Scalar Casimir effect between Dirichlet spheres or a plate and a sphere

0.34 MB·English

by  Aurel Bulgac

FZJ-IKP-TH-2005-28
Scalar Casimir effect between Dirichlet spheres or a plate and a sphere
Aurel Bulgac,1 Piotr Magierski,1,2 and Andreas Wirzba3
1Department of Physics, University of Washington, Seattle, WA 98195-1560, USA
2Faculty of Physics, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warsaw, Poland
3Institut fu¨r Kernphysik (Theorie), Forschungszentrum Ju¨lich, D-52425 Ju¨lich, Germany
(Dated: February 1, 2008)
We present a simple formalism for the evaluation of the Casimir energy for two spheres and a
sphere and a plane, in case of a scalar fluctuating field, valid at any separations. We compare the
exact results with various approximation schemes and establish when such schemes become useful.
Theformalism canbeeasily extendedtoanynumberofspheresand/orplanesinthreeorarbitrary
dimensions,withavarietyofboundaryconditionsornonoverlappingpotentials/nonidealreflectors.
PACSnumbers: 03.65.Sq,03.65.Nk,03.70.+k,21.10.Ma
6
0
0 I. INTRODUCTION forcecalculationforthetwo-spherecaseincorporatesthe
2 sphere-plate geometry as special case. Note that, with

n theexceptionofthenumericalresultsofRefs.[19,20](see
In1948theDutchphysicistHendrikCasimirpredicted
a also [21]), which do not yet extend to large separations,
the existence of a very peculiar effect, the attraction
J noexactresultexists inthe literature forthe Casimiref-
between two metallic uncharged parallel plates in vac-
3 fect for a real scalar field under Dirichlet boundary con-
uum [1]. The existence of such an attraction has been
1 ditions, neither between a sphere and a plate nor be-
confirmed experimentally with high accuracy only re-
tween two spheres. For small separations between the
2 cently [2, 3]. However, nearly all modern experiments
sphere and the plate, the above-mentioned proximity-
v (thenotedexceptionsarethetwo-cylinderworkofRef.[4]
force approximation [6, 7] can be applied. Its justifica-
6 and the two-plate experiment of Ref. [5]) study the at-
5 tionatsmallseparationshas beenprovidedbymany au-
traction between a metallic sphere and a metallic plate
0 thors using various theoretical techniques. For instance,
which are much simpler to align than two plates, but
1 in Refs.[17, 18] semiclassical methods in the framework
1 much harder to calculate. In fact, with the exception of of the Gutzwiller trace formula [26] have been used, in
5 the proximity-force approximation [6, 7], which is only
Ref.[27] the proximity-force approximation for the elec-
0 applicable for vanishing separation, there does not ex-
tromagneticcasehasbeenderivedfromthemultiplescat-
/ ist a theoretical prediction for the Casimir energy of the
h tering expansion of Ref.[28], in Refs.[19, 20] the world-
-t sphere-plate system as function of the distance. line approach in the framework of the Feynman path in-
p The origin of this attractive force can be traced back tegralhasbeenapplied,andinRefs.[22,23,24,25]aray-
e tothemodificationinthespectrumofzeropointfluctua- dynamicalapproachin terms ofoptical paths (i.e. closed
h
tionsoftheelectromagneticfieldwhentheseparatedmir- but not necessarily periodic orbits) has been employed.
:
v rors are brought into close distance. Similar phenomena Here, we will present an evaluation of the scalar
Xi areexpectedtoexistforvariousother(typicallybosonic) Casimir problem that is based on quantum mechanics,
fields [8, 9] and the corresponding forces are referred to without any semiclassical approximation made before-
r
a asCasimirorfluctuation-inducedinteractions. Arelated hand. It utilizes the Krein formula [29, 30] as a bridge
interaction arises when the space is filled with fermions, between the spectral density on the one hand and the
which is particularly relevant to the physics of neutron problemofscatteringofapointparticle betweenspheres
stars[10,11,12,13,14,15]andquarkgluonplasma[16]. in three dimensions (or disks in two dimensions) on the
Since the Casimir effect between a sphere and a plate other hand. It has to be emphasized that in this case
is the experimentally interesting but theoretically dif- the Casimir calculation is not plagued by the removal of
ficult case, because of the electromagnetic nature of diverging ultra-violet contributions. This is related to
the fluctuating fields, the corresponding Casimir ef- thefactthattheKreinformulaisexact,andthatthede-
fect for a real scalar field between two spheres or one terminantsoftheS-matrixandthecorrespondinginverse
sphere and a plate came into the focus of theoretical re- multi-scatteringmatrixforasystemofN nonoverlapping
search [17, 18, 19, 20, 21, 22, 23, 24, 25]. Therefore we spheres(or disks), which both aremanifestly known,are
willhere focus onthe exactandsemiclassicalcalculation finite [31, 32, 33]. In this work we do not consider the
of this scalar Casimir effect between spheres or spheres material dependent stress of, e.g., the deformation of a
and plates in three spatial dimensions, where we assume single spherical shell which was discussed in Ref.[34].
Dirichletboundaryconditions onthe spheresandplates. The paper is organized as follows: first, we formulate
ThisscenariocanbetriviallyextendedtothecaseofNeu- theproblemintermsofthedensityofstatesofthescalar
mann or other boundary conditions, to two-dimensional fieldandrelateittotheS-matrixofthesystemusingthe
systems of disks and/or lines and to analogous systems Krein formula. In Section III we focus on the particu-
in arbitrary dimensions. We will show that the Casimir lar realizationof the Casimireffect for the sphere-sphere
2
and the sphere-plate system. Section IV is devoted to the volume V of the entire space. This redundant diver-
investigationsofthelarge-distancelimitbetweenspheres gencecanbehandledeasilybyconsideringthesmoothed
where the asymptotic expressions are derived and dis- quantities first in a very big box, the volume of which is
cussed. Section V discusses the link between the pre- subsequently taken to infinity[31].
sented approachand the semiclassicalmethods based on Now we will use the Krein formula [29, 30] which
the Gutzwiller trace formula. Finally, the numerical re- provides a link between the (N-body) scattering matrix
sults, approximate expressions and conclusions are pre- S (ε)ofapoint-particlescatteringoffN spheresandthe
N
sented in Sections VI and VII. For the sakeof complete- change in the density of states due to the presence of N
ness,the derivationofproximity-forceapproximationfor scatterers, namely
thesphere-sphereandsphere-platesystemisgiveninAp-
pendix A and a comparison to the two-dimensional two- δg ε, a , ~r =g ε, a , ~r g (ε)
i ij i ij 0
{ } { } { } { } −
disk and disk-line systems can be found in Appendix B.
1 d
=(cid:0) lndet(cid:1)S ε(cid:0), a , ~r .(cid:1) (2)
N i ij
2πi dε { } { }
(cid:0) (cid:1)
II. THE MODIFIED KREIN FORMULA NotethatlndetS ε, a , ~r /2iisnothingelsethan
N i ij
{ } { }
the total phase shift of the scattering problem. The
(cid:0) (cid:1)
Our main goal is to reach a qualitative understand- geometry-dependentpartofthedensityofstatescannow
ing of the scalar Casimir energy at zero temperature be extracted from the genuine multi-scattering determi-
in the case of more complicated geometries than the nant. Inthiswaythecalculationismappedontoaquan-
original two-plate system. For that purpose let us con- tum mechanical billiard problem that classically corre-
siderthefluctuatingrealscalarfieldbetweenN nonover- sponds to the hyperbolic (or even chaotic)point-particle
lapping, nontouching, impenetrable spheres of radii a scattering off N spheres [33] (or N disks in two dimen-
i
(i=1,...,N). It is assumed that that the scalar field is sions [35, 36, 37, 38, 39, 40, 41, 42]).
noninteracting andis subject to Dirichlet boundary con- As shown in Refs.[31, 32, 33], the determinant of the
ditions on the surfaces of the spheres. The spheres are N-scatterer S-matrix, SN ε, ai , ~rij , factorizes into
{ } { }
positionedatfixedrelativedistancesr =L +a +a > the product of the determinants of the single-scatterer
ij ij i j (cid:0) (cid:1)
a +a betweentheircenters;L isthentheshortestrela- S-matrices and the ratio of the determinants of the in-
i j ij
tivedistancebetweentheirsurfaces,and~r isthecenter- verse multi-scattering matrix [31, 32, 33] of Korringa–
ij
to-center distance vector, which includes also the infor- Kohn–Rostoker (KKR) type [43, 44, 45, 46] M(k) =
mation about the spatial orientation. In order to calcu- M(k, a , ~r ) in the complex wave number (k = ~k )
i ij
{ } { } | |
late the Casimir energy we shall represent the smoothed plane:
bosonic density of states of the scalar field as a function
of the energy ε (smoothing is over an energy interval∆ε N detM(k∗)†
larger than the level spacing in the big volume V of the detSN ε, ai , ~rij = detS1(ε,ai) .
entire system): { } { } (i=1 ) detM(k)
(cid:0) (cid:1) Y (3)
N The formula (3) holds in the case when the scattering
g ε, a , ~r = g (ε)+ g (ε,a ) modesarefreemasslessfieldsaswellasinthecaseoffree
i ij 0 W i
{ } { } nonrelativistic fields with a mass m. Both cases imply
i=1
(cid:0) (cid:1) +g (ε,Xa , ~r ), (1) different energy dispersion relations, ε = ~ω = ~ck in
C { i} { ij} themasslesscaseorε=~2k2/(2m)inthenonrelativistic
where g ε, a , ~r is the total density of states of scenario, respectively.
i ij
the scalar fi{eld}, g{ (ε}) is the density of states in the ab- Although the involved matrices are infinite-
0
sence of(cid:0)all scatterers,(cid:1)and g (ε,a ) is the correction to dimensional, all determinants are well-defined, as
W i
the density of states arising from the presence of one long as the number of spheres is finite and the spheres
sphere (sphere i). Clearly N g (ε,a ) is the correc- do neither overlap nor touch. This follows from the
i=1 W i
tion due to the N spheres infinitely far apart from each trace-class property [47, 48] of the matrices SN 11,
other that sums up the excPluded volume effects, surface Si 11andM 11whichwasshowninRefs.[31,32,3−3].1
− −
contributions and Friedel oscillations caused by each of Inserting the exact expression (3) into the original
theobstaclesseparately. Finallyg ε, a , ~r ) isthe Kreinformula(2),usingthedecomposition(1)andiden-
C i ij
remaining part,whichis ofcentralinte{res}tt{o us}here. It tifying the “Weyl-type” density of states with the phase
(cid:0) (cid:1)
vanishesinthelimitofinfinitelyseparatedscatterersand
istheonlyterminthedensityofstateswhichreflectsthe
relative geometry-dependence of the problem. Only this
term contributes to the Casimir energy. 1 Trace-class operators (or matrices) are those, in general, non-
Hermitian operators (matrices) of a separable Hilbert space
Strictly speaking, only the smoothed level densities
which have an absolutely convergent trace in every orthonor-
gW(ε) and gC(ε) are finite, whereas the level densities malbasis. Especiallythedeterminantdet(11+zA)existsandis
g(ε) and g0(ε) are infinite, as they are proportional to anentirefunctionofz,ifAistrace-class.
3
shift of the corresponding single scatterer Using the explicit formulas for the KKR–type matrix
from Ref. [31, 32, 33], one can compute numerically
1 d
gW(ε,ai)= 2πi dεlndetS1(ε,ai), (4) EC for various arrangements of hard spherical (or circu-
lar in 2D) scatterers. For a point-particle scattering off
one finds a new Krein-type exact formula [13] which di- N nonoverlapping nontouching spheres (under Dirichlet
rectly links the geometry-dependent part of the density boundary conditions)
of states with the inverse multi-scattering matrix
g ε, a , ~r = d −1ImlndetM k(ε), a , ~r Mljmj′,l′m′ =δjj′δll′δmm′ +(1−δjj′)√4πi2m+l′−l
C i ij i ij
{ } { } dε π { } { } 2 j (ka )
(2l+1)(2l′+1) aj l j
or (cid:0) (cid:1) 1 (cid:0) (cid:1) ×p (cid:16)aj′(cid:17) h(l′1)(kaj′)
ε, a , ~r = ImlndetM k(ε), a , ~r ∞ l′
NC(cid:0) { i} { ij}(cid:1) −π (cid:0) { i} { ij}((cid:1)5) × Dlm′′,m′′(j,j′)h(l′1′)(krjj′)Ylm′′−m′′ rˆj(jj)′
fortheintegratedgeometry-dependentpartofthedensity lX′′=0m′X′=−l′ (cid:0) (cid:1)
of states √2l′′+1il′′ l′′ l′ l l′′ l′ l (9)
ε × 0 0 0 m m′′ m′′ m
(cid:18) (cid:19)(cid:18) − − (cid:19)
ε, a , ~r = dε′g ε′, a , ~r . (6)
C i ij C i ij
N { } { } { } { }
Z0 is the inverse multi-scattering matrix [33]. Here j,j′ =
(cid:0) (cid:1) (cid:0) (cid:1)
The exact formula (5) is the central expression of this 1,2,...,N are the labels of the N spheres, l,l′,l′′ =
paper. The Casimir energy itself follows via the integral 0,1,2,... aretheangularmomentumquantumnumbers,
and m,m′,m′′ the pertinent magnetic quantum num-
∞ ∞
EC = dε12εgC(ε)=−12 dεNC(ε) bers. Dlm′′,m′′(j,j′) is a Wigner rotation matrix which
Z0 Z0 transformsthe localcoordinate systemfrom sphere j′ to
1 ∞ (1)
= dεIm lndetM(k(ε)) the one of sphere j, jl(kr) and hl (kr) are the spher-
2π Z0 ical Bessel and Hankel functions of first kind respec-
= 1 ∞(1+i0+)dε lndetM(k(ε)) tively, Ylm′′−m′′ rˆj(jj)′ is a spherical harmonic (where rˆj(jj)′
is the unit vector, measured in the local frame of sphere
4πi
"Z0 j, pointing fr(cid:0)om (cid:1)sphere j to sphere j′), and the 3j-
∞(1−i0+) symbols [49] result from the angular momentum cou-
dε lndetM(k∗(ε))† . (7)
−Z0 # pling.
By definition, the inverse multi-scattering matrix in-
Inthemasslesscaseε=~ck,thisexpressioncanbeWick- corporates the pruning rule that two successive scatter-
rotated (i.e., k ik4 for the first term and k ik4 ings have to take place at different scatterers (see the
for the second te→rm of the last relation) to give 2→ − (1 δjj′) term in Eq.(9)). This alternating pattern
−
~c ∞ between a single-scatterer T-matrix and the successive
C = dk4 lndetM(ik4), (8) propagation to a new scatterer [31, 44, 45] distinguishes
E 2π
Z0 the KKR-type method from the multiple scattering ex-
since detM(k) = detM(( k)∗)† and therefore pansion of Refs.[27, 28].
detM(ik )=detM(ik )† if k re−al. 3
4 4 4
III. THE TWO-SPHERE AND THE
SPHERE-PLATE PROBLEM
2 TheWick-rotations areallowed,sincedetM(k)haspolesinthe
lowercomplexk-planeonly,whereasdetM(k∗)† haspolesinthe
upper half-plane [31, 32, 33]. Furthermore, the integrals over In the case of two spheres the system possesses a con-
thecirculararcsvanish,sincelndetM(k)andlndetM(k∗)† are tinuous symmetry, i.e. C in crystallography group
exponentiallysuppressedforImk→±∞,respectively;see,e.g., ∞v
theory notation [50], associated with rotations with re-
thesemiclassicalexpression(40).
3 For the same reason, one can show the corollary that all the specttotheaxisjoiningthecentersofthespheres(andan
correspondingintegralsoveroddpowersofk havetovanish: additionalreflectionsymmetrywithrespecttoanyplane
0 = ~c ∞dkk2n+1ImlndetM(k) containing this symmetry axis). As a consequence the
2πZ0 KKR-matrix is separable with respect to the magnetic
= i(−1)n4~πc Z0∞dk4k42n+1hlndetM(ik4)−lndetM(ik4)†i. q|mua|notnulmy)n[3u3m].bIenr tmhe(ignlofbaaclt,diotmdaeipne,nitdsisognivietns masodulus
Thus,theCasimirenergyovermodeswithanonrelativisticdis-
fipthneeritsfeieornumpεipoe=nric~in2Ctkea2gs/rim(a2timirone)ffilenicmtteigtsr,tauatdseiseedt.goi.nzetRrhoeef,.Fu[e1nr3lme,si1s5mt].hoemreenetxuimstsina (cid:18)MM1211 MM1222(cid:19)lm,l′m=′ δmm′ Al2δl1′ll(′m) Al1δl2′ll(′m)! (10)
4
with multi-scattering determinantdetMo| of the sphere-plate
system is just given as
Ajj′(m) =( 1)l′il′−l a2jjl(kaj) (2l+1)(2l′+1)
ll′ − a2j′h(l′1)(kaj′)p detMo| k,a,Lo| = detMoo k,a,r=2(Lo|+a) D .
× il′′(2l′′+1) l0′′ 0l l0′ l0′′ ml lm′ h(l′1′)(kr). Note that(cid:16)the shor(cid:17)test surface-t(cid:16)o-surface distance(cid:17)i(cid:12)(cid:12)(cid:12)n(t1h5e)
l′′ (cid:18) (cid:19)(cid:18) − (cid:19) symmetric two-sphere case is given by Loo = r 2a,
X −
(11) whereas the shortest surface-to-surface distance in the
sphere-plate case is Lo| = 1r a, see Fig.1.
2 −
If the spheres have moreover the same radius a1 = a2 The exact expressions for the Casimir energy of the
≡
a, there exist also a two-fold reflection symmetry with two-sphere Dirichlet problem, the symmetric two-sphere
respect to the vertical symmetry plane. This additional Dirichletproblemandthesphere-plateDirichletproblem
symmetry makes the total symmetry of the system to are given by the following integrals, respectively:
be D in the crystallography group notation [50] 4,
∞h
which is a simply product of the C∞v group and the ECo1o2(a1,a2,L)
inversion (and rotation by π) with respect to the point ~c ∞
= dk lndetMo1o2(ik ,a ,a ,r=L+a +a ),
of intersection between the symmetry axis and vertical 2π 4 4 1 2 1 2
symmetry plane. Z0 ~c ∞
Therefore the global domain of the two-sphere system oo(a,L)= dk lndetMoo(ik ,a,r=L+2a),
EC 2π 4 4
can be split into two half-domains, separated by this Z0
plane, see Fig.1, and all the (scattering) wave functions o|(a,L)
EC
can decomposedinto symmetric and antisymmetric ones ~c ∞
with respect to the vertical symmetry plane. The sym- = 2π dk4lndetMoo ik4,a,r=2(L+a) D.
metricwavefunctionsaresubjecttoNeumannboundary Z0
(cid:0) (cid:1)(cid:12) (16)
conditionsandtheantisymmetricaresubjecttoDirichlet (cid:12)
boundaryconditionsonthissymmetryplane. Thusthere
Inpractice,theseexpressionshavetobenumericallyinte-
exist two KKR matrices in the the half-domain [33], one grateduptoanuppervaluekmax whichshouldbechosen
corresponding to the Neumann case (N) and the other, 4
large enough, such that the numerical value of the inte-
withthe additionalminussign,totheDirichletcase(D):
gral is stable for some specified range of decimal places.
For the sphere-plate case, kmax 10/L specifies a good
Mlol′o(m) = δll′ +A(llm′), (12) choice. In order that the ev4alua∼tion of the determinant
Mlol′o(m)(cid:12)(cid:12)(cid:12)DN = δll′ −A(llm′), (13) oufppthere vmaalutreixofMkloli′on(cid:0)dmu;ckes,aa,rm=a2x(iLm+ala)v(cid:1)a(cid:12)lDueislmsatxabfloer, tl,hle′
(cid:12) and m (with m l,l′ lmax), namely(cid:12)[31, 46]
(cid:12) | |≤ ≤
FwuhretrheerAml(olm′r)e t≡he KAKl1l2′R((cid:12)md)(cid:12)eat1e=ram2≡inaant=deAtMl2l1′o(mo()k(cid:12)a,1a=,ar2)≡a. lmax ≥ 2ek4maxa≈14a/L. (17)
(cid:12) (cid:12) ≡
detMo1o2(k,a1=a,a2=a,(cid:12)r)ofthefulldomainf(cid:12)actorizes ForsmallvaluesoftheseparationL,themaximalangular
into the product of the determinants of these Neumann momentum and therefore the size of the KKR matrices
and Dirichlet KKR matrices (whichscalewithl2 l wherethe lastfactorresultsfrom
×
the separablem quantumnumber)becomesrapidlyvery
∞
detMoo(k,a,r)= detMoo(m)(k,a,r) large and limits the range of applicability of this numer-
ical computation of the exact integral to medium and
m=−∞
Y large values of L, say, L > 0.1a, chiefly because of the
∞
= detMoo(m)(k,a,r) detMoo(m)(k,a,r) handling of the 3j-symbols which scale with l4.
N D
m=Y−∞ (cid:12) (cid:12)
≡detMoo(k,a,r)|NdetMoo(cid:12)(cid:12)(k,a,r)|D . ((cid:12)(cid:12)14) IV. LARGE-DISTANCE LIMIT
Thus the two-sphere system contains the Dirichlet
sphere-plate system as special case, namely in the sym- As shown in Ref.[13], it is possible to obtain signifi-
metriclimita =a =a,thesphere-platesystemisequal cantly simpler expressionsfor the (integrated) density of
1 2
the Dirichlet case in the half-domain, and the pertinent states in the limit of very large separation or very small
scatterers. If the wave length λ = 2π/k is much larger
than the radii of the scatterers one can show that the
KKR–matrix M(k) is given by
4 Note that Fig.1 is rotated by 90 degree relative to the conven-
tionsofRef.[50],suchthatourverticalsymmetryplaneiscalled [M(k)]jj′ δjj′ (1 δjj′)f (k)exp(ikrjj′) (18)
j
“horizontal”there. ≈ − − rjj′
5
vertical symmetry plane
half−domain I half−domain II
oo
L
a a
Lo|
r
FIG.1: Twoidenticalspheresofradiusaatacenter-to-centerseparationr. Theverticalsymmetryplane,thetwohalf-domains
and the surface-surface separation Loo in theglobal domain and Lo| in one of thehalf-domains are shown.
(see[51]fortheanaloginthe2Dcase),wheretheindices we get the asymptotic result
j,j′ =1,...,N denote the scatterers, rjj′ is the distance
j (ka)
between their centers and fj(k) is the s-wave scattering Moo(m)(k,a,r) δ l1
amplitude on the j–th scatterer. l1l2 ∼ l1l2 ± h(1)(ka)
l2
In the case of two identical spheres of radius a, with
(2l +1)(2l +1) exp(ikr)
their centers locatedatthe distance r apart,one obtains 1 2 δ . (23)
× il1+l2 ikr m0
p
detM(k) = 1 a2 exp[ik(2r 2a)] (19) Since for kr 1 the only nontrivial matrix Moo(0) is
− r2 − separable in l≫and l , the corresponding determiln1la2nt is
1 2
= detM(k)|N detM(k)|D simply given by
a a
= 1+ eik(r−a) 1 eik(r−a) .(20)
r − r detMoo(k,a,r)
n on o ∞ 2
exp(2ikr) j (ka)
As usual, the determinant of this two-sphere system in 1 ( 1)l(2l+1) l
the global domain factorizes into two sub-determinants ∼ − (ikr)2 " − h(1)(ka)#
calculated for one of the half-domains, one subject to Xl=0 l
exp(2ikr)
Neumann boundary conditions and the other (with the 1 [ iA(ka)]2. (24)
≡ − (kr)2 −
minus sign) subject to Dirichlet boundary conditions.
Asmentioned,thisexpressioncanbederivedfromEqs. for the identical two-sphere case and by
(12-13) in the case of large center-to-center separations
of spheres, kr 1: we can make use of the asymptotic detMo|(k,a,L) detMoo k,a,r=2(L+a)
≫ ≡ D
expression for the spherical Hankel function [52] ∞
1 exp(ikr) ( 1)l(2l+(cid:0)1) jl(ka) (cid:1)(cid:12)(cid:12)
exp(ikr) ∼ − ikr − h(1)(ka)
h(1)(kr) , (21) Xl=0 l
l ∼ il+1kr exp(ikr)
1 ( iA(ka)) (25)
≡ − kr −
whichisactuallyexactforh(1)(kr), suchthattheinverse
0 for the sphere-plate case. Here
multi-scattering matrix becomes
∞
j (ka)
j (ka) (2l +1)(2l +1) A(ka) ( 1)l(2l+1) l (26)
Mlo1ol2(m)(k,a,r)∼δl1l2 ± h(l11)(ka) p 1il1+l2 2 ≡Xl=0 − h(l1)(ka)
l2
∞ is the multipole expansion of the scattering amplitude.
exp(ikr) l l l l l l
(2l+1) 1 2 1 2 . (22) For ka 1, the ratio j (ka)/h(1)(ka) becomes [52]
× ikr m m 0 0 0 0 ≪ l l
l=0 (cid:18) − (cid:19)(cid:18) (cid:19)
X
j (ka) i(ka)2l+1e−ika
l
=
Using now the orthogonality relation for the 3j– h(1)(ka) 12 32 (2l 1)2 (2l+1)
symbols [49] l × ×···× − ×
+ (ka)2l+2+δl,0 . (27)
O
∞
l=0(2l+1)(cid:18)ml11 ml22 ml(cid:19)(cid:18)ml1′1 ml2′2 ml (cid:19)=δm1m′1δm2m′2, Tinhguosntlhyeadnodmtinhaenlt>eff0e(cid:0)cttecrommsecsafnrobm(cid:1)etnheegsle-cwtaevde.sCcaotnteser--
X
6
quently, the scattering amplitude becomes The large-distance scaling is therefore proportional to
a/L2, in contrast to the Casimir-Polder energy between
j (ka)
iA(ka) = i 0 + (ka3) a molecule and a conducting plane which scales like
− − h(1)(ka) O a3/L4 [53]. The difference is associated with the fact
0
(cid:0) (cid:1)
= kaexp( ika)+ (ka3) , (28) that the Casimir-Polderenergy results from the induced
− O dipole moment, whereas in the scalar scattering the
which implies that (24) becomes (19) an(cid:0)d (25(cid:1)) becomes monopoletermgivesthe dominantcontribution. Infact,
the Dirichlet part of (20). if one omits the s-wave scattering term and starts in-
Ifthedeterminantintheglobaldomain(19)isinserted steadwiththethep-waveterminthescatteringfunction
into the modified Krein formula (5) we get the following A(ka),thescalarCasimirenergyforthesphere-platesys-
result for the integrated geometry-dependentpart of the tem would show a large-L behaviour
density of states in the s-wave limit of the two-sphere
case [13] (note r =Loo+2a):
πa3 90 1
Nso−owave(ε)= π(Looa+2 2a)2 sin[2k(Loo+a)]+O (ka)3 . Epo−| wave(a,L) = κ L4 (cid:18)π4(cid:19) 1+La 1+2aL 2 (35)
(cid:0) (2(cid:1)9) (cid:0) (cid:1)(cid:0) (cid:1)
Analogously one gets the s-wave limit for the Dirich- which is compatible with the a3/L4 scaling of the
let sphere-plate system by inserting the Dirichlet deter- Casimir-Polder energy. Thus the correct large-distance
minant of the half-domain, namely the second term of behaviour of the scalar Casimir energy has nothing to
Eq.(20), into (5): do with missing diffraction contributions (see Refs. [39,
40, 41, 42]) to the semiclassical trace formula as conjec-
a
o| (ε)= sin k(2Lo|+a) + (ka)3 tured in Ref. [17] and repeated in Ref. [23]. It is rather
Ns−wave 2π(Lo|+a) O
based on the replacement of the semiclassical summa-
h i (cid:0) (3(cid:1)0)
tion, whichwe will discuss in the next sectionandwhich
[note r=2(Lo|+a)].
isvalidwhenmanypartialamplitudescontribute,bythe
Now, the Casimir energy for two identical Dirichlet
leading term(s) in the multipole expansion [13, 51].
spheres in the large L = Loo limit follows simply by in-
serting oo , Eq.(29), into the integral (7) and per- In summary, the asymptotic expressions for the
formingNths−ewWavieck-rotation as in (8): Casimir energy are given by
a2
Eso−owave(a,L)=−~c4π(L+a)(L+2a)2 oo(L a) 4 90 κπa2 3.6958κπa2 (36)
EC ≫ ∼ × π4 L3 ≈ L3
πa2 90 4
=κ , (31)
L3 (cid:18)π4(cid:19) 1+ La 1+ 2La 2 inthe symmetric two-spherescalarDirichlet caseandby
where (cid:0) (cid:1)(cid:0) (cid:1)
~c π4 ~cπ2
κ=−16π2 90 =−1440 (32) o|(L a) 2 90 κπa 1.8479κπa (37)
EC ≫ ∼ × π4 L2 ≈ L2
is the prefactor of the Casimir energy
~cπ2 A in the sphere-plate scalar Dirichlet case.
||(L)= (33)
E −2 720 L3
×
of the corresponding scalar (Dirichlet) two-plate system
where A is the area of the plates. Instead of performing
theWickrotation,onecancomputetheseintegralsalong V. THE SEMICLASSICAL APPROXIMATION
therealaxistoo. Inthiscaseonewouldhavetoincludea
convergence factor exp( ηε) and take the limit η +0
− → The semiclassical approximation of the scalar two-
at the end of the calculation (in a similar manner to the
sphere problem in the frameworkof the Gutzwiller trace
Feynman prescription for propagators).
formula [26] was pioneeredin Refs.[17, 18]. Here we will
Similarly, the Casimir energy for the Dirichlet sphere-
focus on the link between the scattering approach and
plate case in the large L=Lo| limit follows by inserting
these semiclassical methods.
o| , Eq.(30), into the integral (7) and performing
Ns−wave The sphere-plate system at surface-to-surface separa-
the Wick-rotation:
tion L is a special case of the sphere-sphere case for two
a
o| (a,L)= ~c spheres of radii a and a at center-to-center separation
Es−wave − 4π(L+a)(2L+a) 1 2
R = L+ a + a in the limit a . As shown in
1 2 2
πa 90 2 Ref.[13], the integrated density of→sta∞tes for the two-
=κ . (34)
L2 π4 1+ a 1+ a spheresystemfollowssemiclassicallyfromtheGutzwiller
(cid:18) (cid:19) L 2L
(cid:0) (cid:1)(cid:0) (cid:1)
7
trace formula [26] If the semiclassicalexpression (38) is inserted into the
Casimir-energy integral (see [13])
o1o2(a ,a ,L,k)
Nsc 1 2 ∞ d
1 ∞ exp(ni2kL) 1 = 1~c dkk (k)
= Im Esc 2 dkNsc
π n=1 n |det([M(a1,a2,L)]n−11)|1/2 Z0 ∞
1 X∞ 1 exp(ni2kL) = −21~c dkNsc(k), (41)
= Im Z0
π n Λ (a ,a ,L)n+Λ (a ,a ,L)n 2
+ 1 2 − 1 2
nX=1 | − | one gets for the scalar sphere-sphere case, after a Wick
∞ 1 sin(n2kL) rotation (as in the transition from (7) to (8)): 6
= , (38)
nπ Λ (a ,a ,L)n+Λ (a ,a ,L)n 2
+ 1 2 − 1 2
n=1 | − | ∞
X o1o2(a ,a ,L)= 1~c dk o1o2(a ,a ,L,k)
Esc 1 2 −2 Nsc 1 2
where the periodic orbit is the bouncing-orbit between Z0
the spheres and the summation is over the repeats of ∞ 1 ∞dk exp( n2k L)
mthaistroixrboift.thTehsephmeraet-rsipxhMer(eas1y,sat2e,mL)anisdtΛh+e(ma1o,nao2d,rLo)m=y =−12~cnX=1nπ |Λ+(a1,Ra02,L)n4+Λ−−(a1,a42,L)n−2|
1/Λ−(a1,a2,L) is the double–degenerate leading eigen- ~c a1a2 ∞ 1
value of this matrix, i.e.: ≈−16π L2(a +a +L) n4
1 2
n=1
X
2 L 1 a2+a2
1 1 L2 1 (n2 1) L 1 2 (n2 1)
Λ±(a1,a2,L) = 1+2L(cid:18)a1+a2(cid:19)+2a1a2 ×(cid:20) − 3a1+a2 − − 3 a1a2(a1+a2) − (cid:21)
~c a a π4
1 1 L2 2 1 2 1
±s(cid:18)1+2L(cid:18)a1+a2(cid:19)+2a1a2(cid:19) −1, ≈−16π L2(a1+a2+L)(cid:18)90(cid:19)"
(39) 2L 5 1 L(a2+a2) 5 1
1 2 .
− a1+a2 (cid:18)π2−3(cid:19)− a1a2(a1+a2)(cid:18)π2−3(cid:19)#
which is identical with Eq.(3.11) of Ref.[17].
(42)
In fact, the semiclassical expression (38) is consistent
with the semiclassical limit to the exact expression (5):
Here we applied the following identity
In Ref.[33] it was argued for the two-sphere case and in
Ref.[31] it was shown for any N-disk case that semiclas- Λ (a ,a ,L)n+Λ (a ,a ,L)n 2
+ 1 2 − 1 2
sically −
L L L2 2 L2
=4n2 + + + (n2 1)
∞ 1 einklp−iνpπ/2 "a1 a2 a1a2 3a1a2 −
detM(k) exp ,
→ "−Xp nX=1n |det([Mp]n−11)|1/2# + 1 L2 + L2 (n2 1)+ L3 ,(43)
where l , M and ν are the total geometrical len(g4t0h), 3(cid:18)a21 a22(cid:19) − O(cid:18)a3i (cid:19)#
p p p
the monodromymatrixandthe Maslovindex ofthep-th
primitive periodic orbit, respectively. The r.h.s. of (40)
is the Gutzwiller-Voros zeta function [54]. Note that for suchthat
our scalar Dirichlet case there exists only one orbit, the
bouncing-orbitforthetwo-sphere(two-disk)systemwith Naos1yom2 ≈ −π1Imln 1− a41ra22 ei2k(r−a1−a2)
l = 2L, and that the Maslov index is simply ν = 4 h i
bpecause of the two Dirichlet reflections (per repeatp). 5 ≈ 4aπ1ar22 sin[2k(r−a1−a2)].
Thenextterminthe1/krexpansionoftheHankelfunction(21)
generates thecorrection
5 Fortheasymptoticcasekai≫1thescatteringamplitude(26)of Naos1yom2 ≈ 4aπ1ar22 1+ ar1 + ar2 sin[2k(r−a1−a2)]
ttahhneegupBlraeersvsimeoluoasmnsedenctHtuioamnnksseuilmmfpulbniyficteaisonunnin[d5te2erg]rtaahnle:dDtehbeyreeapplapcreomxiemntatoiofnthoef ≈ 4πr(r(cid:16)a−1aa21−a2)sin(cid:17)[2k(r−a1−a2)]
whichisthen=1termoftheGutzwillerformula(38),consistent
A(kai)= ∞(−1)l(2l+1) jl(kai) ≈ikaiexp(−i2kai). withtheasymptoticlimitL>ai≫1/k. ThisisEq.(13)of[13]
Xl=0 h(l1)(kai) 2 6 iTnhtehseeccaosnedal1in=eoaf2E.q.(42),afterthetrivialintegrationoverk4,
Thisimplies isidenticalwithEq.(3.20)ofRef.[17]ifthelatterisdividedbya
detMo1o2≈1− (eik2irk)r2A(ka1)A(ka2)≈1− a41ra22 ei2k(r−a1−a2), ofafc1toirnsotfetawdoo,fa1s/t2h.ezeromodesareweightedtherewithafactor
8
which is exact for the case n=1, and used where (compare with Eqs. (38) and (39))
∞ 1 π4 ∞ 1 π2
= and = . r r 2
n4 90 n2 6 Λ±(a,r) 1 1 1
nX=1 nX=1 ≡ a − ±r(cid:16)a − (cid:17) −
Particularly,in the case of two identical spheres of ra- Lo| Lo| 2
diusaatcenter-to-centerseparationR=L+2aonegets = 1+2 1+2 1
a ±s a −
a simplified expression (cid:18) (cid:19)
= lim Λ (a =a,a ,L=Lo|) (51)
± 1 2
oo(a,L) o1o2(a,a,L)= a2→∞
Esc ≡Esc
~c a2 π4 5 1 L L2
1 2 + under r =2(Lo|+a). Note that
−16πL2(2a+L) 90 − π2−3 a O a2
(cid:18) (cid:19)(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
πa2 5 1 L L2 Λ±(a,r)2 = Λ±(a1 =a,a2 =a,L=Loo) (52)
=κ 1 2 + . (44)
L2(2a+L) − π2−3 a O a2
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) for r =Loo+2a, such that for two identical spheres the
It is remarkable that the leading term is exactly equal integrated density of states is semiclassically
to the plate-based prediction of the proximity-force ap-
proximation for two identical spheres [17] (more details 1 ∞ 1 exp(in2(r 2a)k)
oo = Im − . (53)
in Appendix A): Nsc π n Λ (a,r)2n+Λ (a,r)2n 2
+ −
n=1 | − |
X
πa2
oo = κ . (45)
EplatePFA L2(2a+L)
VI. RESULTS
As mentioned above, the sphere-plate system is a spe-
cial case of the two-sphere system:
The results and approximations, discussed in the pre-
o|(a,L) lim o1o2(a,a ,L)= vious sections for the Casimir energy 0|(a,L) for the
Esc ≡a2→∞Esc 2 scalar Dirichlet sphere-plate case in unEitCs of κπa/L2 =
~c a π4 1 5 1 L + [L/a]2 −~cπ3a/(1440L2) are shown in Fig.2 as function of the
−16π L2 90 − π2 − 3 a O ratio L/a. This figure should be compared with Fig.8
(cid:18) (cid:19)(cid:20) (cid:18) (cid:19) (cid:21)
πa 5 1 L (cid:0) (cid:1) of the world-line approach of Ref.[19] and with Fig.4 of
=κ 1 + [L/a]2 . (46) the optical approachof Ref.[23] which both only present
L2 − π2 − 3 a O
(cid:20) (cid:18) (cid:19) (cid:21) data for L/a 4. The circles (a) represent the numeri-
(cid:0) (cid:1) ≤
This expression agrees with Eq.(11) of Ref. [55]. Note callycalculatedexactexpression(16)forthesphere-plate
that system between L=0.1a and L=512a (the line is only
πa there to guide the eye), the curve (b) shows the s-wave
o|(a,L)< κ . (47)
−Esc − L2 approximation (34), the line (c) represents the asymp-
totic limit 1.847of (37), the curve (d) represents the nu-
Moreover, the leading κπa/L2 behaviour of (46) agrees
mericallycalculated(Wick-rotated)integral(41)overthe
with the leading terms of the plate-based and sphere-
semiclassical expression (50) including all repeats, and
based proximity-force approximations for the Dirichlet
the line (e) shows the analytical semiclassical formula
sphere-plateproblem,respectively(seeRefs.[19,22,23])
(46) valid modulo (L/a)2 corrections. The curve (f)
O
πa 1 is the result of the plate-based proximity-force approx-
o| = κ , (48) (cid:0) (cid:1)
EplatePFA L2 1+L/a imation (48), and the curve (g) represents the result of
the sphere-based proximity-force approximation (49).
o| = κπa 1 3L 6 L 2 Our numerically calculateddata (a) agree for L/a 1
EspherePFA L2 ( − a − a with the in Ref.[19] published data of the world-line≤ap-
(cid:0) (cid:1) proach, within the quoted (statistical) error bars which
1 1+ L ln(1+a/L) . (49) are already sizable at L/a = 1. We have included these
× − a ) world-linedata,whichcovertherangefromL=a/256to
h (cid:0) (cid:1) i L=4a,inFig.2andmarkedthembystars(h). Notethat
Moredetails aboutthe proximity-forceapproximationof
their central values are systematically on the low side in
the Dirichlet sphere-plate system can be found in Ap-
comparisontoours. AllourdatabeyondL=4aarepre-
pendix A.
dictions. In the meantime, after the first version of this
The sphere-plate result (46) can also be derived from
paper wassubmitted, there appearedin [56]new data in
theDirichletpartoftheidentical-two-sphereresultusing
theworld-lineapproachwithimprovedsystematicswhich
o| = oo covertheextendedrangefromL=a/512toL=16aand
Nsc Nsc |D which have smaller, but still sizable error bars for L a
∞
1 1 exp[in(r 2a)k] ≥
= Im − (50) (see the crosses (k) in Fig.2). In the region of overlap,
π n Λ+(a,r)n+Λ−(a,r)n 2 thus now also for the points L/a = 2,4,8,16, the new
n=1 | − |
X
9
c
1.8 a
k
1.6
b
a2 1.4
3 L
π0
c4
~4
−1
/
1.2
)
L h
,
a
(
o|C 1
d
E
e
0.8
g f
0.001 0.01 0.1 1 10 100 1000
L/a
FIG.2: PredictionsforthescalarCasimirenergyE0|(a,L)ofthesphere-plateconfigurationwithDirichletboundaryconditions
C
are shown in unitsof κπa/L2 as function of theratio L/a. The points and curves are explained in thetext,see Section VI.
world-linedatadonicelyagreewithourswhenthequoted The semiclassical calculation, starts out at a value
statistical error bars are taken into account, although 1 (in units of κπa/L2) for L/a 0, which was pre-
→
their central values are now systematically higher. It dictedbytheproximity-forceapproximation(s)[6,7]and
shouldalsoberemarkedthatoursmallestpointL=0.1a confirmed in, e.g., Refs. [17, 18, 19, 22, 23]. Then,
isalreadyaffectedbyasizabletruncationerrorinthein- for intermediate values of L/a, however, the semiclas-
tegration. Ofcoursethisproblemisamatterofnumerics sical results, even though smaller than 1, are supe-
and not a matter of principle. rior to the results of the proximity-force approximation
Note that the s-wave approximation becomes a rea- which become ambiguous [19, 22, 23]. For L a, the
≫
sonable approximation to the exact data from L 4a contributions of the repeats of the bouncing-orbit are
≥
(it works very nicely for L > 15a in agreement with the stronglysuppressedandthenumericallycalculatedsemi-
estimate (17)) and, moreover, that the exact data in- classical expression converges to the one-bounce result
deed converge to the predicted asymptotic value 180/π4 ~ca/(16πL2) = (90/π4) κπa/L2 which is smaller by
− ×
of Eq.(37) and do not show any Casimir-Polder a3/L4 a factor 1/2 than the exact asymptotic answer (37).
scaling. Itisalsointerestingthattheexactdata,atleast
for the L values for which they could be calculated, are
larger than 1 (in units of κπa/L2), whereas the semi- VII. CONCLUSIONS
classical approximation and the proximity-force approx-
imations are strictly less than 1 (in the same units). It We presented here an exact calculation of the scalar
should be noted that (at least) the upper error ranges Casimir energy for the case of two spheres and a sphere
of the old world-line data of Ref. [19], the complete new and a plane. It is based on a new Krein-type formula
world-line data of Ref. [56] (with the exception of the which directly expresses the geometry-dependentpartof
lower error ranges of the points L/a = 1/64,1/128) and the density of states by the inverse multi-scattering ma-
– below L 0.1a – the results of the improved optical trix of the pertinent scatteringproblem. Thus the corre-
∼
approach of Ref. [23] are larger than 1 as well. spondingCasimirenergyfollowsfromtheenergy-integral
10
overthemulti-scatteringphaseshift(thelogarithmofthe FG03-97ER41014,from the Polish Committee for Scien-
multi-scattering determinant). The calculation is there- tific Research(KBN) under ContractNo. 1P03B05927
forenotplaguedbysubtractionsofthesingle-spherecon- andbytheForschungszentrumJu¨lichundercontractNo.
tributionsorbyaremovalofdivergingultra-violetcontri- 41445400(COSY-067) is gratefully acknowledged.
butions. Theasymptoticlimit(37),thepresenteds-wave
approximation (34) and all data with L>4a are totally
new results. Moreover, contrary to claims in the liter-
ature, the Casimir-Polder scaling of the scalar Casimir
effect is excluded by our numerical and analytical calcu-
APPENDIX A: AMBIGUITY IN THE
lations. PROXIMITY-FORCE APPROXIMATION
The two-sphere and sphere-plane cases are only two
examples, and the formalism presented can be easily ex-
tended to any number of spheres and planes as well (or The proximity-force approximation (PFA) for the
disks and lines in two-dimensions). We have exemplified Casimirenergy Coftwoarbitrarysmoothsurfaces(with
E
the calculation of the scalar Casimir energy only for the Dirichletboundaryconditionsfor thescalar-fieldcase)is
case of Dirichlet boundary conditions. One can replace givenby thesurfaceintegralovertheCasimirenergyper
theDirichletwithNeumannboundaryconditionsorwith areawhichbelongstoanequivalentparallel-platesystem
anyotherconditionseasily,orevenreplacethescatterers that locally follows the two surfaces [6, 7, 19, 22, 23] ,
witharbitrarynonoverlappingpotentials/nonidealreflec- i.e.
tors.
Asidefromtheexactresultswehavealsodiscussedsev-
eral approximation schemes, the large separation limit, PFA = dσ ǫ[z(σ)]. (A1)
E
the semiclassical limit and the proximity-force approxi- ZZA
mation. The exact results (which are easy to calculate
andaredefinitelysimplertoevaluatethaninapathinte- Here A is the area of one of the opposing surfaces which
gralapproach)shouldbelookeduponastestexamplesfor arelocallyseparatedbythe(surface-dependent)distance
other approximate methods. These results already show z(σ)andǫ[z(σ)]isthecorrespondingCasimirenergyper
that the proximity formula and the semiclassical/orbit area. In general, the plate segment dσ is tangential to
approaches are limited to small separations only, typi- only one of the surfaces and therefore the local distance
cally much smaller than the curvature radii of the two vector~z(σ) is perpendicular onlyto this surfaceandnot
surfaces. to the other one. Thus is not uniquely defined,
PFA
E
One can make the argument that the dominant mo- since the area A can be either one of the two opposing
menta of the fluctuating fields contributing to the surfaces (or even one ficticious surface somewhere inbe-
CasimirenergyataseparationLareoftheorderk 1/L tween). Particularly, for the case of a sphere of radius
≈
(see Eq.(17)) and thus for large separations only the s- a and a plate at shortest surface-to-surface separation
wave scattering is important. This is the main reason L we get the following expression for the “sphere-based
whythesemiclassicalapproximation(whichisvalidwhen PFA”[19, 22, 23],where the localdistance vector~z(~a)is
many partial amplitudes contribute) fails at large sepa- perpendicular to the sphere,
rations.
Just the opposite is true at small separations, where
1
semiclassics is pretty good and so is the proximity for- o| = κ a2dΩ(2)
mula, when the separations are smaller by one or, re- EspherePFA ZZhalf−sphere |~z(~a)|3
spectively, two orders of magnitude than the curvature π/2 [cos(θ)]3
radii. The same type of analysis can be straightfor- = κa22π dθ sin(θ)
wardly extended to cylinders, or even objects with less Z0 [L+a−acos(θ)]3
sTy-mmmatertircyes(ainppweahriicnhgcinastehethinevceorsreremspuolntid-sincagttienrdinivgidmuaa-l = κπa 1 3L 6 L 2
L2 ( − a − a
trix (see Ref.[31, 44, 45]) become somewhat more com-
(cid:0) (cid:1)
plicated, due to the loss of spherical symmetry).
1 1+ L ln(1+a/L) . (A2)
× − a )
h (cid:0) (cid:1) i
Acknowledgments
The coefficient κ is againthe prefactor κ= ~cπ2 of the
A.W. thanks Holger Gies and Antonello Scardicchio Casimirenergy ||(L)= ~cπ2 A ofthe co−rr1e4s4p0onding
for useful discussions at QFEXT’05. We are grateful to E −2×720 L3
scalar (Dirichlet) two-plate system.
HolgerGiesforsupplyinguswiththenewworld-linedata
[56] prior to publication and for helpful comments. Sup- Onthe otherhand, the “plate-basedPFA” [19,22, 23]
port from the Department of Energy under grant DE- (i.e., the local distance vector ~z(σ) is perpendicular to