Casimir energy with a Robin boundary: the multiple-reflection cylinder-kernel expansion

Z H Liu¹ and S A Fulling^2,3

¹ Department of Physics, Texas A&M University, College Station, TX 77843-4242, USA
² Departments of Mathematics and Physics, Texas A&M University, College Station, TX 77843-3368, USA

³ Author to whom any correspondence should be addressed.

Email: zliu@physics.tamu.edu and fulling@math.tamu.edu

Received 1 May 2006
Published 20 October 2006

Abstract. We compute the vacuum energy of a massless scalar field obeying a Robin boundary condition ((∂/∂ x)  =  β ) on one plate and the Dirichlet boundary condition (  =  0) on a parallel plate. The Casimir energy density for general dimension is obtained as a function of a (the plate separation) and β by studying the cylinder kernel (alias an exponential ultraviolet cutoff); we construct an infinite-series solution as a sum over classical paths and observe that the method of construction has broader applications. The total Casimir energy is finite after subtraction of divergences associated with the individual plates, which do not affect the force between the plates. The series for the total energy is an alternative to the integral formula of Romeo and Saharian, with which it agrees numerically.

Contents

• 1. Introduction
• 2. Notation and main theorem
• 3. Vacuum energy densities of a Robin-Dirichlet slab and a single plate
  • 3.1. Vacuum energy density for two parallel plates
  • 3.2. Vacuum energy density for a single plate
• 4. Total vacuum energy of a slab with Robin and Dirichlet boundary conditions
• 5. Conclusion
• Acknowledgments
• Appendix A. Proof of theorem 1
  • A.1. Proof of equation (2.16)
  • A.2. Proof of (2.17), etc
• Appendix B. Boundary divergences in the total energy
  • B.1. Regularized energy for a single plate
  • B.2. Regularized energy for the slab
• Appendix C. Why we can discard terms with n ≥ 4
• References

1. Introduction

The scalar Casimir effect for parallel plates with Robin boundary conditions has been studied thoroughly by Romeo and Saharian [1] (see also [2]–[4]). Our re-examination of that problem here has several (loosely related) points of novelty. (i) Instead of zeta-function or dimensional regularization, we employ an explicit exponential cutoff on the normal-mode frequencies, which is implemented by calculating with an elliptic Green function associated with the system—the cylinder kernel [5]–[9]. (ii) Instead of dealing directly with the eigenvalues and eigenfunctions of the system, we take the dual approach of constructing the Green function as a sum over classical paths, which in this problem means following the repeated reflections of rays from the plates. The method for incorporating Robin boundary behaviour into this scheme was presented in [10]. (iii) We calculate a local energy density and study carefully its relation to the renormalized total energy, calculated in the same regularization scheme. Please note that the constant denoted by β here (see (2.5)) is called –1/β in [1].

We start the discussion by comparing the cylinder kernel with the better-known heat kernel. Consider the Laplacian operator in a bounded region Ω of d-dimensional Euclidean space, with some self-adjoint boundary condition, and let _n(x) and ω_n² be the corresponding eigenfunctions and eigenvalues. (In this section the spatial integrations are understood to be over Ω.) The local heat kernel is defined by

The global heat kernel is the trace over the local one,

The local cylinder kernel is defined by

Then the global cylinder kernel is

The local heat kernel and cylinder kernel can be viewed as the Green functions of certain differential-equation problems: the heat kernel solves the heat equation in the sense that

is the unique solution of the initial-value problem

It has a well-known asymptotic expansion of the form

The cylinder kernel can be characterized similarly:

is the unique bounded solution of the problem

The counterpart of (1.7) for T is [8, 11, 12]

Both (1.7) and (1.10), as well as equation (1.13) later, come in both local and global versions.

The heat-kernel expansion has been a powerful tool to investigate the divergence structure of the vacuum energy, but it does not contain the nonlocal geometrical information needed to compute the finite part. The cylinder-kernel coefficients in (1.10) do incorporate that information. Formally, we can relate the total Casimir energy to the global cylinder kernel by taking the t-derivative,

and the simplest definition of the vacuum energy density is

(Other definitions differ from (1.12) by total divergences, so that classically they yield the same total energy when surface energies are taken into account [2, 3, 8, 13]. Definition (1.12) corresponds to ξ  =  1/4 in a standard notation. The other definitions (other choices of ξ) will not be treated in this paper. Note that E is independent of ξ.)

In reality, the definitions of Casimir energy and vacuum energy density in (1.11) and (1.12) contain divergent terms. But the coefficients of the divergent terms are simple, local objects that can be absorbed by renormalization, or at least cancelled when calculating forces between rigid bodies. The finite Casimir energy is given by the term of order t in (1.10)

We will discuss the structure of the divergent terms in detail in later sections, since it depends on the dimension.

The paper is organized as follows. In section 2, we set up the notation and show how to construct the integral kernels for a slab with a Robin boundary. In section 3, we calculate the energy density, and in section 4 the total energy. Some implications and motivations are discussed in section 5, and some fine points in appendices.

2. Notation and main theorem

The cylinder kernel of the free massless scalar field in R^d is

where

The cylinder kernel (2.1) is the Green function of the equation (1.9) when Ω is the free space R^d with no boundary. Our initial considerations apply to Green functions for more general problems than (1.9). We know from the method of images that the Green function associated with a Dirichlet problem (u(t,0)  =  0) in a half-space (0,∞)× R^d–1 is

and the Green function associated with a Neumann problem (∂/∂ xu(t,0)  =  0) in the half-space is

But when it comes to the Robin problem

the elementary method of images no longer applies. In ([10] theorem 2), it was shown that one can still construct the Green function for the Robin problem from the Green function for all of R^d, G, by adding an integral correction to the Green function for the corresponding Neumann problem

All this leads us to the more general problem of how to construct the Green function for a slab, Ω  =  (0,a)× R^d–1, or even (a₁,a₂)× R^d–1, with any kind of boundary conditions. It is helpful to define four families of operators D, N, R and this way:

The functions

respectively satisfy Dirichlet, Neumann and Robin boundary conditions at x  =  a. Furthermore, G_D and G_N are Green functions (in particular, they have the correct Dirac-delta boundary behaviour as t→0) both in the region to the left of a and in the region to the right of a, whereas G_R has that property to the right of a.

A Robin condition at a right-hand boundary, to be physically similar to equation (2.5), must be of the form

(The inward normal derivative must have the positive sign.) Then

is the correct Green function for the region left of a.

Theorem 1. Let T(t,x,x_⊥,y,y_⊥) be the cylinder kernel on all of R^d; then the corresponding cylinder kernel of the slab (0,a)× R^d–1 with Robin boundary condition equation (2.5) at x  =  0 and Dirichlet boundary condition at x  =  L is

Here

where

is a Laguerre polynomial ([14], chapter 22) ([15], sections 7.4 and 8.97). Two notational abbreviations have been adopted: the variables (t,x_⊥,y_⊥) are suppressed because they undergo no alteration, and it is understood that the integral terms are to be omitted whenever n  =  0.

We provide the proof of theorem 1 in appendix A; it is parallel to the construction of the wave kernel in ([10] section 5). We now comment on the structure of the formula, which is a sum over classical paths from (x,x_⊥) to (y,y_⊥), including integrations over time delays at the Robin boundary. The terms can be thought of as wave pulses in a generalized sense. Terms (2.17) experience an even number of reflections, starting at the left; terms (2.18) experience an even number of reflections, starting at the right. When y  =  x these terms are constant and are equal in pairs (i.e., D_aR₀T(x,x) and R₀D_aT(x,x) are identical, etc, because (2.1) is a function of |x–y|); these classical paths are periodic orbits (when y_⊥  =  x_⊥ also) and will contribute the spatially uniform Casimir energy associated with the finiteness of L. Terms (2.19) experience an odd number of reflections, starting at the right; terms (2.20) experience an odd number of reflections, starting at the left. When (y,y_⊥)  =  (x,x_⊥) these paths are bounce orbits (closed but not periodic) that contribute the localized vacuum energy of interaction of a quantum field with the boundaries.

Because of the simplicity of the slab geometry, the series solution equation (2.16) is exact, in principle; no stationary-phase approximations, for instance, have been needed. In practice, it may become necessary to truncate the sum, considering only short paths.

Clearly the construction is more general than stated in theorem 1. (i) It applies to other Green functions, such as the wave kernel [10] and heat kernel [16]. (ii) Setting β  =  0 in the theorem's formulae yields T_ND , the cylinder kernel for Neumann condition at x  =  0 and Dirichlet condition at x  =  a. (iii) For Neumann condition at x  =  a, T_RN is given by the same formulae with the factors (–1) omitted. (iv) Formulae (2.7)–(2.15) have been formulated to allow one to get the analogue of (2.16) for any standard boundary conditions by replacing R₀ and D_a by the corresponding operators. Except in the case the analogues of (2.17)–(2.20) follow easily. (v) We believe that the same method can be applied in principle to nonflat boundaries and nonconstant β; of course, the result in such a case cannot be any more accurate than whatever classical-path or multiple-reflection approximation is used for the underlying Green functions G and G_D or G_N .

3. Vacuum energy densities of a Robin–Dirichlet slab and a single plate

In this section we calculate the vacuum energy density of a slab with Robin boundary at x  =  0 and Dirichlet boundary at x  =  a for general spatial dimension d. We also consider the density for a scalar field satisfying a Robin or Dirichlet boundary condition for an isolated plate. The single-plate effects completely account for the singular behaviour of the density in the slab.

3.1. Vacuum energy density for two parallel plates

An infinite summation for the energy density follows from theorem 1, the definition of vacuum energy density in (1.12), and the formula (2.1) for T. According to (2.16) the cylinder kernel is a sum of four types of terms,

On the other hand, in the classical-path interpretation the sum can be reorganized by number of reflections as

We now implement the local version of equation (1.13). The first term in equation (3.5), T, corresponds to a path that experiences no reflection at all; the contribution of this term to the vacuum energy density is

This is the anticipated leading divergent term. It is the universal, x-independent formal vacuum energy of infinite empty flat space; speculations about `dark energy' aside, it is universally agreed that this term should be discarded.

The second and third terms, D_LT and R₀T, experience only one reflection on the boundary; they respectively contribute

and

These two terms are `dangerous' since we can see that the energy density contributed by D_aT goes (nonintegrably) to infinity at the Dirichlet boundary (x→ a) and the energy density contributed by R₀T diverges similarly at the Robin boundary (x→ 0).

Next we write down the contributions to vacuum energy density from paths that are reflected at least twice on the boundary, calculated through equation (1.12)

The total vacuum energy density is the sum of (3.7) through (3.12). The contributions from (D_aR₀)ⁿT and (R₀D_a)ⁿT are independent of x; they correspond to the periodic orbits. The contributions from (D_aR₀)ⁿD_LT and (R₀D_a)^n–1R₀T are x-dependent; they correspond to the bounce orbits. All these terms with at least two reflections are integrable. Thus, apart from the universal divergent term T, the dangerous single-reflection terms (3.7) and (3.8) are the only terms that could be called divergent (and at this local stage they are still pointwise finite). They can be related to the situation of a single plate, as we now verify.

3.2. Vacuum energy density for a single plate

Consider a single plate with Dirichlet boundary condition at x  =  a and a single plate with Robin boundary condition at x  =  0. The corresponding cylinder kernels can be constructed on the basis of equation (2.11) and (2.13)—in other words, by considering closed paths with at most one reflection. Namely,

and

Each cylinder kernel again contains the trivial term T, which we discard. The corresponding vacuum energy densities are then

For x inside the slab, these are precisely the `dangerous' terms of the slab case, equations (3.7) and (3.8).

4. Total vacuum energy of a slab with Robin and Dirichlet boundary conditions

In this section we consider the total vacuum energy (per unit (d–1)-dimensional area) of a slab with Robin boundary at x  =  0 and Dirichlet boundary at x  =  a for general spatial dimension d. Within the cylinder-kernel analysis there are two approaches that could be taken. Firstly, one can attempt to integrate the local energy density given by (3.7)–(3.12), arguing away part or all of the divergent integrals of the dangerous terms. It is clear from subsection 3.2 that the singular behaviour of those terms is identical to that of the separate single-plate systems. Therefore, one can formally subtract the two single-plate energies to obtain the finite (per unit area in x_⊥) Casimir energy associated with the interaction of the two plates. Secondly, one can integrate T(t,x,x) (before or after taking its t derivative)with t strictly positive to define a regularized total energy, then take t to 0 and appeal to `renormalization' to discard divergent terms (those proportional to negative powers of t or to  lnt). This is the global version of (1.13). The first method will be followed here, the second in appendix B.

Integrating equations (3.7) and (3.8) over (0,a), we formally obtain the contributions of the two dangerous terms, D_aT and R₀T, to the total energy as

The first is clearly divergent as x→ a, and the second is similarly divergent as x→0. But the divergence due to the Dirichlet plate at x  =  a is cancelled by subtracting the vacuum energy of a single plate in the region (–∞,a)× R^d–1:

In the same way, for the Robin plate at x  =  0 we subtract the vacuum energy of a single plate in the region (0,∞)× R^d–1:

For general n≥ 1, no divergent terms are involved. (D_aR₀)ⁿT and (R₀D_a)ⁿT each contribute to the total energy

(D_aR₀)ⁿD_aT contributes

For general n≥ 2, (R₀D_a)^n–1R₀T contributes

Now we sum up all the terms to obtain the finite total energy

Because we take ξ  =  1/4, there is no additional surface contribution as in [1].

The first term in E can be expressed by the Riemann η-function:

it is the known result for one Neumann and one Dirichlet plate. The integrals in (4.8) can be evaluated in terms of the incomplete gamma function ([14], section 6.5), ([15], section 8.35). The resulting infinite summation presumably can not be converted to a closed form. However, the terms starting with n  =  4 are relatively small and almost cancel each other, so the expression truncated to n ≤ 3 is a good approximation for the total energy. (The proof of this assertion is in appendix C.) Explicitly, the total energy for d  =  3 as a function of b  =  β a (through order n  =  3) is

(Again, the b of Romeo and Saharian [1] is the negative reciprocal of our b.)

Note that at β  =  0 the Robin boundary becomes a Neumann boundary and one recovers

When β→ ∞, the Robin boundary becomes a Dirichlet boundary, so we expect to recover the familiar result

The graph of a³E_RD as a function of b  =  β a is given in figure 1, which (together with numerical calculations for larger b) confirms equations (4.11) and (4.12). (All computations and graphics were done with Mathematica.) The crossover from positive to negative energy occurs near b  =  1.237, or –1/b ≈ –0.81, in agreement with [1]. That reference states that this value marks a change from repulsive to attractive Casimir force, but that is incorrect: The zero of the force function –(∂/∂ a)E_RD occurs at some larger value of a.

5. Conclusion

In Casimir theory—and in the general study of partial differential equations and the spectral theory of differential operators—the Robin boundary condition is of theoretical interest as the simplest step beyond the standard Dirichlet and Neumann problems for any particular geometrical configuration. The Robin condition also has physical applications: it arises naturally in place of the Neumann condition for half of the modes of the electromagnetic field in the presence of a curved boundary, it mocks up in a simple way the effect of a boundary between two media, and it may have cosmological significance in the brane-world scenario ([17, 18] and references therein).

Our numerical results agree with those of Romeo and Saharian [1] to the extent that they have been compared. Because we use different notations to express the Robin boundary condition, the counterpart of b₂ in their notation is our –1/b. For a more direct comparison, we plot in figure 2 the total energy E_RD with respect to –1/b. The result matches ([1], figure 3) very well, including the location of the zero.

The formula in [1] for the total energy is a rather complicated integral. Ours is an infinite sum whose terms fall off fairly rapidly, so reasonable accuracy can be attained by truncating the series. At least in the case where only one of the boundaries is Robin, the individual terms in the series can be evaluated in terms of known special functions, the Laguerre polynomials. The scope of this paper has not allowed us to tackle the case of two Robin boundaries in such detail, nor to study in much depth the questions of how the signs of the energy and the force depend on the parameters. Finally, we have restricted attention to positive Robin constants; the negative case is of more dubious physical significance, and the construction of the cylinder kernel in that case requires different mathematics [10].

We have taken pains to calculate the local energy density (albeit for only the easiest choice of the conformal coupling parameter, ξ  =  1/4) and to conduct the calculation of the total energy in the same framework. It has been known for many years [19] that vacuum energy densities in flat space are pointwise finite (apart from the ubiquitous zero-point energy of every quantized field) but (in general) nonintegrable near boundaries. The Robin condition introduces a new (less singular) divergent term in addition to those familiar from the more elementary conditions. Direct calculations of total energy lead immediately to formal divergences. When an ultraviolet cutoff (in particular, the cylinder-kernel approach) is used, the divergent terms depend on the cutoff parameter t polynomially or logarithmically, and these terms have a close relation to the divergent integrals of the energy density [8]. In odd dimensions the divergences associated with the Robin constant include one of the logarithmic class. `Analytic' regularization schemes (dimensional and zeta functions) automatically remove the polynomial terms. However, it is not clear that this nonchalance is physically justified. The energy density serves as a source in the gravitational field equation, so its singular behaviour at boundaries cannot just be ignored [9, 19]. Also, the traditional approach to Casimir forces, while plausible for predicting attractions between rigid bodies, has been strongly criticized when applied to deformations of bodies [20]–[23]. It may be that the divergent terms in the vacuum energy (or the related divergent integrals of the energy density) can be absorbed into terms in the equations of motion representing the mechanical response of the materials in the bodies, but there is generally no justification for simply setting those terms to zero. In the end a successful physical analysis of a particular system of experimental relevance must be based on a more realistic and complete model, but in the meantime a clear understanding of the (relatively tractable) vacuum-energy calculations is needed in order to diagnose the problems and to determine the limits of validity of the theory.

In the parallel-plate problem we have shown that the only divergent terms are directly associated with the individual plates. Therefore, they are not functions of the plate separation and do not contribute to the force between the plates. (This was, of course, known already, but our treatment of the total energy in the same framework as the energy density removes a certain mysticism from the renormalization and promises to elucidate the physics in more complicated situations in the future.) Within the remaining finite energy equation (4.8), the Neumann term equation (4.9) is a Casimir energy in the strictest sense: it is associated with the discretization of modes and with periodic orbits of the underlying classical system. The β-dependent terms, on the other hand, include contributions from equation (4.6) and (4.7) associated with closed but nonperiodic orbits (those with an odd number of reflections). Unless there is a nonobvious cancellation, these terms do contribute to the force. (We thank an anonymous referee for pointing out this fact.)

The construction of the cylinder kernel as a multiple-scattering expansion is a powerful method for calculating local spectral and vacuum effects, which demands further development.

Acknowledgments

We thank K A Milton for comments on the manuscript. This work is supported in part by National Science Foundation (NSF) Grant No. PHY-0554849.

Appendix A. Proof of theorem 1

A.1. Proof of equation (2.16)

After rearrangement the proposed series is

Because of the falloff of T as a function of x (see (2.1)) the series converges (absolutely). Therefore, it is easy to see that it satisfies the cylinder equation (1.9) inside the slab and the proper boundary condition at t  =  0. Finally, by virtue of (2.11) and (2.13), it satisfies both the Dirichlet condition at x  =  L and the Robin condition at x  =  0.

A.2. Proof of (2.17), etc

When n  =  1,

so (2.17) is satisfied when n  =  1. Suppose that when n  =  m (2.17) is satisfied

Then when n  =  m + 1,

Let θ  =   + η; then

Thus

That means that equation (2.17) is satisfied also when n  =  m + 1. The formulae (2.18)–(2.20) can be proved by induction in the same way.

Appendix B. Boundary divergences in the total energy

Here, we analyse the total energy by the global approach. That is, we integrate T_RD(t,x,x) to get the global cylinder kernel T_RD(t) before taking its t derivative and examining the limit t→0. We concentrate on the case d  =  3 (hence C(d)  =  π^–2), and we discard from the outset the universal divergent term T of equation (3.6).

B.1. Regularized energy for a single plate

For the infinite space to the right of a Robin plate at x  =  0 the integrated cylinder kernel is, from (3.12),

For later comparison with the case of two plates, it is convenient to keep the lower-limit and upper-limit contributions separate.

From the upper limit at ∞ one gets (for β≠0)

The discontinuity at β  =  0 is only apparent, because we shall now see that the contribution from the integral is cancelled by a like term from the lower limit.

From the lower limit 0 one gets

The sine integral function  Si  and cosine integral function  Ci  have Taylor expansions

where γ is Euler's constant. Therefore, the expansion of (B.3) at small t is

The total regularized energy from (B.2) and (B.6) is

Similarly, the integrated cylinder kernel to the left of an isolated Dirichlet plate at x  =  a is

which corresponds to a regularized energy

B.2. Regularized energy for the slab

We must integrate the terms (3.1)–(3.4) from 0 to a. Recall that only the terms D_aT and R₀T contain divergences. In all the other terms the denominator of the integrand remains nonzero even when both t and are zero, and therefore one can differentiate and pass to the limit t→ 0 before integrating; that is, their contributions are precisely those already presented in (4.5)–(4.7).

For the divergent terms we could recycle the calculations equations (B.1) and (B.8), replacing the upper limit ∞ with a. However, the difference would be the negatives of the integrals from a to ∞, and to them the same argument as above applies: these are perfectly finite contributions to the energy, even when t  =  0, and they have already been computed in (4.3)–(4.4).

All that remains to be considered is the sum of the regularized energies equation (B.7) and equation (B.9). (Recall that we have already discarded the ubiquitous t^–4 term.) The terms of order t^–3 cancel, but this is an artefact of our model, since Dirichlet and Neumann plates have divergent surface energies that are equal and opposite. According to the prescription equation (1.13), we should discard all the terms in the series that diverge as t→ 0. In the present case, because there is a logarithmic term in equation (B.7), we encounter the well known scale ambiguity: because the numerical factor inside the argument of the logarithm is arbitrary, the `finite part' of E_R , hence that of E_RD , is defined only up to an arbitrary numerical multiple of β³. Ignoring E_R entirely in calculating E_RD yields the prescription of section 4. The ambiguous β³ term does not depend on a and hence does not affect the force between the plates. It does, of course, depend on β; one must feel some trepidation in ignoring it (or even the power-law divergent terms) in situations where β is allowed to vary.

Appendix C. Why we can discard terms with n ≥ 4

The expression of the total energy in equation (4.8) is an infinite summation, but we shall prove for case d  =  3 that all terms after n  =  3 are quite small, so it is reasonable to discard them. Note that when d  =  3, C(d)  =  1/π² and hence the β-dependent part of the remainder is

where

Let 2β  =  x; then these equations can be written as

and

It is straightforward to show that f_n(x) is a decreasing function and f_n(x)≥ 0 for any x≥0, so

From a mean value theorem for integrals one has

It follows that

From (8.971.2) and (8.971.5) in [15], we get the integral

Telescoping this recursion, we get

But L₀¹(x)  =  1, so

From ([14], (22.14.13)),

hence

Now we continue (C.7)

which is roughly 2% of |E_DD|  =  π²/1440a³. (The actual error in our numerical calculations is at most 0.1%.)

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