Casimir type effects for scalar fields interacting with material slabs

Casimir type effects for scalar fields interacting with material slabs

I V Fialkovsky†‡, V N Markov ¶ and Yu M Pis’mak† † Department of Theoretical Physics, State University of Saint-Petersburg, Russia ‡ Insituto de Física, Universidade de São Paulo, São Paulo, Brazil ¶ Department of Theoretical Physics, Petersburg Nuclear Physics Institute, Russia
Abstract

We study the field theoretical model of a scalar field in presence of spacial inhomogeneities in the form of one and two finite width mirrors (material slabs). The interaction of the scalar field with the defect is described with position-dependent mass term.

For a single layer system we develop a rigorous calculation method and derive explicitly the propagator of the theory, the S-matrix elements and the Casimir self-energy of the slab. Detailed investigation of particular limits of self-energy is presented, and connection to known cases is discussed.

The calculation method is found applicable to the two mirrors case as well. With its help we derive the corresponding Casimir energy and analyze it. For particular values of parameters of the model an obtained result recovers the Lifshitz formula. We also propose a procedure to unambiguously obtain the finite Casimir self-energy of a single slab without reference to any renormalization conditions. We hope that our approach can be applied to calculation of Casimir self-energies in other demanded cases (such as dielectric ball, etc.)

pacs:
03.70.+k, 11.10.-z

1 Introduction

Usual models of interaction of elementary particles are constructed in a framework of the quantum field theory (QFT) in the homogenous infinite space-time [1]. However, as such they can not be applied for description of phenomena of interaction of quantum fields with macroscopic bodies. At least the form of latter must be presented in the model, and it can change essentially both the spectrum and dynamics of the excited states of the model as well as the properties of the ground (vacuum) state of the system.

First quantitative results for such effects were obtained by H. Casimir in 1948. He predicted [2] macroscopical attractive force between two uncharged conducting plates placed in vacuum. The force appears due to the influence of the boundary conditions on the electromagnetic quantum vacuum fluctuations. Nowadays the Casimir effect is verified by experiments with a precision of % (see [7] for a review).

The properties of vacuum fluctuations in curved spaces, the scalar field models with various boundary conditions and their application to the description of real electromagnetic effects were actively studied throughout the last decades (see discussion and references in [4]-[9]). However, it was well understood from the beginning that boundary conditions must be considered just as an approximate description of a complex interaction of quantum fields with the matter. A generalization of the boundary conditions method has been proposed by Symanzik [3]. In the framework of path integral formalism he showed that the presence of material boundaries (two dimensional defects) in the system can be modeled with a surface term added to the action functional. Such singular potentials with -type profile functions concentrated on the defect surface reproduce some simple boundary conditions (namely Dirichlet and Neumann ones) in the strong coupling limit. The additional action of the defect should not violate basic principles of the bulk model such as gauge invariance (if applicable), locality and renormalizability.

The QFT systems with -type potentials are mostly investigated for scalar fields, see for instance [10]. In [11][13] the Symanzik’s approach was used to describe similar problems in complete quantum electrodynamics (QED), and all -potentials consistent with the QED basic principles were constructed.

It seems quite natural to try applying the same method for the description of the interaction of quantum fields with bulk macroscopic inhomogeneities (slabs, finite width mirrors, etc.) and to study Casimir effects in systems of such kind. The Symanzik’s method of adding extra potential terms into the action of a system was used to model the interaction of quantum scalar fields with bulk defects in a number of papers (e.g. [26], [29]-[32], and others). However, most of them were devoted to study of a limiting procedure of transition from a bulk potential of the defect to the surface -potential one, without paying much attention to other properties.

Traditionally, there are several approaches to study electromagnetic Casimir effect between material slabs. The electromagnetic field and the material bodies can be treated macroscopically and by employing a dissipation fluctuation theorem one obtains the field correlation functions that are needed to construct the Maxwell stress tensor. In alternative approach the presence of dielectric bodies is described by means of a spatially varying permittivity that is a complex function of the frequency. Within these approaches the influence of dielectric and geometrical properties on the Casimir force between dispersing and absorbing plates has been studied in [22, 24]. A dependence of the Casimir force on thickness of the slabs was considered in [25]. A calculation of the Casimir force in a (multi)layer system have been also performed in [19, 20, 21, 23]. Most recently there was an attempt to construct quantum electrodynamics in the presence of dielectric media (i.e. volume defects of special kind) [18]. From the field theoretical point of view none of these methods were truly successful. Particularly the celebrated Lifshitz approach [14] to the description of the interaction between dielectric bodies still attracts a lot of arguments if dispersion is present in the system and formally is proved only for the interaction over the vacuum gap, see [15]. While in some approaches there could be no action functional constructed [6], there are also general severe problems in construction of path-integral quantization for the QED systems with dielectrics [16] and calculation of the heat kernel coefficients for such systems [28]. Moreover, as yet remains unresolved the mystery of Casimir energy of a dielectric ball which is defined unambiguously only in the dilute limit or for the speed of light being constant across the boundary [17].

On the other hand, existing results for the Casimir energy of a scalar field in presence of a single planar layer of finite width are contradictory. The formulae presented recently in [33] do not coincide with previous calculations made in [26] as discussed in [34]. The only attempt to calculate the propagator in such system was undertaken in [35], where hardly any explicit formulae were after all presented. The system of two interacting slabs has never been considered within QFT models.

Thereby, one can see that the specificity of finite volume effects generated by inhomogeneities in QFT has not been yet adequately explored. Our work is dedicated to clarify the problem, and to solve existing controversy within an accurate and unambiguous approach. On the other hand, considered type of square well potentials has its own range of applicability including transport in graphene and other condensed matter systems [39], and (apparently) superluminal effects in wave propagation [40]. While consideration of the scalar fields may only appear restricting, for the case of planar geometries it is apparently sufficient. It is due to the well known fact that in this case the theory of electromagnetic (EM) field is equivalent to two scalar field theories as and modes of the former field do not mix. The study of EM fields in the geometries, when such decomposition does not appear, will be the scope of the future work.

Thus, we consider a model of massive scalar field interacting with the volume defects — finite width material slabs or mirrors, — whose properties are effectively described by single macroscopical parameter, the coupling constant. By construction the model is renormalizable in the sense applicable to QFT with spatial inhomogeneities given in [3]. Working with renormalizable models has an important advantage that their long distance (macroscopical) properties such as Casimir interaction can be described without explicit knowledge of microscopical structure even if there is a true physical cut-off present in the system, [41]. Such description can be given in terms of just a few effective parameters. In our case, the role of these parameters is played by the coupling constant. Any physical cut-off dependence is disregarded in light of renormalizability of the model.

We develop a purely field theoretical approach and construct explicitly the propagator for the system with a single finite width mirror (Sect. 3.1) and the S-matrix elements (Sect. 3.2). Based on these new results in Section 3.3, we recover the known expression for the self-interaction of a single slab and supply it with the investigation of its asymptotical behavior in different physically interesting regimes (Sections 3.53.7).

In Section 4 the Casimir energy of two interacting finite width slabs described within the scalar field model with step potential is obtained for the first time (Sect. 4.2) and several limiting cases are considered (Sect. 4.3). We also reveal its correspondence to other known results in the literature, and in section 4.4 discuss its connection to the Lifshitz formula. Possible ways to construct divergence free limits for Casimir self-interaction of a solitary body are presented in Sect. 4.5. In the Conclusion the main results are summarized. In the A we briefly discuss a by-side question of comparison of different regularizations within QFT.

2 Statement of the problem

Let us consider a model of a scalar field interacting with a space defect with nonzero volume. Using the Symanzik’s approach, we describe such system by the action functional with an additional mass term being non-zero only inside the defect

 S = S0+Sdef S0=12∫d4xϕ(x)(−∂2x+m2)ϕ(x) Sdef=λ2∫d4xθ(ℓ,x3)ϕ2(x)

where 111We operate in Euclidian version of the theory which appears to be more convenient for the calculations.). In the simplest case, the defect could be considered as a homogenous and isotropic infinite plane layer of the thickness placed in the plane. In this case it is sometimes called ‘piecewise constant potential’. The distribution function is then equal to when , and is zero otherwise. In terms of the Heaviside step-function it can be written as

 θ(ℓ,x3)≡[θ(x3+ℓ/2)−θ(x3−ℓ/2)]/ℓ. (2)

In the framework of QFT such potentials were approached for the first time in [26], and later in [30]-[33]. However, the latest consideration [33] was proved to be inconsistent [34].

To describe all physical properties of the systems it is sufficient to calculate the generating functional for the Green’s functions

 G[J]=N∫Dϕexp{−S[ϕ]+Jϕ},N−1=∫Dϕexp{−S0[ϕ]} (3)

where is an external source, and normalization for the generating functional has been chosen in such a way that .

Introducing in (3) an auxiliary field defined in the volume of the defect only, we can present the defect contribution to as

 exp{−λ2ℓ∫d∫ℓ/2−ℓ/2dx3ϕ2(x)}= (4)
 =C∫Dψexp{∫d∫ℓ/2−ℓ/2dx3(−ψ22+i√κψϕ)}

which is written for the case of a single layer. is an appropriate normalization constant, and , here and below we denote with arrows the components of 4-vectors, i.e. . When generalized to other defects the -integration in (4) should be performed only within their support.

With help of projector onto the volume of the layer acting as , we can perform the functional integration over , and consequently over . As the result we get

 G[J]=[DetQ]−1/2e12J^SLJ,^SL=D−κ(DO)Q−1(OD), (5)
 Q=1+κ(ODO). (6)

Here the unity operator 1, as well as the whole , is defined in the volume of the defect only , and is the standard (Feynman) propagator of the free scalar field. We shall note here that the outlook of (5) completely coincides with the expression for the generating functional in the case of delta-potential defect instead of patchwise constant one subject to appropriate redefinition of the projecting operator . It is also evident that a straightforward generalization is possible for non-constant () with depending on .

3 One mirror system

3.1 Calculation of the propagator

To calculate the propagator defined according to (5) let us first derive an explicit formula for the operator .

For this purpose we make the Fourier transformation on the coordinates parallel to the defect (i.e. , , ). In such mixed - representation the propagator of the system without a defect is given by

 D(x)=∫d3(2π)3eiDE2(x3),DV(x)≡e−√V|x|2√V (7)

with , . Then can be defined through the following operator equation

 W+κDE2W=1. (8)

By construction the free scalar propagator is the Green’s function of the following ordinary differential operator

 KV(x,y)=(−∂2∂x2+V)δ(x−y) (9)

for . Multiplying both sides of (8) with and using obvious relation we get

 KρU=−κ (10)

where and .

The general solution to this (inhomogeneous) operator equation can be written as a sum of its partial solution and the general solution of its homogeneous version. Taking into account the symmetry property of : , which follows from its definition, we arrive at

 U(x,y)=−κDρ(x,y)+ae(x+y)√ρ+b(e(x−y)√ρ+e(y−x)√ρ)+ce−(x+y)√ρ (11)

where , and are some constants. To derive them we introduce into (8) and require its identical validity for all and . It yields

 a=c=−ξκ2eℓ√κ+E22√κ+E2,b=−ξκ(E−√κ+E2)22√κ+E2, (12)
 ξ=1e2ℓ√κ+E2(E+√κ+E2)2−(E−√κ+E2)2.

Substituting into (5) and using (12) we can finally derive the explicit formulae for the modified propagator of the system

 ^S(,x3,y3)=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩S−−,x3<−ℓ/2, y3<−ℓ/2S−∘,x3<−ℓ/2, y3∈(−ℓ/2,ℓ/2)S−+,x3<−ℓ/2, y3>ℓ/2S∘∘,x3∈(−ℓ/2,ℓ/2), y3∈(−ℓ/2,ℓ/2) (13)

where

 S−− =e−E|x3−y3|2E+κξ2EeE(ℓ+x3+y3)(1−e2ℓ√κ+E2) (14) S−∘ =ξ(e(ℓ/2+y3)√κ+E2(√κ+E2−E)+e(3ℓ/2−y3)√κ+E2(√κ+E2+E))eE(x3+ℓ/2) (15) S−+ =2ξ√κ+E2e(√κ+E2+E)ℓ+E(x3−y3) (16) S∘∘ =ξeℓ√κ+E22√κ+E2(2κcosh[(x3+y3)√κ+E2]+ +(√κ+E2−E)2e√κ+E2(|x3−y3|−ℓ)+(E+√κ+E2)2e√κ+E2(ℓ−|x3−y3|)).

We have divided the general expression of the propagator into four parts according to the position of , relative to the defect, and all other cases could be easily derived using the symmetry properties of the propagator.

Despite that the problem of square-well potential has been studied extensively in quantum mechanics, its field theoretical interpretation was lacking. To the best of our knowledge the only attempt to calculate the full propagator for such system was presented in [35]. However, the final explicit expression is lacking there, but instructions for its construction are given to the reader. Following these instructions one can compare [35] with (14)-(3.1) and find them coinciding up to the Wick rotation. One should also note that in Euclidian version of the problem the difference between under- and above-barrier scattering vanishes. The for 2-dimensional problem were also presented in [8].

Fixing position of one of the interfaces of the potential and taking the infinite width limit transforms the potential into a step-function one. The propagator for non-relativistic problem with such potential was calculated earlier, e.g. [37]. Subject to necessary redefinitions of parameters, our results (14)-(3.1) for coincide with ones presented there.

3.2 Scattering on the slab

In the quantum field theory the scattering processes are described by the S-matrix elements between different states. To construct them for one-particle states of particular impulses, we consider the full propagator (13) in real (Minkowski) space, and cut the external lines of the free field propagator

 H(x,y)=D−1(x,z′)^S(z′,z′′)D−1(z′′,y). (18)

The defect potential depends only on thus both the energy () and impulse parallel to the defect () are conserved. We omit corresponding delta-functions in the Fourier representation. In the transverse direction we have

 H(p3,q3) = ∫dx3dy3eip3x3+iq3y3H(x3,y3) = −κ∫ℓ/2−ℓ/2dx3∫ℓ/2−ℓ/2dy3eip3x3+iq3y3(δ(x3−y3)+UM(x3,y3))

where the last equation is due to the definition of and the obvious property that is only non-zero when both its arguments lie in the defect region (). By we denote the operator (11), (12) subject to the inverse Wick rotation.

Further we note, that in the Minkowski space the on-shell condition leaves only two possible values for transversal impulse : , . Due to the symmetry reasons there are only two independent values of depending on the sign of . Thus the non-trivial -matrix elements depend only on one parameter and are given by

 hf=−2ip−8p2√κ−p2eℓ(ip+√κ−p2)(e2ℓ√κ−p2−1)(κ−2p2)+2ip√κ−p2(1+e2ℓ√κ−p2) (19) hb=2ipκ(1−e2ℓ√κ−p2)eipℓ(e2ℓ√κ−p2−1)(κ−2p2)+2ip√κ−p2(1+e2ℓ√κ−p2). (20)

where is the forward scattering amplitude, and — the reflection one.

It is possible to check that the presented -matrix amplitudes satisfy the following analog of the optical theorem

 2 Im(hf)=−12p(|hf|2+|hb|2),2%Im(hb)=−12p(hfh∗b+h∗fhb) (21)

3.3 Casimir Energy

The Casimir energy density per unit area of the defect can be presented with the following relation

 E=−1TSlnG[0]=12TSTrln[Q(x,y)]. (22)

In the second equality we used (5), is the (infinite) time interval and — the surface area of the defect. For the explicit calculations we first make the Fourier transformation as in (7). Then

 E=μ4−d∫dd−12(2π)d−1Trln[Q(;x3,y3)], (23)

where we introduced dimensional regularization and an auxiliary mass parameter to handle the UV-divergencies.

Using the definitions of and we can express the -derivative of the integrand of (23) as . Then for the energy density we get

 E=−μ4−d∫κ0dκ′κ′∫dd−12(2π)d−1TrU(κ′). (24)

We have chosen the lower limit of integration over to satisfy the energy normalization condition . As we show below the integral is convergent at .

The trace of the integral operator is straightforward

 TrU≡∫ℓ/2−ℓ/2dxU(x,x)=2bℓ+4asinh(ℓ√ρ)−ℓκ2√ρ (25)

where we already used that . Using and given in (12), one easily notes that when , thus supporting the above statement.

Next, putting (12) into (25) we can prove directly that

 TrU=−κ∂∂κln[e−ℓ(E+√ρ)4E√ρξ]. (26)

Thus, from (24) and (26) we obtain the following expression for the Casimir energy

 E=μ4−d∫dd−12(2π)d−1ln[e−ℓE4E√ρ(eℓ√ρ(E+√ρ)2−e−ℓ√ρ(E−√ρ)2)] (27)

For the first time the explicit results for the Casimir energy of single slab were obtained in [26], and recently rederived in [27]. Upon using different regularization scheme and notation they coincide with (27). The question of comparison of results obtained in different regularization schemes is addressed in A.

To extract the UV divergencies at , we break up the energy in two parts

 E=Efin+Ediv, (28)

where

 Efin=14π2∫∞0Ξ(p)p2dp, (29)
 Ξ(p)≡ln[e−2ℓE4E√ρ(e2ℓ√ρ(E+√ρ)2−e−2ℓ√ρ(E−√ρ)2)]−λ2E(1−λ4ℓE2),
 Ediv=λμ4−d2(2π)d−1∫dd−1p2E(1−λ4ℓE2). (30)

It easy to check that the first item, , is finite if we remove regularization, while is divergent. Its asymptotic in the limit of removal of regularization is given by

 Ediv = λ(2ℓm2+λ)32π2ℓ(d−4)+ λ64π2ℓ(λ+(2ℓm2+λ)(γE−1−ln(4π)+2lnmμ))+\Or(d−4)

Here is the Euler constant. Explicit calculation of will be used in the considerations of Section 3.7.

3.4 Renormalization procedure

Simple analysis of the dependence of (30) on the parameters , shows that for the renormalization of the model at the one-loop level considered here we must add to the action at least the following field-independent counter-term

 δS=fλ+gλ2ℓ−1, (32)

with bare parameters and (of the mass dimensions two and zero correspondingly) 222 We remind the reader that we consider a free scalar field, thus the counter-term is the only one required.. It allows us to choose these parameters in such a way that the renormalized Casimir energy defined by the full action and considered as the function of the renormalized parameters appears to be finite both in the regularized theory, and also after the removing of the regularization.

Thus, within such ‘minimal addition’ renormalization scheme we obtain for the renormalized Casimir energy the following result

 Er=Efin+λfr+grλ2ℓ−1 (33)

where finite parameters , must be determined with appropriate experiments, or fixed with the normalization conditions. The number of the required conditions is determined by the (in)dependence of the coupling constant on the slab thickness . However, in the most interesting cases considered below only one normalization condition is necessary.

Indeed, let us consider the Casimir pressure. It is the observable quantity defined by

 p=−∂Er∂ℓ. (34)

 pmatt=−∂Efin∂ℓ+grλ2ℓ2. (35)

where only one renormalization parameter is present.

The original definition (2) of the distribution function corresponding to this case can be interpreted as preserving the amount of matter within the slab: . However, remains up to the experimental fixing since there is no other natural normalization condition that could be used in this case, unlike the case discussed in the following section. The plot of the as a function of for different values of is presented at Fig. 1. Note the non-monotonous dependence of the pressure on the width of the slab for comparatively large negative .

3.5 Dirichlet limit

Alternatively, one can consider the density of the matter to be fixed and calculate the pressure under this condition. Then the distribution function has a different normalization condition , which is equivalent to the change of variables in the formulae (29)–(33). In this case the two counter-terms in (33) can be effectively combined into a single one of mass dimension one which gives for the pressure

 pr=−∂Efin∂ℓ−κ~gr. (36)

For fixing we can use what we a call a Dirichlet limit, that also provides a natural way to establish a correspondence between our results and the previous calculations.

One can note that putting and then taking the limit effectively converts the system under consideration into a massless scalar field confined within a finite interval in : subject to Dirichlet boundary conditions at the endpoints.

To investigate the behavior of the finite part of the Casimir energy in this limit, one has to construct carefully the asymptotics of (29). It can be done rigourously with help of the Taylor expansion of the integrand of (29) in powers of and consequent resummation of the same order contributions. This procedure yields

 Efin=−m4ℓ128π2+(π6−49)m34π2−π21440ℓ3+\Or(m−1) (37)

Now we can require that the renormalized Casimir pressure (36) in this limit coincide with one calculated for the case of a massless scalar field subject to the Dirichlet boundary conditions [9]

 pDir=−π2480ℓ4. (38)

This condition fixes the renormalization parameter of (36)

 ~gr=−m2128π2. (39)

The Dirichlet limit procedure presented here looks somewhat similar to the ‘large-mass prescription’ widely used in the calculations of the Casimir energy [6]. However, the two schemes have different physical motivation. While in the large mass prescription it is argued on some general grounds that the Casimir energy must vanish in the large mass limit, in our case we collate a particular limit of our results with a well known (unambiguous) physical situation.

3.6 Vacuum Instability

It is well known that quantum systems might become unstable for sufficiently deep negative potentials. In our case the threshold is given by and for deeper potentials the Casimir energy (29) acquires an imaginary part [26]. However the generation of the imaginary part is not only regulated by the strength of the potential but also by its width. Moreover, the stability of the systems depends on the separation in a non-monotonous way.

From the technical point of view the imaginary part of the Casimir energy (29) is due only to the negative values of the argument of the logarithm in . It is clear that is still real for . However, for bigger values of : , the argument of the logarithm is not positively defined anymore in the region of small : . As a function of the separation it changes sign at every satisfying

 ℓ0=1√|ρ|(π−arctan2E√|ρ|E2−|ρ|). (40)

The maximum separation at which the Casimir energy is still real is given by the minimum of as a function of . It is acquired at and we can conclude that the Casimir energy is real for all separation less or equal to

 ℓ0=1m√α−1(π−arctan2√α−12−α). (41)

while for larger it is complex. The dependence of the Casimir energy on the separation for is presented on Fig. 2.

3.7 Absence of divergences

A particulary interesting case of an unstable system is given by (or in notation of the previous subsection). It is characterized by the absence of divergencies as it immediately follows from (3.3). It is possible to check that this value of corresponds to the case of vanishing second heat kernel coefficient — the situation known to possess no divergencies [4]. Note that in this case the dependence on the auxiliary parameter also disappears.

Using the condition elaborated in the previous section we conclude that acquires an imaginary part for . It is pedagogical to rewrite this condition in CI units

 ℓ>π2ℏmc

then at the r.h.s. we recognize immediately the Compton wave length (up to numerical factor) of the particle corresponding to the quantum field. Thus, at the separations (much) larger then the Compton wave length the Casimir energy becomes complex, and the particles are created in the potential well. Such property has a clear semi-classical interpretation — at the separations less then the Compton wave length, there is merely not enough space between the semi-infinite slabs to host any particles.

In the opposite limit — at short separation, — one can derive the following asymptotics for the Casimir energy

 ECas=m34π2(−x(γE−18+lnx)+3x2lnx)+\Or(x2),x≡ℓm (42)

It reveals that for small separations: the potential is attractive. At the energy has an absolute maximum, the force vanishes (at the level), and the system has an unstable equilibrium position. With larger (but still ) the Casimir interaction give rise to a repulsive force on the semi-infinite slabs. This behavior has particular similarities to one described in [4] for negative second heat kernel coefficient . However, we remind the reader, that in our case =0. This analytical study is supported by numerical evaluation of the Casimir energy presented at the Figure 2.

Summarizing, we can say that at small distances , the gap between the slabs tends to shrink thus restoring the homogeneity of the space filled uniformly with matter. If additional energy is brought to the system be moving the slabs to the distances the system becomes unstable emitting particles and the gap growing. This can also be thought as restoring the most homogenous state of the space-time filled uniformly with the particles.

4 Interaction of two finite width mirrors

4.1 The statement of the problem

The method developed above does not only give in a closed form the main characteristics of a QFT model with a single material slab, but can also be generalized for more complicated geometries.

In particular, let us consider two plane slabs of thickness interacting over the distance . In this case the action can be written as

 S = S0+Sdef S0=12∫d4xϕ(x)(−∂2x+m2)ϕ(x) Sdef=∫d3x(κ2∫−a2−a2−ℓ2dx3ϕ2(x)+κ1∫a1+ℓ1a1dx3ϕ2(x))

we assume independent of but corresponding generalization is possible, .

To obtain a formal expression for the generating functional (3) with given above one can proceed in one of the following ways. In a most direct approach, one introduces auxiliary fields following (4) separately for each of the slabs, and integrate consequently over , , and then over . Alternatively, one can consider the system with one slab as an initial ‘free system’, and just substitute in final formulae (5) and (6) the free field propagator with (13). With the knowledge of the explicit formulae for the later approach is much easier, and the results of both of them naturally coincide

 G[J]=[DetV2]−1/2e12J^S2LJ, (44)
 ^S2L=^S−κ2(^SO2)V2(O2^S) (45)
 V2=1+κ2P2, (46)
 P2=O2^SO2 (47)

here is the propagator of the free scalar field in the presence of solitary slab number 1, , and , as well as other notations used below have the same meaning as in Sect. 3 but related to the slab denoted by the subscript.

The linear (integral) operators , are defined within the support of the layer . Accordingly, in what follows we assume that the free spatial arguments of the Fourier transformation of the integral operators belong to .

4.2 Casimir energy

The energy density per unit area of the layers is given by

 E=μ4−d∫dd−12(2π)d−1Trln[V2(;x3,y3)]. (48)

For the explicit calculation of the energy we follow the technique developed in Sect. 3.3 and introduce operator

 J2=V−12−1. (49)

The defining property of can be expressed via (46) with the following operator equation

 J2+κ2P2(1+J2)=0. (50)

According to (47) does not depend on , which allows us to express the -derivative of the integrand of (48) in the following form

 ∂∂κ2lnV2=P2V2=P2(1+J2)=−J2κ2

and for the Casimir energy density we obtain in complete analogy with the case of one slab

 E=−μ4−d∫κ20dκ2κ2∫dd−12(2π)d−1TrJ2. (51)

To solve (50) and find the operator explicitly we first use the expression of the propagator (13) to calculate (47). Assuming that slabs do not intersect we employ (14) and derive

 P2(x,y)=DE2(x,y)+c1eE(x+y) (52)
 c1=κ1ξ12Ee−2a1E(1−e2ℓ1√ρ1)

Taking this expression into account we deduce that possesses properties analogous of those of (10) and satisfies the following equation

 KρJ2=−κ2.

We search for as a sum of partial solution of this equation and general solution of its homogeneous version

 J2(x,y)=−κ2Dρ(x,y)+Ae(x+y)√ρ2+B(e(x−y)√ρ2+e(y−x)√ρ2)+Ce−(x+y)√ρ2.

To deduce the coefficients , , we require that (50) is identically satisfied for all belonging to , then

 A = −κ2ζ22√ρ2e2(a2+ℓ2)√ρ2(κ2e2a2E+2c1E(E+√ρ2)2), (53) B = −κ2ζ22√ρ2(e2a2E(E−√ρ2)2+2c1κ2E), C = −κ2ζ22√ρ2e−2a2√ρ2(κ2e2a2E+2c1E(E−√ρ2)2),

here is defined through

 ζ−12=ξ−12e2a2E+2c1κ2E(e2ℓ2√ρ2−1). (54)

For the trace of we obviously have

 TrJ2≡∫−a2−a2−ℓ2J2(x,x)dx= (55)
 =2ℓ2B−12√ρ2(ℓ2κ2−Ae−2a2√ρ2(1−e−2ℓ2√ρ2)+Ce2a2√ρ2(1−e2ℓ2√ρ2))

Using explicits for , , (53) for the energy we derive

 E=−μ4−d∫κ20dκ2κ2∫dd−12(2π)d−1ε2, (56)
 ε2=−κ2ℓ22√ρ2(1+2ζ2[e2a2E(E−√ρ2)2+2c1κ2E])
 +κ2ζ22ρ2(1−e2ℓ2√ρ2)(κ2e2a2E+2c1E(2E2+κ2)).

This somewhat cumbersome expression can be substantially simplified since the integration over can be made explicitly. This also makes explicit the symmetry of the layers. One obtains

 E2L=μ4−d∫dd−12(2π)d−1×
 ×log[e−ℓ1(E+√ρ1)−ℓ2(E+√ρ2)(1−κ1κ2ξ1ξ2e−2Er(1−e2ℓ1√ρ1)(1−e2ℓ2√ρ2))16E2ξ1ξ2√ρ1√ρ2]

which finally is rewritten as

 (57)

Here give the self-energy (27) of the solitary layers correspondingly. The third term in (57) represents the interaction of two layers and vanishes in the limit . Taking into account the behavior of (12) one notes that the interaction term is UV finite, and the removal of regularization made in (57) is indeed justified. This is in perfect accordance with the general considerations [36] of the finiteness of the Casimir interaction between disjoint bodies. Thus, the interaction between the slabs is given by unambiguously finite force

 F2L ≡ −∂E2L∂r (58) = −∫d3(2π)3E∏i=1,2κiξi(1−e2ℓi√ρi)e2Er−∏i=1,2κiξi(1−e2ℓi√ρi).

4.3 Interaction of a single slab with a delta-spike

Basing on the general formulae for the Casimir interaction of two layers several limiting cases could be considered.

First of all, the limit with , fixed, brings the interaction of a delta-function spike with a finite-width slab

 EδL=E1+∫d32(2π)3log(1−e−2Erκ1ξ1λ22E+λ2(1−e2ℓ1√ρ1)). (59)

It is worth mentioning that in this system there are two (potentially) observable quantities: the self-pressure of the slab in presence of a delta-spike, and the interaction between two of them. According to this, we retained in (59) the self-energy of the first layer , while discarded the self energy of delta-spike which is geometry independent and by no means is observable. Apart from this, we note that no trace of any ‘sharp limit’ complications [29] arise in this situation. The interaction of a finite width slab and an (infinitely) thin one is not influenced in any sense by the divergencies associated with the limit .

Now we can further consider in (59) the limit with , fixed, to prove that (57) does lead to the standard result for the Casimir interaction of two delta-spikes separated by the distance [6]

 E2δ=∫d32(2π)3log(1−e−2Erλ1λ2(2E+λ1)(2E+λ2)), (60)

where again the distance independent contribution was omitted.

The force acting between two distinct objects as function of separation is plotted on the Fig. 3 for the following cases: two finite-width layers, finite-width layer and delta-spike, two delta-spikes. The most interesting peculiarity of two slabs interaction is that it remains finite in the limit , which can also be seen at the analytical level.

4.4 Connection to the Lifshitz formula

It is also possible to consider the opposite limit of the slabs of infinite width separated by the finite distance (i.e. the interaction between two semi-infinite material slabs).

In this case we put , , , and consider tending to infinity: while keeping finite. Then basing on (57) we derive

 ELif=∫d32(2π)3log[1−e−2Erκ1κ2(E+√ρ1)2(E+√ρ2)2] (61)

where the -independent (divergent) terms are omitted.

Taking into account that

 ρ1,2=E2+κ1,2 (62)

we can rewrite (61) further as

 ELif=∫d32(2π)3log[1−e−2Er(E−√ρ1)