Electromagnetic Field Correlators, Maxwell Stress Tensor, and the Casimir Effect for Parallel Walls

Transcription

1 Brzilin Journl of Physics, vol. 35, no. 3A, September, Electromgnetic Field Correltors, Mxwell Stress Tensor, nd the Csimir Effect for Prllel Wlls F. C. Sntos, J. J. Pssos Sobrinho, nd A. C. Tort Instituto de Físic, Universidde Federl do Rio de Jneiro, Cidde Universitári - Ilh do Fundão - Cix Postl 6858, Rio de Jneiro RJ, Brzil Received on 3 My, 5 We evlute the quntum electromgnetic field correltors ssocited with the electromgnetic vcuum distorted by the presence of two plne prllel conducting wlls nd in the presence of conducting wll prllel to perfectly mgneticlly permeble one. Regulriztion is performed through the generlized zet funtion technique. Results re pplied to rederive the trctive nd repulsive Csimir effect through Mxwell stress tensor. Surfce divergences re shown to cncel out when stresses on both sides of the mteril surfce re tken into ccount. I. INTRODUCTION According to Csimir, the mcroscopiclly observble vcuum energy of quntum field is the regulrized difference between the zero point energies with nd without the externl conditions demnded by the prticulr physicl sitution t hnd. In the cse of the quntized electromgnetic field confined between two infinite prllel conducting wlls seprted by distnce, Csimir s conception of the vcuum energy leds to force per unit re between the wlls given by F A = π c 4 4. Until 997 only one experiment involving Csimir s originl setup hd been performed. Ltely, however, the experimentl observtion of this tiny force with metllic surfces ws significntly improved by series of new experiments due to Lmoreux, Mohideen nd Roy, nd Mohideen 3. The concept of n observble vcuum energy cn be extended to ll quntum fields nd severl types of boundry conditions nd/or pplied externl fields. An updted review of ll these Csimir effects cn be found in the monogrph by Bordg, Mohideen, nd Mostepnnko 4, but see lso, Mostepnenko nd Trunov, nd Plunien et l. 5. The locl pproch to the electromgnetic Csimir effect ws initited by Brown nd Mcly who clculted the renormlized stress energy tensor between two prllel perfectly conducting pltes by mens of Green functions techniques 6. An interesting pproch to the stndrd Csimir effect is the one due to Gonzles 7. Anlyzing the vcuum oscilltions of the electromgnetic field confined by two prllel slbs seprted by gp of size, Gonzles pointed out tht the pprently non-objectionble definition of the vcuum energy given bove could esily led to conceptul errors nd stressed the fct tht in ny Csimir interction clcultion, contributions from both sides of the mteril surfces involved must be tken into ccount. This is so becuse the vcuum pressure lwys pushes the mteril surfces involved, therefore, the repulsiveness or trctiveness of the Csimir force depends on the discontinuity of the relevnt component of the quntized Mxwell stress tensor t the loction of the surfce. The purpose of this pper is to pursue this line of resoning by nlyzing the stresses on prllel plne wlls due to the vcuum distortions cused by the presence of these wlls through the quntized version of the Mxwell stress tensor. Though the pproch chosen here hs mny points in common with Ref. 7 cited bove nd Green functions techniques 8, it is different lterntive in the sense tht it relies on objects known s correltors, which re regulrized vcuum expecttion vlues of products of the electromgnetic field components tken t the sme point of spce nd for equl times. These correltors contin ll the informtion we need bout the locl behvior of the vcuum expecttion vlues of elements of Mxwell stress tensor, in prticulr, their behvior ner both sides of the mteril surfce in question, feture crucil to the obtention of the correct result. The locl behvior of the Mxwell tensor, or of the reltivistic symmetricl stress-energy tensor, is extremely importnt becuse s shown by, for exmple, Deutsch nd Cndels 9 with the help of Green functions techniques, s we pproch the boundries we find strong divergencies tht cnnot be simply removed by usul renormliztion procedures. Besides reviewing the obtention of the electromgnetic Csimir for the stndrd cse of two perfectly conducting prllel wlls, we will lso consider pir of prllel wlls, one of them perfectly permeble. This setup ws first proposed by Boyer who nlyzed them from the viewpoint of rndom electrodymnics nd it is the simplest exmple of repulsive Csimir force. For both cses, the conducting plte nd Boyer s setup we lso construct the symmetricl stress tensor. We will employ Gussin units nd set c = =. II. MAXWELL STRESS TENSOR AND THE ELECTROMAGNETIC FIELD CORRELATORS Our im in this section is to obtin n expression for the quntum version of the electromgnetic force per unit re tht cts on plne mteril wll. Mteril wll here mens perfectly conducting squre surfceε or perfectly permeble one µ whose liner dimension L is much lrger thn others relevnt dimensions envolved such s the distnce between two of those wlls. The physicl interction between ny of the two types of wlls considered here nd the vcuum electromgnetic field is mimicked by the imposition of pproprite boundry conditions on the electromgnetic field

2 658 F. C. Sntos et l. on the loction of the wlls. The Crtesin components of the Mxwell stress tensor in Gussin units re given by T = E i E j 4π δ E + B i B j δ B where i, j = x,y,z. Suppose tht the mteril surfce is plced perpendiculrly to the OZ xis. Upon quntizing the electromgnetic field we cn write the quntum version of. For instnce, ˆT zz = Ê 8π z Ê + ˆB z ˆB, 3 where Ê = Ê x + Êy nd ˆB = ˆB x + ˆB y; Ô Ô denotes vcuum expecttion vlue. The other Crtesin com- ponents of this tensor cn be obtined in n nlogous wy. The quntum mcroscopic force ˆF on the wll cn be evluted by integrting the quntum version of the clssicl result F = T ˆnd, 4 R were ˆn is the outwrd norml t the boundry R nd R is ny region contining ll or prt of the wll. Clssiclly, 4 cn be obtined by integrting the Lorentz force per unit volume cting on chrge nd current distributions nd eliminting the sources in fvor of the fields. From quntum point of view, we see tht the problem of evluting the pressure the mteril surfce due to the distorted zero point oscilltions of the electromgnetic field is reduced to the evlution of the vcuum expecttion vlue of the quntum opertors Ê i r,tê j r,t, ˆB i r,t ˆB j r,t, nd Ê i r,t ˆB j r,t. The evlution of these correltors depends on the specific choice of the boundry conditions. A regulriztion recipe will lso be necessry, for these objects re mthemticlly ill-defined. For the setup envolving conducting pltes these correltors were evluted by Lütken nd Rvndl 6, see lso. They cn be lso obtined from the coincidence limit of the photon propgtor between conducting pltes evluted by Bordg et l 7. In the next section we will evlute nd regulrize these correltors by mens of nlitycl continution techniques 3 similr, though not equl, to the ones employed by Lütken e Rvndl 6. We will lso show how to obtin the corresponding results for nother unusul but intersting setup 3. III. CORRELATORS FOR CASIMIR S SETUP Consider n experimentl setup consisting in two infinite perfectly conducting prllel pltes ε kept t fixed distnce from ech other. We will choose the coordintes xis in such wy tht the OZ direction is perpendiculr to the pltes. One of the pltes will be plced t z = nd the other one t z =. The field must stisfy the following boundry conditions on the pltes: the tngentil components E x e E y of the electric field nd the norml component B z of the mgnetic field must be zero on the pltes. Since there re no rel chrges or currents it will be convenient to work in the Coulomb guge in which Ar,t =, nd Φ =, thus Er,t = Ar,t/ t nd Br,t = Ar,t. These physicl boundry conditions combined with the choice of guge llow us to rewrite the boundry conditions in terms of the components of the vector potentil Ar, t in the following wy: t z = we will hve, nd t z =, A x x,y,,t = ; A y x,y,,t = ; z A zx,y,,t =, 5 A x x,y,,t = ; A y x,y,,t = ; z A zx,y,,t =. 6 The vector potentil opertor Âr, t tht stisfies the wve eqution, the Coulmb guge nd the boundry conditions cn be written s Âr,t = π π + â κ,n n= d κ nπz {â κ,nˆκ ẑsin ω ˆκ in nπz ω sin ẑ κ ω cos nπz } e iκ ρ ωt + h.c, 7 where κ = k x,k y nd ρ is the position rrow on the X Y plne. The norml frquencies re given by ω = ωκ,n = κ + n π, 8 with k x,k y R e n N. The symbol indictes tht the term corresponding to n = for normliztion resons must be multiplied by /. The Fourier coefficients â λ κ,n where λ =, is the polriztion index, re opertors in the photon

3 Brzilin Journl of Physics, vol. 35, no. 3A, September, ocuption number spce nd stisfy â λ κ,n,â λ κ,n = δ λλ δ nn δ κ κ. 9 It is convenient to write the vector potentil in the generl form Âr,t = n= d κ λ= â λ κ,na λ κn re iωκ,nt + h.c, where A λ κn r re the modl functions. The modl functions for ech polriztion stte must obey Helmholtz eqution nd the boundry conditions given bove. In our cse the modl functions re nd A κn r = π nπz sin e iκ ρ ˆκ ẑ, π ω A κn r = π ˆκ inπ nπz π ω ω sin ẑ κ nπz ω cos e iκ ρ. The next step is to evlute the electric field opertor Êr,t. Reclling tht â λ κ,n =, we first write the correltors < Ê i r,tê j r,t in the generl form < Ê i r,tê j r,t = E iα re jαr, 3 α where we hve introduced the modl functions E iα r for the electric field. In our cse nd yield nd, E iκn r = i π E iκn r = i π ωκ,nπ ωκ,nπ κ i nπz sin e iκ ρ ˆκ ẑ i, 4 inπ nπz ωκ,n sin κ nπz ẑ i ωκ,n cos e iκ ρ, 5 respectively. Tking 4 nd 5 into 3, we write ˆκ i = cosφδ ix + sinφδ iy, ẑ i = δ iz e ẑ ˆκ i = sinφδ ix cosφδ iy, where φ is the zimuthl ngle on the X Y plne nd we hve performed ll ngulr integrls. In this wy we end up with + + Ê i r,tê j r,t = π π π π δ π δ n= π n= cos nπz π δ n= sin nπz dκκωκ,n n sin nπz dκκω κ,n dκκ 3 ω κ,n, 6 where δ := δ ixδ jx + δ iy δ jy e δ := δ izδ jz. Eqution 6 is forml expression for the correltor E i r,te j r,t, since it is mthemticlly ill-defined unless regulriztion recipe is prescribed. We will regulrize the integrls in 6 with the help of nlyticl continution methods. Consider, for instnce, the first integrl on the r.h.s. of 6 nd let us rewrite it s follows The first term in 6 cn be rewritten s T = dκκ κ + n π / dκκ κ + n π / s. π π δ n= sin nπz dκκ κ + n π s, 7

4 66 F. C. Sntos et l. the second s T = π π π δ n= n sin nπz dκκ κ + n π s, 8 nd the third one s π T 3 = π δ n= cos nπz dκκ 3 κ + n π s. 9 Let us ssume tht Rs is lrge enough to give precise mthemticl mening to these integrls. After evluting them nd mke use of the nlyticl continution of the results we will tke the limit s. Let us see, for instnce, wht hppens with T. Mking use of the following representtion of Euler bet function 9 dxx µ x + c ν B µ =, ν µ c µ+ν, where Bx,y = ΓxΓy/Γx + y, nd tht holds for R ν + µ < nd Rµ >, we obtin Tking this result into T we obtin dκκ κ + n π / s = nπ 3 s Γs 3/ Γs / = nπ 3 s. s 3 T = s 3 π 3 s δ ζ R s 3 n= where ζ R z is the well-known Riemnn zet function. Tking the limit s, we hve T = π 4 δ 3π + 8 n= nπz n 3 s cos, dξ 3 sinnξ where we mde use of the fct tht ζ R 3 = /, defined ξ := πz/, nd wrote, 3 n 3 cosnξ = 8 d3 sinnξ. 4 dξ3 The sum on the R.H.S. of 3 cn be regulrized in mny wys. A quick though non-rigorous wy is to write T = π 4 δ 3π + 8 dξ 3 sinnξ, 5 n= nd express the summnd in terms of exponentil functions of imginry rgument thereby trnsforming ech one of the sums into the eucliden spce. In this wy we obtin = n=sinnξ i n= = i n= expinξ expnξ E n= n= exp inξ exp nξ E 6 where we hve mde the substitution iξ ξ E in the first cn sum nd iξ ξ E in the second. Ech one of the sums bove be esily performed nd the result is sinnξ = cotξ 7 n=

5 Brzilin Journl of Physics, vol. 35, no. 3A, September, 5 66 It follows tht T = π 4 δ 3π + 8 dξ 3 cotξ. 8 Treting the two other terms in 6 in similr mnner we obtin T = π π 4 δ + 8 dξ 3 cotξ, 9 Notice tht some cre must be tken when we ply this procedure to the third term. This is so becuse the term corresponding to n = in T 3 is not zero. In fct its contribution is: T 3 n = = π π δ dκκ s, 3 nd T 3 = 4 π 4 δ 3π 8 dξ 3 cotξ. 3 which diverges when the regulriztion is removed. However this term is non-physicl nd cn be sfely ignored. Finlly, collecting ll prtil results we hve The function F ξ is defined by nd its expnsion bout ξ = is given by E i r,te j r,t = T + T + T 3 π 4 = 3π δ + δ + δ Fξ. 3 F ξ := 8 dξ 3 cotξ, 33 F ξ 3 8 ξ O ξ. 34 Ner ξ = π which corresponds to z = we mke the replcement ξ ξ π. Notice tht due to the behvior of F ξ ner ξ =,, strong divergences predominte in the behvior of the correltors ner the pltes. By pplying the exctly the sme procedure we obtin the mgnetic field correltors π 4 B i r,tb j r,t = 3π δ + δ δ Fξ. 35 A direct evlution lso shows tht the correltors < E i r,tb j r,t re zero. IV. CORRELATORS FOR BOYER S SETUP The other setup we re interested in is tht one in which perfectly conducting plte is plced t z = nd perfectly permeble plte is plced t z =. This setup ws nlyzed for the first time by Boyer in the contetxt of stochstic electrodynmic nd it is the simplest cse of repulsive Csimir effect tht cn be found in the literture. The boundry conditions now re: the tngentil components E x nd E y of the electric field s well s the norml component B z of the mgnetic field must vnish on the surfce of the plte t z = ; b the tngentil components of B x e B y of the mgnetic field s well s norml conponent E z of the electric field must vnish on the surfce of the plte t z =. These boundry conditions trnslted in terms of the components of the vector potentil red A x x,y,,t = ; A y x,y,,t = ; z A zx,y,,t =, 36

6 66 F. C. Sntos et l. t z =, nd t z = z A xx,y,,t = ; z A yx,y,,t = ; A z x,y,,t =. 37 The pproprite vector potentil opertor Âr,t is given by 3 Âr,t = π π n= + â κ,n ˆκ in + sin n + ω d κ {â κ,nˆκ ẑsin n + ω πz πz ẑ κω cos n + } πz e iκ ρ ωt + h.c., 38 where s before κ = k x,k y nd ρ is the position vector on the X Y plne. The norml frequencies re given ωκ,n = κ + n + π, 39 with k x,k y R nd n N. Notice tht contrry to the cse of two conducting pltes normliztion does not require the we multiply the term corresponding to n = by /. The electric nd mgnetic field correltors for Boyer s setup cn be evluted with the sme technique employed before 3. In fct, it is not hrd to convince ourselves tht it is sufficient to perform the substitution n n + / nd follow the sme steps s before to obtin E i r,te j r,t = T + T + T 3, 4 where, for exmple T = Γ s 3 Γ s π { 3 s δ n + n= 3 s cos n + / πz }, 4 The terms T e T show similr structure. The min difference with respect to the two conducting plte cse is tht now we hve to del with the Hurwitz zet function ζ H z,q which hs series representtion given by 9 ζ H z,q = with Rz >, nd q,,,... In our cse we must set n= ζ H s 3, = n= n + q z, 4 n + 3 s. 43 It follows tht in the limit s we hve π 4 δ T = 3π In the sme wy we obtin for T e T the results π 4 δ T = π {ζ H 3, {ζ H 3, + n= n= n + 3 cos n + n + 3 cos n + } πz. 44 } πz. 45 nd π 4 δ T 3 = {ζ H 3, 3π + n= n + 3 cos n + } πz. 46

7 Brzilin Journl of Physics, vol. 35, no. 3A, September, Notice tht this time we do not hve the divergent contribution corresponding to n = s in the cse of the conducting pltes. Adding the three terms we hve Êi r,tê j r,t π 4 { δ = 3π + δ ζ H 3, + n= n + 3 cos n + } πz. 47 The numericl vlue ζ H 3, cn be obtined from 9 ζ H n,q = B n+q n +, 48 where n N nd B n+ q is Bernoulli polynomil defined by 9 B n x = n p= n! p!n p! B px n k, 49 where B p is Bernoulli number. The relevnt polynomil here is: B 4 x = x 4 x 3 + x 3. 5 With B 4 / = 7/8 /3, it follows tht ζ H 3, = 7/8/. The sum cn be regulrized with the sme technique employed before. In fct, we cn define the function Gξ by Gξ : = = 8 n= n= n + 3 cos n + ξ n + 3 cosn + ξ, 5 where s before ξ := zπ/. We cn write n + 3 cosn + ξ = d3 sinn + ξ 5 dξ3 nd formlly we hve Gξ = 8 dξ 3 sinn + ξ. 53 n= Writing sinn + ξ in terms of exponentils of imginry rgument nd pssing to the eucliden spce we obtin fter some simple mnipultions Gξ = 8 dξ 3 sinξ. 54 Collecting ll prtil results we finlly obtin for Êi r,tê j r,t the result Êi r,tê j r,t = π 4 3π Proceeding in the sme wy in the evlution of ˆB i r,t ˆB j r,t we obtin ˆB i r,t ˆB j r,t = π 4 3π 7 δ + δ + δ Gξ δ + δ δ Gξ Observe tht ner ξ = the function Gξ behves s Gξ = 3 8 ξ O ξ, 57 But ner ξ = π its behvior is slightly different Gξ = 3 8 ξ π O ξ π. 58 Agin, direct clcultion shows tht Ê i r,t ˆB j r,t = for this cse.

8 664 F. C. Sntos et l. Z ε ε,µ ^ ^ ^ n'=-z n=z ^ z= z= ε,µ z=l Z' As before the divergent behvior of the correltors ner the pltes we re interested in is n effect of the distortions of the electromgnetic oscilltions with respect to sitution where the pltes re not present. This fct hs received the ttention of severl uthors, see for exmple 9, 6. V. THE CASIMIR EFFECT FOR CONDUCTING PLATES FIG. : Three-plte setup for the obtention of the Csimir force per unit re. The plte t z = l is uxiliry. In order to pply the bove results to Csimir s originl setup we consider for convenience three prllel perfectly conducting pltes perpendiculr to the OZ xis t z =, z = nd z = l. The quntum version of Mxwell tensor is 4π ˆT = Ê i Ê j δ Ê + ˆB i ˆB j δ ˆB 59 Mking use of the correltor given by 3 we obtin the following prtil results Ê x z,t = Ê y z,t = π 4 3π + F ξ, 6 Ê z z,t π 4 = 3π + F ξ, 6 lso Ê x z,tê y z,t = Ê x z,tê z z,t = Ê y z,tê z z,t =. In the sme wy, mking use of 35 we obtin ˆB xz,t = ˆB yz,t π 4 = 3π F ξ, 6 ˆB z z,t π 4 = 3π F ξ, 63 nd lso ˆB x z,t ˆB y z,t = ˆB x z,t ˆB z z,t = ˆB y z,t ˆB z z,t =. The components of the quntum version of Mxwell tensor cn be esily evluted. For instnce performing the necessry substitutions we hve 8π ˆT zz z,t = Ê z z,t Ê x z,t Ê y z,t + ˆB z z,t ˆB xz,t ˆB yz,t. 64 In the sme wy ˆT zz = π ˆT xx = ˆT yy = Ê 8π z z,t + ˆB z z,t = π Notice how conveniently the divergent prts ner the pltes cncel out yielding finite results.

9 Brzilin Journl of Physics, vol. 35, no. 3A, September, Let us obtin now the Csimir force per unit re between the conducting pltes. Consider Figure V nd the plte t z =. The Csimir force per unit re on this plte is F z A = T zzz L + T zz z R = π π 4l 4, 67 where z L,R mens tht z tends to from the left/right. Tking the limit l we obtin the expected result for the Csimir force per unit re F z π A = The minus sign shows tht the resulting pressure pushes towrds the region between the pltes. If simultneously we tke the limits l, keeping the distnce l constnt. The pressure chnges its sign but it still pushes the plte t z = towrds the one t z = l. In order to clculte the renormlized symmetricl stressenergy tensor ˆΘ µν z ren we evlute the energy density ρz ˆΘ z ren in the region between the pltes s well s the sptil components ˆΘ xx z ren, ˆΘ yy z ren, nd ˆΘ zz z ren. The energy density is given by ρr,t = Ê r,t 8π + ˆB r,t 69 mking use of the correltors given by 3 n5 we obtin π ρ = This result is due to the fct tht the divergent pieces in 3 n5 cncel out yielding finite result for the vcuum energy density. Reclling tht Θ z = T z, see, with help of, 3 n5 the remnescent components of the symmetricl stress-energy tensor re esily obtined. The finl result is ˆΘ µν z ren = π dig,,, 3, 7 74 which is in perfect greement with Brown nd Mcly s results 6. Notice lso tht ˆΘ µ µz ren = g µν ˆΘ µν z ren =, with g µν = dig,,,. VI. THE CASIMIR EFFECT FOR ONE CONDUCTING PLATE AND AN INFINITELY PERMEABLE ONE Let us consider now the setup proposed by Boyer which consists of perfectly conducting plte plced perpendiculrly to the OZ xis t z = nd nother infinitely permeble one prllel to the first plced t z =. The boundry conditions on the conducting plte re s before E x = E y = nd B z =, nd for the infinitely permeble plte: B x = B y = nd E z =. Mking use of the correltor given by 55 the following prtil results: Ê x z,t = Êy z,t π 4 7 = 3π 8 + Gξ, 7 Ê z z,t π 4 = 7 3π 8 + Gξ, 73 nd Êx z,tê y z,t = Êx z,tê z z,t = Êy z,tê z z,t =. By the sme token mking use of the correltor given by 56 we obtin ˆB xz,t = ˆB yz,t π 4 7 = 3π 8 Gξ, 74 ˆB z z,t π 4 = 7 3π 8 Gξ, 75 nd lso ˆB x z,t ˆB y z,t = ˆB x z,t ˆB z z,t = ˆB y z,t ˆB z z,t = 76 Proceeding s in the cse of the conducting pltes we obtin the following results for the componenets of the quntum version of Mxwell tensor nd ˆT xx = ˆT yy = 7 8 π 7 4, 77 ˆT zz = 7 8 π Notice tht it is sufficient to multiply the results obtined for Csimir s setup by the fctor 7/8 in order to obtin the results corresponding to Boyer s setup. In order to obtin the Csimir force per unit re for this setup it is convenient to plce third conducting plte t z = l. Then the Csimir force per unit re on the plte t z = will be given by F z A = T zzz L + T zz z R = 7 π π 4l 4.79 Tking the limit l we obtin Csimir force which pushes the plte t z = towrds the region z > given by F z A = 7 8 π This is the result obtined by Boyer for this setup using stochstic electrodynmic methods nd it is one of the simplest exmple of repulsive Csimir force.

10 666 F. C. Sntos et l. In order to evlute the symmetricl stress-energy tensor we first evlute the Csimir energy density. Mking use of 55 e 56 we hve ρ = 7 8 π As in the cse of the conducting pltes simple clcultion shows tht the stress energy tensor for Boyer s setup is given by ˆΘ µν z ren = 7 8 π dig,,, As before ˆΘ µ µz ren = g µν ˆΘ µν z ren =. VII. CONCLUSIONS In this pper we hve shown how to employ the equl time nd spce electromgnetic field correltors evluted between prllel mteril surfces to rederive results concernig the Csimir energy nd pressure nd the symmetricl trceless stress energy tensor. We hve shown tht for the cses we hd in mind here finite results re obtined only when we consider wht hppens on both sides of the surfce boundry. This considertion provided the mechnism by which precise cncelltions occurred nd finite results were obtined. These cncelltions occur only for the simple geometry considered in this pper. This is in greement with, for exmple, Ref. 9 nd should be considered s concrete exmple of the behvior of quntized fields ner nd on boundry surfces. This behvior depends on the geometry of the boundries nd cn become quite complicted. It cn depend on the locl curvture of the boundry, for instnce. Finlly, it must be mentioned tht the equl time nd spce electromgnetic field correltors clculted here were cn be lso employed to rederive the trctive nd the repulsive Csimir effect by mens of quntum version of the Lorentz force. They cn be lso pplied to the Csimir-Polder interction between n tom nd mteril surfces H. B. G. Csimir, Proc. K. Ned. Akd. Wet. 5, ; Philips Res. Rep M. J. Sprny, Physic 4, S. K. Lmoreux, Phys. Rev. Lett. 78, 5 997; errtum Phys. Rev. Lett. 8, ; U. Mohideen nd A. Roy, Phys. Rev. Lett. 8, M. Bordg, U. Mohideen nd V.M. Mostepnenko, Phys. Rep. 353, ; qunt-ph/ V. M. Mostepnenko nd N. N. Trunov, Sov. Phys. Usp. 3, ; V. M. Mostepnenko nd N. N. Trunov, The Csimir Effect nd its Applictions, Oxford, Clrendon, 997; G. Plunien, B. Müller nd W. Greiner, Phys. Rep. 34, ; S. K. Lmoreux, Am. J. Phys. 67, See lso: M.V. Cougo-Pinto, C. Frin nd A. C. Tort, Rev. Brs. Ens. Fís., in Portuguese. 6 L.S. Brown nd G.J. Mcly, Phys. Rev. 84, A. E. Gonzles, Physic 3A K. A. Milton,The Csimir Effect: Physicl Mnifesttions of Zero-Point Energy World Scientific, Singpore,. 9 D. Deutsch nd P. Cndels, Phys. Rev. D, T.H. Boyer Phys. Rev. A 9, J. D. Jckson, Clssicl Electrodynmics, 3rd. ed., John Wiley, New York 999. G. Brton, Phys. Lett. B 37, M. V. Cougo-Pinto, C. Frin, F. C. Sntos nd A. C. Tort: J. of Phys. A 3 999, H.B.G. Csimir, J. Chim. Phys. 46, T.H. Boyer, Phys. Rev. 8, C.A. Lütken nd F. Rvndl, Phys. Rev. A 3, M. Bordg, D. Robschik nd E. Wieczorek, Ann. Phys. NY 65, E. Elizlde, S. D. Odintsov, A. Romeo, A. A. Bytsenko nd S. Zerbini: Zet Regulriztion Techniques with Applictions, World Scientific, Singpore 994. See lso R. Ruggiero, A. H. Zimermn nd A. Villni, Rev. Brs. Fís. 7, For simple introduction to the Zet function regulriztion technique see: J. J. Pssos Sobrinho nd A. C. Tort, Rev. Brs. Ens. Fís. 3, 4 in Portuguese. 9 I. S. Grdshteyn nd I. M. Ryzhik, Tbles of Integrls, Series nd Products, 5th Edition, Acdemic Press, New York 994. C. Frin, F. C. Sntos nd A. C. Tort, Eur. J. of Phys. 4 N5-N9 3. F. C. Sntos, J. J. Pssos Sobrinho, nd A. C. Tort, Brz. J. of Phys