The Casimir effect for pistons with transmittal boundary conditions
aa r X i v : . [ h e p - t h ] A ug The Casimir e ff ect for pistons with transmittal boundary conditions Guglielmo Fucci ∗ Department of Mathematics, East Carolina University, Greenville, NC 27858 USA (Dated: September 28, 2018)This work focuses on the analysis of the Casimir e ff ect for pistons subject to transmittal boundaryconditions. In particular we consider, as piston configuration, a direct product manifold of the type I × N where I is a closed interval of the real line and N is a smooth compact Riemannian manifold.By utilizing the spectral zeta function regularization technique, we compute the Casimir energy ofthe system and the Casimir force acting on the piston. Explicit results for the force are providedwhen the manifold N is a d -dimensional ball. I. INTRODUCTION
The Casimir e ff ect, which was first predicted theoretically in 1948 as an attraction between perfectlyconducting neutral plates [11], is indisputably one of the most studied subjects in the ambit of quantumfield theory and its interaction with external conditions. The analysis of the Casimir e ff ect has evolvedfrom the simple configuration of two parallel plates considered in the pioneering work of Casimir, to muchmore complex geometries and boundary conditions (the reader may refer, for instance, to [8, 9, 33, 34]and references therein). As a result of the very nature of the Casimir e ff ect, which is a set of phenomenaoriginating from modifications to the vacuum energy of a quantum field due to its interaction with externalconditions, calculation of the Casimir energy of a quantum field often lead to a divergent quantity. In orderto extract meaningful results for the Casimir energy, a suitable regularization procedure is required. In thiswork we employ the spectral zeta function regularization technique [4, 10, 16, 17, 27, 30] which representsone of the most widely used regularization methods.Piston configurations, which were first introduced by Cavalcanti in 2004 [12], have lately been the mainfocus of a large body of research. The reason for this widespread interest is encoded in the particulargeometry that piston configurations have. In the most general case, a piston configuration consists of two D -dimensional manifolds, often referred to as chambers, with a common boundary being a codimension onemanifold representing the piston. One of the main portions of the analysis of piston configurations consistsin the computation of the Casimir force acting on the piston. This force is the result of di ff erences in vacuumenergy in the two chambers. What makes pistons such an attractive geometric configuration for studying the ∗ Electronic address: [email protected]
Casimir e ff ect is the fact that while their Casimir energy might be divergent, the force acting on the pistonitself is, in most cases, well defined. In fact, there exist piston configurations with non-vanishing curvaturein which the Casimir force acquires divergent terms that are proportional to particular geometric invariantsof the piston [18–20]. The Casimir e ff ect has been studied throughout the literature for a plethora of pistonconfigurations, a small sample of this work can be found, for instance, in [2, 15, 26, 29, 31, 32]. Mostof the investigations regarding the Casimir e ff ect for pistons are developed by assuming that the quantumfield propagating in the piston satisfies ordinary boundary conditions, such as Dirichlet, Neumann, Robin,or mixed. Although of enormous theoretical importance, the aforementioned boundary conditions onlymodel idealized situations. In order to overcome this limitation, piston configurations have been analyzedfor quantum fields endowed with more general self-adjoint boundary conditions (see e.g. [1, 21]) in anattempt to describe physical, as opposed to idealized, situations. Another approach which has been utilizedto imitate physical boundary conditions is based on the replacement of the piston itself and the associatedboundary conditions with a smooth potential with compact support [3]. Many of the methods employed toanalyze the Casimir e ff ect for piston configurations have a specific characteristic in common. In describingthe propagation of quantum fields in the piston configuration, one is led to solve an eigenvalue equationendowed with particular boundary conditions in each chamber separately . In this way the behavior of thequantum field in one chamber is completely independent of its behavior in the other chamber. In otherwords, ordinary boundary conditions imposed on the piston do not allow any interaction between the twochambers of the piston. These approaches are clearly unsuitable if one wishes to consider configurationspossessing, for example, a semi-transparent piston where the quantum field propagating in one chamberinfluences the propagation of the field in the other chamber through the piston. In order to allow interactionbetween the two chambers one could impose transmittal boundary conditions on the piston [23–25, 27, 28].In this work we focus on the analysis of the Casimir e ff ect for a scalar field propagating in a pistonconfiguration of the type I × N where I is a closed interval of the real line and N is a smooth compact Rie-mannian manifold. On the piston itself we impose transmittal boundary conditions and utilize the spectralzeta function regularization technique to obtain the Casimir energy of the system and the associated forceon the piston. As we will observe in the next section, transmittal boundary conditions are dependent on aparameter. One of the main goals of this work is to show how the Casimir force on the piston depends onthe parameter describing the transmittal boundary conditions.The outline of the paper is as follows. In the next section we describe in details the spectral zeta functionassociated with the piston configuration and the boundary conditions that are considered in this work. Sec-tion III contains an outline of the analytic continuation of the spectral zeta function which is necessary forthe computation of the Casimir energy and force. In Section IV we analyze the case in which the Laplacianon the piston possesses a zero mode and compute its contribution to the spectral zeta function. In SectionV we evaluate the Casimir energy and force for the piston configuration and in Section VI we utilize theresults to compute numerically the force when the piston is a d -dimensional ball. The last section points tothe main results of this paper and outlines possible further studies related to Casimir pistons with transmittalboundary conditions. II. THE SPECTRAL ZETA FUNCTION AND CASIMIR ENERGY
We consider a generalized piston configuration, analyzed also in [20, 21], based on a product manifoldof the type M = I × N where I = [0 , L ] ⊂ R is a closed interval of the real line and N is a smooth compactRiemannian manifold with or without a boundary ∂ N . We assume, furthermore, that the dimension ofthe base manifold N is d which implies that M is, hence, a manifold of dimension D = d +
1. Thepiston configuration can be constructed from the product manifold M by placing at any point a ∈ (0 , L ) themanifold N a which represents the cross-section of I × N at a . In doing so, the manifold M is divided intwo chambers, denoted by M I and M II , separated by the common boundary N a which describes the pistonitself. The two chambers, by construction, are smooth compact Riemannian manifolds with a boundarywhere ∂ M I = N ∪ N a ∪ ([0 , a ] × N ) and ∂ M II = N a ∪ N L ∪ (( a , L ] × N ).We consider a massless scalar field φ propagating on M . By using a set of coordinates ( x , X ) where x ∈ I and X represent the coordinates on N , the dynamics of the scalar field is described by the eigenvalueequation − d d x + ∆ N ! φ = α φ , (2.1)where ∆ N denotes the Laplacian on the manifold N and φ belongs to the space L ( M ) of square integrablefunctions on M . The eigenvalues α are uniquely determined once appropriate boundary conditions areimposed on the scalar field φ . The general solution to the di ff erential equation (2.1) can be written as aproduct φ = f ( x ) ϕ ( X ) where ϕ ( X ) are the eigenfunctions of the Laplacian on N with eigenvalues λ , that is − ∆ N ϕ = λ ϕ , (2.2)and the function f ( x , λ ) satisfies the following second-order ordinary di ff erential equation − d d x + λ − α ! f λ ( x , α ) = . (2.3)The eigenvalues α are, then, used to construct the spectral zeta function associated with our system asfollows: ζ ( s ) = X α α − s , (2.4)which, according to the general theory of spectral functions [16, 17, 27], is well defined in the region of thecomplex plane ℜ ( s ) > D /
2. In the ambit of the spectral zeta function regularization method, the functiondefined in (2.4) can be utilized to compute the Casimir energy of the system [8–10, 16, 17, 27]. Afterperforming a suitable analytic continuation, ζ ( s ) in (2.4) can be extended to a meromorphic function in theentire complex plane possessing only simple poles. The analytically continued expression is then employedto express the Casimir energy as E Cas ( a ) = lim ǫ → µ − s ζ ǫ − , a ! , (2.5)where µ is a parameter with the dimension of mass. Since ζ ( s ) generally presents a pole at s = − /
2, thelimit in (2.5) leads to E Cas ( a ) =
12 FP ζ − , a ! + ǫ + ln µ ! Res ζ − , a ! + O ( ǫ ) . (2.6)The force acting on the piston positioned at x = a is given by F Cas ( a ) = − ∂∂ a E Cas ( a ) . (2.7)It is clear, from (2.6) and (2.7), that the Casimir force on the piston is free from divergences, and hence welldefined, if the residue of the spectral zeta function at s = − / a .As we have previously mentioned, the eigenvalues α in (2.1) can be uniquely determined once appro-priate boundary conditions are satisfied. In this work, we impose ordinary boundary conditions, namelyDirichlet and Neumann, at the two end-points of the piston configuration N and N a . The piston itself N a is, instead, endowed with transmittal boundary conditions which can be described as following [23, 24, 28]:Let M = M I ∪ Σ M II be a d -dimensional compact Riemannian manifold with Σ being a codimension-one common boundary. If D I and D II denote the Laplace operators on M I and M II acting on φ I ∈ V | M I and φ II ∈ V | M II , respectively, then the transmittal boundary conditions are defined as B U φ = B U φ = { φ I | Σ − φ II | Σ } ⊕ (cid:8)(cid:0) ∇ m I φ I (cid:1) | Σ + (cid:0) ∇ m II φ II (cid:1) | Σ + U φ I | Σ (cid:9) . (2.8)In the above expression U is an endomorphism of V | Σ and ∇ m I and ∇ m II are the exterior normal derivativeson M I and M II , respectively, to the boundary Σ .In order to construct the spectral zeta function of our system we need to know the eigenvalues α . Al-though the eigenvalues cannot be explicitly found in general, the boundary conditions will provide implicitequations to obtain them. The general solution to the ordinary di ff erential equation (2.3) in the first chamber M I is a simple linear combination of trigonometric functions f I ,λ ( x , α ) = a I sin (cid:18) p α − λ x (cid:19) + b I cos (cid:18) p α − λ x (cid:19) . (2.9)In the second chamber, namely M II , we find a similar general solution to (2.3) which can be written, forlater convenience, as f II ,λ ( x , α ) = a II sin (cid:20) p α − λ ( L − x ) (cid:21) + b II cos (cid:20) p α − λ ( L − x ) (cid:21) . (2.10)The general solutions (2.9) and (2.10) are then required to satisfy transmittal boundary conditions B U φ = N a , namely f I ,λ ( a , α ) = f II ,λ ( a , α ) f ′ I ,λ ( a , α ) = f ′ II ,λ ( a , α ) + U f I ,λ ( a , α ) , (2.11)with the prime indicating di ff erentiation with respect to the variable x , together with a set of boundaryconditions at the two endpoints N and N L of the piston configuration. In this work we consider three typesof boundary conditions. The first ones, which we denote by the name Dirichlet-Dirichlet are f I ,λ (0 , α ) = f II ,λ ( L , α ) = . (2.12)The subsequent conditions take the form f ′ I ,λ (0 , α ) = f ′ II ,λ ( L , α ) = , (2.13)which we designate as Neumann-Neumann boundary conditions. The last set of boundary conditions, whichwe call mixed , fall into two subsets, defined by the equations f I ,λ (0 , α ) = f ′ II ,λ ( L , α ) = , and f ′ I ,λ (0 , α ) = f II ,λ ( L , α ) = . (2.14)In the fist case one imposes Dirichlet conditions at N and Neumann ones at N L while in the second casethe conditions are reversed.By imposing Dirichlet-Dirichlet boundary conditions (2.12) and the transmittal boundary conditions(2.11) on the general solutions (2.9) and (2.10) we obtain b I = b II = a I sin (cid:18) p α − λ a (cid:19) = a II sin (cid:20) p α − λ ( L − a ) (cid:21) a I p α − λ cos (cid:18) p α − λ a (cid:19) = − a II p α − λ cos (cid:20) p α − λ ( L − a ) (cid:21) + Ua I sin (cid:18) p α − λ a (cid:19) , (2.15)which has a non-trivial solution for the coe ffi cients a I and a II if the determinant of the coe ffi cient matrixvanishes identically, namely Ω DD λ ( α, a ) = p α − λ sin (cid:18) p α − λ L (cid:19) − U sin (cid:18) p α − λ a (cid:19) sin (cid:20) p α − λ ( L − a ) (cid:21) = . (2.16)The above equation implicitly determines the eigenvalues α for the Dirichlet-Dirichlet case. For the case ofNeumann-Neumann boundary conditions we obtain a I = a II = b I cos (cid:18) p α − λ a (cid:19) = b II cos (cid:20) p α − λ ( L − a ) (cid:21) − b I p α − λ sin (cid:18) p α − λ a (cid:19) = b II p α − λ sin (cid:20) p α − λ ( L − a ) (cid:21) + Ub I cos (cid:18) p α − λ a (cid:19) , (2.17)which one can get by imposing transmittal boundary conditions at N a . The system (2.17) has a non-trivialsolution for b I and b II if Ω NN λ ( α, a ) = p α − λ sin (cid:18) p α − λ L (cid:19) − U cos (cid:18) p α − λ a (cid:19) cos (cid:20) p α − λ ( L − a ) (cid:21) = , (2.18)which is the equation determining the eigenvalues α in the Neumann-Neumann case. The exact sameprocedure can be followed to obtain an implicit equation for the eigenvalues in the case of mixed boundaryconditions. In fact, by imposing mixed boundary conditions of the Dirichlet-Neumann type at the endpointsof M and transmittal boundary conditions at N a and by subsequently setting to zero the determinant of thecoe ffi cient matrix of the ensuing linear system we obtain Ω DN λ ( α, a ) = p α − λ cos (cid:18) p α − λ L (cid:19) − U sin (cid:18) p α − λ a (cid:19) cos (cid:20) p α − λ ( L − a ) (cid:21) = . (2.19)For the other type of mixed boundary conditions, namely Neumann-Dirichlet, we obtain, instead, the fol-lowing equation for the eigenvalues α Ω ND λ ( α, a ) = p α − λ cos (cid:18) p α − λ L (cid:19) − U cos (cid:18) p α − λ a (cid:19) sin (cid:20) p α − λ ( L − a ) (cid:21) = . (2.20)The equations (2.16), and (2.18)-(2.20) can be used to write an expression for the spectral zeta functionin terms of a contour integral, valid in the semi-plane ℜ ( s ) > D /
2, as follows [6, 7, 27] ζ ( j ) ( s , a ) = π i X λ d ( λ ) Z γ j κ − s ∂∂κ ln Ω ( j ) λ ( κ, a ) d κ , (2.21)where the index j denotes the type of boundary conditions under consideration and γ j represents a con-tour that encloses, in the counterclockwise direction, all the real zeroes of the appropriate implicit equationfor the eigenvalues Ω ( j ) λ ( κ, a ) =
0. In addition, d ( λ ) indicates the degeneracy of the eigenvalues λ of theLaplacian on the manifold N . According to (2.5) we need to analyze the spectral zeta function in a neigh-borhood of s = − / s = − / ℜ ( s ) ≤ D / III. ANALYTIC CONTINUATION OF THE SPECTRAL ZETA FUNCTION
The first step of the desired analytic continuation is performed by exploiting the replacement κ → z λ and by deforming the integration contour γ j in (2.21) to the imaginary axis [27]. This procedure allows usto rewrite the spectral zeta function as ζ ( j ) ( s , a ) = X λ d ( λ ) ζ ( j ) λ ( s , a ) , (3.1)where ζ ( j ) λ ( s , a ) is represented as a real integral of the form ζ ( j ) λ ( s , a ) = sin( π s ) π λ − s Z ∞ z − s ∂∂ z ln Ω ( j ) λ ( i λ z , a )d z . (3.2)The functions Ω ( j ) λ ( i λ z , a ) can be obtained from (2.16), and (2.18)-(2.20) and explicitly read Ω DD λ ( i λ z , a ) = λ p + z sinh (cid:18) λ p + z L (cid:19) − U sinh (cid:18) λ p + z a (cid:19) sinh (cid:20) λ p + z ( L − a ) (cid:21) , (3.3) Ω NN λ ( i λ z , a ) = λ p + z sinh (cid:18) λ p + z L (cid:19) − U cosh (cid:18) λ p + z a (cid:19) cosh (cid:20) λ p + z ( L − a ) (cid:21) , (3.4) Ω DN λ ( i λ z , a ) = λ p + z cosh (cid:18) λ p + z L (cid:19) − U sinh (cid:18) λ p + z a (cid:19) cosh (cid:20) λ p + z ( L − a ) (cid:21) , (3.5) Ω ND λ ( i λ z , a ) = λ p + z cosh (cid:18) λ p + z L (cid:19) − U cosh (cid:18) λ p + z a (cid:19) sinh (cid:20) λ p + z ( L − a ) (cid:21) . (3.6)In order for the contour deformation to be well defined, one needs to make sure that no zeroes of (2.16), and(2.18)-(2.20) lie on the imaginary axis. However, it is not di ffi cult to realize that the solutions to the implicitequations for α in (2.16), and (2.18)-(2.20) are simple and can be either real or imaginary [21, 35, 36].Since we only want to consider boundary conditions that lead to a self-adjoint boundary value problem,we restrict our analysis to values of the parameter U for which the zeroes of (2.16), and (2.18)-(2.20) arereal and positive. This assumption also allows the contour deformation leading to (3.2) to be well defined.A discussion of the case in which purely imaginary zeroes are present can be found in [35]. To find therange of allowed values of U , we start by noticing that the equations (2.16), and (2.18)-(2.20) have purelyimaginary zeroes if (3.3)-(3.6) have real zeroes. In particular, if we denote by λ > − ∆ N , then (3.3) has no real zeroes if λ sinh ( λ L ) > U sinh ( λ a ) sinh [ λ ( L − a )] . (3.7)The last inequality holds for all a ∈ (0 , L ) if U < U DD where U DD = λ sinh ( λ L ) h sinh (cid:16) λ L (cid:17)i . (3.8)Similarly, the function in (3.4) has no real zeroes if U < U NN with U NN = λ tanh ( λ L ) . (3.9)Lastly, the functions describing mixed boundary conditions (3.5) and (3.6) have no real zeroes for values of U < U M where one finds that U M = λ tanh ( λ L ) . (3.10)For the majority of compact Riemannian manifolds N , and appropriate boundary conditions, the lowesteigenvalue λ cannot be explicitly found. However, lower bounds for the smallest eigenvalue of the Lapla-cian can be found (see e.g. [13, 14]). Such lower bounds can then be utilized in the formulas (3.8)-(3.10)to obtain an upper bound on the allowed values of U . We would like to point out that if λ = ∆ N has a zero mode. We can, finally, conclude thatwhen U < U ( j ) then the contour deformation can be performed without any problems and leads to the for-mulas (3.1) and (3.2). By analyzing the behavior of z − s ∂ z ln Ω ( j ) λ ( i λ z , a ) for z → z → ∞ it is not verydi ffi cult to show (see e.g. [21]) that the integral representation (3.2) is valid in the strip 1 / < ℜ ( s ) <
1. Inorder to extend the region of convergence of the integral (3.2) to ℜ ( s ) ≤ /
2, we subtract, and add, in therepresentation (3.2), a suitable number of terms of the asymptotic expansion of ln Ω ( j ) λ ( i λ z , a ) for λ → ∞ uniform in z = k /λ .By using the explicit expressions (3.3) through (3.6) and the exponential form of the hyperbolic func-tions, one can prove that Ω ( j ) λ ( i λ z , a ) = e λ √ + z L (cid:18) λ p + z − U (cid:19) h + exp ( j ) ( z , λ, a ) i , (3.11)where exp ( j ) ( z , λ, a ) represents exponentially small terms as λ → ∞ . From the expression (3.11) it is notdi ffi cult to obtain the following oneln Ω ( j ) λ ( i λ z , a ) = λ p + z L − ln 2 + ln (cid:18) λ p + z (cid:19) + ln − U λ √ + z ! + ln h + exp ( j ) ( z , λ, a ) i . (3.12)By exploiting the small- x asymptotic expansion of ln(1 − x ) we find the needed uniform asymptotic expan-sion ln Ω ( j ) λ ( i λ z , a ) ∼ λ p + z L − ln 2 + ln (cid:18) λ p + z (cid:19) − ∞ X n = U n n n λ n (1 + z ) n , (3.13)where we have discarded the exponentially small terms.By subtracting and adding N terms of the uniform asymptotic expansion (3.13) in the integral represen-tation (3.2) we obtain, according to (3.1), the following expression ζ ( j ) ( s , a ) = Z ( j ) ( s , a ) + N X k = − A k ( s , a ) . (3.14)The function Z ( j ) ( s , a ) is analytic in the region of the complex plane ℜ ( s ) > ( d − N − / Z ( j ) ( s , a ) = sin( π s ) π X λ d ( λ ) λ − s Z ∞ z − s ∂∂ z " ln Ω ( j ) λ ( i λ z , a ) − λ p + z L + ln 2 − ln (cid:18) λ p + z (cid:19) + N X n = U n n n λ n (1 + z ) n d z . (3.15)The terms A k ( s , a ) are, instead, meromorphic functions of s in the entire complex plane and can be writtenas A − ( s , a ) = sin( π s ) π X λ d ( λ ) λ − s Z ∞ z − s ∂∂ z (cid:18) λ p + z L (cid:19) d z , (3.16) A ( s , a ) = sin( π s ) π X λ d ( λ ) λ − s Z ∞ z − s ∂∂ z (cid:20) ln (cid:18) λ p + z (cid:19)(cid:21) d z , (3.17)and, for k ≥ A k ( s , a ) = sin( π s ) π X λ d ( λ ) λ − s − k U k k k Z ∞ z − s ∂∂ z (cid:20) (1 + z ) − k (cid:21) d z . (3.18)By performing the elementary integrals (3.16)-(3.18) and by using the following definition for the spectralzeta function associated with the Laplacian ∆ N ζ N ( s ) = X λ d ( λ ) λ − s , (3.19)we can finally write the analytically continued expression for the spectral zeta function as ζ ( j ) ( s , a ) = Z ( j ) ( s , a ) + L √ π Γ ( s ) Γ s − ! ζ N s − ! + ζ N ( s ) − Γ ( s ) N X k = U k k + Γ (cid:16) k + (cid:17) Γ s + k ! ζ N s + k ! . (3.20)As it is rendered manifest in the above expression, the meromorphic structure of the spectral zeta functionis completely determined by the terms in (3.20) proportional to the zeta function ζ N ( s ). IV. PRESENCE OF ZERO MODES
The analytic continuation outlined in the previous sections holds if ∆ N has non-vanishing eigenvalues.If, instead, λ = ∆ N with multiplicity d (0), then the process of analytic continuation ofthe spectral zeta function ζ ( s , a ) needs to be somewhat amended since we can no longer utilize the uniform0asymptotic expansion in (3.13). When a zero mode is present, it is convenient to separate its contribution tothe spectral zeta function from the one of the non-vanishing eigenvalues as follows ζ ( j ) ( s , a ) = d (0)2 π i Z γ j κ − s ∂∂κ ln Ω ( j )0 ( κ, a ) d κ + X λ> d ( λ )2 π i Z γ j κ − s ∂∂κ ln Ω ( j ) λ ( κ, a ) d κ . (4.1)The analytic continuation of the contribution coming from the non-vanishing eigenvalues follows exactlythe same process described earlier and, therefore, will not be repeated here. We direct, instead, our attentionto the analytic continuation of the first integral on the right-hand-side of (4.1). When λ = ff erential equation (2.3) becomes − d d x − α ! f λ ( x , α ) = , (4.2)whose general solution in chamber I and chamber II is simply f I , ( x , α ) from (2.9) and f II , ( x , α ) from(2.10), respectively. To these solutions we need to impose appropriate boundary conditions. By applyingDirichlet-Dirichlet boundary conditions with transmittal ones on the piston itself, we obtain the followingequation for the eigenvalues α Ω DD ( α, a ) = . (4.3)The equations that determine the eigenvalues when the other boundary conditions are imposed, namelyNeumann-Neumann and mixed, are found to be Ω NN ( α, a ) = , Ω DN ( α, a ) = , and Ω ND ( α, a ) = . (4.4)By performing the replacement α → iz and by deforming the contour to the imaginary axis we obtain,for the zero mode contribution to the spectral zeta function, the integral representation ζ ( j )0 ( s , a ) = d (0) sin( π s ) π λ − s Z ∞ z − s ∂∂ z ln Ω ( j )0 ( iz , a )d z , (4.5)which, just like earlier, is well defined in the strip 1 / < ℜ ( s ) < Ω ( j )0 ( iz , a ) canbe shown to have the form Ω DD ( iz , a ) = z sinh ( zL ) − U sinh ( za ) sinh [ z ( L − a )] , (4.6) Ω NN ( iz , a ) = z sinh ( zL ) − U cosh ( za ) cosh [ z ( L − a )] , (4.7) Ω DN ( iz , a ) = z cosh ( zL ) − U sinh ( za ) cosh [ z ( L − a )] , (4.8) Ω ND ( iz , a ) = z cosh ( zL ) − U cosh ( za ) sinh [ z ( L − a )] . (4.9)Now, the equations (4.3) and (4.4) have, in principle, both real and imaginary solutions. For reasons alreadyexplained in the previous sections, we need to find the rage of values of the parameter U for which (4.3)1and (4.4) have only real solutions. This is achieved for values of U for which (4.6) through (4.9) have noreal solutions. The allowed values of U satisfy the inequality U < U , ( j ) where the quantity U , ( J ) can beobtained from the equations (3.8)-(3.10) by taking the limit λ →
0. In more details, they read, U , DD = L , U , NN = , and U , M = L . (4.10)Hence, when U < U , ( j ) , the contour deformation can be performed and leads to the representation (4.5)for the zero mode contribution to the spectral zeta function. The process of analytic continuation begins bysplitting the integral representation (4.5) [21] as ζ ( j )0 ( s , a ) = d (0) sin( π s ) π λ − s Z z − s ∂∂ z ln Ω ( j )0 ( iz , a )d z + d (0) sin( π s ) π λ − s Z ∞ z − s ∂∂ z ln Ω ( j )0 ( iz , a )d z . (4.11)The advantage of rewriting (4.5) as the sum in (4.11) lies in the fact that the first integral is analytic in thesemi-plane ℜ ( s ) <
1, and hence no further manipulation is necessary for the purpose of analyzing ζ ( j )0 ( s , a )at s = − /
2. The second integral is analytic for ℜ ( s ) > /
2, and therefore, requires to be extended tothe left of the abscissa of convergence ℜ ( s ) = /
2. To perform the analytic continuation of the secondintegral in (4.11) we proceed as before by subtracting and adding a suitable number of terms of the large- z asymptotic expansion of ln Ω ( j )0 ( iz , a ) from the second integral in (4.11). From the expressions (4.6)-(4.9) itis not very di ffi cult to find the relationln Ω ( j )0 ( iz , a ) = zL − ln 2 + ln z + ln (cid:18) − U z (cid:19) + ln h + exp , ( j ) ( z , λ, a ) i , (4.12)where exp , ( j ) ( z , λ, a ) denotes exponentially decaying terms. The desired large- z asymptotic expansion isobtained by expanding ln(1 − U / z ) for small values of U / z and by discarding exponentially small termsin (4.12). This procedure leads toln Ω ( j )0 ( iz , a ) ∼ zL − ln 2 + ln z + ∞ X n = U n n n z n . (4.13)By subtracting and adding from the second integral in (4.11), N terms of the asymptotic expansion(4.13) and by then performing the remaining trivial integrals we obtain, for the zero mode contribution tothe spectral zeta function, the expression ζ ( j )0 ( s , a ) = Z ( j )0 ( s , a ) + d (0) sin( π s ) π L s − + s − N X k = (cid:18) U (cid:19) k s + k , (4.14)where Z ( j )0 ( s , a ) is an analytic function for ℜ ( s ) > − ( N + / Z ( j )0 ( s , a ) = d (0) sin( π s ) π Z ∞ z − s ∂∂ z ln Ω ( j )0 ( iz , a ) − Θ ( z − zL − ln 2 + ln z + N X n = U n n n z n , (4.15)with Θ ( z ) denoting the Heaviside step-function. Once again, the meromorphic structure of ζ ( j )0 ( s , a ) iscompletely encoded in the terms in square parentheses in (4.14).2 V. COMPUTATION OF THE CASIMIR ENERGY AND FORCE
The analytic continuation of the spectral zeta function ζ ( j ) ( s , a ) in (3.20) can be used, at this point, tocompute the Casimir energy of the system. According to the definition given in (2.5) we need to analyzethe structure of ζ ( j ) ( s , a ) in a neighborhood of s = − /
2. In order to obtain an expression valid in aneighborhood of s = − / ffi cient to set N = D in the analytic continuation (3.20), namely ζ ( j ) ( s , a ) = Z ( j ) ( s , a ) + L √ π Γ ( s ) Γ s − ! ζ N s − ! + ζ N ( s ) − Γ ( s ) D X k = U k k + Γ (cid:16) k + (cid:17) Γ s + k ! ζ N s + k ! . (5.1)The representation (5.1) is now well defined for ℜ ( s ) > −
1. To compute ζ ( j ) ( s , a ) at s = − / s = ǫ − / ǫ expansion.Since Z ( j ) ( s , a ) is analytic for ℜ ( s ) > − s = − /
2. To perform thesmall- ǫ expansion for the remaining terms on the right-hand-side of (5.1) we need to take into account themeromorphic structure of the spectral zeta function ζ N ( s ) which, according to the general theory [22, 27], is ζ N ( ǫ − n ) = ζ N ( − n ) + ǫζ ′ N ( − n ) + O ( ǫ ) , (5.2) ζ N ǫ + d − k ! = ǫ Res ζ N d − k ! + FP ζ N d − k ! + O ( ǫ ) , (5.3) ζ N ǫ − n + ! = ǫ Res ζ N − n + ! + FP ζ N − n + ! + O ( ǫ ) , (5.4)where n ∈ N and k = { , . . . , d − } . It is important, at this point, to mention that the residues of the spectralzeta function ζ N ( s ) are proportional to the coe ffi cient of the small- t asymptotic expansion of the trace of theheat kernel associated with the Laplace operator ∆ N [22, 25, 27], that is Γ d − k ! Res ζ N d − k ! = A N k , Γ − n + ! Res ζ N − n + ! = A N d + n + . (5.5)By substituting s = ǫ − / L √ π Γ (cid:16) ǫ − (cid:17) Γ ( ǫ − ζ N ( ǫ − = L ζ N ( − πǫ + L π h ζ ′ N ( − + (2 ln 2 − ζ N ( − i + O ( ǫ ) . (5.6)For the next term, we utilize the expression (5.4) with n = ζ N ǫ − ! = ǫ Res ζ N − ! +
12 FP ζ N − ! + O ( ǫ ) . (5.7)In the sum appearing in (5.1) we need to separate the contribution of the term with k = k = − U √ π Γ (cid:16) ǫ − (cid:17) Γ ( ǫ ) ζ N ( ǫ ) = U πǫ ζ N (0) + U π h ζ ′ N (0) + − ζ N (0) i + O ( ǫ ) , (5.8)3which can be obtained thanks to (5.2). For the terms of the sum with k = { , . . . , D } we exploit (5.3) to get − U k k + Γ (cid:16) k + (cid:17) Γ (cid:16) ǫ + k − (cid:17) Γ (cid:16) ǫ − (cid:17) ζ N ǫ + k − ! = U k k + √ π Γ (cid:16) k + (cid:17) ǫ Γ k − ! Res ζ N k − ! + U k k + √ π Γ (cid:16) k + (cid:17) Γ k − ! ( FP ζ N k − ! + " − γ − + Ψ k − ! Res ζ N k − !) . (5.9)The results obtained above allow us to write the expression for the Casimir energy of the piston asfollows E ( j )Cas ( a ) = ǫ + ln µ ! " L π ζ N ( − +
12 Res ζ N − ! + U π ζ N (0) + D X k = U k k + √ π Γ (cid:16) k + (cid:17) Γ k − ! Res ζ N k − ! + Z ( j ) − , a ! + L π h ζ ′ N ( − + (2 ln 2 − ζ N ( − i +
12 FP ζ N − ! + U π h ζ ′ N (0) + − ζ N (0) i + D X k = U k Γ (cid:16) k − (cid:17) k + √ π Γ (cid:16) k + (cid:17) ( FP ζ N k − ! + " − γ − + Ψ k − ! Res ζ N k − !) + O ( ǫ ) . (5.10)It is clear from this expression that the Casimir force for the piston configuration is, in general, not a well-defined quantity as has already been observed before [9]. The ambiguity in the energy is proportional tothe coe ffi cients, which encode geometric information of the manifold N , of the asymptotic expansion ofthe heat kernel associated with the Laplacian on N as one can easily infer from the relations (5.5) and thefollowing one [22, 27] A N d + p = ( − p ζ N ( − p ) p ! , (5.11)with p ∈ N . To complete the result about the Casimir energy we would like to consider the contribution toit coming from possible zero modes of the Laplacian on N . From the expression (4.14) for ζ ( j )0 ( s , a ), we set N = s = ǫ − /
2, and compute the small- ǫ expansion of the resulting formula to obtain E ( j )Cas , ( a ) = d (0)8 π ǫ + ln µ ! U + Z ( j )0 − , a ! + d (0)4 π ( L + , (5.12)which, just like before, is in general not a well-defined quantity.Although the Casimir energy is, in general, an ambiguous quantity, the force acting on the piston is welldefined. In fact, it is easy to show, according to the definition (2.7), that the Casimir force acting on thepiston is simply F ( j )Cas ( a ) = − (cid:16) Z ( j ) (cid:17) ′ − , a ! . (5.13)4Analogously, if the Laplacian on N has zero modes then their contribution to the Casimir force on the pistoncan be obtained from (2.7) and (5.12), more explicitly F ( j )Cas , ( a ) = − (cid:16) Z ( j )0 (cid:17) ′ − , a ! . (5.14) VI. A d -DIMENSIONAL SPHERE AS PISTON In this section we apply the results for the Casimir force obtained earlier to the case of a piston configu-ration where the piston itself is assumed to be a unit d -dimensional sphere. For simplicity we also assumethat the length of the piston configuration is L =
1. For a d -dimensional sphere, the eigenvalues of theLaplacian on N are explicitly known and can be written as λ = l + d − , (6.1)where l ∈ N . The eigenfunctions on N are found to be hyperspherical harmonics with degeneracy d ( l ) = (2 l + d −
1) ( l + d − l !( d − . (6.2)By using (6.1) and (6.2) the spectral zeta function on the manifold N can be expressed as ζ N ( s ) = ∞ X l = (2 l + d −
1) ( l + d − l !( d − l + d − ! − s , (6.3)which, in turn, can be written as a linear combination of Hurwitz zeta functions [5, 6, 18, 19] ζ N ( s ) = d − X α = e α ζ H s − α − , d − ! , (6.4)with the coe ffi cients e α determined according to the formula( l + d − l !( d − = d − X α = e α l + d − ! α . (6.5)By using the eigenvalues (6.1) and their degeneracy (6.2) we can now analyze explicitly the Casimir force onthe piston (5.13) in the cases of Dirichlet-Dirichlet, Neumann-Neumann, and mixed boundary conditions.In what follows we set, for definiteness, d =
2. Obviously the analysis can be easily carried out in anydimension d . A. Dirichlet-Dirichlet boundary conditions
First, we consider the case in which Dirichlet boundary conditions are imposed at the endpoints ofthe piston configuration x = x =
1. Transmittal boundary conditions are, instead, imposed on the5 - - - Figure 1: Dirichlet boundary conditions at x = x =
1, and transmittal boundary conditions at x = a . The valuesalong the y -axis provide the magnitude (in units for which h = c =
1) of the Casimir force on the piston. piston itself at x = a . For a two-dimensional spherical piston N of unit radius the lowest eigenvalue of theLaplacian can be found to be, from (6.1), λ = /
2. This implies, according to the constraint (3.8), that theallowed values of the parameter U satisfy, in this case, the inequality U <
12 sinh(1 / / ≃ . . (6.6)Figure 1 displays the Casimir force acting on the piston positioned at x = a with a ∈ (0 , ff erent thickness represent the graph of the Casimir force for di ff erent values of the parameter U . Figure1 shows, in particular, the Casimir force on the piston when U = {− , − . , − . , . , . , . } . The thickerthe line the closer the value of U for that line is to the upper limit U DD ≃ . U are negative the piston is repelled from both endpoints of the pistonconfiguration while when 0 < U < U DD the piston is always attracted to the closest endpoint. This impliesthat by changing the sign of the parameter U in the transmittal boundary condition one can change theCasimir force from repulsive to attractive. The cuto ff value is U = de facto no piston configuration. B. Neumann-Neumann boundary conditions
We focus, now, on the Neumann-Neumann case. Namely, we impose Neumann boundary conditionsat x = x =
1, and impose transmittal boundary conditions on the piston. In this case the parameter U needs to satisfy the inequality U < U NN which for a spherical piston with d = - - - Figure 2: Neumann boundary conditions at x = x =
1, and transmittal boundary conditions at x = a . The valuesalong the y -axis provide the magnitude (in units for which h = c =
1) of the Casimir force on the piston. according to (3.9), U NN =
12 tanh ! ≃ . . (6.7)Figure 2 shows the Casimir force acting on the piston positioned at x = a with a ∈ (0 ,
1) for values of U inthe set U = {− , − . , − . , − . , . , . } . Just like the previous case, the thicker lines represent the graphof the Casimir force on the piston for values of U closer to the upper limit U NN ≃ . ffi cult to observe that for negative values of the parameter U the Casimir force tend to move the piston to the closest endpoint while for values of U in the interval0 < U < U NN the piston gets shifted away from the endpoints of the piston configuration. Once again, wecan conclude that also in this case a sign change in U changes the attractive or repulsive nature of the forceon the piston. C. Mixed boundary conditions
We analyze, lastly, the case of mixed boundary conditions. This case, as already mentioned earlier,contains two types of boundary conditions. In one instance Dirichlet boundary conditions are imposed at x = x = U in the transmittal boundary conditions must satisfythe inequality U < U M where, according to (3.10), U M =
12 tanh(1 / ≃ . . (6.8)7 - - - Figure 3: Dirichlet boundary conditions at x = x =
1. In addition, transmittalboundary conditions are imposed at x = a . The values along the y -axis provide the magnitude (in units for which h = c =
1) of the Casimir force on the piston. - - - Figure 4: Neumann boundary conditions at x = x =
1. In addition, transmittalboundary conditions are imposed at x = a . The values along the y -axis provide the magnitude (in units for which h = c =
1) of the Casimir force on the piston.
In figure 3 we have the Casimir force on the piston in the Dirichlet-Neumann case and in figure 4 wehave the Casimir force for the Neumann-Dirichlet case. Once again, the thicker the line the closer U is to the upper limit U M in (6.8). More precisely, the graphs, for both cases, were obtained for U = {− , − . , − . , . , . , } As we can clearly see from the Dirichlet-Neumann graphs in figure 3, for negative values of U theCasimir fore on the piston is always positive. This implies that the piston is attracted to the endpoint at x =
1. When 0 < U < U M , instead, the piston is always attracted to the x = U changes the sign of the Casimir force. VII. CONCLUSIONS
In this work we have analyzed the Casimir energy, and the corresponding force, for a massless scalar fieldpropagating on a piston configuration of the type I × N . The field is assumed to satisfy Dirichlet or Neumannboundary conditions at the endpoints of the piston configuration and transmittal boundary conditions on thepistons itself. A regularization scheme based on the spectral zeta function has been utilized to obtain explicitexpressions for the Casimir energy and force acting on the piston. For this configuration we analyzed threetypes of boundary conditions which we denoted by Dirichlet-Dirichlet, Neumann-Neumann, and mixed.The spectral zeta function associated with the piston configuration has been analytically continued to aneighborhood of the point s = − /
2. This procedure allowed us to explicitly evaluate the Casimir energyand force for a general piston N . The general results found in this work have then been used to analyzethe Casimir energy and force for the three types of boundary conditions in the case in which the piston is a d -dimensional sphere. Numerical results have been shown for a two-dimensional spherical piston with unitradius. Obviously, by using the general formulas one could obtain explicit results for any given dimension d and also, by suitably rescaling the spectral zeta function ζ N ( s ), for any specified radius of the sphere.As we have already mentioned earlier, this work is focused on the analysis of the Casimir energy andforce for a piston configuration endowed with transmittal boundary conditions on the piston and simpleDirichlet or Neumann boundary conditions at the endpoints. It seems natural that the next step in this in-vestigation would consist in considering more general boundary conditions at the endpoints of the pistonconfiguration. Such generalized boundary conditions can be written as a linear combination, through realcoe ffi cients, of the values of the field and its derivative at the given endpoint (see for instance [21]). Un-fortunately, finding a suitable range of values for the real coe ffi cients characterizing the general boundaryconditions and the parameter U that leads to a self-adjoint boundary value problem proves to be a prohibitivetask within the formalism employed in this work. It would be very interesting to understand whether themethod developed in [1] could be more appropriate to analyze the Casimir e ff ect in this more general case.A number of generalizations to this work can be envisaged. Apart from considering more general bound-ary conditions, as mentioned above, it would be very interesting to analyze piston configurations possessingdi ff erent types of geometry. For instance, one could consider a piston configuration constructed from threeconcentric spheres (or three coaxial cylinders) with Dirichlet or Neumann boundary conditions imposedon the innermost and outermost surfaces and transmittal boundary conditions imposed on the one between9the two. One could also consider warped piston configurations of the type I × f N where, I = [0 , L ], and f is a suitable warping function as considered in [20]. In this case transmittal boundary conditions wouldbe imposed on the piston represented by the manifold N positioned at a ∈ (0 , L ). It would be particularlyintriguing to understand how the presence of both the warping function and the parameter U influence theCasimir force on the piston and if, for a given warping function f , one could find non-vanishing values of U for which the piston experiences no Casimir force. We hope to report on some of these generalizationsin future works. [1] Asorey M. and Mu˜noz Casta˜neda J. M., Attractive and repulsive Casimir vacuum energy with general boundaryconditions, Nucl. Phys. B , 852 (2013)[2] Barton G., Casimir piston and cylinder, perturbatively,
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