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Casimir Effect for a Perfectly Conducting Wedge
Annals of Physics PH5594 annals of physics 251, 157179 (1996) article no. 0111
Casimir Effect for a Perfectly Conducting Wedge I. Brevik* and M. Lygren *Department of Applied Mechanics and -Department of Physics, Norwegian University of Science and Technology, N-7034 Trondheim, Norway Received February 23, 1996; revised May 21, 1996
The Casimir surface force on a perfectly conducting wedge with opening angle : is calculated, by means of Schwinger's source theory. The wedge geometry is an attractive system to study, since it is nontrivial enough to demonstrate the main features of quantum field theory applied to media, and yet simple since it avoids the divergences associated with curved boundaries. We regularize the energy-momentum tensor by subtracting its value in free space. The surface pressure diverges as r &4 near to the vertex; the physical reason for this is the idealized boundary conditions implying the neglect of a finite skin depth. Our final results agree with those of Deutsch and Candelas (1979) and others, developed along different lines. An inconsistency in earlier calculations is pointed out and corrected. Possible experimental consequences are discussed. 1996 Academic Press, Inc.
I. INTRODUCTION In quantum field theory it is generally desirable to consider models that are simple but yet nontrivial enough to show significant physical effects. The purpose of the present paper is to analyse the fluctuating electromagnetic field, primarily at zero temperature, in the vacuum wedge region shown in Fig. 1, and to calculate the surface force (Casimir force) on its surfaces. The surfaces, as a simple working hypothesis, are taken to be perfectly conducting. Our study is motivated by the following considerations: (1) The formalism is relatively simple, since it is restricted to Minkowski space. Yet its range of applicability extends to the exterior region of a straight cosmic string. It is known that the geometry outside such a string is that of locally flat space, with an angle 8=8?G+ being removed. Here G is the gravitational constant and + &10 22 gcm is the mass per unit length. The relationship between the Minkowski space wedge angle : and the cosmic string deficit angle 8 is 8=2?&2:. We can thus gain valuable insight into string physics by limiting the formalism to that of Minkowski space. Perfect conductivity of the wedge walls corresponds to perfect conductivity of the surface of the cosmic string. A system of the last-mentioned kind (typically a superconducting cosmic string) has recently been analysed in [1]. 157 0003-491696 18.00 Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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BREVIK AND LYGREN
(2) It is desirable to calculate the electromagnetic force on the wedge walls using Schwinger's source theory [2]. The energy-momentum tensor has been found for the same geometry by Deutsch and Candelas using a different approach [3]. Schwinger's theory is more transparent, since it operates directly with the electric and magnetic fields. Moreover, the paper of Deutsch and Candelas, and also other papers such as the treatment of Helliwell and Konkowski [4], contain a disturbing mathematical problem: use is made of the integration formula
|
dp p 1&=J 2& ( pr)=
0
2 1&=1(&1+=) 1(&+1&=2) r 2&=1 2(=2) 1(&+=2)
(1)
in the limit when = 0 (this is Eq. (12) in [4]; it is equivalent to Eq. (5.32) in [3]). The integral is however convergent only when =>1; cf. Eq. (6.574.2) in [5]. When = 0 the integral simply diverges. Confronted with such a behaviour one may wonder: does this mathematical inconsistency imply that the previous calculations of the surface force density F are wrong? As we shall see below, this is not so. Use of Schwinger's theory leads to the expression F(r)=
?2 3 +11 2 4 720? r : 2
\
+\
?2 &1 , :2
+
(2)
which is in agreement with [3] (the cusp of the wedge is taken to lie at the origin r=0). It is reassuring that our different method of calculation supports the earlier calculations. (3) The simplicity of the model, implying planar boundary surfaces, makes it relatively easy to see where the formal infinities in the theory come from. We note that there are the three following idealized conditions still inherent in our model: (i) (ii) (iii)
the two surfaces of the wedge are sharp; the conductivity of the material is infinite; there is the cusp at the origin.
None of these conditions can be satisfied in practice. There is always a skin depth $; the electromagnetic field does not ``see'' the precise position of the boundary. When estimating numerical quantities it is convenient to use SI units; we then have $=(2+|_) 12, where _ is the conductivity of the material. Let us take copper as an example, for which _=6.0_10 7 (0m) &1, and let us put +=+ 0 . The main frequency range contributing to the Casimir force at zero temperature lies around ca, where a is the local separation between the surfaces. Typically, at1 +m, and so $t10 nm. Material surfaces of extension less than this cannot be regarded as sharp. In particular, mathematical divergences encountered because of the cusp can be judged as unphysical. The necesssity of being aware of ``imperfect'' boundary conditions was stressed also by Deutsch and Candelas [3]. These were the planar surfaces; if the surfaces are instead curved, then there appear additional divergences in the formalism.
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CASIMIR EFFECT FOR A WEDGE
159
The outline of the paper is as follows: Section II develops Schwinger's formalism as it appears in the present geometry, and gives the expressions for the two scalar Green functions. In Section III the field products are worked out in regularized form. In Section IV we calculate the diagonal components of the energy-momentum tensor ( S +& ) and its regularized version ( 3 +& ), following two different ways of approach. The first and more complete way works when : is arbitrary, whereas the second, more elegant way, works when p=?: is an integer. Agreement with results of previous works [3, 4] are found at the end. In Section V we discuss some possibilities of testing the theory experimentally, whereas in the final Section VI we summarize our results and give some remarks on the earlier literature. When developing the main theory, we adopt HeavisideLorentz units and put and c equal to unity.
II. BASIC FORMALISM. GREEN'S FUNCTIONS A. Basic Formalism We first have to outline the main ingredients of the Schwinger source theory, when applied to the wedge geometry of Fig. 1. The formalism has strong similarities with the one given recently by Brevik and Nyland, for the case of a single compact dielectric sylinder [6]. This formalism in turn was developed along the same lines as followed previously by DeRaad and Milton, for the case of a perfectly conducting cylindric shell [7]. The basic formalism was originally introduced by Schwinger, DeRaad, and Milton [8]. This kind of theory is strongly related to the fundamental treatise on electromagnetism found in Stratton [9]. The starting point is to express the electric field E(x) in terms of the polarization source P(x$):
|
E(x)= dx$ 1{ (x, x$) P(x$).
(3)
Here 1{ (x, x$) is a dyad with Fourier transform 1{ (r, r$, |): 1{ (x, x$)=
|
&
d| &i|{ e 1{ (r, r$, |), 2?
(4)
{ meaning t&t$. From Maxwell's equations, {_{_1{ (r, r$, |)&| 2 1(r, r$, |)=| 2 1{ $(r&r$),
(5)
where 1{ is the unit dyad. We introduce another dyad 1{ $: 1{ $=1{ +1{ $(r&r$),
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BREVIK AND LYGREN
which has the virtue of being divergence-free. When r{r$, as we shall assume, then 1{ and 1{ $ are equivalent. The effective electric and magnetic field products are as given, for example, in [1, 6, 8]: i( E i (r) E j (r$)) | =1 $ij (r, r$, |), i( H i (r) H j (r$)) | =& =
(7)
1 ({_1{ $_{$) ij |2
1 = = $ 1 $ (r, r$, |). | 2 ikl jmn k m ln
(8)
Here the angle brackets denote the quantum mechanical expectation value. Since 1$ is divergence-free, it is conveniently expanded in cylindrical coordinates. We let % denote the polar angle, so that %=0 and %=: are the wedge surfaces, and let the cusp coincide with the z axis. The boundary conditions are that the electric field is normal, and the magnetic field is tangential, to the surfaces (see Fig. 1). Taking these into account, we can write the expansion for 1$in the region % # [0, :] as 1{ $(r, r$, |)=
2 $ : : m=0
|
&
dk {_z^f m(r, r$) cos(mp%) 2?
_
i + {_{_z^g m(r, r$) sin(mp%) e ikz, |
&
(9)
where k is the axial wave number. The functions f m and g m implicitly carry the second tensor index. The prime on the summation sign means that the m=0 term is taken with half weight. Finally, we have defined p as p=?:.
(10)
We note that the following operator relations are useful: {_z^ {_{_z^
cos mp% sin sin (mp%)&% (mp%) , (mp%) e ikz =e ikz r^ r cos sin r cos cos (mp%) e ikz =e ikz sin
_ cos r^ ik (mp%) _ sin r
%ik {_{_{_z^
&
mp% sin cos (mp%)&z^ (mp%) d m , r cos sin
cos cos (mp%) e ikz =&e ikz {_z^ (mp%)(d m +k 2 ), sin sin
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&
(11)
(12) (13)
CASIMIR EFFECT FOR A WEDGE
Fig. 1.
161
Sketch of the wedge geometry. Boundary conditions are illustrated for the fields.
where d m is the operator dm =
1 m 2p 2 r & 2 . r r r r
(14)
The expansion (9) satisfies the relations % } {_1{ $ | s =0,
%_1{ $ | s =0
(15)
at the wedge surfaces automatically, as it should according to the boundary conditions. It can also be shown by explicit calculation that (9) satisfies the Maxwell equations (5). The calculation is somewhat involved and will not be given here. Instead, we focus attention on the single scalar Green function F m that can be introduced in this problem. Because there is no boundary condition in the radial direction (in contradistinction to the cases studied in [6] and [7]), we need only to introduce one single function of this kind. It is related to the functions f m and g m in (9) by 1 f m =& 2 (d m &k 2 ) f m , | (16) 2 {$_z^ cos(mp%$) e &ikz$F m(r, r$), f m = :
and i g m =& |
2
: {$_{$_z^ sin(mp%$) e
&ikz$
F m(r, r$).
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BREVIK AND LYGREN
From Maxwell's equations we obtain the following equations for F m : 1 1 d D m F m(r, r$)= $(r&r$), 2 m | r
(18)
where D m is the differential operator D m =d m +q 2,
(19)
and q is the transverse wave number, q 2 =| 2 &k 2.
(20)
We are free to define an auxiliary Green function Gm , given by 1 d m Gm = $(r&r$). r
(21)
Thus we get the following governing equation for Fm :
_
D m Gm &
q2 1 F m = $(r&r$). 2 | r
&
(22)
It is seen that this equation corresponds to those for a cylinder [6, 7] with the d m operator slightly modified: m mp. B. The Scalar Green Functions We now impose boundary conditions on the scalar Green function. The function will be required to be bounded as r 0 and to describe outgoing waves as r . The boundary conditions at the wedge surfaces are automatically satisfied by the expansion (9) for 1{ $. Consider first Eq. (21) for the auxiliary Green function. When rewritten in the form
\
1 2 1 p 2m 2 & 2 + Gm = $(r&r$) r 2 r r r r
+
(23)
it becomes apparent that the solution is Gm =&
1 r< 2mp r >
\ +
mp
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163
CASIMIR EFFECT FOR A WEDGE
for m{0 (actually m>0 in view of (9)), and 1 r< G0 =& ln 2 r>
(25)
for m=0. Here r <(r > ) is the lesser (greater) of r and r$. Consider next Eq. (22). Rewriting it using (19) and (14), we see that the solution contains the ordinary Bessel and Hankel functions of order mp and argument qr. We take into account the discontinuity in slope across r=r$, and use the known value for the Wronskian, W[J &(x), H &(x)]=2i?x, to fix the multiplicative coefficient. The final solution is Fm(r, r$)=
|2 i? Gm(r, r$)+ J mp(qr < ) H mp(qr > ) . q2 2
_
&
(26)
We recall that this expression refers to a vacuum region, in the angular interval % # [0, :].
III. EFFECTIVE FIELD PRODUCTS A. Field Products in the Presence of the Wedge In evaluating the field products, we let two points x and x$ be separated in both radial and angular directions. The separation in radial direction is necessary to avoid the delta function $(r&r$). The separation in angular direction is useful (not necessary) since it will simplify the calculations. Before considering the effective field products explicitly, we express the dyads 1{ $ and {_1{ $_{$ in terms of the function Fm : 1{ $(r, r$, |) =
2 :$ 2 :| m=0
|
&
dk [&({_z^ )({$_z^ )(d m &k 2 ) F m(r, r$) cos(mp%) cos(mp%$) 2?
+({_{_z^ )({$_{$_z^ ) F m(r, r$) sin(mp%) sin(mp%$)] e ik(z&z$),
(27)
{_1{ $(r, r$, |)_{$ =
2 :$ 2 :| m=0
|
&
dk [({_{_z^ )({$_{$_z^ )(d m &k 2 ) F m(r, r$) cos(mp%) cos(mp%$) 2?
&({_z^ )({$_z^ )(d m &k 2 )(d $m &k 2 ) F m(r, r$) sin(mp%) sin(mp%$)] e ik(z&z$).
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BREVIK AND LYGREN
It is here convenient to note at once that the function G0 does not contribute to the dyad 1{ $, so that we can use the expression for Gm , m>0, in our calculations. When m=0 the nonvanishing term in 1{ $ contains ({_z^ )({$_z^ )(d 0 &k 2 ) F 0(r, r$) e ik(z&z$),
(29)
and the contribution from G0 is proportional to 2 (d 0 &k 2 ) G0 & k G0 , r r$ r r$
(30)
in view of the effective relation d 0 G0 =0, Eq. (21), which holds when the two points are kept separated in the radial direction. Then using (25) we see that (30) vanishes. We now consider the field products. We shall henceforth put z=z$ and t=t$. It is convenient to introduce the symbol L for the integral operator L=
2 :
|
&
d| 2?
|
&
dk :$ . 2? m=0
(31)
Some calculation results in the following expressions for the effective field products: i ( E z (r, %) E z (r$, %$)) =
|
&
=L
d| z^ } 1{ $ } z^ 2? i?
_2 q J 2
pm
&
(qr < ) H pm(qr > ) sin(mp%) sin(mp%$) ,
(32)
i ( E % (r, %) E % (r$, %$)) =
|
&
=L +
d| % } 1{ $ } % 2? i?
_ 2 {| J$ (qr 2
mp
<
) H$mp(qr > )
k 2m 2p 2 J mp(qr < ) H mp(qr > ) cos(mp%) cos(mp%$) , q 2rr$
=
&
(33)
i ( E r (r, %) E r (r$, %$)) =
|
&
=L
d| r^ } 1{ $ } r^ 2? i? | 2m 2p 2 J mp(qr < ) H mp(qr > ) 2 q 2rr$
_ {
=
&
+k 2J$mp(qr < ) H$mp(qr > ) sin(mp%) sin(mp%$) ,
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CASIMIR EFFECT FOR A WEDGE
i ( H z (r, %) H z (r$, %$))
|
=&
&
=L
i?
d| z^ } ({_1{ $_{$) } z^ 2?| 2
_2 q J 2
mp
&
(qr < ) H mp(qr > ) cos(mp%) cos(mp%$) ,
(35)
i ( H % (r, %) H % (r$, %$))
|
=&
&
=L
d| % } ({_1{ $_{$) } % 2?| 2
i? k 2m 2p 2 J mp(qr < ) H mp(qr > ) 2 q 2rr$
_ {
=
&
(36)
&
(37)
+| 2J$mp(qr < ) H$mp(qr > ) sin(mp%) sin(mp%$) , i ( H r (r, %) H r (r$, %$))
|
=&
&
=L
d| r^ } ({_1{ $_{$) } r^ 2?| 2
i? | 2m 2p 2 J (qr ) H mp(qr > ) 2 q 2rr$ mp <
_ {
=
+k 2J$mp(qr < ) H$mp(qr > ) cos(mp%) cos(mp%$) .
In these expressions the differences (r&r$) and (%&%$) are arbitrary; they are not necessarily small. We also note that the contributions from the function G m cancel out in all field products; this being closely connected to the effective relations d m Gm =0. From (22) therefore follows another effective relation: d m F m =&q 2F m .
(38)
There are also other field products, such as ( E i E k ) and (H i H k ) with i{k. They will be present in the off-diagonal part of the spatial energy-momentum tensor. This tensor reads, in a vacuum region, Sik =&E i E k &H i H k + 12 $ ik(E 2 +H 2 ).
(39)
It is rather remarkable that the off-diagonal products ( E i E k ) and ( H i H k ) do not vanish by themselves, but that their sum ((E i E k ) +( H i H k ) ), present in (39), does so. This can be verified by explicit calculation, along the same lines as above, although the calculation will not be shown here. Similarly it can be shown that the
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BREVIK AND LYGREN
Poynting vector vanishes. The only components of the four-dimensional energymomentum tensor that turn out to be nonvanishing are thus the diagonal stress components S ii (no sum) and the energy density component. B. The Infinite Vacuum Expressions In any physical application the above expressions for the field products have to be regularized by subtracting off the corresponding expressions pertaining to the case of an infinite vacuum, i.e., to the wedge being removed. Probably the most striking formal property due to the presence of the wedge is that the order of the Bessel functions becomes modified: m mp. In the free space, only integral orders m do occur. In the following we let the superscript zero refer to the infinite vacuum field. We obtain i ( E z (r, %) E z (r$, %$)) 0 =i ( H z (r, %) H z (r$, %$)) 0 =
: i? 2 L q J m(qr < ) H m(qr > ) cos m(%&%$) , 2? 2
_
&
(40)
i ( E % (r, %) E % (r$, %$)) 0 =i ( H % (r, %) H % (r$, %$)) 0 =
: i? L | 2J$m(qr < ) H$m(qr > ) 2? 2
_ {
+
k 2m 2 J (qr ) H m(qr > ) cos m(%&%$) , q 2rr$ m <
=
&
(41)
i ( E r (r, %) E r (r$, %$)) 0 =i ( H r (r, %) H r (r$, %$)) 0 =
: i? | 2m 2 L J m(qr < ) H m(qr > ) 2? 2 q 2rr$
_ {
=
&
+k 2J$m(qr < ) H$m(qr > ) cos m(%&%$) .
(42)
The relations obeyed by the off-diagonal components of the stress tensor are the same as noted in the previous subsection.
IV. ENERGY-MOMENTUM TENSOR A. On the Calculation of ( S %% ) We now consider the diagonal spatial components of the energy-momentum tensor, i.e., the stress tensor with the opposite sign. We let henceforth the two points
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CASIMIR EFFECT FOR A WEDGE
approach each other, r r$, but keep both (r&r$) and (%&%$) different from zero. We write ( E 2r ) instead of ( E r(r, %) E r(r$, %$)) | r r$ , etc., to simplify the notation. The most important component of S ik , as far as the surface force is concerned, is S %% . From (39) we have (S %%(r)) = 12 [( E 2r ) &( E 2% ) +( E 2z ) +( H 2r ) &( H 2% ) +( H 2z ) ] r r$ .
(43)
Inserting the product expressions from the previous section, we get ( S %%(r)) =L =
?
1
_ 4 {q +rr$ % %$ & r r$ = J 2
? 2:(2?) 2
_ :$
|
d| &
(qr < ) H mp(qr > ) cos mp(%&%$)
&
r r$
dk &
1
_{q +rr$ % %$ & r r$= J 2
m=0
|
mp
(qr < ) H mp(qr > ) cos mp(%&%$)
mp
&
. r r$
(44) This expression has to be regularized by subtracting its value in vacuum: ( S 0%% (r)) = =
: ? 2 1 L & q + J m(qr < ) H m(qr > ) cos m(%&%$) 2? 2 rr$ % %$ r r$
_{
? (2?) 3
_ :$ m=0
|
=
&
_{
1 & J (qr ) H m(qr > ) cos m(%&%$) rr$ % %$ r r$ m <
q2 +
r r$
|
d|
&
dk &
=
&
. (45) r r$
Comparison between (44) and (45) shows that ( S %% (:=?)) =( S 0%% ). Thus we only need to calculate ( S %% ); the contact term immediately follows from it. Now, making a complex frequency rotation, | i|^, implying the following effect on the variable q: q 2 =&(|^ 2 +k 2 )#&\ 2,
(46)
we obtain ( S %% (r)) =
1 :? 2
|
0
d|^
|
0
1
_{&\ +rr$ % %$& r r$= ) cos mp(%&%$) & ,
dk :$
2
m=0
I mp(\r < ) K mp( \r >
r r$
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BREVIK AND LYGREN
I mp and K mp being modified Bessel functions [10]. We perform the integration in polar coordinates, noting that \ is the radius in the |^k plane. Integrating over angles from 0 to ?2 we get ( S %% (r)) =
1 :$ 2?: m=0 _
|
d\ \ 0
1
_{&\ +rr$ % %$& r r$= I 2
mp
(\r < ) K mp( \r > ) cos mp(%&%$)
&
. r r$
(48) It is convenient to split this expression into two parts S I and S II , defined by 1 $ : S I =& 2?: m=0 S II =
1 :$ 2?: m=0 _
|
|
0
d\ \ 3I mp(\r < ) K mp( \r > ) cos mp(%&%$)| r r$ ,
(49)
d\ \ 0
1
_rr$ % %$& r r$& I
mp
( \r < ) K mp( \r > ) cos mp(%&%$)| r r$ .
(50)
We also introduce the notation z=
r< . r>
(51)
We are going to calculate ( S %% ) =S I +S II using two different methods. The first method is the more general one, valid for all values of the opening angle. The second one is valid when p is an integer. B. Evaluation of ( 3 %% ) for Arbitrary : We first perform the integrations in (49) and (50), keeping the points separated in both r and % directions. Thereafter we do the summation and take the limit r r$. The integral over \ in (49) can be expressed in terms of a hypergeometric function, and further manipulated so as to involve only elementary functions. Equation (50) can be handled similarly. Details are given in Appendix A and we confine ourselves here to giving the final expression for the tensor component ( 3 %% ), where ( 3 +& ) in general is defined to be the regularized energy-momentum tensor: ( 3 +&(r)) =( S +&(r)) &( S 0+&(r)).
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CASIMIR EFFECT FOR A WEDGE
We get for the %% component 3 ?2 +11 ( 3 %%(r)) =& 2 4 720? r : 2
\
+\
?2 &1 . :2
+
(53)
When :=? the wedge degenerates into a plane sheet, and the expression vanishes as it should according to its construction. C. Alternative Evaluation of ( 3 %% ) When p Is an Integer A more simple and elegant way of calculating ( S %% ) and ( 3 %% ) can be given if p is an integer. The method involves use of the generalized Graf addition theorem for modified Bessel functions. Details are given in Appendix B. Using (B.6) we can write Eq. (48) simply as ( S %%(r)) =
1 (2?) 2
|
_
d\ \ &\ 2 +
0
1 & rr$ % %$ r r$
&
p&1
: K 0(\R n )| r r$ ,
(54)
n=0
where from (B.2) R n =[r$ 2 +r 2 &2r$r cos((%&%$)+2?np)] 12.
(55)
We now use the formula, valid for +>&1,
|
1++ 2
_ \ +&
x +K 0(ax) dx=2 +&1a &+&1 1
0
2
(56)
(cf. Eq. (11.4.22) in [10]), to get ( S %%(r)) =
1 1 p&1 4 1 & : & 2+ (2?) 2 n=0 Rn rr$ % %$ r r$ R 2n
_
\
+ &
.
(57)
r r$
Taking the limits of the terms in the square bracket, we get after some calculation 4 R 2n
} 1 1 \rr$ % %$ & r r$ + R } &
=& r r$
2 n r r$
1 , 4r 4 sin 4(?np)
(58)
1 . 2r 4 sin 4(?np)
(59)
=&
Here the n=0 term in (57) is seen to be divergent. This divergence, however, is cancelled by the divergence in the contact term ( S 0%% ) which is to be subtracted off. From [11] we have the summation formula p&1
: n=1
( p 2 +11)( p 2 &1) 1 = , sin (?np) 45 4
whereby we recover finally the result (53) for the tensor component ( 3 %% ).
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BREVIK AND LYGREN
D. The Complete Regularized Tensor ( 3 +& ) The remaining components of ( 3 +& ) are found using the same procedure as above. We therefore give here only the results. We define the tensor components in the following way: ( 3 +& ) =( 3 rr , 3 %% , 3 zz , &w),
(61)
where w is the electromagnetic energy density. We get (3 +&(r)) =
1 ?2 +11 720? 2r 4 : 2
\
+\
?2 &1 diag(1, &3, 1, 1). :2
+
(62)
We see that the trace of the energy-momentum tensor is zero, ( 3 ++ ) =0, as it should for the electromagnetic field. Moreover, the energy density w is negative (if :
?2 , 720a 4
(63)
in accordance with the standard result in Casimir theory for two plates. We note that there are two physical conditions to be satisfied here if the medium is a real one: first, we must have ar* 0 , where * 0 is a characteristic wavelength of the absorption spectrum of the medium. Typically, * 0 amounts to a few tenths of nanometers. Secondly, we must have, in dimensional units, aRck B T in order to be able to apply zero temperature theory. In other words, the number of significant Matsubara frequencies is large. Typically, aR10 +m. Equation (63) usually holds in the interval 0.05 +mRaR10 +m. For further discussions on this point the reader may consult, for instance, Refs. [12, 13, 14].
V. ON POSSIBLE EXPERIMENTAL CONSEQUENCES It would be highly desirable to be able to subject the above theory to experimental tests. Probably the idea that lies most closely at hand is the possibility of measuring the surface force. If the wedge is cut out from a massive perfect conductor (of infinite permittivity), then there are only field fluctuations present in the opening, 0 % :. The regularized surface force density F(r) on the horizontal surface in Fig. 1 is then equal to &( 3 %%(%=0)) which means, according to (53), that F(r) becomes equal to the expression given previously in Eq. (2). The force is attractive. The r &4 divergence at the origin is obviously a consequence of the over-idealization of our boundary conditions: in reality there is always, as mentioned earlier, a finite skin depth $ which smears out the physical effect from the mathematically singular
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CASIMIR EFFECT FOR A WEDGE
cusp at the origin. Similarly, the r &4 divergent result obtained if we calculate the moment of force (per unit length in the z direction), 0 rF(r) dr, is a mathematical artifact which is not rooted in the real physics. [It may be noted that this divergence is not related to our model of the medium as being perfectly conducting at all frequencies. We might take dispersion into account by introducing a highfrequency cutoff for |^ in the integral (47), but would always encounter the same kind of r &4 divergence in the result.] The idea of measuring the force (or force moment) on a solid metal block would however be difficult to carry out in practice. Perhaps a better idea is to imagine the wedge to be a thin foil. Then the field fluctuations in both regions, % # [:, 2?] as well as % # [0, :], have to be taken into account. To avoid the singularity at the origin we may assume that a small pivot (with its axis vertically into the paper) has been inserted in the horizontal surface in Fig. 1, at a distance dr$ from the origin. Then the moment of force about the pivot reads, per unit length into the paper, in dimensional units, M# =
|
rF(r) dr d
3c 1440? 2d 2
_\
?2 +11 :2
+\
?2 ?2 &1 & +11 :2 (2?&:) 2
+ \
+\
?2 &1 (2?&:) 2
+& .
(64)
The moment of force will try to diminish the minor angle of Fig. 1. In practice, it is natural to choose :R1. Then, the vacuum fluctuations will try to turn the horizontal surface in the upward direction, about the pivot. What magnitude can we expect for the quantity M ? Let us choose :t0.01 rad (0.57%), and dt100 +m. Then, the minimum active opening between the two foils becomes :dt1 +m, which is much larger than the skin depth $t10 nm for copper, and yet small enough to justify the use of zero temperature theory. Formula (64) then yields Mt6_10 &7 dyn cmcm as a typical force moment on the movable foil. It is questionable whether such a small quantity can be measured. Let us round off this section by making some general remarks. It is almost surprising how few measurements of the Casimir effect there really exist. There are classic experiments of the attractive forces between metal plates, by Sparnaay [15], for the region 0.5&2 +m. By using a capacitive method to determine the interplate distance, deflections as small as 1&10 nm could be measured. There are also well known force experiments by Sabisky and Anderson [16], testing the theoretical prediction (originally due to Lifshitz) accurately over decades of distances. More recent work has been done by the groups of Overbeek [17], Deryagin [18], and Israelachvili [19]. Useful reviews can be found in [19, 20]. However, it turns out to be difficult to obtain satisfactory accuracy in force measurements between plates. There are interfering effects present, such as surface roughness (for a recent theoretical work analysing this kind of effect, see Bordag et al. [21]). Probably the growing atomic force microscope technique [22, 20] can
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BREVIK AND LYGREN
prove to be an effective tool in the Casimir force context [23]. The experimental paper of Onofrio and Carugno [24], in which the use is proposed to dynamically detect Casimir forces between two conducting plates, seems also promising. After all it may be so, however, that force measurements are not the most profitable way of approach, experimentally. Recently Sukenik et al. [25], instead of measuring forces, measured the deflection of a ground-state atomic beam passing through a micron-sized cavity; in this manner they proved clear evidence for the existence of the CasimirPolder force. Their cavity was actually wedge-shaped, with dimensions of the same order of magnitude as we assumed above when estimating the magnitude of the force moment M. It must be noted however, that the Casimir Polder force, although closely related to the Casimir effect, is not really the same as the force between macroscopic surfaces.
VI. FURTHER COMMENTS A. Analogy with Cosmic String As mentioned in Section I, there exists an analogy between our wedge and a cosmic string. Let us briefly illustrate this analogy mathematically. The expression for the line element outside the string is ds 2 =&dt 2 +dr 2 +(1&4G+) 2 r 2 d% 2 +dz 2.
(65)
We can imagine (65) to represent the geometry of locally flat space, with a wedge of angle 8=8?G+ being removed. Let us introduce the symbol ;, defined as ;=(1&4G+) &1 =(1&82?) &1.
(66)
Then, the electromagnetic energy-momentum tensor outside an infinitely thin cosmic string, as calculated by Frolov and Serebriany [26], can be expressed as ( 3 +& ) =
1 ( ; 2 +11)( ; 2 &1) diag(1, &3, 1, 1). 720? 2r 4
(67)
Comparison with (62) immediately tells us what the mathematical analogy is: ; corresponds to p=?: in the wedge geometry. Hence 8 corresponds to 2?&2:. Another point worth noticing is a comparison with the theory of the outside region of a superconducting cosmic string [1]. In that paper, which similarly to the present paper was based upon Schwinger's source theory, the string was assumed to have a finite radius. Omitting the finite-radius terms, we obtain from Eqs. (26) and (27) in [1] the following expressions for the scalar string Green functions (we needed two functions there):
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CASIMIR EFFECT FOR A WEDGE
|2 i?; J m; (qr < ) H m; (qr > ) , G mF (r, r$)+ q2 2
_ | i?; J G (r, r$)+ G (r, r$)= q _ 2 F m(r, r$)=
m
G m
2
m;
(qr < ) H mp (qr >
& ) . &
(68) (69)
These are remarkably similar to the scalar Green function for the wedge, Eq. (26) above. B. Finite Temperatures The above theory refers to zero temperature, T=0. As alread mentioned, this theory usually gives an adequate approximation, even at room temperature, if the distance a between surfaces is small, at1 +m. Under some circumstances it can be of importance to consider the finite temperature version of the theory. We shall not go into detail about this point here, but confine ourselves to giving the formal expression for the azimuthal diagonal component of the energy-momentum tensor, called ( S %% ) T, at finite T. The starting point is the T=0 expression (47) for ( S %% ), after imposition of the complex frequency rotation. Applying standard rules [12] we obtain ( S %%(r)) T =
2k B T :?
|
dk :$ :$
0
n=0 m=0
1
_{&\ +rr$ % %$ 2 n
I mp( \ n r < ) K mp(\ n r > ) cos mp(%&%$) & r r$
=
&
,
(70)
r r$
where |^ n =2?nk B T,
\ 2n =|^ 2n +k 2.
(71)
|^ n are the Matsubara frequencies. Also the other components of the energymomentum tensor are thereby known. C. Comments on a Previous Calculation As mentioned in Section I, Deutsch and Candelas (DC) [3] make use of an integration formulacf. Eq. (1) abovewhich is wrong for the actual choice of parameters. It seems to be worthwile to clear up this confusing point. The core of the problem is that DC use Eq. (1) to derive Eq. (5.35) in [3]. We shall now use a different method, very similar to our calculations above, to derive the same equation. We start from Eqs. (5.27) and (5.30) in [3] for the two scalar Green functions, taking the limits t$, z$ t, z (not r$ r). After frequency rotation,
{
i D(x, x$) = 2 :$ 2? : m=0 N(x, x$)
=
|
_
0
ds
|
|
du uJ mp(ur < ) J mp(ur > ) 0
&
d|
|
&
dk e &s(|
2 +k 2 +u 2 )
sin mp%
{cos mp%
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sin mp%$ . cos mp%$
=
(72)
174
BREVIK AND LYGREN
Making use of the idea of evaluating the integrals over | and k in a (2&=) dimensional space [4], we have
|
ds
0
|
d|
&
|
dk e &s(|
2 +k 2 +u 2 )
&
=
? = = 1 1 1& , u= 2 2
\+ \ +
(73)
whereby the integral over u can be done [5]:
|
u 1&=J mp(ur < ) J mp(ur > ) du
0
=
z mp1(mp+1+=2) = = F mp+1& , 1& ; mp+1; z 2 . 2 r 1(mp+1) 1(=2) 2 2
\
=&1 2&= >
+
(74)
Here z=r <r > , and F is the hypergeometric function. Taking = 0, and using Eq. (A.7), we get
{
i D(x, x$) :$ z mp[cos mp(%&%$)cos mp(%+%$)]. = 2 2 2?:r >(1&z ) m=0 N(x, x$)
=
(75)
Finally using Eq. (A.12) to calculate the two sums, we obtain lim z1
p 1 x $ z mp cos mpx= csc 2 , : 2 1&z m=0 8 2
(76)
whereby we recover Eq. (5.35) in [3]:
{
i D(%, %$) ?(%&%$) ?(%+%$) csc 2 = csc 2 . 2 2 16: r 2: 2: N(%, %$)
=
_
&
(77)
In this manner we have suceeded to avoid using Eq. (1) at all. We have not got any explanation for the fact that although DC use Eq. (1) their final answer is correct.
VII. CONCLUSIONS Let us summarize as follows: (1) The final expression for the regularized energy-momentum tensor ( 3 +& ) =( S +& ) &( S 0+& ) is given by (62). In particular, the electromagnetic energy density w in the wedge region is negative (if :
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CASIMIR EFFECT FOR A WEDGE
(2) Within the framework of Schwinger's source theory, our final result is obtained via two different routes, the first holding for arbitrary values of :, the second holding when p=?: is an integer. Our result for the energy-momentum tensor is in agreement with Deutsch and Candelas [3], although we avoid using Eq. (1) which is actually wrong for the parameter choices made in [3]. (3) As regards possible experimental consequences, the expression (2) for the surface force diverges as r &4. This divergence is merely a mathematical artifact, however, since the infinite conductivity model neglects the finite skin depth $ which is always present in real materials. Imagining the wedge to be a thin foil, typical magnitudes of the moment of force, per unit length, lie around 10 &7 dyn cmcm (Section V). The wedge geometry has actually turned out to be a convenient device for carrying out modern cavity experiments on the CasimirPolder force [25]. (4) The transition to finite temperature theory is accomplished by replacing the integral over imaginary frequencies by a sum over discrete Matsubara frequencies. We give the formal expression for ( S %% ) T in Eq. (70), although we do not study finite temperature effects in detail. (5) For some purposes it is useful to note the formal analogy between wedge geometry and the outside geometry of a straight cosmic string. This analogy is illustrated in Section VI.A.
APPENDIX A: ( S %% ) WHEN : IS ARBITRARY We shall evaluate S I and S II , given by Eqs. (49) and (50) in the text. The starting point is Eq. (6.576.5) in [5], from which we have
|
\ *I &( \r < ) K &( \r > ) d\
0
z &1 =
1
*
1
*
\2+2 +&+ 1 \2+2+ F 2
1&* *+1 >
r
1(&+1)
\
1 * 1 * + +&, + ; &+1; z 2 . 2 2 2 2
+
(A.1)
From this we derive
|
\ 3I &(\r < ) K &( \r > ) d\=
0
|
\I &(\r < ) K &( \r > ) d\= 0
4z &1(2+&) F(2+&, 2; &+1; z 2 ), r 4>1(&+1)
(A.2)
z& F(1+&, 1; &+1; z 2 ). r 2>
(A.3)
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176
BREVIK AND LYGREN
The hypergeometric functions can be transformed according to Eq. (15.3.3) in [10]: F(a, b; c; z)=(1&z) c&a&b F(c&a, c&b; c; z),
(A.4)
and further manipulation can be done using Eq. (15.4.1) in [10]: m
(&m) n (b) n z n , (c) n n!
(A.5)
F(2+&, 2; &+1; z 2 )=
&(1&z 2 )+(1+z 2 ) , (1+&)(1&z 2 ) 3
(A.6)
F(1+&, 1; &+1; z 2 )=
1 . 1&z 2
(A.7)
F(&m, b; c; z)= : n=0
with m=0, 1, 2, ... . We then get
The integrals (A.2) and (A.3) then become
|
\ 3I &(\r < ) K &( \r > ) d\=
0
|
\I &(\r < ) K &( \r > ) d\=
0
4z & [&(1&z 2 )+(1+z 2 )], r (1&z 2 ) 3
(A.8)
z& . r 2>(1&z 2 )
(A.9)
4 >
The r and r$ derivatives in (50) can in view of (A.9) be transferred into % and %$ derivatives in order to simplify the calculation (this is an advantage of our method keeping % and %$ different): z& cos &(%&%$) r r$ r 2>(1&z 2 ) =&z & +
{
_ % r
2 sin &(%&%$) r (1&z 2 ) 2
3 > <
1 6r < + 5 cos &(%&%$) . 3 2 % %$ r > r <(1&z ) r >(1&z 2 ) 3
=
&
(A.10)
Similarly, the main part of (49) can in view of (A.8) be rewritten as
|
d\ \ 3I &( \r < ) K &( \r > ) cos &(%&%$)
0
=
4z & (1&z 2 ) sin &(%&%$)+(1+z 2 ) cos &(%&%$) . r (1&z 2 ) 3 % 4 >
_
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&
(A.11)
177
CASIMIR EFFECT FOR A WEDGE
Since z<1 we can do the summations using the Fourier formulas
1+z p , 2(1&z p )
(A.12)
mp pz p :$ z sin mp(%&%$)| % %$ = , % m=0 (1&z p ) 2
(A.13)
mp p 2z p(1+z p ) :$ z cos mp(%&%$)| % %$ = . % %$ m=0 (1&z p ) 3
(A.14)
:$ z mp cos mp(%&%$)| % %$ = m=0
Inserting these formulas into (49) we get S I =&
p 2pz p(1&z 2 ) (1+z 2 )(1+z p ) + . 2 3 ? r (1&z ) (1&z p ) 2 1&z p 2 4 >
_
&
(A.15)
Subtracting off the term S 0I corresponding to p=1 (i.e., to :=?), and taking the limit z 1, we obtain 3 I #S I &S 0I | z 1 =&
1 ( p 2 +11)( p 2 &1). 720? 2r 4
(A.16)
Correspondingly, S II =
p p 2z p&1(1+z p ) 2z(1+z p ) pz p&1(1+z 2 ) + + , (A.17) ? 2r 4>(1&z 2 ) (1&z 2 )(1&z p ) 2 (1&z p ) 3 (1&z 2 ) 2 (1&z p )
_
&
and 2 ( p 2 +11)( p 2 &1). 3 II #S II &S 0II | z 1 =& 720? 2r 4
(A.18)
[When evaluating the limits above the support from an analytic computer program such as Mathematica or Maple turns out to be very convenient.] Finally forming the sum of 3 I and 3 II , we get 3 ( p 2 +11)( p 2 &1), (3 %%(r)) #3 I +3 II =& 720? 2r 4 in accordance with Eq. (53) in the text.
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(A.19)
178
BREVIK AND LYGREN
APPENDIX B: ADDITION THEOREM FOR MODIFIED BESSEL FUNCTIONS There exists a very useful generalization of the addition theorem for Bessel functions: p&1
: e i&n Z &(kR n )=p n=0
J mp(kr 1 ) Z &+mp(kr 2 ) e imp,,
:
(B.1)
m=&
where R n =[r 21 +r 22 &2r 1 r 2 cos(,+2?np)] 12,
(B.2)
and e 2in =
r 2 &r 1 e &i(,+2?np) . r 2 &r 1 e i(,+2?np)
(B.3)
The function Z & denotes any linear combination of J & and N & (Neumann function). The proof of (B.1) can be effectuated by starting from the usual addition theorem for p=1, Eq. (9.1.79) in [10], which can be written
e i&0 Z &(kR 0 )= :
J l (kr 1 ) Z &+l (kr 2 ) e il,.
(B.4)
l=&
The validity of (B.1) now follows by rewriting its left hand side using (B.4). More details are given by Davies and Sahni [11]. In our context we need the generalized addition theorem for modified Bessel functions. We let Z & =H & and r i ir i , whereby R n iR n . Then (B.1) becomes transformed to p&1
: e i&n K &(kR n )=p n=0
:
I mp(kr 1 ) K &+mp(kr 2 ) e imp,.
(B.5)
m=&
For &=0, this equation can be simplified. Since I &n(z) K &n(z)=I n(z) K n(z) for n equal to an integer or zero, we get p&1
: K 0(kR n )=2p :$ I mp(kr 1 ) K mp(kr 2 ) cos(mp,). n=0
m=0
This theorem is used in Section IV.C.
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(B.6)
CASIMIR EFFECT FOR A WEDGE
179
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.
I. Brevik and T. Toverud, Class. Quant. Grav. 12 (1995), 1229. J. Schwinger, ``Particles, Sources, and Fields,'' Vol. I, AddisonWesley, Reading, MA, 1970. D. Deutsch and P. Candelas, Phys. Rev. D 20 (1979), 3063. T. M. Helliwell and D. A. Konkowski, Phys. Rev. D 34 (1986), 1918. I. S. Gradshteyn and I. M. Ryzhik, ``Table of Integrals, Series, and Products,'' Academic Press, London, 1980. I. Brevik and G. H. Nyland, Ann. Phys. (N.Y.) 230 (1994), 321. L. L. DeRaad, Jr. and K. A. Milton, Ann. Phys. (N.Y.) 136 (1981), 229. J. Schwinger, L. L DeRaad, Jr., and K. A. Milton, Ann. Phys. (N.Y.) 115 (1978), 1; K. A. Milton, L. L. DeRaad, Jr., and J. Schwinger, Ann. Phys. (N.Y.) 115 (1978), 388. J. A. Stratton, ``Electromagnetic Theory,'' McGrawHill, New York, 1941. M. Abramowitz and I. A. Stegun (Eds.), ``Handbook of Mathematical Functions,'' National Bureau of Standards, Washington, DC, 1964; reprinted by Dover, New York, 1972. P. C. W. Davies and V. Sahni, Class. Quant. Grav. 5 (1988), 1. E. M. Lifshitz and L. P. Pitaevskii, ``Statistical Physics,'' Part 2, Chap. VIII, Pergamon Press, Oxford, 1980. Yu. S. Barash and V. L. Ginzburg, Electromagnetic fluctuations and molecular forces in condensed matter, Chap. 6, in ``The Dielectric Function of Condensed Systems'' (L. V. Keldysh, D. A. Kirzhnitz, and A. A. Maradudin, Eds.), North-Holland, Amsterdam, 1989. I. Brevik and J. S. Ho% ye, Physica A 153 (1988), 420. M. J. Sparnaay, Nature (London) 180 (1957), 334; Physica (Utrecht) 24 (1958), 751. E. S. Sabisky and C. H. Anderson, Phys. Rev. A 7 (1973), 790. P. H. G. M. van Blokland and J. Th. G. Overbeek, J. Chem. Soc. Faraday Trans. I 74 (1978), 74. B. V. Deryagin, Y. I. Rabinovich, and N. V. Churaev, Nature (London) 272 (1978), 313. J. Israelachvili, ``Intermolecular and Surface Forces,'' 2nd ed., Academic Press, 1992. M. J. Sparnaay, Chap. 9A in ``Physics in the Making'' (A. Sarlemijn and M. J. Sparnaay, Eds.), Elsevier Science, Amsterdam, 1989. M. Bordag, G. L. Klimchitskaya, and V. M. Mostepanenko, Phys. Lett. A 200 (1995), 95. G. Binnig, C. F. Quate, and Ch. Gerber, Phys. Rev. Lett. 56 (1986), 930. M. Bordag, G. L. Klimchitskaya, and V. M. Mostepanenko, Surface Science 328 (1995), 129. R. Onofrio and G. Carugno, Phys. Lett. A 198 (1995), 365. C. I. Sukenik, M. G. Boshier, D. Cho, V. Sandoghdar, and E. A. Hinds, Phys. Rev. Lett. 70 (1993), 560. V. P. Frolov and E. M. Serebriany, Phys. Rev. D 35 (1987), 3779.
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Casimir Effect for a Perfectly Conducting Wedge I. Brevik* and M. Lygren *Department of Applied Mechanics and -Department of Physics, Norwegian University of Science and Technology, N-7034 Trondheim, Norway Received February 23, 1996; revised May 21, 1996
The Casimir surface force on a perfectly conducting wedge with opening angle : is calculated, by means of Schwinger's source theory. The wedge geometry is an attractive system to study, since it is nontrivial enough to demonstrate the main features of quantum field theory applied to media, and yet simple since it avoids the divergences associated with curved boundaries. We regularize the energy-momentum tensor by subtracting its value in free space. The surface pressure diverges as r &4 near to the vertex; the physical reason for this is the idealized boundary conditions implying the neglect of a finite skin depth. Our final results agree with those of Deutsch and Candelas (1979) and others, developed along different lines. An inconsistency in earlier calculations is pointed out and corrected. Possible experimental consequences are discussed. 1996 Academic Press, Inc.
I. INTRODUCTION In quantum field theory it is generally desirable to consider models that are simple but yet nontrivial enough to show significant physical effects. The purpose of the present paper is to analyse the fluctuating electromagnetic field, primarily at zero temperature, in the vacuum wedge region shown in Fig. 1, and to calculate the surface force (Casimir force) on its surfaces. The surfaces, as a simple working hypothesis, are taken to be perfectly conducting. Our study is motivated by the following considerations: (1) The formalism is relatively simple, since it is restricted to Minkowski space. Yet its range of applicability extends to the exterior region of a straight cosmic string. It is known that the geometry outside such a string is that of locally flat space, with an angle 8=8?G+ being removed. Here G is the gravitational constant and + &10 22 gcm is the mass per unit length. The relationship between the Minkowski space wedge angle : and the cosmic string deficit angle 8 is 8=2?&2:. We can thus gain valuable insight into string physics by limiting the formalism to that of Minkowski space. Perfect conductivity of the wedge walls corresponds to perfect conductivity of the surface of the cosmic string. A system of the last-mentioned kind (typically a superconducting cosmic string) has recently been analysed in [1]. 157 0003-491696 18.00 Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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BREVIK AND LYGREN
(2) It is desirable to calculate the electromagnetic force on the wedge walls using Schwinger's source theory [2]. The energy-momentum tensor has been found for the same geometry by Deutsch and Candelas using a different approach [3]. Schwinger's theory is more transparent, since it operates directly with the electric and magnetic fields. Moreover, the paper of Deutsch and Candelas, and also other papers such as the treatment of Helliwell and Konkowski [4], contain a disturbing mathematical problem: use is made of the integration formula
|
dp p 1&=J 2& ( pr)=
0
2 1&=1(&1+=) 1(&+1&=2) r 2&=1 2(=2) 1(&+=2)
(1)
in the limit when = 0 (this is Eq. (12) in [4]; it is equivalent to Eq. (5.32) in [3]). The integral is however convergent only when =>1; cf. Eq. (6.574.2) in [5]. When = 0 the integral simply diverges. Confronted with such a behaviour one may wonder: does this mathematical inconsistency imply that the previous calculations of the surface force density F are wrong? As we shall see below, this is not so. Use of Schwinger's theory leads to the expression F(r)=
?2 3 +11 2 4 720? r : 2
\
+\
?2 &1 , :2
+
(2)
which is in agreement with [3] (the cusp of the wedge is taken to lie at the origin r=0). It is reassuring that our different method of calculation supports the earlier calculations. (3) The simplicity of the model, implying planar boundary surfaces, makes it relatively easy to see where the formal infinities in the theory come from. We note that there are the three following idealized conditions still inherent in our model: (i) (ii) (iii)
the two surfaces of the wedge are sharp; the conductivity of the material is infinite; there is the cusp at the origin.
None of these conditions can be satisfied in practice. There is always a skin depth $; the electromagnetic field does not ``see'' the precise position of the boundary. When estimating numerical quantities it is convenient to use SI units; we then have $=(2+|_) 12, where _ is the conductivity of the material. Let us take copper as an example, for which _=6.0_10 7 (0m) &1, and let us put +=+ 0 . The main frequency range contributing to the Casimir force at zero temperature lies around ca, where a is the local separation between the surfaces. Typically, at1 +m, and so $t10 nm. Material surfaces of extension less than this cannot be regarded as sharp. In particular, mathematical divergences encountered because of the cusp can be judged as unphysical. The necesssity of being aware of ``imperfect'' boundary conditions was stressed also by Deutsch and Candelas [3]. These were the planar surfaces; if the surfaces are instead curved, then there appear additional divergences in the formalism.
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CASIMIR EFFECT FOR A WEDGE
159
The outline of the paper is as follows: Section II develops Schwinger's formalism as it appears in the present geometry, and gives the expressions for the two scalar Green functions. In Section III the field products are worked out in regularized form. In Section IV we calculate the diagonal components of the energy-momentum tensor ( S +& ) and its regularized version ( 3 +& ), following two different ways of approach. The first and more complete way works when : is arbitrary, whereas the second, more elegant way, works when p=?: is an integer. Agreement with results of previous works [3, 4] are found at the end. In Section V we discuss some possibilities of testing the theory experimentally, whereas in the final Section VI we summarize our results and give some remarks on the earlier literature. When developing the main theory, we adopt HeavisideLorentz units and put and c equal to unity.
II. BASIC FORMALISM. GREEN'S FUNCTIONS A. Basic Formalism We first have to outline the main ingredients of the Schwinger source theory, when applied to the wedge geometry of Fig. 1. The formalism has strong similarities with the one given recently by Brevik and Nyland, for the case of a single compact dielectric sylinder [6]. This formalism in turn was developed along the same lines as followed previously by DeRaad and Milton, for the case of a perfectly conducting cylindric shell [7]. The basic formalism was originally introduced by Schwinger, DeRaad, and Milton [8]. This kind of theory is strongly related to the fundamental treatise on electromagnetism found in Stratton [9]. The starting point is to express the electric field E(x) in terms of the polarization source P(x$):
|
E(x)= dx$ 1{ (x, x$) P(x$).
(3)
Here 1{ (x, x$) is a dyad with Fourier transform 1{ (r, r$, |): 1{ (x, x$)=
|
&
d| &i|{ e 1{ (r, r$, |), 2?
(4)
{ meaning t&t$. From Maxwell's equations, {_{_1{ (r, r$, |)&| 2 1(r, r$, |)=| 2 1{ $(r&r$),
(5)
where 1{ is the unit dyad. We introduce another dyad 1{ $: 1{ $=1{ +1{ $(r&r$),
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(6)
160
BREVIK AND LYGREN
which has the virtue of being divergence-free. When r{r$, as we shall assume, then 1{ and 1{ $ are equivalent. The effective electric and magnetic field products are as given, for example, in [1, 6, 8]: i( E i (r) E j (r$)) | =1 $ij (r, r$, |), i( H i (r) H j (r$)) | =& =
(7)
1 ({_1{ $_{$) ij |2
1 = = $ 1 $ (r, r$, |). | 2 ikl jmn k m ln
(8)
Here the angle brackets denote the quantum mechanical expectation value. Since 1$ is divergence-free, it is conveniently expanded in cylindrical coordinates. We let % denote the polar angle, so that %=0 and %=: are the wedge surfaces, and let the cusp coincide with the z axis. The boundary conditions are that the electric field is normal, and the magnetic field is tangential, to the surfaces (see Fig. 1). Taking these into account, we can write the expansion for 1$in the region % # [0, :] as 1{ $(r, r$, |)=
2 $ : : m=0
|
&
dk {_z^f m(r, r$) cos(mp%) 2?
_
i + {_{_z^g m(r, r$) sin(mp%) e ikz, |
&
(9)
where k is the axial wave number. The functions f m and g m implicitly carry the second tensor index. The prime on the summation sign means that the m=0 term is taken with half weight. Finally, we have defined p as p=?:.
(10)
We note that the following operator relations are useful: {_z^ {_{_z^
cos mp% sin sin (mp%)&% (mp%) , (mp%) e ikz =e ikz r^ r cos sin r cos cos (mp%) e ikz =e ikz sin
_ cos r^ ik (mp%) _ sin r
%ik {_{_{_z^
&
mp% sin cos (mp%)&z^ (mp%) d m , r cos sin
cos cos (mp%) e ikz =&e ikz {_z^ (mp%)(d m +k 2 ), sin sin
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&
(11)
(12) (13)
CASIMIR EFFECT FOR A WEDGE
Fig. 1.
161
Sketch of the wedge geometry. Boundary conditions are illustrated for the fields.
where d m is the operator dm =
1 m 2p 2 r & 2 . r r r r
(14)
The expansion (9) satisfies the relations % } {_1{ $ | s =0,
%_1{ $ | s =0
(15)
at the wedge surfaces automatically, as it should according to the boundary conditions. It can also be shown by explicit calculation that (9) satisfies the Maxwell equations (5). The calculation is somewhat involved and will not be given here. Instead, we focus attention on the single scalar Green function F m that can be introduced in this problem. Because there is no boundary condition in the radial direction (in contradistinction to the cases studied in [6] and [7]), we need only to introduce one single function of this kind. It is related to the functions f m and g m in (9) by 1 f m =& 2 (d m &k 2 ) f m , | (16) 2 {$_z^ cos(mp%$) e &ikz$F m(r, r$), f m = :
and i g m =& |
2
: {$_{$_z^ sin(mp%$) e
&ikz$
F m(r, r$).
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(17)
162
BREVIK AND LYGREN
From Maxwell's equations we obtain the following equations for F m : 1 1 d D m F m(r, r$)= $(r&r$), 2 m | r
(18)
where D m is the differential operator D m =d m +q 2,
(19)
and q is the transverse wave number, q 2 =| 2 &k 2.
(20)
We are free to define an auxiliary Green function Gm , given by 1 d m Gm = $(r&r$). r
(21)
Thus we get the following governing equation for Fm :
_
D m Gm &
q2 1 F m = $(r&r$). 2 | r
&
(22)
It is seen that this equation corresponds to those for a cylinder [6, 7] with the d m operator slightly modified: m mp. B. The Scalar Green Functions We now impose boundary conditions on the scalar Green function. The function will be required to be bounded as r 0 and to describe outgoing waves as r . The boundary conditions at the wedge surfaces are automatically satisfied by the expansion (9) for 1{ $. Consider first Eq. (21) for the auxiliary Green function. When rewritten in the form
\
1 2 1 p 2m 2 & 2 + Gm = $(r&r$) r 2 r r r r
+
(23)
it becomes apparent that the solution is Gm =&
1 r< 2mp r >
\ +
mp
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(24)
163
CASIMIR EFFECT FOR A WEDGE
for m{0 (actually m>0 in view of (9)), and 1 r< G0 =& ln 2 r>
(25)
for m=0. Here r <(r > ) is the lesser (greater) of r and r$. Consider next Eq. (22). Rewriting it using (19) and (14), we see that the solution contains the ordinary Bessel and Hankel functions of order mp and argument qr. We take into account the discontinuity in slope across r=r$, and use the known value for the Wronskian, W[J &(x), H &(x)]=2i?x, to fix the multiplicative coefficient. The final solution is Fm(r, r$)=
|2 i? Gm(r, r$)+ J mp(qr < ) H mp(qr > ) . q2 2
_
&
(26)
We recall that this expression refers to a vacuum region, in the angular interval % # [0, :].
III. EFFECTIVE FIELD PRODUCTS A. Field Products in the Presence of the Wedge In evaluating the field products, we let two points x and x$ be separated in both radial and angular directions. The separation in radial direction is necessary to avoid the delta function $(r&r$). The separation in angular direction is useful (not necessary) since it will simplify the calculations. Before considering the effective field products explicitly, we express the dyads 1{ $ and {_1{ $_{$ in terms of the function Fm : 1{ $(r, r$, |) =
2 :$ 2 :| m=0
|
&
dk [&({_z^ )({$_z^ )(d m &k 2 ) F m(r, r$) cos(mp%) cos(mp%$) 2?
+({_{_z^ )({$_{$_z^ ) F m(r, r$) sin(mp%) sin(mp%$)] e ik(z&z$),
(27)
{_1{ $(r, r$, |)_{$ =
2 :$ 2 :| m=0
|
&
dk [({_{_z^ )({$_{$_z^ )(d m &k 2 ) F m(r, r$) cos(mp%) cos(mp%$) 2?
&({_z^ )({$_z^ )(d m &k 2 )(d $m &k 2 ) F m(r, r$) sin(mp%) sin(mp%$)] e ik(z&z$).
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(28)
164
BREVIK AND LYGREN
It is here convenient to note at once that the function G0 does not contribute to the dyad 1{ $, so that we can use the expression for Gm , m>0, in our calculations. When m=0 the nonvanishing term in 1{ $ contains ({_z^ )({$_z^ )(d 0 &k 2 ) F 0(r, r$) e ik(z&z$),
(29)
and the contribution from G0 is proportional to 2 (d 0 &k 2 ) G0 & k G0 , r r$ r r$
(30)
in view of the effective relation d 0 G0 =0, Eq. (21), which holds when the two points are kept separated in the radial direction. Then using (25) we see that (30) vanishes. We now consider the field products. We shall henceforth put z=z$ and t=t$. It is convenient to introduce the symbol L for the integral operator L=
2 :
|
&
d| 2?
|
&
dk :$ . 2? m=0
(31)
Some calculation results in the following expressions for the effective field products: i ( E z (r, %) E z (r$, %$)) =
|
&
=L
d| z^ } 1{ $ } z^ 2? i?
_2 q J 2
pm
&
(qr < ) H pm(qr > ) sin(mp%) sin(mp%$) ,
(32)
i ( E % (r, %) E % (r$, %$)) =
|
&
=L +
d| % } 1{ $ } % 2? i?
_ 2 {| J$ (qr 2
mp
<
) H$mp(qr > )
k 2m 2p 2 J mp(qr < ) H mp(qr > ) cos(mp%) cos(mp%$) , q 2rr$
=
&
(33)
i ( E r (r, %) E r (r$, %$)) =
|
&
=L
d| r^ } 1{ $ } r^ 2? i? | 2m 2p 2 J mp(qr < ) H mp(qr > ) 2 q 2rr$
_ {
=
&
+k 2J$mp(qr < ) H$mp(qr > ) sin(mp%) sin(mp%$) ,
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(34)
165
CASIMIR EFFECT FOR A WEDGE
i ( H z (r, %) H z (r$, %$))
|
=&
&
=L
i?
d| z^ } ({_1{ $_{$) } z^ 2?| 2
_2 q J 2
mp
&
(qr < ) H mp(qr > ) cos(mp%) cos(mp%$) ,
(35)
i ( H % (r, %) H % (r$, %$))
|
=&
&
=L
d| % } ({_1{ $_{$) } % 2?| 2
i? k 2m 2p 2 J mp(qr < ) H mp(qr > ) 2 q 2rr$
_ {
=
&
(36)
&
(37)
+| 2J$mp(qr < ) H$mp(qr > ) sin(mp%) sin(mp%$) , i ( H r (r, %) H r (r$, %$))
|
=&
&
=L
d| r^ } ({_1{ $_{$) } r^ 2?| 2
i? | 2m 2p 2 J (qr ) H mp(qr > ) 2 q 2rr$ mp <
_ {
=
+k 2J$mp(qr < ) H$mp(qr > ) cos(mp%) cos(mp%$) .
In these expressions the differences (r&r$) and (%&%$) are arbitrary; they are not necessarily small. We also note that the contributions from the function G m cancel out in all field products; this being closely connected to the effective relations d m Gm =0. From (22) therefore follows another effective relation: d m F m =&q 2F m .
(38)
There are also other field products, such as ( E i E k ) and (H i H k ) with i{k. They will be present in the off-diagonal part of the spatial energy-momentum tensor. This tensor reads, in a vacuum region, Sik =&E i E k &H i H k + 12 $ ik(E 2 +H 2 ).
(39)
It is rather remarkable that the off-diagonal products ( E i E k ) and ( H i H k ) do not vanish by themselves, but that their sum ((E i E k ) +( H i H k ) ), present in (39), does so. This can be verified by explicit calculation, along the same lines as above, although the calculation will not be shown here. Similarly it can be shown that the
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166
BREVIK AND LYGREN
Poynting vector vanishes. The only components of the four-dimensional energymomentum tensor that turn out to be nonvanishing are thus the diagonal stress components S ii (no sum) and the energy density component. B. The Infinite Vacuum Expressions In any physical application the above expressions for the field products have to be regularized by subtracting off the corresponding expressions pertaining to the case of an infinite vacuum, i.e., to the wedge being removed. Probably the most striking formal property due to the presence of the wedge is that the order of the Bessel functions becomes modified: m mp. In the free space, only integral orders m do occur. In the following we let the superscript zero refer to the infinite vacuum field. We obtain i ( E z (r, %) E z (r$, %$)) 0 =i ( H z (r, %) H z (r$, %$)) 0 =
: i? 2 L q J m(qr < ) H m(qr > ) cos m(%&%$) , 2? 2
_
&
(40)
i ( E % (r, %) E % (r$, %$)) 0 =i ( H % (r, %) H % (r$, %$)) 0 =
: i? L | 2J$m(qr < ) H$m(qr > ) 2? 2
_ {
+
k 2m 2 J (qr ) H m(qr > ) cos m(%&%$) , q 2rr$ m <
=
&
(41)
i ( E r (r, %) E r (r$, %$)) 0 =i ( H r (r, %) H r (r$, %$)) 0 =
: i? | 2m 2 L J m(qr < ) H m(qr > ) 2? 2 q 2rr$
_ {
=
&
+k 2J$m(qr < ) H$m(qr > ) cos m(%&%$) .
(42)
The relations obeyed by the off-diagonal components of the stress tensor are the same as noted in the previous subsection.
IV. ENERGY-MOMENTUM TENSOR A. On the Calculation of ( S %% ) We now consider the diagonal spatial components of the energy-momentum tensor, i.e., the stress tensor with the opposite sign. We let henceforth the two points
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CASIMIR EFFECT FOR A WEDGE
approach each other, r r$, but keep both (r&r$) and (%&%$) different from zero. We write ( E 2r ) instead of ( E r(r, %) E r(r$, %$)) | r r$ , etc., to simplify the notation. The most important component of S ik , as far as the surface force is concerned, is S %% . From (39) we have (S %%(r)) = 12 [( E 2r ) &( E 2% ) +( E 2z ) +( H 2r ) &( H 2% ) +( H 2z ) ] r r$ .
(43)
Inserting the product expressions from the previous section, we get ( S %%(r)) =L =
?
1
_ 4 {q +rr$ % %$ & r r$ = J 2
? 2:(2?) 2
_ :$
|
d| &
(qr < ) H mp(qr > ) cos mp(%&%$)
&
r r$
dk &
1
_{q +rr$ % %$ & r r$= J 2
m=0
|
mp
(qr < ) H mp(qr > ) cos mp(%&%$)
mp
&
. r r$
(44) This expression has to be regularized by subtracting its value in vacuum: ( S 0%% (r)) = =
: ? 2 1 L & q + J m(qr < ) H m(qr > ) cos m(%&%$) 2? 2 rr$ % %$ r r$
_{
? (2?) 3
_ :$ m=0
|
=
&
_{
1 & J (qr ) H m(qr > ) cos m(%&%$) rr$ % %$ r r$ m <
q2 +
r r$
|
d|
&
dk &
=
&
. (45) r r$
Comparison between (44) and (45) shows that ( S %% (:=?)) =( S 0%% ). Thus we only need to calculate ( S %% ); the contact term immediately follows from it. Now, making a complex frequency rotation, | i|^, implying the following effect on the variable q: q 2 =&(|^ 2 +k 2 )#&\ 2,
(46)
we obtain ( S %% (r)) =
1 :? 2
|
0
d|^
|
0
1
_{&\ +rr$ % %$& r r$= ) cos mp(%&%$) & ,
dk :$
2
m=0
I mp(\r < ) K mp( \r >
r r$
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(47)
168
BREVIK AND LYGREN
I mp and K mp being modified Bessel functions [10]. We perform the integration in polar coordinates, noting that \ is the radius in the |^k plane. Integrating over angles from 0 to ?2 we get ( S %% (r)) =
1 :$ 2?: m=0 _
|
d\ \ 0
1
_{&\ +rr$ % %$& r r$= I 2
mp
(\r < ) K mp( \r > ) cos mp(%&%$)
&
. r r$
(48) It is convenient to split this expression into two parts S I and S II , defined by 1 $ : S I =& 2?: m=0 S II =
1 :$ 2?: m=0 _
|
|
0
d\ \ 3I mp(\r < ) K mp( \r > ) cos mp(%&%$)| r r$ ,
(49)
d\ \ 0
1
_rr$ % %$& r r$& I
mp
( \r < ) K mp( \r > ) cos mp(%&%$)| r r$ .
(50)
We also introduce the notation z=
r< . r>
(51)
We are going to calculate ( S %% ) =S I +S II using two different methods. The first method is the more general one, valid for all values of the opening angle. The second one is valid when p is an integer. B. Evaluation of ( 3 %% ) for Arbitrary : We first perform the integrations in (49) and (50), keeping the points separated in both r and % directions. Thereafter we do the summation and take the limit r r$. The integral over \ in (49) can be expressed in terms of a hypergeometric function, and further manipulated so as to involve only elementary functions. Equation (50) can be handled similarly. Details are given in Appendix A and we confine ourselves here to giving the final expression for the tensor component ( 3 %% ), where ( 3 +& ) in general is defined to be the regularized energy-momentum tensor: ( 3 +&(r)) =( S +&(r)) &( S 0+&(r)).
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(52)
169
CASIMIR EFFECT FOR A WEDGE
We get for the %% component 3 ?2 +11 ( 3 %%(r)) =& 2 4 720? r : 2
\
+\
?2 &1 . :2
+
(53)
When :=? the wedge degenerates into a plane sheet, and the expression vanishes as it should according to its construction. C. Alternative Evaluation of ( 3 %% ) When p Is an Integer A more simple and elegant way of calculating ( S %% ) and ( 3 %% ) can be given if p is an integer. The method involves use of the generalized Graf addition theorem for modified Bessel functions. Details are given in Appendix B. Using (B.6) we can write Eq. (48) simply as ( S %%(r)) =
1 (2?) 2
|
_
d\ \ &\ 2 +
0
1 & rr$ % %$ r r$
&
p&1
: K 0(\R n )| r r$ ,
(54)
n=0
where from (B.2) R n =[r$ 2 +r 2 &2r$r cos((%&%$)+2?np)] 12.
(55)
We now use the formula, valid for +>&1,
|
1++ 2
_ \ +&
x +K 0(ax) dx=2 +&1a &+&1 1
0
2
(56)
(cf. Eq. (11.4.22) in [10]), to get ( S %%(r)) =
1 1 p&1 4 1 & : & 2+ (2?) 2 n=0 Rn rr$ % %$ r r$ R 2n
_
\
+ &
.
(57)
r r$
Taking the limits of the terms in the square bracket, we get after some calculation 4 R 2n
} 1 1 \rr$ % %$ & r r$ + R } &
=& r r$
2 n r r$
1 , 4r 4 sin 4(?np)
(58)
1 . 2r 4 sin 4(?np)
(59)
=&
Here the n=0 term in (57) is seen to be divergent. This divergence, however, is cancelled by the divergence in the contact term ( S 0%% ) which is to be subtracted off. From [11] we have the summation formula p&1
: n=1
( p 2 +11)( p 2 &1) 1 = , sin (?np) 45 4
whereby we recover finally the result (53) for the tensor component ( 3 %% ).
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170
BREVIK AND LYGREN
D. The Complete Regularized Tensor ( 3 +& ) The remaining components of ( 3 +& ) are found using the same procedure as above. We therefore give here only the results. We define the tensor components in the following way: ( 3 +& ) =( 3 rr , 3 %% , 3 zz , &w),
(61)
where w is the electromagnetic energy density. We get (3 +&(r)) =
1 ?2 +11 720? 2r 4 : 2
\
+\
?2 &1 diag(1, &3, 1, 1). :2
+
(62)
We see that the trace of the energy-momentum tensor is zero, ( 3 ++ ) =0, as it should for the electromagnetic field. Moreover, the energy density w is negative (if :
?2 , 720a 4
(63)
in accordance with the standard result in Casimir theory for two plates. We note that there are two physical conditions to be satisfied here if the medium is a real one: first, we must have ar* 0 , where * 0 is a characteristic wavelength of the absorption spectrum of the medium. Typically, * 0 amounts to a few tenths of nanometers. Secondly, we must have, in dimensional units, aRck B T in order to be able to apply zero temperature theory. In other words, the number of significant Matsubara frequencies is large. Typically, aR10 +m. Equation (63) usually holds in the interval 0.05 +mRaR10 +m. For further discussions on this point the reader may consult, for instance, Refs. [12, 13, 14].
V. ON POSSIBLE EXPERIMENTAL CONSEQUENCES It would be highly desirable to be able to subject the above theory to experimental tests. Probably the idea that lies most closely at hand is the possibility of measuring the surface force. If the wedge is cut out from a massive perfect conductor (of infinite permittivity), then there are only field fluctuations present in the opening, 0 % :. The regularized surface force density F(r) on the horizontal surface in Fig. 1 is then equal to &( 3 %%(%=0)) which means, according to (53), that F(r) becomes equal to the expression given previously in Eq. (2). The force is attractive. The r &4 divergence at the origin is obviously a consequence of the over-idealization of our boundary conditions: in reality there is always, as mentioned earlier, a finite skin depth $ which smears out the physical effect from the mathematically singular
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CASIMIR EFFECT FOR A WEDGE
cusp at the origin. Similarly, the r &4 divergent result obtained if we calculate the moment of force (per unit length in the z direction), 0 rF(r) dr, is a mathematical artifact which is not rooted in the real physics. [It may be noted that this divergence is not related to our model of the medium as being perfectly conducting at all frequencies. We might take dispersion into account by introducing a highfrequency cutoff for |^ in the integral (47), but would always encounter the same kind of r &4 divergence in the result.] The idea of measuring the force (or force moment) on a solid metal block would however be difficult to carry out in practice. Perhaps a better idea is to imagine the wedge to be a thin foil. Then the field fluctuations in both regions, % # [:, 2?] as well as % # [0, :], have to be taken into account. To avoid the singularity at the origin we may assume that a small pivot (with its axis vertically into the paper) has been inserted in the horizontal surface in Fig. 1, at a distance dr$ from the origin. Then the moment of force about the pivot reads, per unit length into the paper, in dimensional units, M# =
|
rF(r) dr d
3c 1440? 2d 2
_\
?2 +11 :2
+\
?2 ?2 &1 & +11 :2 (2?&:) 2
+ \
+\
?2 &1 (2?&:) 2
+& .
(64)
The moment of force will try to diminish the minor angle of Fig. 1. In practice, it is natural to choose :R1. Then, the vacuum fluctuations will try to turn the horizontal surface in the upward direction, about the pivot. What magnitude can we expect for the quantity M ? Let us choose :t0.01 rad (0.57%), and dt100 +m. Then, the minimum active opening between the two foils becomes :dt1 +m, which is much larger than the skin depth $t10 nm for copper, and yet small enough to justify the use of zero temperature theory. Formula (64) then yields Mt6_10 &7 dyn cmcm as a typical force moment on the movable foil. It is questionable whether such a small quantity can be measured. Let us round off this section by making some general remarks. It is almost surprising how few measurements of the Casimir effect there really exist. There are classic experiments of the attractive forces between metal plates, by Sparnaay [15], for the region 0.5&2 +m. By using a capacitive method to determine the interplate distance, deflections as small as 1&10 nm could be measured. There are also well known force experiments by Sabisky and Anderson [16], testing the theoretical prediction (originally due to Lifshitz) accurately over decades of distances. More recent work has been done by the groups of Overbeek [17], Deryagin [18], and Israelachvili [19]. Useful reviews can be found in [19, 20]. However, it turns out to be difficult to obtain satisfactory accuracy in force measurements between plates. There are interfering effects present, such as surface roughness (for a recent theoretical work analysing this kind of effect, see Bordag et al. [21]). Probably the growing atomic force microscope technique [22, 20] can
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BREVIK AND LYGREN
prove to be an effective tool in the Casimir force context [23]. The experimental paper of Onofrio and Carugno [24], in which the use is proposed to dynamically detect Casimir forces between two conducting plates, seems also promising. After all it may be so, however, that force measurements are not the most profitable way of approach, experimentally. Recently Sukenik et al. [25], instead of measuring forces, measured the deflection of a ground-state atomic beam passing through a micron-sized cavity; in this manner they proved clear evidence for the existence of the CasimirPolder force. Their cavity was actually wedge-shaped, with dimensions of the same order of magnitude as we assumed above when estimating the magnitude of the force moment M. It must be noted however, that the Casimir Polder force, although closely related to the Casimir effect, is not really the same as the force between macroscopic surfaces.
VI. FURTHER COMMENTS A. Analogy with Cosmic String As mentioned in Section I, there exists an analogy between our wedge and a cosmic string. Let us briefly illustrate this analogy mathematically. The expression for the line element outside the string is ds 2 =&dt 2 +dr 2 +(1&4G+) 2 r 2 d% 2 +dz 2.
(65)
We can imagine (65) to represent the geometry of locally flat space, with a wedge of angle 8=8?G+ being removed. Let us introduce the symbol ;, defined as ;=(1&4G+) &1 =(1&82?) &1.
(66)
Then, the electromagnetic energy-momentum tensor outside an infinitely thin cosmic string, as calculated by Frolov and Serebriany [26], can be expressed as ( 3 +& ) =
1 ( ; 2 +11)( ; 2 &1) diag(1, &3, 1, 1). 720? 2r 4
(67)
Comparison with (62) immediately tells us what the mathematical analogy is: ; corresponds to p=?: in the wedge geometry. Hence 8 corresponds to 2?&2:. Another point worth noticing is a comparison with the theory of the outside region of a superconducting cosmic string [1]. In that paper, which similarly to the present paper was based upon Schwinger's source theory, the string was assumed to have a finite radius. Omitting the finite-radius terms, we obtain from Eqs. (26) and (27) in [1] the following expressions for the scalar string Green functions (we needed two functions there):
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CASIMIR EFFECT FOR A WEDGE
|2 i?; J m; (qr < ) H m; (qr > ) , G mF (r, r$)+ q2 2
_ | i?; J G (r, r$)+ G (r, r$)= q _ 2 F m(r, r$)=
m
G m
2
m;
(qr < ) H mp (qr >
& ) . &
(68) (69)
These are remarkably similar to the scalar Green function for the wedge, Eq. (26) above. B. Finite Temperatures The above theory refers to zero temperature, T=0. As alread mentioned, this theory usually gives an adequate approximation, even at room temperature, if the distance a between surfaces is small, at1 +m. Under some circumstances it can be of importance to consider the finite temperature version of the theory. We shall not go into detail about this point here, but confine ourselves to giving the formal expression for the azimuthal diagonal component of the energy-momentum tensor, called ( S %% ) T, at finite T. The starting point is the T=0 expression (47) for ( S %% ), after imposition of the complex frequency rotation. Applying standard rules [12] we obtain ( S %%(r)) T =
2k B T :?
|
dk :$ :$
0
n=0 m=0
1
_{&\ +rr$ % %$ 2 n
I mp( \ n r < ) K mp(\ n r > ) cos mp(%&%$) & r r$
=
&
,
(70)
r r$
where |^ n =2?nk B T,
\ 2n =|^ 2n +k 2.
(71)
|^ n are the Matsubara frequencies. Also the other components of the energymomentum tensor are thereby known. C. Comments on a Previous Calculation As mentioned in Section I, Deutsch and Candelas (DC) [3] make use of an integration formulacf. Eq. (1) abovewhich is wrong for the actual choice of parameters. It seems to be worthwile to clear up this confusing point. The core of the problem is that DC use Eq. (1) to derive Eq. (5.35) in [3]. We shall now use a different method, very similar to our calculations above, to derive the same equation. We start from Eqs. (5.27) and (5.30) in [3] for the two scalar Green functions, taking the limits t$, z$ t, z (not r$ r). After frequency rotation,
{
i D(x, x$) = 2 :$ 2? : m=0 N(x, x$)
=
|
_
0
ds
|
|
du uJ mp(ur < ) J mp(ur > ) 0
&
d|
|
&
dk e &s(|
2 +k 2 +u 2 )
sin mp%
{cos mp%
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sin mp%$ . cos mp%$
=
(72)
174
BREVIK AND LYGREN
Making use of the idea of evaluating the integrals over | and k in a (2&=) dimensional space [4], we have
|
ds
0
|
d|
&
|
dk e &s(|
2 +k 2 +u 2 )
&
=
? = = 1 1 1& , u= 2 2
\+ \ +
(73)
whereby the integral over u can be done [5]:
|
u 1&=J mp(ur < ) J mp(ur > ) du
0
=
z mp1(mp+1+=2) = = F mp+1& , 1& ; mp+1; z 2 . 2 r 1(mp+1) 1(=2) 2 2
\
=&1 2&= >
+
(74)
Here z=r <r > , and F is the hypergeometric function. Taking = 0, and using Eq. (A.7), we get
{
i D(x, x$) :$ z mp[cos mp(%&%$)cos mp(%+%$)]. = 2 2 2?:r >(1&z ) m=0 N(x, x$)
=
(75)
Finally using Eq. (A.12) to calculate the two sums, we obtain lim z1
p 1 x $ z mp cos mpx= csc 2 , : 2 1&z m=0 8 2
(76)
whereby we recover Eq. (5.35) in [3]:
{
i D(%, %$) ?(%&%$) ?(%+%$) csc 2 = csc 2 . 2 2 16: r 2: 2: N(%, %$)
=
_
&
(77)
In this manner we have suceeded to avoid using Eq. (1) at all. We have not got any explanation for the fact that although DC use Eq. (1) their final answer is correct.
VII. CONCLUSIONS Let us summarize as follows: (1) The final expression for the regularized energy-momentum tensor ( 3 +& ) =( S +& ) &( S 0+& ) is given by (62). In particular, the electromagnetic energy density w in the wedge region is negative (if :
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CASIMIR EFFECT FOR A WEDGE
(2) Within the framework of Schwinger's source theory, our final result is obtained via two different routes, the first holding for arbitrary values of :, the second holding when p=?: is an integer. Our result for the energy-momentum tensor is in agreement with Deutsch and Candelas [3], although we avoid using Eq. (1) which is actually wrong for the parameter choices made in [3]. (3) As regards possible experimental consequences, the expression (2) for the surface force diverges as r &4. This divergence is merely a mathematical artifact, however, since the infinite conductivity model neglects the finite skin depth $ which is always present in real materials. Imagining the wedge to be a thin foil, typical magnitudes of the moment of force, per unit length, lie around 10 &7 dyn cmcm (Section V). The wedge geometry has actually turned out to be a convenient device for carrying out modern cavity experiments on the CasimirPolder force [25]. (4) The transition to finite temperature theory is accomplished by replacing the integral over imaginary frequencies by a sum over discrete Matsubara frequencies. We give the formal expression for ( S %% ) T in Eq. (70), although we do not study finite temperature effects in detail. (5) For some purposes it is useful to note the formal analogy between wedge geometry and the outside geometry of a straight cosmic string. This analogy is illustrated in Section VI.A.
APPENDIX A: ( S %% ) WHEN : IS ARBITRARY We shall evaluate S I and S II , given by Eqs. (49) and (50) in the text. The starting point is Eq. (6.576.5) in [5], from which we have
|
\ *I &( \r < ) K &( \r > ) d\
0
z &1 =
1
*
1
*
\2+2 +&+ 1 \2+2+ F 2
1&* *+1 >
r
1(&+1)
\
1 * 1 * + +&, + ; &+1; z 2 . 2 2 2 2
+
(A.1)
From this we derive
|
\ 3I &(\r < ) K &( \r > ) d\=
0
|
\I &(\r < ) K &( \r > ) d\= 0
4z &1(2+&) F(2+&, 2; &+1; z 2 ), r 4>1(&+1)
(A.2)
z& F(1+&, 1; &+1; z 2 ). r 2>
(A.3)
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176
BREVIK AND LYGREN
The hypergeometric functions can be transformed according to Eq. (15.3.3) in [10]: F(a, b; c; z)=(1&z) c&a&b F(c&a, c&b; c; z),
(A.4)
and further manipulation can be done using Eq. (15.4.1) in [10]: m
(&m) n (b) n z n , (c) n n!
(A.5)
F(2+&, 2; &+1; z 2 )=
&(1&z 2 )+(1+z 2 ) , (1+&)(1&z 2 ) 3
(A.6)
F(1+&, 1; &+1; z 2 )=
1 . 1&z 2
(A.7)
F(&m, b; c; z)= : n=0
with m=0, 1, 2, ... . We then get
The integrals (A.2) and (A.3) then become
|
\ 3I &(\r < ) K &( \r > ) d\=
0
|
\I &(\r < ) K &( \r > ) d\=
0
4z & [&(1&z 2 )+(1+z 2 )], r (1&z 2 ) 3
(A.8)
z& . r 2>(1&z 2 )
(A.9)
4 >
The r and r$ derivatives in (50) can in view of (A.9) be transferred into % and %$ derivatives in order to simplify the calculation (this is an advantage of our method keeping % and %$ different): z& cos &(%&%$) r r$ r 2>(1&z 2 ) =&z & +
{
_ % r
2 sin &(%&%$) r (1&z 2 ) 2
3 > <
1 6r < + 5 cos &(%&%$) . 3 2 % %$ r > r <(1&z ) r >(1&z 2 ) 3
=
&
(A.10)
Similarly, the main part of (49) can in view of (A.8) be rewritten as
|
d\ \ 3I &( \r < ) K &( \r > ) cos &(%&%$)
0
=
4z & (1&z 2 ) sin &(%&%$)+(1+z 2 ) cos &(%&%$) . r (1&z 2 ) 3 % 4 >
_
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&
(A.11)
177
CASIMIR EFFECT FOR A WEDGE
Since z<1 we can do the summations using the Fourier formulas
1+z p , 2(1&z p )
(A.12)
mp pz p :$ z sin mp(%&%$)| % %$ = , % m=0 (1&z p ) 2
(A.13)
mp p 2z p(1+z p ) :$ z cos mp(%&%$)| % %$ = . % %$ m=0 (1&z p ) 3
(A.14)
:$ z mp cos mp(%&%$)| % %$ = m=0
Inserting these formulas into (49) we get S I =&
p 2pz p(1&z 2 ) (1+z 2 )(1+z p ) + . 2 3 ? r (1&z ) (1&z p ) 2 1&z p 2 4 >
_
&
(A.15)
Subtracting off the term S 0I corresponding to p=1 (i.e., to :=?), and taking the limit z 1, we obtain 3 I #S I &S 0I | z 1 =&
1 ( p 2 +11)( p 2 &1). 720? 2r 4
(A.16)
Correspondingly, S II =
p p 2z p&1(1+z p ) 2z(1+z p ) pz p&1(1+z 2 ) + + , (A.17) ? 2r 4>(1&z 2 ) (1&z 2 )(1&z p ) 2 (1&z p ) 3 (1&z 2 ) 2 (1&z p )
_
&
and 2 ( p 2 +11)( p 2 &1). 3 II #S II &S 0II | z 1 =& 720? 2r 4
(A.18)
[When evaluating the limits above the support from an analytic computer program such as Mathematica or Maple turns out to be very convenient.] Finally forming the sum of 3 I and 3 II , we get 3 ( p 2 +11)( p 2 &1), (3 %%(r)) #3 I +3 II =& 720? 2r 4 in accordance with Eq. (53) in the text.
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(A.19)
178
BREVIK AND LYGREN
APPENDIX B: ADDITION THEOREM FOR MODIFIED BESSEL FUNCTIONS There exists a very useful generalization of the addition theorem for Bessel functions: p&1
: e i&n Z &(kR n )=p n=0
J mp(kr 1 ) Z &+mp(kr 2 ) e imp,,
:
(B.1)
m=&
where R n =[r 21 +r 22 &2r 1 r 2 cos(,+2?np)] 12,
(B.2)
and e 2in =
r 2 &r 1 e &i(,+2?np) . r 2 &r 1 e i(,+2?np)
(B.3)
The function Z & denotes any linear combination of J & and N & (Neumann function). The proof of (B.1) can be effectuated by starting from the usual addition theorem for p=1, Eq. (9.1.79) in [10], which can be written
e i&0 Z &(kR 0 )= :
J l (kr 1 ) Z &+l (kr 2 ) e il,.
(B.4)
l=&
The validity of (B.1) now follows by rewriting its left hand side using (B.4). More details are given by Davies and Sahni [11]. In our context we need the generalized addition theorem for modified Bessel functions. We let Z & =H & and r i ir i , whereby R n iR n . Then (B.1) becomes transformed to p&1
: e i&n K &(kR n )=p n=0
:
I mp(kr 1 ) K &+mp(kr 2 ) e imp,.
(B.5)
m=&
For &=0, this equation can be simplified. Since I &n(z) K &n(z)=I n(z) K n(z) for n equal to an integer or zero, we get p&1
: K 0(kR n )=2p :$ I mp(kr 1 ) K mp(kr 2 ) cos(mp,). n=0
m=0
This theorem is used in Section IV.C.
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(B.6)
CASIMIR EFFECT FOR A WEDGE
179
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