The proximity force approximation for the Casimir energy as a derivative expansion
aa r X i v : . [ h e p - t h ] N ov The proximity force approximation for the Casimir energy as a derivative expansion
C´esar D. Fosco , , Fernando C. Lombardo , and Francisco D. Mazzitelli , Centro At´omico Bariloche, Comisi´on Nacional de Energ´ıa At´omica, R8402AGP Bariloche, Argentina Instituto Balseiro, Universidad Nacional de Cuyo, R8402AGP Bariloche, Argentina and Departamento de F´ısica Juan Jos´e Giambiagi, FCEyN UBA,Facultad de Ciencias Exactas y Naturales, Ciudad Universitaria,Pabell´on I, 1428 Buenos Aires, Argentina - IFIBA (Dated: today)The proximity force approximation (PFA) has been widely used as a tool to evaluate the Casimirforce between smooth objects at small distances. In spite of being intuitively easy to grasp, it isgenerally believed to be an uncontrolled approximation. Indeed, its validity has only been tested inparticular examples, by confronting its predictions with the next to leading order (NTLO) correctionextracted from numerical or analytical solutions obtained without using the PFA. In this article weshow that the PFA and its NTLO correction may be derived within a single framework, as the firsttwo terms in a derivative expansion. To that effect, we consider the Casimir energy for a vacuumscalar field with Dirichlet conditions on a smooth curved surface described by a function ψ in front ofa plane. By regarding the Casimir energy as a functional of ψ , we show that the PFA is the leadingterm in a derivative expansion of this functional. We also obtain the general form of correspondingNTLO correction, which involves two derivatives of ψ . We show, by evaluating this correction termfor particular geometries, that it properly reproduces the known corrections to PFA obtained fromexact evaluations of the energy. I. INTRODUCTION
In the last years, there have been important theoreticaland experimental advances in the analysis of the Casimireffect [1].Until the recent development of theoretical methodsthat allowed for the exact evaluation of the Casimir en-ergy for several geometries, the interaction between dif-ferent bodies has been mostly computed using the socalled proximity force approximation (PFA) [2]. Thisapproximation, expected to be reliable as long as the in-teracting surfaces are smooth, almost parallel, and veryclose, makes use of Casimir’s expression for the energyper unit area for two parallel plates at a distance a apart.For the case of a single massless scalar field and Dirichletconditions (the case we deal with in this paper) it is givenby: E pp ( a ) = − π a . (1)The PFA then approximates the interaction between twoDirichlet surfaces separated by a gap of spatially varyingwidth z , as follows: E PFA = Z Σ dσ E pp ( z ) , (2)where Σ is one of the two surfaces. Quite obviously, thisformula does not take into account the non-parallelismof the surfaces. Moreover, the result may depend on theparticular surface Σ chosen to perform the integral.As the PFA was believed to be an uncontrolled approx-imation, its accuracy has been assessed only in some ofthe particular geometries where it was possible to com-pute the Casimir energy numerically or analytically. On general grounds, denoting by L a typical length associ-ated to the curvature of one of the surfaces (assumedmuch smaller than the curvature of the second one) andby a the minimum distance between surfaces, one expectsthat: E C = E PFA (cid:26) γ a L + O (cid:20)(cid:16) a L (cid:17) (cid:21)(cid:27) , (3)where γ is a constant, whose numerical value fixes the ac-curacy of the PFA in each particular geometry (the situa-tion could be more complex, since the corrections to PFAmay contain non-analytic corrections as (cid:0) a L (cid:1) n log (cid:0) a L (cid:1) ).One can write similar expressions for geometries that in-volve two surfaces of similar curvature.In this paper we explore the following simple idea.The Casimir energy can be thought as a functional ofthe shape of the surfaces of the interacting bodies. Asthe PFA should be adequate for almost plane surfaces,a derivative expansion [3] of this functional should re-produce, to lowest order, the PFA. Moreover, the termsinvolving derivatives of the functions that describe theshape of the surfaces should contain the corrections tothe PFA. We will show that this is indeed the case, andthat it is possible to find a general formula to computethe first corrections to PFA for rather arbitrary surfaces.Just to avoid some technical complications, we considera massless scalar field in the presence of a curved surfacein front of a plane. We will assume that the quantum fieldsatisfies Dirichlet boundary conditions on both surfaces.Generalizations to other boundary conditions and to theelectromagnetic field will be analyzed in a forthcomingwork.This paper is organized as follows. In Section II, wedescribe the model and derive a formal expression for theCasimir energy in the geometry described above. Then,in Section III, we perform a derivative expansion in theexpression for the Casimir energy to obtain the main re-sult of the paper: a general formula for the interactionenergy between an arbitrary curved surface and a plane,containing up to two derivatives of the function ψ thatdescribes the curved surface. The leading term of theexpansion corresponds to the PFA, while the term withderivatives is the first non trivial correction.In Section IV we present some examples: a sphere, acylinder, or a corrugated surface in front of a plane. Weshow, by comparing with existing analytical results, thatthe derivative expansion of the Casimir energy describescorrectly both the PFA and its first correction for all ofthese geometries. We also compute the derivative ex-pansion of the Casimir energy for geometries involvingparabolic mirrors. Section V contains the conclusions ofour work. II. FORMAL EXPRESSION FOR THE VACUUMENERGY
We shall consider a model consisting of a massless realscalar field ϕ in 3 + 1 dimensions, coupled to two mirrorswhich impose Dirichlet boundary conditions. In our Eu-clidean conventions, we use x , x , x , x to denote thespacetime coordinates, x being the imaginary time.The mirrors occupy two surfaces, denoted by L and R , defined by the equations x = 0 and x = ψ ( x , x ),respectively.Following the functional approach to the Casimir ef-fect, we introduce Z , which may be interpreted as thezero temperature limit of a partition function, for thescalar field in the presence of the two mirrors. It may bewritten as follows: Z = Z D ϕ δ L ( ϕ ) δ R ( ϕ ) e − S ( ϕ ) , (4)where S is the free real scalar field Euclidean action S ( ϕ ) = 12 Z d x ( ∂ϕ ) , (5)while δ L ( δ R ) imposes Dirichlet boundary conditions onthe L ( R ) surface.Exponentiating the two delta functions by introduc-ing two auxiliary fields, λ L and λ R , we obtain for Z anequivalent expression: Z = Z D ϕ D λ L D λ R e − S ( ϕ ; λ L ,λ R ) , (6)with S ( ϕ ; λ L , λ R ) = S ( ϕ ) (7) − i Z d xϕ ( x ) (cid:2) λ L ( x k ) δ ( x ) + λ R ( x k ) δ ( x − ψ ( x k )) (cid:3) where we have introduced the notations x k ≡ ( x , x , x )and x k ≡ ( x , x ). Integrating out ϕ , we see that Z , corresponding tothe field ϕ in the absence of boundary conditions factorsout, while the rest becomes an integral over the auxiliaryfields: Z = Z Z D λ L D λ R e − R d x k R d y k P α,β λ α ( x k ) T αβ λ β ( y k ) , (8)where α, β = L, R and we have introduced the objects: T LL ( x k , y k ) = h x k , | ( − ∂ ) − | y k , i (9) T LR ( x k , y k ) = h x k , | ( − ∂ ) − | y k , ψ ( y k ) i (10) T RL ( x k , y k ) = h x k , ψ ( x k ) | ( − ∂ ) − | y k , i (11) T RR ( x k , y k ) = h x k , ψ ( x k ) | ( − ∂ ) − | y k , ψ ( y k ) i (12)where we use a “bra-ket” notation to denote matrix ele-ments of operators, and ∂ is the four-dimensional Lapla-cian. Thus, for example, h x | ( − ∂ ) − | y i = Z d k (2 π ) e ik · ( x − y ) k . (13)The vacuum energy of the system, E vac , subtracting thezero-point energy of the free field (contained in Z ), is: E vac = lim T →∞ (cid:0) Γ T (cid:1) = 12 T Tr log T , (14)where T is the extent of the time dimension (or β − , inthe thermal partition function setting), Γ ≡ − log ZZ andthe trace is meant to act on both discrete and continuousindices.Note that E vac still contains ‘self-energy’ contributions,due to the vacuum distortion produced by each mirror,even when the other is infinitely far apart. This piece(irrelevant to the force between mirrors) shall be sub-tracted, in order to obtain a finite Casimir energy, in thecalculations below. III. DERIVATIVE EXPANSION
We present here a derivation of the first two terms in aderivative expansion of the Casimir energy for the systemdefined in the previous section.To that end, and for calculational purposes, it is con-venient to consider first a simplified situation: we split ψ into two components, ψ ( x k ) = a + η ( x k ) , (15)where a (assumed to be greater than zero), is the spatialaverage of ψ , and therefore a constant, whereas η containsthe varying piece of ψ . The simplified case amounts toexpanding up to the second order in η . Since the deriva-tives of ψ equal the derivatives of η , to find the termswith up to two derivatives of ψ , it is sufficient to expandΓ up to the second order in η , keeping up to the secondorder term in an expansion in derivatives:Γ( a, η ) = Γ (0) ( a ) + Γ (1) ( a, η ) + Γ (2) ( a, η ) + . . . (16)where the index denotes the order in derivatives. Eachterm will be a certain coefficient times the spatial integralover x k of a local term, depending on a and derivativesof η .So far this is a perturbative expansion in η and itsderivatives. However, to the same order in derivatives, itis quite straightforward to include the corrections whichare of the same order in derivatives but of arbitrary orderin η . Indeed, to do this, in the terms obtained in (16),one just has to replace a by ψ and also η by ψ , beforeperforming the spatial integrals. This procedure accountsfor all the terms of higher order in η , and the same orderin derivatives, that contribute to the respective order thederivative expansion. Formally, this procedure may berepresented as follows:Γ ( l ) ( ψ ) = Γ ( l ) ( a, η ) (cid:12)(cid:12) a → ψ,η → ψ (17)for each term in (16).Let us calculate the different terms in the derivativeexpansion for Γ, following this procedure.Expanding first the matrix T in powers of η T = T (0) + T (1) + T (2) + . . . , (18)we obtain Γ = Γ (0) + Γ (1) + Γ (2) + . . . , whereΓ (0) = 12 Tr log T (0) Γ (1) = 12 Tr log h ( T (0) ) − T (1) i Γ (2) = 12 Tr log h ( T (0) ) − T (2) i −
14 Tr log h ( T (0) ) − T (1) ( T (0) ) − T (1) i , (19)where, in Γ ( l ) , we need to keep up to l derivatives of η .The zeroth-order term is thus simply obtained by re-placing first ψ by a constant, a , and then subtracting thecontribution corresponding to a → ∞ , to get rid of thedivergent self-energies. This yields,Γ (0) ( a ) = 12 Tr log (cid:2) − ( T (0) LL ) − T (0) LR ( T (0) RR ) − T (0) RL (cid:3) (20)where the T (0) αβ elements are identical to the ones on wouldhave for the two flat parallel mirrors at a distance a apart.As mentioned above, we have then to replace a by ψ atthe end of the calculation. After evaluating the trace, weobtain:Γ (0) = T Z d x k Z d k k (2 π ) log[1 − e − k k a ] . (21)We then replace a → ψ to extract the zeroth orderCasimir energy, E (0)vac = 12 Z d x k Z d k k (2 π ) log[1 − e − k k ψ ( x k ) ]= − π Z d x k ψ ( x k ) , (22) which equals the PFA approximation to the vacuum en-ergy.The first order term in the derivative expansion Γ van-ishes identically, while for the second order one we havetwo contributions:Γ (2) = Γ (2 , + Γ (2 , (23)where, Γ (2 , = 12 Tr log h ( T (0) ) − T (2) i (24)andΓ (2 , = −
14 Tr log h ( T (0) ) − T (1) ( T (0) ) − T (1) i , (25)where we have to keep up to two derivatives of η .The form of those terms can be obtained in a quitestraightforward fashion; indeed, we first note that, inFourier space, and before expanding to second order inmomentum (derivatives), they have the structure:Γ (2 ,j ) = T Z d k (2 π ) f (2 ,j ) ( k ) | ˜ η ( k ) | (26)( j = 1 , k = ( k , k ), ˜ η is the Fourier transformof η , and the f (2 ,j ) kernels are the k → f (2 , ( k ) = − Z d p (2 π ) | p | | p + k | − e − | p + k | a f (2 , ( k ) = − Z d p (2 π ) | p || p + k | e − | p + k | a (1 + e − | p | a )(1 − e − | p | a )(1 − e − | p + k | a ) . Besides, we need to subtract an a -independent self-energycontribution, obtained by taking a → ∞ in the expres-sions above. Putting together the two terms above, andsubtracting the a → ∞ limit, the total contribution toΓ (2) adopts the form:Γ (2) = T Z d k (2 π ) f (2) ( k ) | ˜ η ( k ) | (27)with: f (2) ( k ) = − Z d p (2 π ) | p | | p + k | (1 − e − | p | a )( e | p + k | a −
1) (28)where we just need to extract its k term in a Taylorexpansion at zero momentum. Namely f (2) ( k ) ≃ χ k ,where χ = 12 (cid:2) ∂ f (2) ( k ) ∂k (cid:3) k → = − Z d p (2 π ) | p | (1 − e − | p | a ) lim k → ∂ ∂k h | p + k | ( e | p + k | a − i . The resulting integral may be exactly calculated, χ = − π a . (29)Thus, Γ (2) ( a, η ) = − T π Z d k (2 π ) k a | ˜ η ( k ) | = − T π Z d x k a ( ∂ α η ) , (30)where, to obtain the second order contribution in deriva-tives to the vacuum energy, we need to replace a → ψ , η → ψ , and cancel the T factor, obtaining: E (2)vac = Γ (2) ( ψ ) T = − π Z d x k ( ∂ α ψ ) ψ , (31)where the index α runs from 1 to 2.Putting together the terms up to second order, theexpression for the energy becomes: E DE ≡ E (0)vac + E (2)vac = − π Z d x k ψ (cid:20) ∂ α ψ ) (cid:21) . (32)This is the main result of this paper. The first term is thePFA for the Casimir energy. The second term containsthe first non-trivial correction to PFA for an arbitrarysurface. We could have guessed the form of both terms inthe final formula by using dimensional and symmetry ar-guments. The global factor could also be determined byconsidering the particular case of parallel plates. There-fore, the calculation presented above, besides confirmingthe general arguments, provides the relative weight be-tween both terms, which turns out to be 2 /
3, regardlessof the form of the surface.
IV. EXAMPLES
We provide here some applications of the general for-mula for the Casimir interaction energy.
A. A corrugated surface in front of a plane
Let us first consider a corrugated surface in front of aplane. For simplicity we assume sinusoidal corrugationsin the direction of x ψ ( x ) = a + ǫ sin (cid:18) πx λ (cid:19) , (33)where a is the mean distance to the flat surface, ǫ is theamplitude, and λ the wavelength of the corrugation. Weassume a square plane of side L , which is much largerthan any other length in the problem.The derivative expansion for the Casimir energy isgiven by E DE = − π "Z d x k a + ǫ sin πx λ ) × (cid:18) πλ (cid:19) ǫ cos πx λ ! . (34) In this case, the derivative expansion is an expansion inpowers of a/λ and ǫ/λ , i.e. λ is largest relevant distancein the problem. In order to compare with previous resultsin the literature [4], we will further assume that ǫ ≪ a .In this limit we obtain E DE ≃ − π L a (cid:20) (cid:16) ǫa (cid:17) + 4 π (cid:16) ǫλ (cid:17) (cid:21) . (35)This expression coincides with the small a/λ expansionof the result obtained in Ref.[4]. Indeed, in that work theinteraction energy was written as E vac L = − π a − ǫ a G TM (cid:16) aλ (cid:17) , (36)where G TM ( x ) can be written in terms of Polylogarithmfunctions [5]. One can readily compute the small argu-ment expansion of G T M to obtain G TM ( x ) ≃ π
480 + π x . (37)After inserting this expansion into Eq. (36), the resultcoincides with the derivative expansion Eq. (35). B. A sphere in front of a plane
We now consider a sphere of radius R at a distance a from a plane. The evaluation of the Casimir energy inthe electromagnetic case for this configuration has beenperformed in Refs. [6, 7], while the evaluation for scalarfields has been previously reported in Ref. [8]. See also[9, 10] for asymptotic expansions in the scalar and elec-tromagnetic cases near the proximity limit.For this geometry, we expect the derivative expansionto be adequate in the limit a ≪ R . It is worth notingthat the surface of the sphere cannot be described by asingle valued function x = ψ ( x , x ). Note that even ifwe consider an hemisphere, the derivatives of ψ will bedivergent on the equator. For these reasons, the deriva-tive expansion will not converge. In spite of this, we willsee that it still gives quantitative adequate results evenbeyond the lowest order approximation.In order to avoid these problems, we will consider onlythe region of the sphere which is closer to the plane. Thisis the usual approach when computing the Casimir en-ergy using the PFA. The final result will not depend onthe part of the sphere considered. Denoting by ( ρ, ϕ )the polar coordinates in the ( x , x ) plane the function ψ reads ψ ( ρ ) = a + R − r − ρ R ! . (38)This function describes an hemisphere when 0 ≤ ρ ≤ R . As mentioned above, the derivative expansion will bewell defined if we restrict the integrations to the region0 ≤ ρ ≤ ρ M < R .Inserting this expression for ψ into the derivative ex-pansion for the Casimir energy, one can perform explic-itly the integrations and obtain an analytic expression E DE ( ρ M , a, R ). We do not present this rather long ex-pression here, but only the leading terms in an expansionin powers of a/R , which is given by E (0)vac ≃ − π Ra h − aR i (39) E (2)vac ≃ − π a , (40)and therefore E DE ≃ − π Ra (cid:18) aR (cid:19) . (41)It noteworthy that, up to this order, the result does notdepend on ρ M . Moreover, the result is in agreement withthe asymptotic expansion obtained from the exact for-mula for this configuration [9], and with the former nu-merical evaluation in [11].It is interesting to remark that E (0)vac includes part ofthe next to leading order corrections. It is correct to keepthe second term in Eq. (39) only when the contributioncoming from E (2)vac is also taken into account. C. A cylinder in front of a plane
Let us now consider a cylinder of radius R and length L ≫ R at a distance a from a plane. The Casimir energyfor this configuration was first evaluated in the PFA inRef.[12]. The exact result was first derived in Ref.[13].The caveats mentioned in the above subsection also applyfor this geometry. We will consider the function ψ givenby ψ ( x ) = a + R − r − x R ! , (42)with − x M < x < x M < R in order to cover the part ofthe cylinder which is closer to the plane. The calculationis similar to the previous case, and the final result is E DE ≃ − π L √ R / a / (cid:18) aR (cid:19) . (43)Once more, up to this order, the result does not dependon x M . Moreover, it is in agreement with the asymp-totic expansion obtained from the exact formula for thecylinder-plane geometry and numerical findings [14–17]. D. A parabolic cylinder in front of a plane
We compute here the Casimir interaction energy be-tween a parabolic cylinder of length L in front of a plane.The surface is defined by the function ψ ( x ) = a + x R , (44) with − x M < x < x M < R . Once more, we only con-sider the portion of the curved surface which is closer tothe plane (note that the functions defining the cylinderEq.(42) and the parabolic cylinder Eq.(44) coincide up tofirst order in x /R ). The integrations needed to computethe derivative expansion of the Casimir energy are verysimple. Expanding the result in powers of a/R we obtain E DE ≃ − π L √ R / a / (cid:18) aR (cid:19) . (45)The final answer is independent of x M and the leadingorder coincides with that of the cylinder in front of aplane. E. A paraboloid in front of a plane
As a final example we consider a paraboloid, definedby ψ ( ρ ) = a + ρ R , (46)with 0 < ρ < ρ M < R , in front of a plane.The approximation for the vacuum energy reads E DE ≃ − π Ra (cid:18) aR (cid:19) . (47)As in all the previous examples, the result does not de-pend on the region of integration defined by ρ M . More-over, the leading order is equal to that of the sphere infront of a plane, as expected from the fact that the func-tions describing both surfaces Eqs. (38) and (46) coincidein the region closer to the plane. V. CONCLUSIONS
We have shown that the PFA can be thought of as akinto a derivative expansion of the Casimir energy with re-spect to the shape of the surfaces. Our main result, givenin Eq. (32), shows that the lowest order (the “effectivepotential”) reproduces the PFA. Moreover, when the firstnon trivial correction containing two derivatives of ψ isalso included, the general formula gives the NTLO cor-rection to PFA for a general surface.Several remarks are in order: to begin with, at least forthe surfaces considered in this paper, the PFA becomesa well defined and controlled approximation scheme: theleading corrections are small when | ∂ α ψ | ≪ ψ . Moreover,the derivatives of ψ diverge when the surface becomesperpendicular to the plane, and therefore it is clear thatthe derivative expansion will not converge. In spite ofthis, it is remarkable that Eq. (32) describes the inter-action energy for these configurations including the firstnon trivial correction to PFA. Strictly speaking, for thesegeometries we are computing the interaction energy be-tween a plane and a large curved surface which, in theregion closest to the plane, has a cylindrical or sphericalshape.We expect the main idea presented in this paper to begeneralizable in several directions, as for instance for ascalar field satisfying Neumann or Robin boundary con-ditions, and also to the electromagnetic field satisfyingperfect conductor boundary conditions on the surfaces. In all these cases, we expect the derivative expansion tobe of the form E DE = − π Z d x k ψ (cid:2) β + β ( ∂ α ψ ) (cid:3) , (48)where the constants β i will depend on the kind of fieldsand boundary conditions considered.Other interesting generalizations would be to considertwo curved surfaces, and the case of imperfect boundaryconditions. Moreover, as the applications of the PFAare not restricted to the Casimir energy, the derivativeexpansion could also be useful to compute gravitational[18], electrostatic [19] or even nuclear forces [20]. ACKNOWLEDGEMENTS
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