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Casimir Energies in Light of Quantum Field Theory N. Graham Department of Physics and Astronomy, University of California at Los Angeles, Los Angeles, CA 90095 R. L. Jaffe, V. Khemani, M. Quandt, and M. Scandurra Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 H. Weigel 3 Institute for Theoretical Physics, Tu¨bingen University Auf der Morgenstelle 14, D–72076 Tu¨bingen, Germany 0 0 We study the Casimir problem as the limit of a conventional quantum field theory coupled to a 2 smooth background. The Casimir energy diverges in the limit that the background forces the field n to vanish on a surface. Weshow that this divergence cannot be absorbed into a renormalization of a theparametersofthetheory. Asaresult,theCasimirenergyofthesurfaceandotherquantitieslike J thesurface tension, which are obtained by deforming the surface, cannot be defined independently 0 ofthedetailsofthecouplingbetweenthefieldandthematteronthesurface. Incontrast,theenergy 3 density away from the surface and the force between rigid surfaces are finite and independent of thesecomplications. 3 v PACSnumbers: 03.65.Nk,03.70.+k,11.10.Gh 5 0 2 The vacuum energyof fluctuating quantumfields that in continuum quantum field theory without boundaries 7 are subject to boundary conditions has been studied in- (QFT) provides the only physical way to regulate, dis- 0 2 tensely over the half-century since Casimir predicted a cuss, and eventually remove divergences. Therefore we 0 force between grounded metal plates[1, 2, 3]. The plates propose to embed the Casimir calculation in QFT and / change the zero-point energies of fluctuating fields and study its renormalization. After renormalization, if any h t thereby give rise to forces between the rigid bodies or quantity is still infinite in the presence of the boundary - p stresses on isolated surfaces. The Casimir force between condition, it will depend in detail on the properties of e grounded metal plates has now been measured quite ac- the material that provides the physical ultraviolet cutoff h curately and agrees with his prediction [4, 5, 6]. andwillnotexistintheidealizedCasimirproblem. Sim- : v Casimirforcesarisefrominteractionsbetweenthefluc- ilarsubtletieswereaddressedinthecontextofdispersive Xi tuating fields and matter. Nevertheless, it is traditional media in Ref. [11]. to study idealized “Casimir problems” where the physi- It is straightforward to write down a QFT describing r a calinteractionsarereplacedabinitiobyboundarycondi- the interaction of the fluctuating field φ with a static tions. In this Letter we study under what circumstances background field σ(x) and to choose a limit involving this replacementis justified. Arealmaterialcannotcon- the shape of σ(x) and the coupling strength between strain modes of the field with wavelengths much smaller φ and σ that produces the desired boundary conditions than the typical length scale of its interactions. In con- on specified surfaces. We have developed the formal- trast, a boundary condition constrains all modes. The ism required to compute the resulting vacuum energy in sum over zero point energies is highly divergent in the Ref. [12]. Here we focus on Dirichlet boundary condi- ultraviolet and these divergences depend on the bound- tions on a scalar field. Our methods can be generalized ary conditions. Subtraction of the vacuum energy in the tothephysicallyinterestingcaseofconductingboundary absenceofboundariesonlyremovestheworstdivergence conditions on a gauge field. (quartic in three space dimensions). Ideally,weseekaCasimirenergythatreflectsonlythe The factthat the energy ofa fluctuating field diverges effects of the boundary conditions and not on any other when a boundary condition is imposed has been known features of σ(x). Therefore we do not specify any action for many years[7, 8]. Schemes have been proposed to for σ except for the standard counterterms induced by cancelthese divergencesby introducing new, ad hoc sur- the φ-σ interaction. The coefficients of the counterterms face dependent counterterms[9] or quantum boundary are fixed by renormalization conditions applied to per- functions[10]. We are not interested in such a formal turbative Green’s functions. Once the renormalization solution to the problem. The method of renormalization conditionshavebeenfixedbythedefinitionsofthephys- 2 ical parameters of the theory, there is no ambiguity and renormalization. Then we present two examples, leav- no further freedom to make subtractions. Moreover, the ing the details to Ref. [12]. We begin with the simplest renormalization conditions are independent of the par- Casimir problem: two Dirichlet points on a line, where ticular choice of background σ(x), so it makes sense to we can compare our results with standard calculations compare results for different choices of σ(x), ie. differ- that assume boundary conditions from the start [2]. We ent geometries. Having been fixed in perturbation the- find that the renormalizedCasimir energy is infinite but ory,thecountertermsarefixedonceandforallandmust theCasimirforceisfiniteintheDirichletlimit. Weshow serve to remove the divergences that arise for any phys- how the QFT approach resolves inconsistencies in the ically sensible σ(x). The Yukawa theory with coupling standard calculation. Next we study the Dirichlet circle g in three space dimensions gives a textbook example: in two dimensions. We demonstrate explicitly that the The gψ¯σψ coupling generates divergences in low order renormalized Casimir stress on the circle diverges in the Feynmandiagramsproportionaltoσ2,σ4 and(∂σ)2 and sharp limit. We show that the divergence is associated thereforerequiresonetointroduceamass,aquarticself- witha simple Feynmandiagramandwillpersistinthree coupling, and a kinetic term for σ. This is the only con- dimensions (the “Dirichlet sphere”) and beyond. textinwhichonecanstudythe fluctuationsofafermion We define the bare Casimir energy to be the vac- coupled to a scalar background in three dimensions. uum energy of a quantum field φ coupled to a back- In this Letter we study the vacuum fluctuations of a ground field σ by (φ,σ), minus the vacuum energy real scalar field φ coupled to a scalar background σ(x) Lint in the absence of σ. This quantity can be written as with coupling λ, (φ,σ) = 1λσ(x)φ2(x,t). In the limitwhereσ(x)bLecinotmesaδ-fun2ctiononsomesurface the sum over the shift in the zero-point energies of all the modes of φ relative to the trivial background σ = 0, S aφnmduwshtevraenλish→on∞,.itWiseecaasyllttohivsetrhifeyDthiraitchalleltmliomdiet.s oItf Ebare[σ] = ~2 n(ωn[σ]−ωn(0)). Equivalently, using the consists of the sharSp limit, where σ(x) gets concentrated effective actioPn formalism of QFT, Ebare[σ] is given by the sum of all 1-loop Feynman diagrams with at least on , followed by the strong coupling limit, λ . In S → ∞ one external σ field. The entire Lagrangianis general,wefindthatthedivergenceofthevacuumenergy inthe Dirichletlimit cannotbe renormalized. Generally, even the sharp limit does not lead to a finite Casimir = 1∂ φ∂µφ m2φ2 λφ2σ(x)+ [σ], (1) µ CT energy except in one dimension, where the sharp limit L 2 − 2 − 2 L exists but the Casimir energy diverges as λlnλ in the strong coupling limit. where [σ]is thecountertermLagrangianrequiredby CT L This divergenceindicates thatthe Casimir energy of a renormalization. Combining its contribution to the en- scalarfieldforcedtovanishonasurfaceinanydimension ergywithE [σ]yieldstherenormalizedenergyE [σ]. bare cas isinfinite. However,allisnotlost. Theunrenormalizable We have taken the dynamics of the background field divergences are localized on , so quantities that do not σ(x) to include only the φ-σ coupling and the countert- S probe are well defined. For example, it is straightfor- ermsrequiredbyrenormalizationtheory. Thefrequencies wardtoSshowthatthe vacuumenergy density awayfrom ω[σ] aredeterminedby 2φ(x)+(m2+λσ(x))φ(x) = is well defined in the Dirichlet limit, even though the {ω2[σ]φ}(x). This is a reno−rm∇alizable quantum field the- S energy density on diverges[13]. We expect that this is ory, so E [σ] will be finite for any smooth σ and fi- cas S true in general. The forces between rigid bodies are also nite λ. We use the method developed in Ref. [12] to finite in the Dirichletlimit. But any quantitywhose def- compute the Casimir energy of the background config- initionrequiresadeformationorchangeinareaof will uration exactly while still performing all the necessary S pick up an infinite contribution from the surface energy renormalizations in the perturbative sector. The inter- density and therefore diverge. For example, we will see ested reader should consult Ref. [14] for an introduction explicitlythatthevacuumcontributiontothestress ona to the method andRef. [15]for applications. We assume (generalized)Dirichletsphere in two or more dimensions that the background field σ(x) is sufficiently symmetric is infinite. to allowthe scatteringamplitude to be expandedinpar- The remainder of this Letter is organized as follows: tialwaves,whichwelabelbyℓ. Weexpresstherenormal- First we briefly review our computational method and ized Casimir energy as a sum overbound states ω plus ℓj discuss the structure of the counterterms required by an integral over continuum modes with ω =√k2+m2, ∞ ω dk d Ecas[σ]= Nℓ 2ℓj +Z 2πω(k)dk [δℓ(k)]N+EFND+ECT (2) Xℓ Xj 0   3 whereNℓdenotesthemultiplicity,δℓ thescatteringphase nite limit as σ δS(x) and λ and that the result shift and 1dδℓ the continuum density of states in the coincideswitht→hatfoundinbou→nd∞aryconditioncalcula- π dk ℓth partial wave. The subscript N on δ indicates that tions. We also find a finite and unambiguous expression ℓ the first N terms in the Born expansion of δ have been for the renormalized Casimir energy density where σ(x) ℓ subtracted. These subtractions are compensated exactly is nonzero, as long as it is nonsingular and the coupling by the contribution of the first N Feynman diagrams, strength is finite. But as we approach the sharp limit, EN = N E(i). In eq. (2) E is the contribution of the renormalized energy density on diverges, and this FD i=1 FD CT S the counPterterm Lagrangian, LCT. Both EFND and ECT divergence cannot be renormalized. depend on the ultraviolet cutoff 1/ǫ, but EN +E re- By analyzing the Feynman diagrams that contribute FD CT mains finite as ǫ 0. One can think of ǫ as the stan- to the effective energy we can deduce some general re- → dard regulator of dimensional regularization, although sults about possible divergences in the Casimir energy our methods are not wedded to any particular regular- and energy density in the sharp limit. In particular, the ization scheme. After subtraction, the k-integration in divergencesthatoccurintheCasimirenergyinthesharp eq. (2) converges and can be performed numerically for limit come from low-order Feynman diagrams. Specif- any choice of σ(x). It is convenient for computations to ically, using dimensional analysis it is possible to show rotate the integration contour to the imaginary k-axis that in n space dimensions the Feynman diagram with giving m external insertions of σ is finite in the sharp limit if m>n. ∞ dt t Although less sophisticated methods can be used to Ecas[σ]=Xℓ NℓmZ 2π √t2−m2 [βℓ(t)]N+EFND+ECT, orebntoarinmathliezaetnioenrgiysduennnseitcyesasatrpy,oianstsfaarwaasywfreomknSowwohnerlye (3) our method canbe used to define andstudy the Casimir where t = ik. The real function βℓ(t) is the log- energydensity where σ(x) is nonzeroandtherefore on arithm of th−e Jost function for imaginary momenta, in the Dirichlet limit. S β (t) lnF (it). Efficient methods to compute β (t) ℓ ℓ ℓ Consider, as a pedagogical example, a real, massive ≡ and its Born series can be found in Ref. [12]. The renor- scalarfieldφ(t,x)inonedimension,constrainedtovanish malized Casimir energy density for finite λ, ǫ (x), can cas at x = a and a. The standard approach, in which also be written as a Born subtracted integral along the the boun−dary conditions are imposed a priori, gives an imaginary k-axis plus contributions from counterterms energy [2] and low order Feynman diagrams [12]. In less than three dimensions only the lowest order m 2a ∞ √t2 m2 Feynman diagram diverges, so only a counterterm lin- E2(a)=− 2 − π Z dt e4at− 1 , (4) ear in σ is necessary, = c λσ(x). Since the tadpole m − LCT 1 e graph is also local, we can fix the coefficient c1 by re- where the tilde denotes the imposition of the Dirichlet quiring a complete cancellation, E(1) + E = 0. In boundary condition at the outset. From this expression FD CT three dimensions it is necessaryto subtract two terms in oneobtainsanattractiveforcebetweenthetwoDirichlet the Born expansion of β (t) and add back the two low- points, given by ℓ est order Feynman graphs explicitly. The counterterm Lagrangianmust be expanded to include a term propor- dE ∞ dt t2 2 tionaltoσ2,LCT =c1λσ(x)+c2λ22 σ2(x). Thenewterm F(a)=−d(2ea) =−Zm π √t2−m2(e4at−1). (5) cancelsthedivergencegeneratedbythevacuumpolariza- e tiondiagramE(2),butitdoesnotcompletelycancelE(2), In the massless limit, we have E (a) = π/48a and FD FD 2 − becauseE(2) itisnotsimplyproportionalto d3xσ2(x). F(a)= π/96a2. FD − e To fix c we can only demand that it cancelRs E(2) at a These results are not internally consistent, suggesting 2 FD e specified momentum scale p2 =M2. Different choices of that the calculation has been oversimplified: As a , →∞ M correspondtodifferentmodelsfortheself-interactions E2(a) m/2,indicatingthattheenergyofanisolated →− “Dirichlet point” is m/4. As a 0 we also have a ofσ andgiverisetofinitechangesintheCasimirenergy. e − → Weuseeq.(3)andtheanalogousexpressionfortheen- single Dirichlet point, but E2(a) as a 0. Also → ∞ → ergydensitywithbackgroundsthatarestronglylocalized notethatE (a)iswelldefinedasm 0,butweknowon 2 e → about , but not singular, to see how the Dirichlet limit generalgroundsthatscalarfieldtheorybecomesinfrared S e isapproached. ItisstraightforwardtorelatetheCasimir divergent in one dimension when m 0. energy density at the point x to the Green’s function at Westudythisproblembycoupling→φ(t,x)tothestatic x in the background σ, and then to show that it is fi- backgroundfieldσ(x)=δ(x+a)+δ(x a)withcoupling nite as long as σ(x)=0. Thus we find that the Casimir strengthλasineq.(1). Anelementrar−ycalculationgives energy density at any point away from goes to a fi- the renormalized Casimir energy for finite λ, S 4 ∞ dt 1 λ λ2 E (a,λ)= tln 1+ + (1 e−4at) λ (6) 2 Z 2π√t2 m2 (cid:26) (cid:20) t 4t2 − (cid:21)− (cid:27) m − The same method can be applied to an isolated point Note, however, that E (a,λ) diverges like λlogλ as 2 giving, λ . ThustherenormalizedCasimirenergyinasharp →∞ backgrounddivergesastheDirichletboundarycondition ∞ dt tln 1+ λ λ isimposed,aphysicaleffectwhichismissedifthebound- E (λ)= 2t − 2 (7) 1 Z 2π √(cid:2)t2 m(cid:3)2 ary condition is applied at the outset. m − For any finite coupling λ, the inconsistencies noted in E (a) do not afflict E (a,λ): As a , E (a,λ) The Casimir energy density for x = a can be calcu- 2 2 2 → ∞ → 6 ± 2E (λ), and as a 0, E (a,λ) E (2λ). Also E (a,λ) lated assuming Dirichlet boundary conditions from the e 1 → 2 → 1 2 diverges logarithmically in the limit m 0 as it should. startsimply by subtracting the density in the absence of → Theforce,obtainedbydifferentiatingeq.(6)withrespect boundarieswithoutencounteringanyfurtherdivergences to 2a, agrees with eq. (5) in the limit λ . [2], →∞ m ∞ dt√t2 m2 m2 ∞ cos nπ(x a) ǫ (x,a) = − a − for x <a 2 −8a −Z π e4at 1 − 4a ((cid:2)nπ)2+m2(cid:3) | | m − nX=1 2a m2 p ǫ (x,a) = K (2mx a) for x >a. (8) 2 0 −2π | − | | | The Casimir energy density for finite λ was computed let boundary condition, φ(a) = 0, is assumed from the in Ref. [12]. In the limit λ it agrees with eq. (8) start. As in one dimension, nothing can be said about → ∞ except at x = a where it contains an extra, singular the totalenergybecauseǫ˜(r) is notdefined atr=a, but ± contribution. Ifoneintegrateseq.(8)overallx,ignoring unlike that case the integralof ǫ˜(r) now divergeseven in thesingularitiesatx= a,oneobtainseq.(4). Including the sharp limit for finite λ. ± the contributions at a gives eq. (6). To understand the situation better, we take σ(x) to ± This simple example illustrates our principal results: be a narrow Gaußian of width w centered at r = a In the Dirichlet limit the renormalized Casimir energy and explore the sharp limit where w 0 and σ(x) → → divergesbecausetheenergydensityonthe“surface,”x= δ(r a). For w = 0, σ does not vanish at any value − 6 adiverges. Howeverthe Casimirforceandthe Casimir of r, so [G (r,r,it)] no longer falls exponentially at ± ℓ 0 energy density for all x= a remain finite and equal to large t, and subtraction of the first Born approxima- 6 ± theresultsobtainedbyimposingtheboundaryconditions tion to G (r,r,it) is necessary. As in one dimension, ℓ a priori, eqs. (5) and (8). the compensating tadpole graphcanbe canceledagainst A scalar field in two dimensions constrainedto vanish the counterterm, c λσ(x). The result is a renormal- 1 on a circle of radius a presents a more complex prob- ized Casimir energy density, ǫ(r,w,λ), and Casimir en- ∞ lem. We decompose the energy density in a shell of ergy,E(w,λ)= drǫ(r,w,λ), both of which are finite. 0 widthdr ataradiusr intoasumoverangularmomenta, However as w R 0 both ǫ(a,w,λ) and E(w,λ) diverge, ∞ → ǫ(r) = ǫ (r), where ǫ (r) can be written as an in- indicating that the renormalized Casimir energy of the ℓ=0 ℓ ℓ tegral oPver imaginary momentum t = ik of the partial Dirichlet circle is infinite. − waveGreen’sfunctionatcoincidentpointsGℓ(r,r;it)and The divergenceoriginatesinthe orderλ2 Feynmandi- itsradialderivatives. Firstsupposewefixσ(x)=δ(r a) agram. Westudythisdiagrambysubtractingthesecond − and consider r = a. It is easy to see that the difference Born approximation to G (r,r,it) and adding back the 6 ℓ [Gℓ(r,r,it)]0 betweenthefullGreen’sfunctionGℓ(r,r,it) equivalent diagram explicitly. Then the ℓ-sum and t- and the free Green’s function G(0)(r,r,it) vanishes expo- integralno longerdivergeinthe sharplimit. Inthe limit ℓ nentially as t . For finite λ, both the t-integral and w 0, the diagram contributes →∞ → the ℓ-sum are uniformly convergent so λ can be taken under the sum and integral. The res→ulti∞ng energy λ2a2 Λ p lim E(2) = dpJ2(ap) arctan (9) density, ǫ˜(r), agreeswiththatobtainedwhenthe Dirich- w→0 FD − 8 Z0 0 2m 5 whichdivergeslikelnΛ. Thedivergenceoriginatesinthe high momentum components in the Fourier transformof σ(r)=δ(r a),notinthehighenergybehaviorofaloop − [1] H. B. G. Casimir, Kon.Ned. Akad.Wetensch. Proc. 51, integral, and therefore cannot be renormalized. Taking 793 (1948). λ only makes the divergence worse. Because it [2] V.M. Mostepanenko and N.N. Trunov, The Casimir Ef- → ∞ varies with the radius of the circle, this divergence gives fectanditsApplication,ClarendonPress,Oxford(1997), aninfinitecontributiontothesurfacetension. Thisdiver- K. A. Milton, The Casimir Effect: Physical Manifesta- genceonlygetsworseinhigherdimensions(incontrastto tions Of Zero-Point Energy, River Edge, USA: World the claim of Ref. [16]). For example, for σ(r)=δ(r a) Scientific (2001). − [3] M.Bordag,U.MohideenandV.M.Mostepanenko,Phys. in three dimensions the renormalized two point function Rept. 353, 1 (2001) [arXiv:quant-ph/0106045]. is proportional to an integral over p of a function pro- [4] S. K. Lamoreaux, Phys. Rev.Lett. 78, 5 (1997). portional to λ2a4p2j02(pa)lnp at large p. The integral [5] U. Mohideen and A. Roy, Phys. Rev. Lett. 81, 4549 diverges like ΛlnΛ. Such divergences cancel when we (1998) [arXiv:physics/9805038]. compute the force between rigid bodies, but not in the [6] G. Bressi, G. Carugno, R.Onofrio, and G.Ruoso, Phys. case of stresses on isolated surfaces. Rev. Lett.88, 041804 (2002)[arXiv:quant-ph/0203002] In summary, by implementing a boundary condition [7] P.CandelasandD.DeutschPhys.Rev.D20(1979)3063. as the limit of a less singular background,we are able to [8] P. Candelas, Ann.Phys. 143 (1982) 3063. [9] K. Symanzik,Nucl. Phys. B190 (1981) 1. study the divergences that arise when a quantum field is [10] A. Actor, Ann. Phys. 230 (1994) 303, Fort. Phys. 43 forcedto vanishonaprescribedsurface. For allcaseswe (1995) 141. have studied, the renormalized Casimir energy, defined [11] G. Barton, J. Phys.A 34, 4083 (2001). in the usual sense of a continuum quantum field theory, [12] N. Graham, R. L. Jaffe, V. Khemani, M. Quandt, diverges in the Dirichlet limit. Physical cutoffs (like the M. Scandurra and H. Weigel, Nucl. Phys. B 645, 49 plasma frequency in a conductor) regulate these diver- (2002) [arXiv:hep-th/0207120]. gences, which are localized on the surface. On the other [13] Theenergydensityawayfrom S isinterestinginitsown right; since it can be negative, it may induce interest- handthe energydensity awayfromthe surfacesorquan- ing effects in general relativity: F. J. Tipler, Phys. Rev. tities like the force between rigid bodies, for which the Lett. 37, 879 (1976), S. W. Hawking, Phys. Rev. D 46, surfaces can be held fixed, are finite and independent of 603 (1992), L. H. Ford and T. A. Roman, Phys. Rev. the cutoffs. Observables that require a deformation or D 51, 4277 (1995), K. D. Olum, Phys. Rev. Lett. 81, change in area of cannot be defined independently of 3567 (1998), [arXiv:gr-qc/9805003]. K. D. Olum and N. S the other material stresses that characterize the system. Graham, [arXiv:gr-qc/0205134]. Similarstudies areunderwayforfluctuating fermionand [14] For a comprehensive review see: N.Graham, R.L. Jaffe andH.Weigel[arXiv:hep-th/0201148]inM.Bordag,ed., gauge fields, leading to Neumann and mixed boundary Proceedings of the Fifth Workshop on Quantum Field conditions with the same types of divergences. Theory Under the Influence of External Conditions,Intl. J. Mod. Phys. A 17 (2002) No. 6 & 7. Acknowledgments We gratefully acknowledge discus- [15] E. Farhi, N. Graham, P. Haagensen and R. L. Jaffe, sions with G. Barton, E. Farhi and K. D. Olum. N. G. Phys. Lett. B 427, 334 (1998) [arXiv:hep-th/9802015], and R. L. J. are supported in part by the U.S. De- E. Farhi, N. Graham, R.L. Jaffe, and H. Weigel, Phys. partment of Energy (D.O.E.) under cooperative re- Lett. B 475, 335 (2000) [arXiv:hep-th/9912283], Nucl. Phys. B 585, 443 (2000) [arXiv:hep-th/0003144], Nucl. search agreements #DE-FG03-91ER40662 and #DF- Phys. B 630, 241 (2002) [arXiv:hep-th/0112217]. FC02-94ER40818. M.Q.andH.W.aresupportedbythe [16] K. A. Milton, arXiv:hep-th/0210081. Deutsche Forschungsgemeinschaft under contracts Qu 137/1-1and We 1254/3-2.