

Acknowledgement
We where Dϕ is the appropriate measure, and S[ϕ] is the classical action associated with the scalar field. The quantity Z(J) can be regarded as the functional integral representation for the imaginary time evolution operator ϕ 2 |U(t 2 , t 1 )|ϕ 1 with the boundary conditions ϕ(t 1 , x) = ϕ 1 ( x) and ϕ(t 2 , x) = ϕ 2 ( x). The quantity Z(J) gives the transition amplitude from the initial state |ϕ 1 > to a final state |ϕ 2 > in the presence of some scalar source of compact suport. As usual W (J), the generating functional of the connected correlation functions shall be given by W (J) = lnZ(J). In a free theory the partition function Z(J) and also W (J) can be calculated exactly. In the presence of the waveguide we must assume that the path integral must be taken over the space of functions that vanish in the boundaries of the waveguide. Since our aim is to discus the non-interacting theory, let us assume that J = 0. Thus, defining the d-dimensional laplacian by ∆, we have: where ζ ′ (0) = d ds ζ(s)| s=0 and the operator (−∆ + m 2 ) has the spectrum (k 2 0 + k 2 1 + ( nπ a ) 2 + ( n ′ π b ) 2 + m 2 ), k 0 , k 1 ∈ ℜ, n, n ′ ∈ N. (A.6) We can use the same procedure to define the local generalized zeta function which is related with the effective action. The local generalized zeta function is given by ζ −∆+m 2 (s, x, y) = N ′ dk 0 dk 1 ∞ n,n ′ =1 sin 2 ( nπx a ) sin 2 ( n ′ πy b ) (k 2 0 + k 2 1 + ( nπ a ) 2 + ( n ′ π b ) 2 + m 2 ) s , To give a precise meaning of the derivative one can use for example the Liouville's concept of fractional derivative [75]. This result show that the one-loop vacuum fluctuations determines the effective action in the one-loop level.