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FINITE TEMPERATURE CASIMIR EFFECT IN KALUZA-KLEIN SPACETIME L.P.TEO 9 0 0 Abstract. Inthisarticle,weconsiderthefinitetemperatureCasimireffectin 2 Kaluza-Kleinspacetimeduethethevacuumfluctuationofmasslessscalarfield with Dirichlet boundary conditions. We consider the general case where the n extra dimensions (internal space) can be any compact connected manifoldor a orbifoldwithoutboundaries. Usingpistonanalysis,weshowthattheCasimir J forceisalwaysattractiveatanytemperature,regardlessofthegeometryofthe 5 internalspace. Moreover,themagnitudeoftheCasimirforceincreasesasthe 1 sizeoftheinternalspaceincreasesanditreducestotheCasimirforcein(3+1)- dimensionalMinskowskispacetimewhenthesizeoftheinternalspaceshrinks ] to zero. In the other extreme where the internal space is large, the Casimir h force can increase beyond all bound. Asymptotic behaviors of the Casimir t - force in the low and high temperature regimes are derived and itis observed p that the magnitude of the Casimir force grows linearly with temperature in e thehightemperatureregime. h PACSnumbers: 04.50.Cd,11.10.Wx,11.10.Kk,04.62.+v. [ 1 v 5 1. Introduction 9 1 Oneoftheinterestingpredictionsofstringtheoryisthatweliveinuniversewith 2 nine or ten spacedimensions, three ofwhich arevisible. In fact, spacetime with an . 1 extradimensioncurleduptoatinycirclehasalreadybeenpostulatedbyKaluzaand 0 Klein[1,2]around1920’sinanattempttounifytwoofthefundamentalinteractions 9 – gravitational and electromagnetic forces. Besides the motivation coming from 0 string theory, the interest in universe with more than three spatial dimensions is : v also stimulated by the developments in particle physics and cosmology. Different i X spacetimemodelsthatcontainextradimensionshavebeenproposedintheendeavor tofindasatisfactoryexplanationforthelargehierarchybetweensomefundamental r a scales,aswellastoaccountforthedarkenergythatacceleratestheexpansionofthe universe [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 16, 13, 14, 17, 18, 19, 21, 20, 22, 23, 24]. One of the proposed form of the dark energy is the cosmological constant – a constant energy density physically equivalent to the vacuum energy or Casimir energy. Postulated in 1948 [25], Casimir effect has penetrated into different areas of physics such as quantum field theory, condensed matter physics, atomic and molecular physics, gravitation and cosmology, and mathematical physics [26]. In the scenarios of extra dimensional physics, Casimir effect has been studied in the contextofstringtheory[27,28,29,30],darkenergyandcosmologicalconstant[15, 16,17,18,19,21,20,24,31,32,33,34,35,36,37,38],aswellasstabilizationofextra dimensions [39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54]. Recently, Casimir force acting on a pair of parallel plates in Kaluza-Klein spacetime model Keywordsandphrases. HigherDimensionalFieldTheory,CasimirEffect,FiniteTemperature, Kaluza-KleinSpacetime, MasslessScalarField. 1 2 L.P.TEO with n extra dimensions compactified to a n-torus Tn = (S1)n and in Randall- Sundrum spacetime model have been calculated and analyzed for massless scalar fieldwithDirichletboundaryconditionsonapairofparallelplatesin[55,56,57,58, 59, 60] and in [61, 62, 63] respectively. For electromagnetic field, the Casimir force acting on a pair of parallel and perfectly conducting plates in (4+1)-dimensional Kaluza-Kleinspacetime is studied in [24, 64, 65, 66, 67]. AlthoughitwaspointedoutnotlongafterthediscoveryofCasimirthatthermal corrections have to be taken into account in the determination of Casimir effect, majority of the work done for Casimir effect, especially those related to spacetime with extra dimensions, are at zero temperature. This might due to a few reasons. One of them being the mathematical techniques required for the computations of Casimir effect at finite temperature is more complicated than those used for zero temperature. Another might be the fact that the thermal corrections will only be significant at plate separation larger than 1µm. Therefore in the present ex- perimentally accessible measurements of Casimir force [68, 69, 70, 71], the small separation required for the detectability of Casimir force renders the thermal correction at room temperature negligible. A third reason may be attributed to the controversialstateofthe resultsobtainedforfinite temperatureCasimirforcefrom thepointofviewofthermodynamics[72]. Inanyway,bynomeanstheseissuescan mask the importance of taking the thermal correctioninto account when consider- ingCasimireffect. Infact,forspacetimewithmorethanfourdimensions,thefinite temperatureCasimirenergywasfirstcalculatedin[73]for(d+1)-dimensionalrect- angularcavitiesusingdimensionalregularization. Someotherstudiesthatconsider finite temperature Casimir effect for higher dimensional spacetime can be found in [41,44,74,75,76,77,78,79,80,81,82,83,84,85]. Inthe contextofKaluza-Klein spacetime model, the finite temperature Casimir force has only been investigated in [56] for a pair of parallel plates in spacetime of the form M4 S1, where M4 is × the (3+1)–dimensionalMinkowskispacetime,andit wasclaimedthat the Casimir force can become repulsive under certain conditions. Recently we re-calculate the Casimir force for the case considered in [56] but using a different regularization setupcalled the pistonapproach,and we find that the Casimirforce shouldalways be attractive [86]. As a matter of fact, even at zero temperature, except for results for parallel plates, contradictory results for Casimir force appear due to the employment of different renormalization procedures. The issue on renormalizability of surface divergence terms was brought up in [87, 88] and is still under discussions [50, 89, 90, 91, 92, 93]. In [94], Cavalcanti proposed a new geometric setup called pistonwhichcanavoidtheproblem. HeshowedthattheCasimirforceactingonthe piston is free of surface divergence and can therefore be calculated unambiguously. Since then, Casimir piston has attracted considerable interest [59, 95, 96, 97, 98, 99,100,101,102,103,104,105,106,107,108]. Infact,thepistonscenariohasbeen used in the early work on Casimir force between parallel plates as a regularization procedure [109, 110]. Inthispaper,westudythefinitetemperatureCasimireffectinageneralKaluza- Klein spacetime of the form M4 Nn, where the internal space Nn can be any n- × dimensionalcompactconnectedmanifoldororbifold. Toensurethatthe spacetime M4 Nn isconnected,it isnecessaryto assumethatNn is connected. Otherfrom × this,wedonotmakeanyadditionalassumptionsonthetopologyorgeometryofthe FINITE TEMPERATURE CASIMIR EFFECT IN KALUZA-KLEIN SPACETIME 3 internalspace. Inordertoobtainafinite unambiguousresultfortheCasimirforce, piston setup is used. We consider massless scalar field with Dirichlet boundary conditions and derive exact and explicit formulas for the Casimir force acting on the piston. From this, we draw the conclusion that the Casimir force is always attractive regardless of the temperature. Moreover the Casimir force reduces to the Casimir force in M4 when the size of the internal space goes to zero. This showsthatwhentheinternalspacebecomesnegligible,thefourdimensionaltheory is recovered. Asymptotic behaviors of the Casimir force for different limits such as lowandhightemperature,smallplateseparation,smallandlargeextradimensions areworkedout. Inthehightemperatureregime,itisnoticedthatthemagnitudeof the Casimir force grows linearly with temperature. This implies that the Casimir force might become significant when the temperature is very high. Partoftheresultsinthispaperhasalreadybeenannouncedinourpreviousletter [86]. After this work is done, we notice the closely related work [111] by Kirsten and Fulling where the zero temperature case is discussed in detail and the finite temperaturecaseappearsasabriefremarkinthe conclusion. Inthe followingsthe units with ~=c=k =1 are used throughout. B 2. Basic Formalism We consider Kaluza-Klein spacetime of the form M4 Nn, where M4 is the × (3+1)-dimensionalMinkowskispacetimeandNn is the internalspaceofdimension n, assumed to be compact and connected. We are interested in the Casimir effect due to massless scalar field ϕ with Lagrangianand action 1 = gµν∂ ϕ∂ ϕ, S = g dd+1x. µ ν L 2 L | | Z p Here ds2 =g dxµdxν =η dxαdxβ G dxadxb µν αβ ab − is the spacetime metric with η = diag(1, 1, 1, 1) the usual (3+1)-D metric αβ − − − on M4; ds2 = G dxadxb a Riemannian metric on Nn; g = ( 1)n 1det[g ] is N ab | | − − µν the absolute value of the determinant of the matrix [g ]n+3 and d=n+3 is the µν µ,ν=0 total space dimension. The field ϕ satisfies the Laplace equation 1 (2.1) ∂ g gµν∂ ϕ=0. µ ν g | | | | p Our aim is to calculate the Cpasimir force acting on a pair of parallel plates due to the vacuum fluctuations of the massless scalar field with Dirichlet boundary conditions on the plates. As has been pointed out in [60], a correct regularization approachto this problemis the piston setup[94,109], wherethe Casimirforcewas firstcomputedforthecasethetwoplatesareembeddedinaclosedcylinder[99]and then the limit where the surrounding cylinder is brought to infinity is evaluated. However since it is enough for us to consider the force acting on one of the plates, it will be sufficient to treatthe plate concernedas a movable piston inside a closed cylinder(seeFIG.1)andfindtheCasimirforcebeforelettingoneendofthecylinder approachinfinity. In fact, since inrealitywe canneverhaveinfinite parallelplates, it will be interesting by its own to consider the Casimir force acting on a movable pistoninside aclosedcylinder. We workwith the fullgeneralitywherethe cylinder is allowed to have arbitrary cross section. Mathematically speaking, this means 4 L.P.TEO Region I Region II a L − a 1 Figure1: Amovablepistoninsideaclosed cylinderdividesthecylinderintotworegions. thatthe crosssectionofthe cylinder is a simply connecteddomainΩ onthe plane. According to [94], the Casimir energy of the piston system is the sum E (a;L ;T)=EI (a;T)+EII (L a;T)+Eext(T) Cas 1 Cas Cas 1− Cas of the Casimir energies of Region I, Region II and the exterior region, and the Casimir force acting on the piston is given by ∂ (2.2) F (a;L ;T)= E (a;L ;T). Cas 1 Cas 1 −∂a Beingindependent ofthe pistonposition, the Casimir energyofthe exteriorregion would not contribute to the Casimir force. To compute the Casimir energy in Regions I and II, it suffices to compute the Casimir energy inside the cylinder I Ω Nn, where I = [0,L], and letting L equal to a and L a respectively. 1 × × − Usingmodesumapproach,thefinitetemperatureCasimirenergyinsidethecylinder I Ω Nn is given by × × 1 (2.3) Ecyl (L;T)= Tlog = ω+T log 1 e ω/T , Cas − Z 2 − − where X X (cid:16) (cid:17) e−2ωT = Z 1 e−Tω − canberegardedasthepartitionofagrYandcanonicalensemble,T isthetemperature andthesummationsin(2.3)runthroughallnonzeroeigenfrequenciesωofthefieldϕ satisfyingtheequation(2.1),withDirichletboundaryconditionsontheboundaries of the cylinder I Ω Nn. Using separation of variables, it is immediate to find × × that a complete set of solutions to (2.1) with Dirichlet boundary conditions on I Ω Nn is given by × × πkx1 ϕk,j,l(xα,xa)=e−iωk,j,ltsin φj(x2,x3)Φl(xa), L (2.4) 2 πk ω = +ω2 +ω2 , k,j N,l N 0 , k,j,l s L Ω,j N,l ∈ ∈ ∪{ } (cid:18) (cid:19) where xα = (t,x1,x2,x3) and xa = (x4,...,xn+3) are the coordinates on M4 and Nn respectively; φ (x2,x3),j = 1,2,... is a complete set of eigenfunctions j of the Laplace operator ∂2 + ∂2 on Ω with Dirichlet boundary conditions ∂(x2)2 ∂(x3)2 FINITE TEMPERATURE CASIMIR EFFECT IN KALUZA-KLEIN SPACETIME 5 φ = 0 and eigenvalues ω2 arranged so that 0 < ω2 ω2 ...; and Φj(|x∂Ωa),l = 0,1,2,... is a comΩp,jlete set of eigenfunctions oΩf,1th≤e LaΩp,2lac≤e operator l 1 ∂ √GGab ∂ onNnwitheigenvaluesω2 arrangedsothat0=ω2 <ω2 √G∂xa ∂xb N,l N,0 N,1 ≤ ω2 .... Notice that since Ω is simply connected, the Laplace operator with N,2 ≤ Dirichlet boundary conditions does not have zero eigenvalue. On the other hand, since Nn is a compact connected manifold, its Laplace operator has a single zero eigenvalue which corresponds to constant functions. IfinsteadofDirichletboundaryconditions,weconsiderNeumannboundarycon- ditions,weneedtoreplacethesinefunctionin(2.4)bycosinefunctionandletkruns fromzerotoinfinity;andreplaceφ (x2,x3)withψ (x2,x3),j =0,1,2,...whichare j j eigenfunctions of the Laplace operators on Ω with Neumann boundary conditions ∂ψj =0,wherenistheunitvectornormalto∂Ω,andwitheigenvaluesω2 . In ∂n Ω,j ∂Ω this(cid:12)case there is a zero eigenvalue corresponding to constant functions. Therefore (cid:12) fork(cid:12)=j =l=0,thetermωk,j,l iszeroandhastobeomittedfromthesummation in (2.3). In the following,we only consider Dirichlet boundary conditions since the case of Neumann boundary conditions can easily be derived analogously. 3. Casimir force acting on the piston or a pair of parallel plates Since the first summation in the definition of the Casimir energy given by (2.3) is divergent, we introduce a cut-off and compute the small λ-expansion of 1 (3.1) ECcyals(λ;L;T)= 2 ωk,j,le−λωk,j,l +T log 1−e−ωk,j,l/T , kX,j,l kX,j,l (cid:16) (cid:17) uptothe termλ0. Thecalculationscanbedoneexplicitlyusingzetafunctions and heat kernels [112, 113, 114] and we leave it to the appendix. The result is (3.2) n+2 Γ(n+4 i) c log[λµ] ψ(1) log2+1 Ecyl (λ;L;T)= − cyl,i + − − c Cas Γ n+3 i λn+4 i 2√π cyl,n+4 i=0 2− − X T ζ(cid:0) (0;(cid:1)L)+log(µ2)ζ (0;L) , − 2 c′yl,T cyl,T where µ is a normalizati(cid:0)on constant with dimension leng(cid:1)th 1, − (3.3) ζcyl,T(s;L)= ∞ ∞ ∞ ∞ ωk2,j,l+[2πpT]2 −s k=1j=1l=0p= XXX X−∞(cid:0) (cid:1) isathermalzetafunction,andfori=0,1,...,n+4,c areheatkernelcoefficients cyl,i of the Laplace operator on I Ω N with Dirichlet boundary conditions. The × × dependence of c on L is linear (see appendix), i.e. cyl,i L 1 c = c c , cyl,i Ω N,i Ω N,i 1 2√π × − 2 × − wherec areheatkernelcoefficientsofΩ N andareindependentofL. Notice Ω N,i that as λ× 0+, Ecyl (λ;L;T) contains div×ergence terms of order λ j for j = → Cas − 1,2,...,n+4. The leading divergent term Γ(n+4)c Γ(n+4)vol(I Ω Nn) Γ n+23 λcny+l,40 = Γ n+23 (4×π)n+2×3 λ−n−4 (cid:0) (cid:1) (cid:0) (cid:1) 6 L.P.TEO is the bulk divergence and is usually subtracted away. However,the regularization of the other divergent terms is a highly nontrivial issue. Naive zeta regularization method set all these divergent terms to zero and might lead to physicality issue. A regularization procedure close in spirit to the piston scenario was introduced in [115]. However, since our main interest is the Casimir force, we shall not deal further with this issue. Upon substituting into the definition of Casimir force ∂ F (a;L ;T)= Ecyl(a;T)+Ecyl(L a;T) Cas 1 − ∂a Cas Cas 1− (cid:16) ∂ (cid:17) = lim Ecyl(λ;a;T)+Ecyl(λ;L a;T) , −λ 0+ ∂a Cas Cas 1− → (cid:16) (cid:17) wefindthatsinceallλ 0+ divergenttermsarelinearinL,theircontributionsto → the Casimir force cancel each other and therefore the λ 0+ limit is well-defined → and is given by T ∂ F (a;L ;T)= ζ (0;a)+ζ (0;L a) . Cas 1 2 ∂a c′yl,T c′yl,T 1− (cid:8) (cid:9) Here we have also used the fact that ζ (0;L)= c /(2√πT) is linear in L. cyl,T cyl,n+4 Using (A.12), we have the explicit formula: (3.4) F (a;L ;T)=F (a;T) F (L a;T), Cas 1 C∞as − C∞as 1− where F (a;T)= lim F (a;L ;T) C∞as Cas 1 L1→∞ T ∂ ∞ ∞ ∞ ∞ 1 = exp 2ka ω2 +ω2 +[2πpT]2 2 ∂a k − Ω,j N,l (3.5) Xk=1Xj=1Xl=0p=X−∞ (cid:16) q (cid:17) ω2 +ω2 +[2πpT]2 ∞ ∞ ∞ Ω,j N,l = T . − exp 2aq ω2 +ω2 +[2πpT]2 1 Xj=1Xl=0p=X−∞ Ω,j N,l − (cid:16) q (cid:17) Notice that the expression (3.5) is always negative and is an increasing function of a. This means that when one end of the closed cylinder is moved to extremely distant place, the Casimir force acting on the piston (or two parallel plates inside an infinitely long cylinder) is always attractive and the magnitude of the force decreases as the plate separations increases. Moreover,the magnitude of the force decreases in exponential rate. For a finite piston inside a closed cylinder, (3.4) then shows that the Casimir force always tend to pull the piston away from the equilibrium position x1 = L /2 towards the nearer end and the magnitude of the 1 Casimir force increasesas the piston is farther awayfromthe equilibrium position. Eq. (3.5) can also be regarded as a high temperature expansion of the Casimir force. It shows that in the high temperature regime, the Casimir force is linear in T with leading term ω2 +ω2 FC∞a,sT≫1(a;T)∼−T ∞ ∞ exp 2aq ωΩ2,j +ωN2,l 1, Xj=1Xl=0 Ω,j N,l − (cid:16) q (cid:17) corresponding to the p = 0 term. Since this term is independent of the Planck constant ~, it is sometimes known as the classical limit [116, 117, 118, 119]. FINITE TEMPERATURE CASIMIR EFFECT IN KALUZA-KLEIN SPACETIME 7 ForthelowtemperaturebehavioroftheCasimirforce,(A.14)showsthatF (a;T) C∞as can be written as the sum of the zero temperature Casimir force F (a;T = 0) C∞as plusthetemperaturecorrection∆ F (a;T),wherethezerotemperatureCasimir T C∞as force F (a;T =0) is given by C∞as ω2 +ω2 1 ∞ ∞ ∞ Ω,j N,l F (a;T =0)= K 2ka ω2 +ω2 C∞as − 2πa q k 1 Ω,j N,l (3.6) kX=1Xj=1Xl=0 (cid:16) q (cid:17) 1 ∞ ∞ ∞ ω2 +ω2 K 2ka ω2 +ω2 ; − π Ω,j N,l 0 Ω,j N,l Xk=1Xj=1Xl=0(cid:0) (cid:1) (cid:16) q (cid:17) and the temperature correction to the Casimir force ∆ F (a;T) is given by T C∞as ω2 +ω2 p ω2 +ω2 T ∞ ∞ ∞ Ω,j N,l Ω,j N,l ∆TFC∞as(a;T)=− π q p K1 q T  j=1l=0p=1 (3.7) XXX π2 ∞ ∞ ∞ k2   + . a3 ωk,j,l k=1j=1l=0 ωk,j,l e T 1 XXX − (cid:16) (cid:17) Althoughthe attractivepropertyofthe Casimirforceatanytemperaturehasbeen observed from the compact expression (3.5), it is interesting to remark that the expression for zero temperature Casimir force given by (3.6) also manifests the attractive property at zero temperature. To investigate the behavior of the Casimir force when the cross section Ω and the internal space Nn is small or large compared to a, we define the size variables r and R in the following way: r := Area(Ω), R:=(Vol(Nn))n1 , so that Ω has area r2 andpNn has volume Rn. If we rescale the domain Ω (resp. Nn) to Ω/r (resp. Nn/R)1, then the eigenvalues ω2 (resp. ω2 ) on Ω (resp. ′ ′ Ω,j N,l Nn) is related to the eigenvalues ω2 (resp. ω2 ) on Ω/r (resp. Nn/R) Ω/r′,j N/R′,l ′ ′ by ω2 = ω2 /(r )2 (resp. ω2 = ω2 /(R)2). Therefore we can define Ω,j Ω/r′,j ′ N,j N/R′,j ′ dimensionlessvariablesω =rω andω =Rω sothattheyareindependent Ω′,j Ω,j N′ ,l N,l of the relative size of Ω and Nn. The expression for Casimir force (3.5) can then be rewritten as (3.8) ω′ 2 ω′ 2 Ω,j + N,l +[2πpT]2 ∞ ∞ ∞ r R FC∞as(a;T)=−TXj=1Xl=0p=X−∞exp 2ar(cid:16) ωΩ′,(cid:17)j 2+(cid:16) ωN′ ,(cid:17)l 2+[2πpT]2 1. r r R !− (cid:16) (cid:17) (cid:16) (cid:17) Since the function x x 7→ ex 1 − is a decreasing function, it follows immediately from (3.8) that as the size of the internal space decreases (i.e. R decreases), the magnitude of the Casimir force 1Thisisequivalent torescalethemetricGabdxadxb to(r′)−2Gabdxadxb. 8 L.P.TEO decreases. Inthelimittheinternalspacevanishes,allthetermswithl =0vanishes 6 and the Casimir force reduces to ω′ 2 Ω,j +[2πpT]2 (3.9) FC3Das,∞(a;T)=−T Xj∞=1p=X∞−∞exp 2ar(cid:16) ωrΩ′,(cid:17)j 2+[2πpT]2 1, r r !− (cid:16) (cid:17) whichistheCasimirforceactingonapairofparallelplatesinsideaninfinitelylong cylinder with cross section Ω due to massless scalar field with Dirichlet boundary conditions [99]. Taking similar limit to (3.6), one also finds that in the limit of vanishing internal space, the zero temperature Casimir force reduces to the corresponding zero temperature Casimir force in (3+1)-dimensions given by [99] FC3Das,∞(a;T =0)=−2π1a ∞ ∞ ωΩk,jK1(2kaωΩ,j)− π1 ∞ ∞ ωΩ2,jK0(2kaωΩ,j). k=1j=1 k=1j=1 XX XX Another interesting property we can read from the expression of Casimir force given by (3.5) or (3.8) is the effect of increasing the number of extra dimensions. Thenumberofextradimensionscanbeincreasedbyaddinganotherextracompact space n′ of dimension n to Nn so that the internal space becomes Nn n′, ′ N ×N a compact manifold of dimension n+n. The spectrum of the Laplace operator ′ on Nn n′ can be written as ω2 + ω2 , l,l = 0,1,2,..., where ω2 and ω2 ar×e Nthe spectrums of the LapNl,alce opNe,rl′ators′ on Nn and n′ respeNct,lively. SiNn,cle′ each term in the summation of (3.5) is negative, it is thNen immediate to deduce that adding the dimension of the internal space increase the magnitude of the Casimir force. Moreover, as the size of n′ shrinks to zero, the Casimir force whentheinternalspaceisNn n′ reducesNtotheCasimirforcewhentheinternal ×N space is Nn. This shows that adding extra dimensions increase the Casimir force hierarchically. Now we consider the case where the area of the cross section of Ω is large compared to the plate separation, i.e. the ratio r/a is large. As is derived in the appendix B, when r/a is large, the leading term of the Casimir force is of order r2. Divide the Casimir force by the area r2 of Ω and taking the limit r , we → ∞ obtain the Casimir force density on a pair of infinite parallel plates. Its high and low temperature expansions are given respectively by (3.10) F (a;T) k (a;T)= lim C∞as FCas r r2 →∞ 3 = ζR(3)T + T ∞ ωN2,l+[2πpT]2 2 K 2ka ω2 +[2πpT]2 − 8πa3 4π32a23 Xk=1l∈N∪X{0},p∈Zq k  23 (cid:18) q N,l (cid:19) (p,l)6=(0,0)   5 ω2 +[2πpT]2 2 T ∞ N,l K 2ka ω2 +[2πpT]2 , − 2π23a12 Xk=1l∈N∪X{0},p∈Z(cid:16)q k12 (cid:17) 25 (cid:18) q N,l (cid:19) (p,l)=(0,0) 6 FINITE TEMPERATURE CASIMIR EFFECT IN KALUZA-KLEIN SPACETIME 9 and (3.11) k (a;T) FCas π2 3 ∞ ∞ ωN2,l 1 ∞ ∞ ωN3,l π2T4 = K (2kaω ) K (2kaω ) − 480a4 − 8π2a2 k2 2 N,l − 4π2a k 1 N,l − 90 k=1l=1 k=1l=1 XX XX + πT ∞ ∞ ∞ k2 exp p ω2 + πk 2 T2 ∞ ∞ ωN2,lK pωN,l . 2a3 p −Ts N,l a − 2π2 p2 2 T Xk=1Xl=0Xp=1 (cid:18) (cid:19) Xl=1Xp=1 (cid:16) (cid:17)   Neither of these expressionsshow manifestly that the Casimir force density is negative. However, since they are obtained as limits of a negative Casimir force, the Casimir force density acting on a pair of infinite parallel plates in the presence of extra dimensional space is always attractive regardless of the geometry of the internal space. In the limit the extra dimensions vanish, the Casimir force (3.10) and (3.11) reduce to FC3Das,k(a;T)=− ζR8π(3a)3T + √a2T23 52 ∞ ∞ kp 32 K23 (4πpkTa) kX=1Xp=1(cid:16) (cid:17) 4√2πT27 ∞ ∞ p52 K (4πpkTa) − a21 k21 52 k=1p=1 XX or 3D, π2 π2T4 πT ∞ ∞ k2 πpk k(a;T)= + exp , FCas −480a4 − 90 2a3 p − Ta k=1p=1 (cid:18) (cid:19) XX which are the well known results for the Casimir force density on a pair of infinite parallel plates in (3+1)-dimensional Minskowski spacetime due to massless scalar field with Dirichlet boundary conditions. As is discussedabove,in the limit the internalspace vanishes, the Casimir force alwaysreducestothecorrespondingCasimirforcein(3+1)-dimensionalspacetime. Therefore,wecanwritetheCasimirforceF (a;T)asthesumoftheCasimirforce C∞as in(3+1)-dimensionalspacetime F3D(a;T)plus the correctionterm ∆ F (a;T) Cas N C∞as due to the presence of the extra dimensional compact space Nn. As can be read from (3.5), the correction term is given by ω2 +ω2 +[2πpT]2 ∞ ∞ ∞ Ω,j N,l (3.12) ∆NFC∞as(a;T)=−T exp 2aq ω2 +ω2 +[2πpT]2 1. Xj=1Xl=1p=X−∞ Ω,j N,l − (cid:16) q (cid:17) Here the l =0 term has been omitted. It will be interesting to investigate whether there is a bound for this correction term. For this purpose, it suffices to consider thebehaviorofthiscorrectiontermwhenthe sizeRoftheextradimensionalspace is large. In fact, this question is also important since the model with large extra dimensions, which is also known as ADD model [8, 10], has aroused considerable interest as an alternative to explain the weakness of gravity compared to other forces. As before, using the re-scaling ω = ω /R, then by the same method N,l N′ ,l as we derive the asymptotic behavior of the Casimir force when r/a is large in appendix B, but with the roles of Ω and Nn interchanged, we find that when R/a 10 L.P.TEO is large, the leading term of ∆ F (a;T) is proportional to Rn – the volume of N C∞as Nn, and is given explicitly by (3.13) ∆ F (a;T) N C∞as n+1 =Rn T ∞ ∞ ∞ ωΩ2,j +[2πpT]2 2 K 2ka ω2 +[2πpT]2 (2nπn+21an+21 Xk=1Xj=1p=X−∞q k  n+21 (cid:18) q Ω,j (cid:19)  n+3 ω2 +[2πpT]2 2 T ∞ ∞ ∞ Ω,j K 2ka ω2 +[2πpT]2 − 2n−1πn+21an−21 Xk=1Xj=1p=X−∞(cid:16)q kn−21 (cid:17) n+23 (cid:18) q Ω,j (cid:19)) +O Rn 1 − n+2 n+4 =Rn(−(cid:0) 2n+1(cid:1)nπn++221an+22 k∞=1j∞=1 ωkΩn+,22j2 Kn+22(2kaωΩ,j)− 2nπn1+22an2 k∞=1j∞=1 ωkΩn,22j Kn2(2kaωΩ,j) XX XX n−1 + 2n−2T1πn−2n−123a3 k∞=1j∞=1p∞=1k2qωΩ2,jp+(cid:0)πak(cid:1)2 2 Kn−21 TpsωΩ2,j +(cid:18)πak(cid:19)2 XXX n+2 n+2    T 2 ∞ ∞ ωΩ,j 2 K pωΩ,j +O Rn 1 . − 2n2πn+22 Xj=1Xp=1(cid:18) p (cid:19) n+22 (cid:16) T (cid:17)) (cid:0) − (cid:1) ThisshowsthatthecorrectiontermoftheCasimirforceduetothepresenceofextra dimensions can increase beyond all bounds when the size of the extra dimensions is increased. In the zero temperature limit, the second expression in (3.13) shows that the leading correction term of the zero temperature Casimir force when R/a is large is given by n+2 ∆ F (a;T =0)=Rn n+1 ∞ ∞ ωΩ,2j K (2kaω ) N C∞as (−2n+1πn+22an+22 k=1j=1 kn+22 n+22 Ω,j XX n+4 − 2nπn1+22an2 k∞=1j∞=1 ωkΩn,22j Kn2(2kaωΩ,j))+O Rn−1 . XX (cid:0) (cid:1) For a pair of infinite parallel plates, similar methods (see (B.6) and (B.7)) show that the leading behavior of the Casimir force density when R/a is large is given by (3.14) k (a;T) FCas Rn (n+2)Γ n+23 ζR(n+3) T + 21−2nTn+25 ∞ ∞ p n+23 K (4πkpTa) ∼ (− ((cid:0)4π)n(cid:1)+23 an+3 an+23 Xk=1Xp=1(cid:16)k(cid:17) n+23 5−n n+7 n+5 2 2 πT 2 ∞ ∞ p 2 K (4πkpTa) +O(Rn 1) − an+21 k=1p=1kn+21 n+25 ) − XX