Casimir Energy for concentric δ - δ ′ spheres
Ines Cavero-Pelaez, Jose M. Munoz-Castaneda, Cesar Romaniega
CCasimir energy for concentric δ - δ (cid:48) spheres In´es Cavero-Pel´aez ∗ Centro Universitario de la Defensa, Zaragoza, 50019,Spain. Departamento de F´ısca Te´orica, Facultad de Ciencias, Universidad de Zaragoza,
J. M. Munoz-Castaneda † and C. Romaniega ‡ Departamento de F´ısica Te´orica, At´omica y Optica,Universidad de Valladolid, Valladolid, 47011, Spain.
We study the vacuum interaction of a scalar field and two concentric spheres defined by a singularpotential on their surfaces. The potential is a linear combination of the Dirac- δ and its derivative.The presence of the delta prime term in the potential causes that it behaves differently when itis seen from the inside or from the outside of the sphere. We study different cases for positiveand negative values of the delta prime coupling, keeping positive the coupling of the delta. As aconsequence, we find regions in the space of couplings, where the energy is positive, negative orzero. Moreover, the sign of the δ (cid:48) couplings cause different behaviour on the value of the Casimirenergy for different values of the radii. This potential gives rise to general boundary conditions withlimiting cases defining Dirichlet and Robin boundary conditions what allows us to simulate purelyelectric o purely magnetic spheres. I. INTRODUCTION
Casimir forces are measurable effects arising whenvacuum fluctuations of quantum fields are modifiedby external conditions such as bodies with differentgeometries or boundaries. Among many others, someexamples can be found in Refs. [1–3]. A large amountof studies for different geometries have been carriedout over the years, where a great deal of the workhas focused on the interaction energy between bodies[4–6]. This makes sense since it is feasible to setupan experiment that measures forces between objects.The interpretation of the Casimir interaction energy isclearer and less controversial than that of the Casimirself-energy of a single body, where surface divergencesare still an open subject [7–9]. As in the original setupproposed by Casimir [10], most of the systems studiedpresent two objects outside each other, even thoughother configurations like cavities are experimentallyrealizable. In this context, a lot of work has beenfocused on systems as long cylinders [11], configurationsof spheres [12–14] or Casimir-Polder interactions with apolarizable particle [15, 16].In the line with what is mentioned above, the systemsstudied with the practical formulation based on func-tional determinants proposed in 2006 by Kenneth andKlich [17–19] also focus on separated interacting bodies.Even though there is no restriction on the dispositionof the objects as long as they do not overlap, most ofthe attention has been directed towards bodies outsideeach other. However, if one body is inside the other, ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] the so-called TGTG formula is still valid, althoughinterior and exterior scattering must be considered. Toour knowledge, this was firstly discussed in Ref. [20]for the electromagnetic field. With this formalism newresults were obtained: corrections to the proximity forceapproximation (PFA) [21], analysis of the torque andalignment of a spheroid inside a cavity [22] and thestability of certain collections of objects [23], where someprevious plausible configurations for stable levitationwere discarded. In Ref. [14] it is studied the scalar andelectromagnetic fields interacting in the presence of twobodies outside each other or one inside the other whosecenters are separated a certain distance. Bimonte [24]particularized the formula to the interaction between twoperfectly conducting spheres in different arrangements,including the concentric case.In this work we compute the interaction energybetween two partially transparent concentric spheresfor a scalar field using the TGTG formula. In doingso, we take advantage of the spherical symmetry ofthe system. The properties of each sphere enter in theinteraction energy only through its T -matrix [25], whichcan be easily calculated for spherical bodies. We mimicthe spheres by a generalization of the Dirac δ sphericalshell, the so-called δ - δ (cid:48) interaction [26]. Configurationsbased on the δ -potential have widely appeared in theliterature, just to name some: δ sphere in two and threedimensions [27–30], concentric [31] and non-concentric δ spheres [14] for both scalar and electromagnetic fieldsand the interaction between two δ lattices [32–34]. Theaddition of the δ (cid:48) term to the potential that defines theplates was firstly considered in Ref. [35] in the contextof Casimir physics. This is useful, essentially, in twoaspects. Firstly, Robin boundary conditions can beobtained as a finite limit as was shown in Ref. [35].Secondly, although it is still not well understood, the signof the force depends strongly on the boundary, switching a r X i v : . [ h e p - t h ] S e p from attractive to repulsive forces [36]. Basically, theonly general result concerning this issue is restricted tomirror symmetric bodies, originally proposed in Ref. [17]and extended in Ref. [37]. As we shall prove, we cangain insight in the latter with this interaction. We areable to identify the configurations in which the energy ispositive or negative as a function of the parameters thatdefine the potential on the spheres. For parallel plates,this has already been proved in Ref. [35].Specifically, the semitransparent δ - δ (cid:48) spheres will bedefined by the potentials¯ V i ( r ) = a i δ ( r − r i ) + b i δ (cid:48) ( r − r i ) , a i , b i ∈ R i = 1 , , (1)where r and r are the radii of the inner and outer sphere( r < r ), respectively. The definition of the previous po-tential is given by suitable matching conditions imposedon the scalar field [26]. These conditions come from theoriginal work of Kurasov [38] in one dimensional systems.The main advantage of these singular potentials is thatthey are often exactly solvable and therefore, provide agood insight for some of the relevant quantum proper-ties . Given the above, the action that governs the dy-namics of the massless scalar field interacting with thisbackground is S ( ϕ ) = (cid:90) d y (cid:2) ( ∂ϕ ) − V ( x ) ϕ (cid:3) , (2)where V ( x ) = V ( x ) + V ( x )= (cid:88) i =1 λ ,i δ ( x − x i ) + 2 λ ,i δ (cid:48) ( x − x i ) . (3)We have chosen units such that (cid:126) = c = 1 and introduceda mass parameter µ in order to work with dimensionlessquantities, x ≡ rµ, x i ≡ r i µ, ϕ ≡ φµ , (4) λ ,i ≡ a i µ , λ ,i ≡ b i . (5)The paper is organized as follows. In section 2 we give aninterpretation of the TGTG formula based on the modesummation approach when applied to the case of concen-tric spheres. In section 3 we study the solutions of thefield modes for the potential under consideration and cal-culate the relevant elements of the TGTG formula, whichallow us to give a simple expression of the interaction en-ergy shown in section 4 together with some numerical Despite its apparent simplicity, there is a collection of applica-tions in a variety of areas in modern physics, see Ref. [39] andreferences quoted therein. results. Finally, we discuss these results and comparethem with limiting cases. We finish in section 5 with theconclusions.
II. SCATTERING FORMALISMINTERPRETATION
The scattering approach to the computation of Casimirinteraction energies between two bodies has been used inmany calculations since more than half a century. It isworth to mention the original work of Balian and Du-plantier in the 1970s [40, 41]. Most modern forms ofcalculating these energies have been developed by otherauthors already mentioned [17, 19, 20, 42, 43]. Theirmethod has become very popular since it is free of di-vergences and it allows to obtain numerical results in asimple way. Examples of that are the interaction be-tween a compact object and a plane [44] or more specif-ically, between a sphere and plane [45]. In those cases,the authors calculate the Casimir energy by computingthe transition matrices of the scattered waves on the ob-jects separately (the Lippmann-Schwinger operators ofthe bodies [46]), and the translation matrices from oneobject’s origin to the other describing the propagation ofthe wave between them. In particular if we denote by T i , i = 1 ,
2, the Lippmann-Schwinger operators for eachbody, and U i,j , i, j = 1 ,
2, the free Green function thatrepresents the translation from the center of body i to thecentre of body j the so-called TGTG formula is given by E C = 12 π (cid:90) ∞ dχ Tr ln( I − T U T U ) . (6)Here the integration is over the imaginary frequency.Concerning our system of two concentric spheres, inEq. (6) we denote with subindex 1, quantities referredto the interior sphere, and the subindex 2 refers to theanalogues for the exterior sphere.As we have previously stated, our case correspondsto the interaction of a scalar field on a backgroundof concentric spheres with singular potential on theirsurfaces given by Eq. (1). The great advantage of usingEq. (6) is that we only need to have informationabout each of the bodies individually. It is sufficient toknow the shape of the incident and scattered waves andhow they scatter on their surfaces, something that isdetermined by the boundary conditions on the spheres.The interaction between the spheres is due to thescalar quantum vacuum fluctuations in the intermediateregion, that means in the exterior of 1 and interior of2. Due to the spherical symmetry of the problem, itis convenient to use spherical coordinates so that weexpand the waves in the spherical harmonics from theirorigins. Since they share origin, the transition matricesbecome diagonal identities [24] (maybe multiplied by aconstant depending on the normalization used).The components of the Lippmann-Schwinger operators T , of each object that will appear in the TGTG formulafor our system will account only for the quantum vacuumfluctuations in the intermediate region between spheres.Since we have one body inside the other, they representscattering produced by the exterior and interior sides ofthe spheres respectively. It is of note that the inner andthe outer side of the δ - δ (cid:48) sphere do not produce the sameinteraction. Each object will contribute with differentcomponents of the T -operator. Specifically: • The scattering problem concerning the interiorsphere is described by incoming waves coming frominfinity and reflected towards infinity after interact-ing with the exterior side of the sphere. • The case concerning the exterior sphere is the scat-tering of waves generated in the interior of thesphere and reflected back after interacting with theinterior side of the sphere.
III. GENERAL SCATTERING SOLUTIONSAND T -OPERATORS. Let’s consider now a single sphere defined by the po-tential given in section I, V ( x ) = λ δ ( x − x ) + 2 λ δ (cid:48) ( x − x ) , x ∈ R + . (7)Infinitesimal variations of the action in (2) impose thatthe scalar field ϕ ( t, x ) satisfies the equation of motion − ∂ µ ∂ µ ϕ ( t, x ) − V ( x ) ϕ ( t, x ) = 0 , (8)where µ is an index that can take the values { , , , } .Since the potential is time independent, the Fouriertransform in time allows us to work at a given frequencythat later on, we integrate over the whole range. Then, ϕ ( t, x ) = (cid:90) ∞−∞ dω ϕ ω ( x ) e − iωt , x ≡ ( x, θ, φ ) . The resulting equation can now be written as[ − ∆ + V ( x )] ϕ ω ( x ) = ω ϕ ω ( x ) , (9)where ∆ is the Laplacian operator. The non-relativisticSchr¨odinger Hamiltonian in Eq. (9) has been recentlystudied in detail in [26], where the potential V ( x ) is de-fined by matching conditions at the sphere of dimension-less radius x = x over the space of field modes as (cid:18) ϕ ( x +0 , θ, φ )˙ ϕ ( x +0 , θ, φ ) (cid:19) = (cid:18) α (cid:101) β α − (cid:19) (cid:18) ϕ ( x − , θ, φ )˙ ϕ ( x − , θ, φ ) (cid:19) , (10)where we have introduced the notation˙ ϕ ( x ) ≡ ∂ϕ∂x , x +0 and x − denotes that we approach x from the rightor from the left respectively and α ≡ λ − λ , (cid:101) β ≡ (cid:101) λ − λ , (cid:101) λ ≡ − λ x + λ . (11)Due to the spherical symmetry of the potential, in Eq. (9)we perform separation of variables that enables to ex-pand the solution in the spherical harmonics Y (cid:96)m ( θ, φ )and write the modes of the field as ϕ ω ( x ) = ∞ (cid:88) (cid:96) =0 (cid:96) (cid:88) m = − (cid:96) ρ (cid:96) ( x ) Y (cid:96)m ( θ, φ ) . (12)Accordingly, the radial modes ρ (cid:96) ( x ) satisfy the differen-tial equation (cid:20) − d dx − x ddx + (cid:96) ( (cid:96) + 1) x − ω (cid:21) ρ (cid:96) ( x ) = 0 . (13)Two independent solutions are ρ reg (cid:96) ( x ) = j (cid:96) ( ωx ) and ρ out (cid:96) ( x ) = h (1) (cid:96) ( ωx ). The former is the spherical Besselfunction regular at the origin and the latter is the spher-ical Hankel function of the first kind, which determinesthe radial part of a purely outgoing wave [47]. Now wemake the scalar field scatters with the sphere in two dif-ferent situations. a. The exterior scattering. As mentioned above, theinterior sphere of our system enters the TGTG formulathrough the component of the Lippmann-Schwinger op-erator that represents the scattering problem with thesource and detector outside the object. In this sense,the general solution of a radial mode in the two regionsseparated by the sphere of radius x is ρ (cid:96) ( x ) = (cid:26) A (cid:96) ρ reg (cid:96) ( x ) x < x a (cid:96) ρ reg (cid:96) ( x ) + b (cid:96) ρ out (cid:96) ( x ) x > x . (14)If we impose matching conditions, given by Eq. (10), onthe surface of the sphere we obtain the system of equa-tions, (cid:18) a (cid:96) ρ reg (cid:96) ( x ) + b (cid:96) ρ out (cid:96) ( x ) a (cid:96) ˙ ρ reg (cid:96) ( x ) + b (cid:96) ˙ ρ out (cid:96) ( x ) (cid:19) = A (cid:96) (cid:18) α (cid:101) β α − (cid:19) (cid:18) ρ reg (cid:96) ( x )˙ ρ reg (cid:96) ( x ) (cid:19) , (15)where we have used the quantities and notation definedin Eq. (11). Then the scattering produced by the spherein this situation can be calculated as T (cid:96)i = − b (cid:96) a (cid:96) . Eliminating A (cid:96) from Eq. (15) we find T (cid:96) ( ω ) = j (cid:96) ( ωx )Λ( ω ) (cid:110)(cid:2) (cid:96) ( α − − x α ˜ β (cid:3) j (cid:96) ( ωx ) − ( α − ωx j (cid:96) +1 ( ωx ) (cid:111) , (16)where Λ( ω ) ≡ j (cid:96) ( ωx ) (cid:2) ( (cid:96) ( α − − α ˜ β x ) h (1) (cid:96) ( ωx ) − α ωx h (cid:96) +1 ( ωx ) (cid:3) + ωx j (cid:96) +1 ( ωx ) h (cid:96) ( ωx ) . (17) b. The interior scattering. For the exterior sphereof our system, we need to obtain the component of the T -operator describing a scattering problem in which boththe source of the incident wave and the detector are insidethe sphere. Hence, we consider now the sphere subject tothe same δ - δ (cid:48) potential at the surface, but the source isnow inside the body, at its origin. Therefore, the generalsolution for the radial part of a field mode is ρ (cid:96) ( x ) = (cid:26) ˜ a (cid:96) ρ reg (cid:96) ( x ) + ˜ b (cid:96) ρ out (cid:96) ( x ) x < x B (cid:96) ρ out (cid:96) ( x ) x > x . (18)The coefficients { B (cid:96) , ˜ a (cid:96) , ˜ b (cid:96) } above must satisfy theboundary conditions obtained by plugging Eq. (18) intoEq. (10), B (cid:96) (cid:18) ρ out (cid:96) ( x )˙ ρ out (cid:96) ( x ) (cid:19) = (cid:18) α (cid:101) β α − (cid:19) (cid:18) ˜ a (cid:96) ρ reg (cid:96) ( x ) + ˜ b (cid:96) ρ out (cid:96) ( x )˜ a (cid:96) ˙ ρ reg (cid:96) ( x ) + ˜ b (cid:96) ˙ ρ out (cid:96) ( x ) (cid:19) . (19)As in the previous case the desired component of the T -operator is given by the ratio of the reflected flux to theemitted wave, but this time inside the sphere, (cid:101) T (cid:96) = − ˜ a (cid:96) ˜ b (cid:96) . The latter can be easily obtained from the Eq. (19): (cid:101) T (cid:96) ( ω ) = h (1) (cid:96) ( ωx )Λ( ω ) (cid:110)(cid:2) (cid:96) ( α − − x α ˜ β (cid:3) h (1) (cid:96) ( ωx ) − ( α − ωx h (1) (cid:96) +1 ( ωx ) (cid:111) . (20) c. On the relation between T and (cid:101) T . If we comparethe numerators in Eqs. (16) and (20) we can see thatthe are related by exchanging j (cid:96) ( ωx ) ↔ h (1) (cid:96) ( ωx ). Thesame property does not hold for the components T (cid:96) ( ω )and (cid:101) T (cid:96) ( ω ). This reciprocity corresponds to exchangingthe incident and reflected wave. But as we have noted,the interior and exterior sides of the sphere are differentso we have also to exchange them. In this sense, takinginto account that the inverse of the matching conditionmatrix appearing in (15) and (19) (cid:18) α (cid:101) β α − (cid:19) − = (cid:18) α − − (cid:101) β α (cid:19) (21)is reached with the coupling transformation { λ , λ } →{− λ , − λ } we conclude that (cid:101) T (cid:96) ( ω ; x , λ , λ ) = T (cid:96) ( ω ; x , − λ , − λ ; j (cid:96) ↔ h (1) (cid:96) ) . (22)This result is quite surprising when we compare withthe one dimensional case that enables to mimic two di-mensional plates as was shown in Ref. [35]. For the one dimensional case and the potential V D = w δ ( x ) +2 w δ (cid:48) ( x ) the role played by T (cid:96) and (cid:101) T (cid:96) in our case is playedby the reflection amplitudes (see Ref. [35]): r R = − ω w − iw ω ( w + 1) + iw , r L = ω w − iw ω ( w + 1) + iw . (23)In this case it is straightforward to notice that thetransformation ( w , w ) (cid:55)→ ( w , − w ) (24)acting on the reflection amplitudes enables us to obtainthe analogue of the three dimensional cas, i. e.: r R ( ω ; w , − w ) = r L ( ω ; w , w ) . (25)Hence, meanwhile the symmetry between reflection am-plitudes in the one dimensional δ - δ (cid:48) potential only re-quires the change of sign of the δ (cid:48) coupling, for the spher-ical three dimensional case it is necessary, in addition, achange of sign of the Dirac- δ coupling as it is shown in(22). This additional requirement implies that the Dirac- δ potentials changes from being a potential well/barrierto a barrier/well. IV. ANALYTIC EXPRESSION ANDNUMERICAL RESULTS FOR THE CASIMIRINTERACTION ENERGY
In order to use Eq. (6) we remind the reader that in ourcase object 1 refers to the interior sphere and thereforethe scattering is produced outside. This is described by T (cid:96) as given in Eq. (16) setting x = x , and ( α, ˜ β ) =( α , ˜ β ). On the other hand, the waves reaching object 2are scattered from the inside and therefore it correspondsto (cid:101) T (cid:96) as in Eq. (20) with x = x , and ( α, ˜ β ) = ( α , ˜ β ).In addition, we need to obtain the expressions for the T -operators for imaginary frequencies in order to use theTGTG formula in its euclidean version, where the T -operators are Hermitian and oscillatory behavior in theintegrals is avoided. Therefore, we define ω = iχ with χ >
0. The Bessel functions with imaginary argumentscan be written in terms of the modified Bessel functionsof the first and second kind [47], j (cid:96) ( iχx ) = i (cid:96) (cid:114) π χx I (cid:96) +1 / ( χx ) h (1) (cid:96) ( iχx ) = − i − (cid:96) (cid:114) πχx K (cid:96) +1 / ( χx ) . Taking into account the equations above and Eqs. (16)and (20) the euclidean rotated components of the re-quired T -operators become T (cid:96) ( iχ )= C I ν ( y ) (cid:104) I ν ( y ) (cid:16) (cid:96) ( α − − α x (cid:101) β (cid:17) + (cid:0) α − (cid:1) y I ν +1 ( y ) (cid:105) Ξ( y ) , ˜ T (cid:96) ( iχ )= C − K ν ( y ) (cid:104) K ν ( y ) (cid:16) (cid:96) ( α − − α x (cid:101) β (cid:17) − (cid:0) α − (cid:1) y K ν +1 ( y ) (cid:105) Ξ( y ) , (26)Ξ( y i ) ≡ I ν ( y i ) (cid:104) K ν ( y i ) (cid:16) (cid:96)α i − (cid:96) − α i x i (cid:101) β i (cid:17) − α i y i K ν +1 ( y i ) (cid:105) − y i I ν +1 ( y i ) K ν ( y i ) , where C = ( − (cid:96) π , ν ≡ (cid:96) + 1 /
2, and y i ≡ χ x i ,for i =1 ,
2. Before computing the quantum vacuum energy, asa consistency test, it is straightforward to observe thatby turning off the δ (cid:48) term λ , i = 0 ( α i = 1 , β i = λ ,i ),equations in (26) become T (cid:96) ( iχ ) = C λ , x I ν ( y )[1 + λ , x K ν ( y ) I ν ( y )] , ˜ T (cid:96) ( iχ ) = C − λ , x K ν ( y )[1 + λ , x K ν ( y ) I ν ( y )] . (27)These expressions are in agreement with the ones for twoconcentric spheres having delta potentials on their sur-faces calculated in [14] and [48]. See also Ref. [28] wherethey calculate the same T through the phase shift.In this sense, the TGTG formula for the interactionenergy when both spheres share center can be written as E C = 12 π ∞ (cid:88) (cid:96) =0 (2 (cid:96) + 1) (cid:90) ∞ dχ ln (cid:104) − T (cid:96) ( iχ ) (cid:101) T (cid:96) ( iχ ) (cid:105) . (28)The expressions in Eqs. (26) and (28) enable us to obtainnumerical results for the quantum vacuum interaction en-ergy between the two concentric spheres. Regarding thepresentation of the numerical plots of the quantum vac-uum energy we consider different possible scenarios bychanging the couplings in the potential. In all the casesshown bellow, we take the coefficient of the δ term to bepositive and allow the coefficient of the δ (cid:48) to change sign.As it has been seen along the paper, the presence of the δ (cid:48) term makes the potential on the spheres behave differ-ently when the scattering is produced from the inside orfrom the outside of the body.First we consider the couplings to be equal in bothspheres, such that λ , = λ , = λ and λ , = λ , = λ .We show the results in Fig.1 for two different values ofthe radii. In the plot on the left we have used x = 1 and x = 2, and x = 1 . x = 2 in the one on the right.The color gradient denotes changes on the energy value.We observe that in both plots there are regions in thespace of couplings where the vacuum energy takes posi-tive, negative and zero values. When λ = 0 we recoverthe case of the interaction between two δ (semitranspar-ent) spheres that is known to be negative. The potentialsin both spheres have the same sign. When λ = 0, and since we have set the couplings of the δ (cid:48) term equal, thesign of the potentials on each sphere is determined by thesign of the δ (cid:48) that we know behaves differently from theinside and outside. Consequently, the interaction energybecomes positive. When both terms are present in thepotential, one of them is dominant over the other. As λ increases, higher absolute values of λ are needed to ob-tain a positive energy. This pattern holds for both plots,although the numerical values depend on the radii.We also observe in Fig.1 that the δ (cid:48) contribution is notsymmetric under λ → − λ . In the right graph, wherethe radii of the spheres do not differ much from eachother (and therefore the situation approaches the paral-lel plates configuration when the radii are large enough)we see that if both couplings of the δ (cid:48) are positive λ > λ <
0. This symme-try fades out as the difference between the values of theradii increases (see plot on the left); that means, whenthe inner sphere becomes comparatively smaller than theouter one. Next we turn off the delta interaction in thepotential by doing λ , = λ , = 0, so that we are leftwith concentric spheres defined by a δ (cid:48) potential aloneon their surfaces. Results are shown in the right graphin Fig. 2 for x = 1 and x = 2. We observe that whenthe couplings have the same sign the interaction energy ispositive (as we mentioned above), while it becomes neg-ative if the couplings have different sign. We comparethis result with the equivalent one from a plane geome-try showed on the left of Fig. 2, where the same patternis obtained. For both geometries the results agree withthe change in sign that the δ (cid:48) introduces when it is ap-proached from inside or outside, or equivalently for pla-nar geometry, from one side or another. We furthermoreobserve again how the spherical geometry introduces anasymmetry on the values of the positive and negative en-ergies compared with parallel plates.We test the numerical results by making the radii of thespheres large while keeping a small constant the differ-ence between them so that we can compare with the par-allel plates geometry. The plots are presented in Fig.3.We see a tendency to recover the behaviour of the Casimirenergy for planar geometry studied in Ref. [35]. The plotsshow the interaction Casimir energy for different valuesof the couplings when these are the same in both bodies. FIG. 1: The quantum vacuum interaction energy obtained from Eq. (28) when λ , = λ , = λ and λ , = λ , = λ . In theLEFT plot: radii x = 1 and x = 2. In the RIGHT plot: radii x = 1 . x = 2FIG. 2: Comparison between the quantum vacuum interaction energy of two δ - δ (cid:48) plane parallel plates and two concentric δ - δ (cid:48) spheres with λ , = λ , = 0. The LEFT plot: two plates separated unit distance. RIGHT plot: spherical shells with x = 1and x = 2 The plot on the left shows the result for parallel plateswhile the one on the right is generated from concentricspheres with large radii keeping values with small differ-ence between them. It can be seen that in this situationthere is a tendency to recover the behaviour of the quan-tum vacuum interaction energy between two plates as thevalues of x and x increase keeping constant the distancebetween them.Finally, in Fig.4 we consider the case in which onesphere is defined by a δ and the other one by a δ (cid:48) inter- action. As expected, the sign of the interaction energychanges from one setup to the other illustrating the in-fluence of having the δ (cid:48) hit from the interior sphere orthe exterior one.We wrap up this section stressing a common feature inthe plots showed. We observe maximum absolute valuesof E C when | λ | = 1. In this case the matching condi-tions (10) are ill defined and they transform into Robinor Dirichlet boundary conditions [35, 39].Again, the δ (cid:48) term makes the matching condition dif- FIG. 3: Effect of the distance on the quantum vacuum interaction energy of two δ - δ (cid:48) -spheres for λ , = λ , = λ and λ , = λ , = λ . LEFT plot: plates separated 0.1 units of distance. RIGHT plot: radii x = 10 and x = 10 . x = 1 and x = 2. LEFT plot: δ vs δ (cid:48) : λ , = λ , = 0 . RIGHT plot: δ (cid:48) vs δ : λ , = λ , = 0. ferent form one side of the body than from the other, ρ (cid:96) ( x − ) = 0 , ˙ ρ (cid:96) ( x +0 ) = − Dρ (cid:96) ( x +0 ) if λ = 1 , ˙ ρ (cid:96) ( x − ) = Dρ (cid:96) ( x − ) , ρ (cid:96) ( x +0 ) = 0 if λ = − , (29)where D = 4 / ( λ − x ) is a constant on the sphere.For example, in Fig. 2 we see that higher values of thepositive energy are achieved for λ , = λ , = 1 (Robin vs Dirichlet) rather than for λ , = λ , = − vs Robin). For negative energies | E C | reaches highervalues for λ , = − λ , = 1 (Robin vs Robin) than for λ , = − λ , = − vs Dirichlet). In Figs. 1and 3 the two local maximum values of E C are reachedfor | λ | = 1 with λ = 0. The same holds in Fig. 4, but | E C | grows with λ in the range considered. Modelingthe spheres in this way, we can study cases where one ofthe spheres behaves purely electric, by imposing Dirichletboundary conditions that correspond to TE modes, andthe other purely magnetic, by imposing Robin boundaryconditions that correspond to TM modes, or any otherpossible combination. V. CONCLUSIONS
We have computed the quantum vacuum interactionenergy between two concentric spheres mimicked byspherically symmetric δ - δ (cid:48) potentials. We have used theTGTG formula stressing the difference between the twoT operators that enter the system denoted by T and (cid:101) T .The analytical expressions given in Eqs. (16) and (20)allowed us to study in detail the physical interpretationof the so-called (cid:101) T -operator in terms of a non-standardscattering problem where the source of incident proba-bility flux is placed in the centre of the sphere instead ofbeing placed at infinity as it happens in most standardscattering problems.In addition, the analytical results from Eqs. (16) and(20) enables us to relate the (cid:101) T -operator with the morecommon T -operator by means of the symmetry transfor-mation given in Eq. (22). The mentioned transformationrequires the change in sign of the coupling of the δ po-tential unlike it happens for the same potential in theone-dimensional case.By using Eqs. (26) and (28) we have been able toobtain numerical results for the quantum vacuum inter-action energy of two concentric δ - δ (cid:48) spheres as a functionof the four free parameters entering in the potential. Asa result, it can be seen, in Figs. 1-4, that the quantumvacuum interaction energy has not a well-defined sign asa function of the parameters { λ ,i , λ ,i } i =1 , . The posi-tive energy values are clearly due to the presence of the δ (cid:48) term since we have considered positive contributions of the δ potentials λ , i >
0. This is due to the fact thatthe δ (cid:48) term behaves differently on one side of the sphereand the other, causing a change of sign and affecting theboundary conditions on the sphere.We observe maximum values of the quantum vacuuminteraction energy for couplings of the δ (cid:48) equal to 1 or −
1. The potential reported could be equivalent to con-sider Robin boundary conditions. Moreover, for certainvalues of the couplings we can achieve purely Dirichelt orNeumann boundary conditions. We have shown that ourresult can also be extrapolated with success to limitingcases as parallel plates with δ - δ (cid:48) potential or concentric δ spheres. Acknowledgments
I.C.P. would like to thank funding from the grantDGA: E21-17R. JMMC and CR are grateful to theSpanish Government-MINECO (MTM2014-57129-C2-1-P) and the Junta de Castilla y Le´on (BU229P18,VA137G18 and VA057U16) for the financial support re-ceived. C.R. is grateful to MINECO for the FPU fellow-ship programme (FPU17/01475). [1] S. K. Lamoreaux.
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