Casimir force between surfaces close to each other
H. Ahmedov1 and I. H. Duru1
arXiv:hepth/0207186v3 9 Feb 2003
1. Feza Gursey Institute, P.O. Box 6, 81220, C ? en
gelk¨ oy, Istanbul, Turkey 1 .
Abstract Casimir interactions ( due to the massless scalar ?eld ?uctuations ) of two surfaces which are close to each other are studied. After a brief general presentation of the technique, explicit calculations are performed for speci?c geometries.
I. Introduction
Experiments to observe and measure Casimir forces have so far been performed with the geometrical setups involving two ( actually disconnected ) surfaces [1]. The original parallel plate Casimir interaction is exact for in?nite plane surfaces [2], which in practice means that valid for planes very close to each other. E?ect for the parallel plane geometry were ?rst veri?ed in 1958 [3]. Recently the experiment for this geometry was improved to a high precision [4]. The Casimir experiments other than the above mentioned ones have been performed for a sphere close to a plane con?guration [5]: which do not give rise the precise alignment problem of the parallel planes. Note that the calculation for the sphereplane geometry gets closed to be exact if the radius of the sphere is small compared to the distance to the plane [6]. Spheresphere geometry has also been studied subject to the similar approximation as the sphereplane problem [7]. The interaction of two cocentric spheres has recently been addressed [8]. Single cavity experiments so far have not been realized [8], which we think would be very interesting: For example inserting the data from quantum
1
Email : hagi@gursey.gov.tr and duru@gursey.gov.tr
1
dots ( i.e. radius ? 10?7 cm ) into the theoretical expression for the vacuum energy of a spherical cavity capable of con?ning electromagnetic ?eld [1], one gets ( in h ? = c = 1 units ) 0.5 · 106 cm?1 = 10eV for the Casimir energy which is of appreciable magnitude [9]. This is comparable to the total energy π2 between the parallel plates of the latest experiment [4], i.e. E = 720 (Area d3 π2 ?1 6 ?1 of Plates)? 720(5·10?3 )3 · (2 · 2)cm ? 10 cm . The purpose of the present work is to study new two surfaces geometries. We calculate the Casimir forces resulting from the vacuum ?uctuations of massless scalar ?elds between surfaces close to each other. For massive ?elds, for any realistic experimental setups the Casimir energies are extremely small. The expressions always involve a factor e??△ , where ? is the mass and △ is the separation; which for electron and for nanometer distances 10 ?7 is e?2,5·10 ·10 ? e?2500 ; thus it is practically zero. After a brief outline of our approach to the surface surface interactions in Section 2, we proceed with speci?c examples, that is coaxial cylinders, cocentric tori, cocentric spheres and coaxial conical surfaces.The case of coaxial cylinders my o?er an experimental test in the light of the recent advances in stable metal nanotubes [10].
II. Casimir energy for the region between two boundaries which are close to each other
We ?rst choose the suitable spatial curvilinear coordinates η j , j = 1, 2, 3 for the geometry we deal with. The corresponding Minkowski metric and the KleinGordon operator are then ds2 = dt2 ? gij dη idη j and ( g ≡ det(gij ) ) △= The Green function is G=
λ1 ,λ2 ,λ3
(1)
?2 1 ? ij √ ? ?√ g g j. 2 ?t g ?η i ?η
(2)
eiω(λ)(t?t ) Φω(λ) (η )Φω(λ) (η ′ ), 2 ω (λ ) 2
′
(3)
where Φω(λ) (η ) and ω 2 (λ) are the eigenfunctions and eigenvalues of the equation for the massless scalar ?eld 1 ? ij √ ? ?√ g g j Φω(λ) (η ) = ω 2 (λ)Φω(λ) (η ). g ?η i ?η (4)
( For massive scalar ?eld one only changes ω 2 by ω 2 + ?2 ; with ? being the mass ). We assume that the above equation is separable in the spatial coordinates η j . Here η and λ stand for the collection of the coordinates η j and the corresponding quantum numbers λj ( which are speci?ed by the boundary conditions ) respectively. The functions Φω(λ) (η ) are normalized with respect to the norm Φ
2
=
A
√ d3 η g Φ(η )2 ,
(5)
where A is the domain of the coordinates η j . The vacuum energy density can be then obtained by calculating the coincidence limit derivatives as: ?2 ?2 ij +g )G(η, η ′)]. T = Reg [ jlim ( ′ i ′ j ′ ′ j t,η →t ,η ?t?t ?η ?η (6)
”Reg” stands for regularization. In the speci?c examples it means that we have to subtract the terms ( in the Plana sum formulas to be employed over the modes ) corresponding to the vacuum energy of the free space, the boundary energy etc. To calculate the Casimir energy one needs the eigenvalues of the problem. The eigenvalues ω 2 (λ) depend on three quantum numbers λj corresponding to the degrees of freedoms in directions η j in which we assume that the equation (4) can be separated. We further assume that after the separation of variables the eigenvalue equations in coordinates η 1 , η 2 can be trivially solved, and the corresponding quantum numbers λ1 , λ2 are easily obtained. This assumption does not introduce a strong restriction. In fact many problems in the literature are of that type. For example when one studies the Casimir energy inside a spherical cavity, only nontrivial problem is the radial equation in which one has to deal with the roots of the Bessel functions to impose the boundary condition [1]. In this work we employ an approximation method to calculate the nontrivial spectral parameter λ3 , which is valid if the problem involves two boundaries in direction η 3 , which are close to each other.
3
After the separation, the problem in hand in η3 can be converted into the Schr¨ odinger form [? d2 + Wλ1 λ2 (η3 )]Φλ3 (η 3 ) = E (λ)Φλ3 (η 3 ). d (η 3 )2 (7)
The form of the ”potential” Wλ1 λ2 (η3 ) and the relation between ω 2 (λ) and E (λ) depend on the choice of coordinate systems. The explicit examples are given in the following sections. The boundary conditions we wish to impose for the type of geometries under investigations are
3 Φλ3 (η0 ) = 0, 3 Φλ3 (η1 ) = 0,
(8)
3 3 where η0 < η1 . In practice these boundary conditions require dealing with the roots of special functions which are quite involved. However if the boundaries are close to each other, instead of (7) we can employ the simpler Schr¨ odinger equation d2 3 0 0 3 [? + Vλ01 λ2 (η3 )]Φ0 (9) λ3 (η ) = E (λ)Φλ3 (η ) d (η 3 )2
where the constant potential in the region is given by Vλ01 λ2 (η3 ) =
3 3 ∞, η 3 = η0 , η 3 = η1 3 3 3 3 η1 ), η 3 ∈ (η0 , η1 ) Wλ1 λ2 ( η0
(10)
The eigenvalue equation (9) has the following solutions E 0 (λ ) = ( and
3 Φ0 λ3 (η ) =
πλ3 2 0 ) + Wλ 1 λ2 △ 2 πλ3 sin( ), △ △
(11)
(12)
3 3 where △ = η1 ? η0 and λ3 = 1, 2, . . .. The system given by (9) is a good approximation if the condition 0 max Wλ1 λ2 (η3 ) ? Wλ  ? min E 0 (λ) 1 λ2 λ3
3 ,η 3 ) η3 ∈(η0 1
(13)
is satis?ed. 4
In the following sections we apply this approximation method to the speci?c geometries.
III. Casimir energy in the region between two close coaxial cylinders.
In the cylindrical coordinates, i.e. with the metric ds2 = dt2 ? dz 2 ? dr 2 ? r 2 dφ2 the eigenvalue problem we have to solve is ?[ 1 ? ? 1 ?2 ?2 r + 2 2 + 2 ]Φ = ω 2 Φ r ?r ?r r ?φ ?z eipz +imφ √ vnm (r ). 2π r (15) (14)
After solving for the trivial coordinates z and φ we have Φ= (16)
p Here vnm (r ) are the normalized wavefunctions corresponding to the radial equation d2 m2 ? 1/4 [? 2 + ]vnm = ?2 (17) nm vnm , 2 dr r with (18) ωpnm = p2 + ?2 nm .
The quantum number n should be determined from the boundary conditions on the coaxial cylinders with the radii r0 < r1 : vnm (r0 ) = 0, vnm (r1 ) = 0. (19)
The solution of (17) satisfying the boundary condition at r0 is given in terms of the Bessel functions as vnm (r ) = √ ?nm r Jm (?nm r0 )Nm (?nm r ) ? Jm (?nm r )Nm (?nm r0 ) , ?nm
r1 r0
(20)
where ?nm to be obtained from the normalization drr vnm(r )2 = 1. 5 (21)
In practice however the above integral is very di?cult to calculate for arbitrary values of r0 and r1 . The spectrum ?nm should be determined from the boundary condition at r1 which is quite involved equation. However if the cylindrical surfaces are close to each other we can rely on the approximation method summarized in the previous section. Instead of the eigenvalue problem (17) we consider the following one d2 0 2 0 + V (r )]vnm = (?0 nm ) vnm 2 dr with the constant potential [? V (r ) =
? ? ?
(22)
∞ , r = r0 , r = r1
m2 ? 1 4 r0 r1
, r ∈ (r 0 , r 1 )
(23)
The above equation is then trivially solved as
0 vnm =
2 sin(?0 nm (r ? r0 )); △
△ ≡ r1 ? r0
(24)
with the spectrum ?0 nm = π 2 n2 m2 ? 1 4 ; n = 1, 2 , 3 , . . . + △2 r0 r1 (25)
For the present speci?c case the condition (13) is valid for △ ? r0 . The Green function of the system is then easy to deal with: G= where
0 ωpnm = 2 p2 + (?0 nm ) . ∞ ∞ ∞
n=1 m=?∞ ?∞
dp
eiωpnm (t?t )+ip(z ?z )+im(φ?φ ) 0 0 vnm (r )vnm (r ′ ) 8π 2 ω pnm
0
′
′
′
(26)
(27)
We insert the above Green function into the coincidence limit formula 1 1 T = Reg [ (28) lim (?t ?t′ + ?r ?r′ + ?z ?z ′ + 2 ?φ ?φ′ )G], ′ ′ ′ ′ 2 t,r,z,φ→t ,r ,z ,φ r where ”Reg” stands for the usual regularization which will be de?ned explicitly. The total vacuum energy per unit height is E=
2π 0
dφ
r1 r0
1 rdrT = 2
∞ ?∞
∞ dp ∞ 0 ωpnm ]. Reg [ 2π m=?∞ n=1
(29)
6
To perform summations we use the Plana formula [1]:
∞ n=0
F (n) =
F (0) + 2
∞ 0
dnF (n) + i
∞ 0
dt
F (it) ? F (?it) . e2πt ? 1
(30)
In the application of the above formula to the summation over the radial quantum number n, the n = 0 term and the second term, corresponding to the surface singularity and free space divergence respectively should be subtracted. Thus the regularization stands for (in n summation ) Reg [
∞
F (n)] =
∞ n=1
F (n) +
n=1
F (0) ? 2
∞ 0
dnF (n).
(31)
Going back to (29) we ?rst perform the sum over m. Note that since the argument of the square root in (29) is always positive we can replace it by the absolute value E = + 1 2
∞ ?∞ ∞
dp 2π
∞ ?∞
dm Reg
∞
∞ n=1 ∞ m=1
 p2 + ( 
dp Reg Reg ?∞ 2π n=0 1 = 2πRE (△, ) + E1 , 2R where R2 = r0 r1 . Here E (△ , 1 1 )= 2R 2
∞ ?∞
πn 1 m2 + p 2 + ( )2 ? = 2 R △ 4R 2 (32)
πn 2 m2 ? 1/4 ) + + △ R2
? → ∞ d2 k Reg [ (2π )2 n=1
1 πn ? →  ]  k 2 + ( )2 ? △ 4R 2
(33)
? → ? → with k being 2dim. vector k = (p, m ); and, R E1 =
∞ ?∞ ∞ ∞ dp Reg [ Reg [ 2π m=1 n=1

πn 1 m2 + p 2 + ( )2 ?  ]]. 2 R △ 4R 2
(34)
It is obvious that employment of Reg ∞ m=1 does not mean that there is an actual regularization in m summation. It simply imply the usage of (31); for as it will be seen below that the ?rst summation on the right hand side of that formula is exactly calculable. 7
To evaluate E1 , we ?rst employ (31) in n summation. The last term, i. e., the 0∞ dn integral term, becomes formally the same as E of (33). Thus we can write 1 E1 = ?2△E (πR, ) + E2 + E3 . (35) 2R The terms E2 and E3 which come from the ?rst and second terms of (31) are 1 E2 = 2 and E3 =
∞ ?∞ ∞ ?∞ ∞ n=1 ∞ dp Reg [ 2π m=1

1 m2 + p2 ? ] 2 R 4R 2
(36)
dp 2π
Reg [
∞
m=0

πn 1 m2 + p 2 + ( )2 ?  ]. 2 R △ 4R 2
(37)
To evaluate E2 , we write (31) as the last term of the right hand side of (30); then after suitable change of variables we arrive at √ 2 √ 2 ∞ 1 ∞ ∞ y ?1 y +1 x3 dx x3 dx 1 √ √ + ) E2 = ? 2 2 ( dy dy πxy πxy 8π R 0 e ?1 e ?1 0 1 + x2 1 1 ? x2 1 (38) We can easily estimate the upper limit of the above integrals. The ?rst one 1 is smaller than 7200 , while the second is smaller than 1 . Thus 6 E2  < 1 . 48π 2 R2 (39)
To evaluate E3 , since △ ? R, we can ?rst approximate it as: E3 ? ? ? ?
∞ ?∞ ∞ ?∞
dp ∞ π n=1 dp ∞ π n=1
∞ ?∞
∞ R ∞ R p2 + π
2 n2 △2
p2 + π
2 n2 △2
dm
m2 R2
e2πm ? 1
? p2 ?
π 2 n2 △2
?
dme?2πm
m2 π 2 n2 2? ? p = R2 △2 (40)
1 = ? 2 2π
πn 2 2 dp ∞ ?K0 (2πRs p + ( △ ) ) . 2π n=1 ?R
We use the formula [11]
∞ 0
√ π ?xb e dλK0 (x λ2 + b2 ) = 2x 8
(41)
for the integration over p, then perform the summation over n. Finally we have 2R 1 E3 ? ? (42) e?2π △ . 4πR△ which is negligible small. To calculate the main contribution (33) and the ?rst term of of (35) we apply (30) and (31): E (a, ?) = ? = ?
? ∞ ?∞ ∞ ?
? → d2 k (2π )2 dkk 2π
π
√ a
∞
k 2 ??2 
dn 2 πn e ?1
π 2 n2 ? k 2 + ?2 = a2 + ?2
π
√ a
∞
dn
k 2 ??2
dn πan ? k 2 + ?2 dkk ∞ 2 ? = √ a 2π π e2πn ? 1 ?2 ?k 2 0 ∞ 2?a y 3 dy √ π2 1 = ? dx 1 + x2 ? 3 2 3 yx 1440a 32π a 0 e ? 1 1 For (33) i. e., with a = △ and ? = E (△ ,
1 , 2R
2 2
π 2 n2 ? k2 a2 e2πn ? 1
(43)
since △ ? R we have (44)
2
1 π2 1 )?? ? 3 2R 1440△ 192R2△
we see that its contribution is 10?2 △ Inspecting (43), for a = πR, ? = 21 R R2 times the second term in the above expression: thus it is also negligible. The ?nal result for the Casimir energy between the close cylinders is then: E=? 15 △2 π3R (1 + ). 720△3 2π 2 R 2 (45)
Note that the inclusion of the second term in the above expression does not contradict our approximation of (23), for the contribution of the ?rst term 3 . It is easy after this approximation in the potential would be of the order △ R3 R to check that in △ → ∞ limit the above result becomes same as parallel plate energy. Finally we like to remark that, for oneboundary geometries, for example for D dimensional ball there are satisfactory techniques to deal with the problem involving the roots of Bessel functions [12]. We may hope that 9
these techniques may also be adopted for geometries with twoboundaries. For boundaries close to each other however, we can rely on the result of (45), R for it gives the correct limit of parallel plates in △ → ∞ limit.
IV. Casimir energy in the region between two tori
Problem di?ers from the previous one by the boundary condition. Instead of (19), the solution of the evalue equation (15) should satisfy Φr=r0 = Φr=r1 = 0, Φz =0 = Φz =L (46)
where L is the circumference of the tori. For △ ? r0 we have Φ= and ωknm = ( m πn 1 2πk 2 ) + ( )2 + ( )2 ? ; L R △ 4R 2 (48) ei
2πkz +imφ h
√ 2π r
2 πn sin( (r ? r0 )) △ △
(47)
where k, m ∈ Z and n = 1, 2, 3, . . .; and R2 = r0 r1 as in the previous section. The total energy between the close tori is then, after employment of Plana formulas E = +
∞ ∞ ∞ 1 ∞ ωknm ] = Reg [ 2 m=?∞ m=?∞ k =?∞ n=1 ∞ ∞ ∞ 0
dk Reg [
∞
ωknm ] (49)
n=1
Reg [
∞
ωknm]
m=?∞ n=1
k =1
Note that unlike the previous case, since the degree of freedom along the tori ( i. e. along zcoordinate ) is also restricted, we have to perform regularizations for the k summation too. The ?rst term on the right hand side of the above equation is exactly the Casimir energy for the coaxial cylinders considered in the previous section. Thus, we rewrite (49) as E = LEc + E ′ (50)
10
In a fashion parallel to the evaluation of (40), taking the advantage of R ? △ ′ we can evaluate Reg ∞ k =1 in E : E′ = ? From
∞ m=?∞
1 ? ∞ ∞ m2 π 2 n2 + ) K0 (2L 8π ?L n=1 m=?∞ R2 △2 m2 π 2 n2 + )+ R2 △2
m2 R2
(51)
K0 (2L
m2 π 2 n2 + ) = 2 R2 △2 + 2π
∞ 0
dmK0 (2L dmJ0 (2L
∞
πRn △
+
e2πm ? 1
πRn △ √
π 2 n2 ) △2
,
(52)
using the approximation
∞ m=?∞
1 e2πm ?1
? e?2πm for m ≥ ) = πR( e
L n ?2π △
we arrive at [11]
π 2 R2 +L2 △
K0 (2L
m2 R2
+
π 2 n2 △2
2L
e + √
?2πn
π 2 R2 + L2
).
(53)
Thus we have E′ ? ? 1 1 R ? 1 1 √ ) ( +√ 2 2 L 2 R2 +L2 π 2 π 2 8 ?L 2L e △ ? 1 π R + L e2π △ ? 1/ (54)
Since L > R ? △ this contribution is negligible small in compared to LEc .
V. Coaxial cylindrical boxes of ?nite height.
Instead of (46), the solution of the evalue equation (15) should satisfy Φr=r0 = Φr=r1 = 0, Φz =0 = Φz =L = 0 (55)
with L being the height of the cylinders. For △ ? r0 we have Φ= sin( πk z )eimφ L √ π rπ 2 πn sin( (r ? r0 )) L△ △ 11 (56)
and ωknm = ( πk 2 m πn 1 ) + ( )2 + ( )2 ? ; L R △ 4R 2 (57)
where m ∈ Z and n, k = 1, 2, 3, . . .; and R2 = r0 r1 as in the previous section. The total energy between the close cylinders is then E = ?
∞ ∞ 1 ∞ 1 ∞ Reg [ ωknm ] = 2 m=?∞ 2 m=?∞ k =1 m=?∞ ∞ 0
dk Reg [
∞
ωknm] (58)
n=1
∞ ∞ 1 ∞ ∞ 1 ∞ Reg [ ωknm ] Reg [ ω0nm ] + 4 m=?∞ 2 m=?∞ n=1 n=1 k =1
The ?rst term on the right hand side of the above equality is equal to LEc , where Ec is the Casimir energy for the coaxial cylinders geometry given L by (45). The third term for △ ≥ 1 can be explicitly calculated using its similarity with (51) of the previous section. Namely we have to multiply (51) by 1 and make a change L → 2L. Using (54) we arrive at 2 E2 ? ? 1 1 R ? 1 1 √ +√ 2 2 ) ( L 2 R2 +4L2 π 2 4 π 32 ?L 4L e △ ? 1 π R + 4L e2π △ ?1 (59)
Since R ? △ we can neglect the second term of the above expression. For L > △ we have L 1 ?4π △ R π ( + )e (60) E2 ? 32L △ 4L which is small due to the exponential factor. Let us consider the second term of (58). For the sake of simplicity we 1 omit the factor ? 4R 2 in the spectrum. Applying the Plana formula to the summation over m we get W = in which ζ (3)R + W ′, 2 16△ (61)
△ζ (3) 1 ? (62) 48R 8πR2 term is very small. The main contributions to the total energy than come from the ?rst terms of (58) and (61): W′ ? E=? π 3 RL Rζ (3) + . 720△3 16△2 12 (63)
Inspecting the above result we observe that the energy is positive around L≤ 3 △ ( within our approximation ). Around this value of the height, the 2 ?E radial force Frad = ? ? is repulsive. The force on the axial direction Faxial = △ ?E ? ?L however, is repulsive for all values of L, which forces the cylinders to become of in?nite length. When L becomes longer than 3 △, the radial force 2 too becomes attractive.
VI. Casimir energy between two close cocentric spheres.
We employ the spherical coordinates ds2 = dt2 ? dr 2 ? r 2 (dθ2 + sin2 θdφ2 ) and insert the solution in terms of the spherical harmonics
l Φ = Ym (θ, φ)
(64)
vln (r ) ; r
l = 0, 1, 2, . . . , ?l ≤ m ≤ l
(65)
into the KleinGordon equation (4). The resulting radial eigenvalue problem we have to deal is [? (l + 1 )2 d2 2 + ]vln (r ) = (ωln )2 vln (r ) dr 2 r2 (66)
subject to the boundary conditions vln (r0 ) = 0, vln (r1 ) = 0. (67)
Here r0 < r1 are the radii of the spheres and n is the radial quantum number to be determined by the boundary conditions. To satisfy the boundary conditions one has to deal with the roots of the radial wave function vnl (r ) which as in the previous section are the Bessel functions ( with m replaced by l + 1/2). However since we are interested in △ ≡ r1 ? r0 ? r0 limit, we can proceed as we have done in the previous section. For the radial wave functions and the eigenvalues we obtain
0 vln (r ) =
2 0 sin(ωln (r ? r0 )), △r 13
(68)
1 2 ) π 2 n2 (l + 2 + ; n = 1, 2 , . . . . (69) = △2 R2 With the above approximated radial eigenfunctions and eigenvalues we can write the Green function as 0 2 (ωln )
eiωln (t?t ) 0 0 l l (θ ′ , φ ′ ) G= vln (r )vln (r ′ )Y m (θ, φ)Ym 0 2 ω ln n=1 l=0 m=?l Integrating the vacuum energy density T = Reg [
∞
∞
l
′
(70)
1 1 1 ′ + ?r ?r ′ + ′ + lim [ ? ? ? ? ?φ ?φ′ ]G] (71) t t θ θ 2 t,r,θ,φ→t′,r′ ,θ′ ,φ′ r2 r 2 sin2 θ
∞ 1 0 ]. (l + ) Reg [ ωln 2 n=1 l=0 ∞
over the volume between two cocentric spheres we get the total energy E= (72)
Applying the Plana formula to the n summation and dropping the n = 0 term and the integration over n we get E=? where F (n) = To use the Plana formula [1] 1 f (k + ) = 2 k =0
∞ ∞ 0
2△ πR2
∞
∞ 1
dnF (n) s3
(73)
1 s= 2
e2 R sn ? 1
∞ 0
△
(74)
dyf (y ) ? i
dy
f (iy ) ? f (?iy ) 1 + e2πy
(75)
we have to get rid o? the poles of the function F (n) at the imaginary axis ns = 2iπm. Thus we work with the function 2△ R Fβ = with β > 0. Then (73) becomes E=? π3R + E′ 360△3 14 (77)
∞
△ 1 s= 2
s3 e2 R x(s+β ) ? 1
(76)
where E′ = Using 1 lim 2π △ β → 0
2△ R ∞ 0
dss3 e2πs + 1
2△ R
dx x2 ? (
2△ 2 1 1 ) ( x(β +is) + x(β ?is) ) R e ?1 e ?1 (78) (79)
π 288△ Thus the total energy in the region between the spheres is E′ ? ? E=? 5△ 2 π 3 R2 (1 + ) 360△3 4π 2 R 2
? 1 we get
(80)
R In △ → ∞ it is obvious that the above energy approaches the parallel plate formula.
VII. Casimir interactions of two close coaxial cones.
The geometry we like to present in this section is two cones with common axis at positive z direction and appeces at the origin. By close cones we mean the appex angles θ1 and θ1 are close to each other, that is △ ≡ θ1 ? θ0 ? sin θ0 sin θ1 ≡ Θ. (81)
In the above approximation the solutions we employe ( in spherical coordinates ) which vanishes at the surfaces θ = θ0 and θ = θ1 are Φω nm = where eimφ ω πn J?nm (ωr ) √ sin( (θ ? θ0 )), r △ π△ ?nm = ( m πn 2 ) + ( )2 △ Θ (82)
(83)
The energy ω in (82) is continuous. The Green function is ( with the cut o? factor β ) e?βω+iω(t?t ) ω ′ ′ ′ Φnm (r, θ, φ)Φω G= nm (r , θ , φ ). 2ω n=1 m=0 m=?l 15
∞ ∞ l
′
(84)
Note that in this section we employ di?erent regularization method than the previous ones. The cut o? method is more suitable for the continuous energy spectra. Inserting the Green function of (84) into the coincidence limit formula and then integrating over θ and φ, we arrive at the vacuum energy density at r : 1 ?2 ? 2 ∞ ∞ Q?1/2+?nm (1 + E= Regβ [( 2 + 2 2 ) 4πr ?r ?β n=1 m=?∞ r
β2 ) 2r 2
]
(85)
Regβ stands for the cut o? regularization, that is we pick the ?nite part of the expression in β → 0 limit. In deriving (85) we used the formula [11]
∞ 0
dωe?βω (Jν (ωr ))2 =
β2 1 Qν ?1/2 (1 + 2 ), πr 2r
(86)
where Qν (x) is Legendre function of the second kind. We rewrite the expression (85) as E= with
∞ ∞ 1 Reg [ O Q?1/2+?nm (1 + y 2)] y 2πr 4 n=1 m=?∞
(87)
y2 + 1 ?2 β ? + . y ≡ √ , O ≡ 1 + 2y ?y 2 ?y 2 2r Applying the Plana formula to the summation over m we arrive at E = E0 + E1 , where E0 =
∞ 1 Reg [ O y πr 4 n=1 ∞ 0
(88)
(89)
dmQ?1/2+?nm (1 + y 2 )]
(90)
and
∞ 1 E1 = 4 Regy [O r n=1 ∞
π Θn △
dm tanh
e2πm ? 1
2 (m )2 ? ( πn ) Θ △
P? 1 +i√( m )2 ?( πn )2 (1 + y 2)]
2 Θ △
(91) 7 s2 ? 8 2
Making use of
π Θn △
? 1, tanh x ≤ 1 and Regy [OP?1/2+is (1 + y 2)] = 16 (92)
we get 1 ∞ E1  ≤ 4  r n=1 That is E1  ≤
∞
π Θn △
dme
?2πm
πn 2 m 2 7 (Θ) ? ( △ ) ( ? ) . 8 2
(93)
Thus E1 is negligible small. To evaluate E0 , we apply the Plana formula to the summation over n in (90). The formula we obtain is E0 = E + where Θ△ E= 2πr 4
∞ 0
2Θ Θ e?2π △ . 4 4π △ r
(94)
Θ Θ△ A ? B, 2πr 4 2πr 4
(95)
dx
∞ x
√ tanh y 2 ? x2 7 dy ( + x2 ? y 2 ) 2 △ y e ?1 4
∞ 0
(96) (97) (98)
A = Regy [O B = Regy [O
dssQ? 1 +s (1 + y 2)]
2
∞ 0
dsQ? 1 +s (1 + y 2)]
2
Changing the variables y = t, x2 = t2 ? k 2 (96) can be rewritten as E= Θ△ 2πr 4
∞ 0
dtt e2△t ? 1
1 0
1 0
√
7 dkk tanh( kt )( ? k 2 t2 ) 2 4 1?k
(99)
Inspecting the integrals over k , that is the terms f1 (t) = and f2 (t) =
0
√
dkk tanh(kt) 1 ? k2
(100)
1
we see that both approach very fast from the value f1 (0) = f2 (0) = 0 to their 2 . respective asymptotic values f1 (t → ∞) = 1 and f2 (t → ∞) = 0.66 = 3 Let us treat the second term in (99) in detail. We approximate f2 (t) as f2 (t) ? { at, t ∈ [0, b] 2 t ∈ [b, ∞) 3 17 (102)
dkk 3 √ tanh(kt) 1 ? k2
(101)
where a and b are both of order 1. The second term in (99) then becomes E2 = ? Θ△ (a 2πr 4
b 0
dtt4 + e2△t ? 1
∞ b
dtt3 ) e2△t ? 1
(103)
Since △ ? 1, we can approximate the denominator of the ?rst integrand as e2△t ? 1 ? 2△t. In the second integral making the change of variables 2△t = s , we can replace the lower boundary as 2b△ ? 0. Thus (103) becomes E2 ? ? 1 Θ△ ab4 ( + 2πr 4 8△ 16△4
∞ 0
Θab4 Θπ 3 dss3 ) = ? ? . es ? 1 16πr 4 720r 4 △3
(104)
Note that the above ”density” is an expression obtained after integrating over Θ and φ. If we divide (105) to the angular integral
θ1 θ0
It is obvious that the ?rst term is negligible compared to the second. Similar treatment shows that that the ?rst term in (99) gives contributions of orders 1 O (△) and O ( △ ) both are small. Inspecting (90) we see that the second and third terms in E0 are also negligible. Thus the ?nal result for our Casimir energy is : Θπ 3 . (105) E ? E2 ? ? 720r 4 △3
sin θ
2π 0
dφ ? 2π Θ△
(106)
we obtain the energy density averaged over the angular variables: E ? E2 = ? π2 + O (△ ? 3 ). 1440r 4△4 (107)
In small △ limit the above result is in perfect agreement with the energy density in the region between two in?nite planes with angle △ between them ( i. e. the wedge problem ) [13] E=? π2 △2 1 ( ? ). 1440r 4△2 △2 π2 (108)
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Acknowledgments The authors thank Alikram Aliev for useful theoretical discussions; and R. O. Onofrio for bringing some of the experiments including theirs into our attention. One of the author ( I. H. Duru )acknowledges the support of Turkish Academy of Sciences ( TUBA ) for its support.
References
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[13] J. S. Dowker and G. Kennedy, J. Phys. A: Gen. Phys., 11, 895 (1978); and D. Deutsch and P. Candelas, Phys. Rev., D20, 3063 (1979); T. H. Boyer, Physical Rewiew, 174, 1764 (1968).
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