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K A Milton Physi(al Manif<69>stations of z<>ro-Point [n<>rgy

TH( CASIMIR (ff(CT
Physical Manifestations of Zero-Point Energy

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TU[ CASIMIR [ff[(J
Physical Manifestations of Zero-Point [nergy
K A Milton
University of Oklahoma, USA
\<5C>\b<> World Scientific NewJersey• London • Singapore • Hong Kong • Bangalore

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CASIMIR EFFECT: PHYSICAL MANIFESTATIONS OF ZERO-POINT ENERGY Copyright© 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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ISBN 981-02-4397-9
Printed in Singapore by World Scientific Printers

To the memory of Hendrik Brugt Gerhard Casimir, who contributed so prodigiously to physics.

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Preface
I first became interested in the Casimir effect [1] in 1975 when I heard
the lectures of Julian Schwinger on the subject, in which he succeeded in deriving the Casimir force between parallel conducting plates without making reference to zero-point energy, a concept foreign to his non-operator version of quantum field theory, source theory. That presentation shortly
appeared as a brief letter [2] . He justified this publication not merely as
a rederivation of this effect in his own language, but as a resolution of the discrepancy between the finite-temperature effect first obtained by Sauer
and Mehra [3, 4, 5, 6] and that obtained from the Lifshitz formula for parallel dielectrics [7, 8, 9] . Because of this discrepancy, Hargreaves [10]
had called it "desirable that the whole general theory be reexamined and perhaps set up anew." Schwinger attempted just that. Unfortunately, he was unaware that the simple error in Lifshitz's paper had been subsequently corrected, so this was a non-issue. Nevertheless, this sparked an interest in the Casimir effect on Schwinger's part, which continued for the rest of his life.
Within a year or so Lester DeRaad and I, his postdocs at UCLA, joined
Schwinger in reproducing the results of Lifshitz [1 1]; aside from a somewhat
speculative treatment of surface tension, this paper contained few results that were new. Lifshitz wrote a somewhat peevish note to Schwinger com­ plaining about the elevation of the temperature error to a significant dis­ crepancy. But what was significant about this paper was the formulation: A general Green's dyadic procedure was developed that could be applied to a wide variety of problems. That procedure was immediately applied to a recalculation of the Casimir effect of a perfectly conducting spherical
vii

viii

Preface

shell, which contrary to the expectation of Casimir [12] had been shown by Boyer [13] to be repulsive, not attractive. We derived a general formula, but then became stuck on its evaluation for a few months; in the meantime, the paper of Balian and Duplantier [14] appeared. DeRaad and I quickly found an even more accurate method of evaluation, and our confirmation of Boyer's result followed [15].
Thus my interest in the Casimir effect was launched. Before I left UCLA I explored the Casimir effect for a dielectric sphere, with inconclusive but seminal results [16] . My interest was rekindled a year or so later, when Ken Johnson proposed adapting the Casimir effect in a bag model of the vacuum [17] . Since he used the estimate from both interior and exterior modes, which seemed hardly applicable to the confinement situation of QCD, I proposed a better estimate based on interior contributions only [18, 19, 20] , and followed it by examining the local Casimir contribution to the gluon and quark condensates [21 ] . I tried to improve the global estimates of these QCD effects shortly after I moved to Oklahoma, by attempting to elucidate the cutoff dependence of the interior modes [22 ] . I also worked out the unambiguously finite results for massless fermions interior and ex­ terior to a perfect spherical bag [23] (in informal collaboration with Ken Johnson) , and, somewhat earlier with DeRaad, computed the more difficult electromagnetic Casimir effect for a conducting cylinder [24].
In the late 1 980s I was interested for a while in the Casimir effect in Kaluza-Klein spaces, particularly when the dimensionality of the compact­ ified space was even [25, 26, 27], growing out of the work of Appelquist and Chodos [28] and Candelas and Weinberg [29]. A few years later I wrote two papers with Ng on the Maxwell-Chern-Simons Casimir effect [30, 31], the second of which signaled a serious problem for the Casimir effect in a two-dimensional space with a circular boundary. This was clarified shortly thereafter in a paper with Bender [32 ] , where we computed the Casimir effect for a scalar field with Dirichlet boundary conditions on a D­ dimensional sphere. Poles occur for arbitrary positive even D. I extended the work to include the TM modes, which exhibited qualitatively similar behavior [33].
It was the still inadequately understood phenomenon of sonolumines­ cence [34] that sparked some of my most recent work in the field. Schwinger, in the last years of his life, suggested that the mechanism by which sound was converted into light in these repeatedly collapsing air bubbles in water

Preface

ix

had to do with the "dynamical Casimir effect" [35, 36] . After his death, I concluded he was wrong [37, 38] . But, probably the most interesting result of this work was a simple finite calculation of the regulated and renor­ malized van der Waals energy of a dielectric sphere. A year later Brevik, Marachevsky, and I, and others [39, 40, 41, 42] , demonstrated that this coincided with the Casimir energy of a dilute dielectric ball, as formulated by me nearly two decades previously [16], suitably regulated and renormal­ ized. At the same time I discovered that the Casimir energy of a dilute cylinder (with the speed of light the same inside and outside) vanished, as did the regulated van der Waals energy for a purely dielectric cylinder [43, 44]. The significance of these null results is still not clear.
This recounting of my personal odyssey through the Casimir world of course does no justice to the many other workers in the field, whose con­ tributions I will attempt to more fully trace in the following. It is rather intended as a guide to the reader so my own personal biases may be dis­ cerned, biases which will be reflected in the following as well. Although I will attempt to survey the field, I will, of necessity, approach it with my own personal viewpoint. I will make some attempt to survey the literature, but I beg forgiveness from those authors whose work I slight or fail to cite. Hopefully, a document with an individual orientation will still have value in the new millennium.
Finally, I must thank the US Department of Energy for partial support of my research over the years, and my various collaborators whose con­ tributions were invaluable. And most of all, I thank my wife, Margarita B anos-Milton, without whose support none of this would have been possi­ ble.
Kimball A. Milton Norman, Oklahoma
April, 2001

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Contents

Preface

vii

Chapter 1 Introduction to the Casimir Effect

1

1 . 1 Van der Waals Forces

1

1.2 Casimir Effect

3

1.3 Dimensional Dependence

8

1.4 Applications

11

1.5 Local Effects

13

1.6 Sonoluminescence

14

1 .7 Radiative Corrections

15

1.8 Other Topics

16

1.9 Conclusion

16

1 . 10 General References

16

Chapter 2 Casimir Force Between Parallel Plates

19

2 . 1 Introduction . . . . . . . .

19

2.2 Dimensional Regularization

20

2.3 Scalar Green's Function

22

2.4 Massive Scalar . . . . .

28

2.5 Finite Temperature . . .

30

2.6 Electromagnetic Casimir Force

36

2.6.1 Variations . . . .

40

2.7 Fermionic Casimir Force . . .

41

2.7.1 Summing Modes

42

2.7.2 Green's Function Method

44

xi

xii

Contents

Chapter 3 Casimir Force Between Parallel Dielectrics

49

3 . 1 The Lifshitz Theory . . . . . . . . . . . . . . . . . . .

49

3.2 Applications

. . . . . . . . . . . . . . . . . . .

53

3.2.1 Temperature Dependence for Conducting Plates

54

3.2.2 Finite Conductivity . . . . . . . . . . . . . . . .

57

3.2.3 van der Waals Forces . . . . . . . . . . . . . . .

57

3.2.4 Force between Polarizable Molecule and a Dielectric

Plate . . . . . . . . . . . . . . . . . . .

59

3.3 Experimental Verification of the Casimir Effect

61

Chapter 4 Casimir Effect with Perfect Spherical

Boundaries

65

4.1 Electromagnetic Casimir Self-Stress on a Spherical Shell

65

4 . 1 . 1 Temperature Dependence

73

4.2 Fermion Fluctuations

75

Chapter 5 The Casimir Effect of a Dielectric Ball:

The Equivalence of the Casimir Effect and

van der Waals Forces

79

5 . 1 Green's Dyadic Formulation

79

5.2 Stress on the Sphere

82

5.3 Total Energy

84

5.4 Fresnel Drag

86

5.5 Electrostriction

88

5.6 Dilute Dielectric-Diamagnetic Sphere

89

5.6.1 Temperature Dependence

92

5. 7 Dilute Dielectric Ball . . . . . .

93

5.7.1 Temperature Dependence

96

5.8 Conducting Ball . . . . . . . .

97

5.9 Van der Waals Self-Stress for a Dilute Dielectric Sphere

99

5.10 Discussion . . . . . . . . . . . . . . . . . . . .

102

Chapter 6 Application to Hadronic Physics: Zero-Point

Energy in the Bag Model

105

6 . 1 Zero-point Energy o f Confined Gluons

107

6.2 Zero-point Energy of Confined Virtual Quarks

112

6.2.1 Numerical Evaluation . . . . .

113

6.2.1 . 1 J = 1/2 Contribution .. . . .

113

Contents

xiii

6.2.1.2 Sum Over All Modes

1 14

6.2.1.3 Asymptotic Evaluation of Lowest J

Contributions

115

6.3 Discussion and Applications . .

116

6.3.1 Fits to Hadron Masses

118

6.4 Calculation of the Bag Constant

120

6.5 Recent Work

123

Chapter 7 Casimir Effect in Cylindrical Geometries

125

7.1 Conducting Circular Cylinder

125

7 . 1 . 1 Related Work . . . . .

132

7.1.2 Parallelepipeds

133

7.1.3 Wedge-Shaped Regions

133

7.2 Dielectric-Diamagnetic Cylinder - Uniform Speed of Light

1 34

7.2.1 Integral Representation for the Casimir Energy

135

7.2.2 Casimir Energy of an Infinite Cylinder when

tl/-Ll = t2/-L2 . . . . . . . . . . . . . . . . . .

137

7.2.3 Dilute Compact Cylinder and Perfectly Conducting

Cylindrical Shell . . . . . . . . . . . .

142

7.3 Van der Waals Energy of a Dielectric Cylinder .

1 46

Chapter 8 Casimir Effect in Two Dimensions:

The Maxwell-Chern-Simons Casimir Effect

149

8.1 Introduction . . . . . . . . . . . .

149

8.2 Casimir Effect in 2 + 1 Dimensions

151

8.2.1 Temperature Effect . . . . .

158

8.2.2 Discussion . . . . . . . . . .

158

8.2.3 Casimir Force between Chern-Simons Surfaces

159

8.3 Circular Boundary Conditions . . . . . . . . .

160

8.3.1 Casimir Self-Stress on a Circle . . . . .

161

8.3.2 Numerical Results at Zero Temperature

171

8.3.3 High-Temperature Limit

1 75

8.3.4 Discussion . . . . . . . . .

176

8.4 Scalar Casimir Effect on a Circle

1 79

Chapter 9 Casimir Effect on a D-dimensional Sphere

183

9.1 Scalar or TE Modes

183

9.2 TM Modes . . . . .

1 90

xiv

Contents

9.2.1 Energy Derivation

193

9.2.2 Numerical Evaluation of the Stress . . . . .

194

9.2.2.1 Convergent Reformulation of (9.52)

195

9.2.3 Casimir Stress for Integer D ::;: 1

197

9.2.4 Numerical results . . . . . . .

198

9.3 Toward a Finite D = 2 Casimir Effect

199

Chapter 10 Cosmological Implications of the Casimir

Effect

201

10.1 Scalar Casimir Energies in M4 X sN

202

10.1.1 N = 1 . . . . . . . . . .

204

10.1.2 The General Odd-N Case . .

205

10. 1 . 3 The Even-N Case . . . . . .

208

10.1.4 A Simple (-Function Technique

216

10.2 Discussion . . . . . . . . . .

219

10.2.1 Other Work

219

10.3 The Cosmological Constant

220

Chapter 11 Local Effects

223

1 1 . 1 Parallel Plates . . . . .

223

1 1 .2 Local Casimir Effect for Wedge Geometry

228

1 1 .3 Other Work . . . . . . . . . . . . . . . . .

229

1 1 .4 Quark and Gluon Condensates in the Bag Model

229

1 1 .5 Surface Divergences

236

Chapter 12 Sonoluminescence and the Dynamical

Casimir Effect

239

12.1 Introduction . . . . . . . . . . . . . .

239

12.2 The Adiabatic Approximation . . . . .

242

12.3 Discussion of Form of Force on Surface

244

12.4 Bulk Energy

247

12.5 Dynamical Casimir Effect

249

Chapter 13 Radiative Corrections to the Casimir Effect 255

13.1 Formalism for Computing Radiative Corrections . .

257

13.2 Radiative Corrections for Parallel Conducting Plates

259

13.2.1 Other Work . . . . . . . . . . . . . . . . .

262

Contents

XV

13.3 Radiative Corrections for a Spherical Boundary

262

13.4 Conclusions . . . . . . . . . . . . . . .

263

Chapter 14 Conclusions and Outlook

265

Appendix A Relation of Contour Integral Method to

Green's Function Approach

269

Appendix B Casimir Effect for a Closed String

273

B.1 Open Strings . . . . . . . . . . . . . . . . . . . .

275

Bibliography

277

Index

293

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Chapter 1
Introduction to the Casimir Effect

1.1 Van der Waals Forces

The understanding of the nature of the force between molecules has a long
history. We will start our synopsis of that history with van der Waals. It
was early recognized, by Herapath, Joule, Kronig, Clausius and others (for
an annotated bibliography see ter Haar [45]) , that the ideal gas laws of Boyle
and Gay-Lussac could be explained by the kinetic theory of noninteracting
point molecules. However, these laws were hardly exact. Van der Waals [46, 47] found in 1873 that significant improvements could be effected by includ­
ing a finite size of the molecules and weak forces between the molecules. At the time, these forces were introduced in a completely ad hoc manner, by placing two parameters in the equation of state,

(P + :2) (v - b) = RT.

(1.1)

O f course, it required the birth o f quantum mechanics to begin to un­
derstand the origin of atomic and interatomic forces. In 1930 London [48, 49] showed that the force between molecules possessing electric dipole mo­
ments falls off with the distance r between the molecules as 1 /r6. The simple argument goes as follows: The interaction Hamiltonian of a dipole moment d with an electric field E is H = - d · E. From this, one sees that
the the interaction energy between two such dipoles, labelled 1 and 2, is

(1.2) where r is the relative position vector of the two dipoles. Now in first

2

Introduction to the Casimir Effect

order of perturbation theory, the energy is given by (Hint), but this is zero
because the dipoles are oriented randomly, (di) = 0. A nonzero result first
emerges in second order,

V.:eff-_

"' 0
mf'O

(OIHintlm)(miHintiO)
E0 - Em

(1.3)

which evidently gives V.,ff "' r-6. This is a short-distance electrostatic effect.
In 1947 Casimir and Polder [50] included retardation. They found that
at large distances the interaction between the molecules goes like ljr7. This result can be understood by a simple dimensional argument. For
weak electric fields E the relation between the induced dipole moment d and the electric field is linear (isotropy is assumed for convenience) ,

d = aE,

(1.4)

where the constant of proportionality a is called the polarizability. At zero temperature, due to fluctuations in d, the two atoms polarize each other. Because of the following dimensional properties:

(1.5)

where L represents a dimension of length, we conclude that the effective potential between the two polarizable atoms has the form

v.: ff e

"'

ala2 r6

nc
r ,

( 1 . 6)

while at high temperatures the ljr6 behavior is recovered,

v.:eff

"'

a1 a2 r6

kT ,

T -4 00.

(1 .7)

The London result is reproduced by noting that in arguing ( 1 .6) we implic­ itly assumed that r » A, where A is a characteristic wavelength associated with the polarizability, that is

a(w) <20> a(O) for w < >c:·

(1.8)

In the opposite limit,

r «A.

( 1 .9)

Casimir Effect

3

These results, with the precise numerical coefficients, will be derived in Chapter 3.

1.2 Casimir Effect

In 1948 Casimir [1] shifted the emphasis from action at a distance between molecules to local action of fields.* That is, the phenomenon discussed above in terms of fluctuating dipoles can equally be thought of in terms of fluctuating electric fields, in view of the linear relation between these quan­ tities. This apparently rather trivial change of viewpoint opens up a whole new array of phenomena, which we refer to as the Casimir effect. Specifi­ cally, in 1948 Casimir considered two parallel conducting plates separated by a distance a. (See Fig. 2 . 1 . ) Although (E) = 0 if there is no charge on the plates, the same is not true of the square of the fields,
( 1 . 10)
and so the expectation value of the energy,

(1.11)

is not zero. This gives rise t o a measurable force on the plates. It is not

possible without a detailed calculation to determine the sign of the force,

however. It turns out in this circumstance to be attractive. Much of the

following chapter will be devoted to a careful derivation of this force; the

results, as found by Casimir, for the energy per unit area and the force per

unit area are

:F

=

-- dda E

7!"2 =-- 240- a4 fie.

( 1 . 12)

The dependence on a is, of course, completely determined by dimensional considerations. Numerically, the result is quite small,

(1 . 1 3)

* This was due to a comment by Bohr in 1947, to the effect that the van der Waals force "must have something to do with zero-point energy." In April 1948 Casimir communicated a new derivation of the force between an atom and a plate, and between two atoms, based on quantum fluctuations to a meeting in Paris [51] . For further history of the development of the Casimir effect, see Refs. [52, 53] .

4

Introduction to the Casimir Effect

and will be overwhelmed by electrostatic repulsion between the plates if each plate has an excess electron density n greater than 1/a2, from which it is clear that the experiment must be performed at the p,m level. Nev­ ertheless, there have many attempts to directly measure this effect, al­ though somewhat inconclusively [54, 55, 56, 57, 58, 59, 60, 61 , 62, 63,
64, 65] . [The cited measurements include insulators as well as conduct­
ing surfaces; the corresponding theory will be given in Chapter 3.] Until recently, the most convincing experimental evidence came from the study of thin helium films [66] ; there the corresponding Lifshitz theory [7, 67, 8] was confirmed over nearly 5 orders of magnitude in the van der Waals po­
tential (nearly two orders of magnitude in distance) . However, the Casimir
effect between conductors has recently been confirmed to about the 5% level by Lamoreaux [68, 69, 70] , to perhaps 1% by Mohideen and Roy [71 , 72, 73] , and by Erdeth [74] , and to about the same level very recently by Chan, Aksyuk, Kleiman, Bishop, and Capasso [75].
In general, let us define the Casimir effect as the stress on the bounding surfaces when a quantum field is confined to a finite volume of space. The boundaries may be described by real material media, with electromagnetic properties such as dielectric functions, in which case fields will exist on both sides of the material interface. The boundaries may also represent the interface between two different phases of the vacuum of a field theory such as quantum chromodynamics, in which case colored fields may only exist in the interior region. The boundaries may, on the other hand, represent the
topology of space, as in higher-dimensional theories (e.g., Kaluza-Klein or strings) , where the extra dimensions may be curled up into a finite geometry
of a sphere, for example. In any case, the boundaries restrict the modes of the quantum fields, and give rise to measurable and important forces which may be more or less readily calculated. It is the aim of the present monograph to give a unified treatment of all these phenomena, which have implications for physics on all scales, from the substructure of quarks to the large scale structure of the universe. Although similar claims of universality of other particle physics phenomena are often made, the Casimir effect truly does have real-world applications to condensed-matter and atomic physics.
There are many ways in which the Casimir effect may be computed. Perhaps the most obvious procedure is to compute the zero-point energy

Casimir Effect

5

in the presence of the boundaries. Although

1
L 2nw
modes

(1 . 14)

is terribly divergent, it is possible to regulate the sum, subtract off the divergences (one only measures the change from the value of the sum when no boundaries are present), and compute a measurable Casimir energy. (A simple version of this procedure is given in Sec. 2.2.) However, a far superior technique is based upon the use of Green's functions. Because the Green's function represents the vacuum expectation value of the (time­ ordered) product of fields, it is possible to compute the vacuum expectation value of (1 . 1 1 ) , for example, in terms of the Green's function at coincident
arguments. Once the energy U is computed as a function of the coordinates
X of a portion of the boundary, one can compute the force on that portion of the boundary by differentiation,

(1 .15)
Similarly, one can compute the stress-energy tensor, T<>"v, from the Green's
function, and thereby compute the stress on a boundary element,

(1 . 1 6)

where n represents the normal to the surface element da, and where the

brackets represent the vacuum expectation value.

The connection between the sum of the zero-point energies of the modes

and the vacuum expectation value of the field energy may be easily given.

Let us regulate the former with an oscillating exponential:

nw 100 dw w 1- <20> """' io _ w2 2

2 a e-iWa T -- - ft

00 a

- iWT
- 27fi e

<EFBFBD> ""'"
a

w<EFBFBD>

2W
-

-

iE'

(1 .17)

where a labels the modes, and, because we assume T goes to zero through positive values, the contour of integration in the second form may be closed in the lower half plane. For simplicity of notation let us suppose we are dealing with massless scalar modes, for which the eigenfunctions and eigen­ values satisfy

(1.18)

6

Introduction to the Casimir Effect

Because these are presumed normalized·, we may write the second form in
(1 . 1 7) as

( 1 . 1 9)

where the Green's function G(x,t;x' , t') satisfies

( : ) <20>\72 +

2 t2

G(x, t; x' , t') = 5(x <20> x')5(t <20> t'),

(1 .20)

and is related to the vacuum expectation value of the time-ordered product of fields according to

G(x,t;x' , t') = *(T¢(x,t)¢(x' , t')).

(1.21)

For a massless scalar field, the canonical energy-momentum tensor is

(1.22)

The second term involving the Lagrangian in ( 1 .22) may be easily shown
not to contribute when integrated over all space, by virtue of the equation
of motion, <20>82¢ = 0 outside the sources, so we have the result, identifying
the zero-point energy with the vacuum expectation value of the field energy,

<EFBFBD> = J La nwa (dx)(T00(x)).

(1.23)

In the vacuum this is divergent and meaningless. What is observable is the

change in the zero-point energy when matter is introduced. In this way we

can calculate the Casimir forces.

Variational forms may also be given. For example, in Chapter 3 we will

derive the following formula for the variation in the electromagnetic energy

when the dielectric function is varied slightly,

j 5U

=

in - 2

(dr)- d2w7f 5E(r, w)fkk (r, r,, w),

( 1 .24)

Casimir Effect

7

Ez

z=O

z=a

Fig. 1.1 Geometry of parallel dielectric surfaces.
where r is the electric Green's dyadic. From this formula one can easily compute the force between semi-infinite parallel dielectrics, as shown in
Fig. 1 . 1 . The celebrated formula of Lifshitz [7, 67, 8] is obtained in this
way for the force per unit area,

(1.25)

Here the frequency integration has been rotated 1r/2 in the complex fre­ quency plane, ( = -iw, ( real, and the following abbreviations have been
used,

""I

K,
=­

E

(1.26)

It is this formula which was spectacularly confirmed for a thin film of helium on a quartz substrate in a beautiful experiment by Sabisky and Anderson
[66] . By taking the appropriate limit,

1 , E3 --+-

E 1 = Ez ---+ 00,

(1.27)

the Lifshitz formula (1.25) reproduces the Casimir result for parallel con­ ductors, (1.12). Furthermore, by regarding the dielectrics to be tenuous

8

Introduction to the Casimir Effect

gases,
(1.28)
N being the density of molecules of the two types and a being the molecular
polarizabilities, the London [49] and Casimir and Polder [50] intermolecular forces, ( 1 .9) and ( 1 .6), respectively, may be immediately inferred:

r « >.,

( 1 .29)

(1.30)

1.3 Dimensional Dependence

We have already noted above that the sign of the Casimir effect cannot be deduced until after the entire calculation is completed. This is be­ cause the starting expressions are purely formal, and require regulariza­ tion and a careful subtraction of infinities before a finite force can be ex­ tracted. Nevertheless, the results quoted above demonstrate that in the one-dimensional geometries considered to this point the Casimir force is strictly attractive, whether one is dealing with conductors or dielectrics,
and whether the helicity of the field is 0 or 1 . [The same i s true for spin 1 /2; see Sec. 2.7.] This is certainly in accord with the interpretation
of the effect as the sum of van der Waals attractions between molecules.
Accordingly, Casimir suggested in 1956 [12] that the Casimir force could
play the role of a Poincare stress in stabilizing a classical model of the electron. In this way he hoped that a value for the fine structure con­ stant could be calculated. Unfortunately, when Boyer did the calcula­
tion in 1968 [13] he found a result which was repulsive; Boyer's calcula­ tion was a tour de force, and has been independently confirmed [76, 14, 15]: The stress on a perfectly conducting spherical shell of radius a is

Ssphere = 2ahe (0.04618 . . .).

(1.31)

The details o f this calculation will b e given i n Chapter 4. Also there will be given the corresponding result for fermions [23]:

Ssphere = 2ahe (0.0204 . . .) .

(1.32)

Dimensional Dependence

9

These nonintuitive results immediately raise the question of what happens at intermediate dimensions. A partial answer was given in Ref. [24] , where the Casimir effect was derived for a right-circular cylinder, with a small, attractive result for the force per unit area

Fcylinder = -4 ahe (0. 00432 . . .) .

( 1 . 33 )

(See Chapter 7.)

Another context in which the dimensional dependence of the Casimir

effect was studied was for Kaluza-Klein theories in 4 + N dimensions,

= = where the extra dimensions were compactified into a sphere (or products
of spheres) . For odd N, and a single sphere sN, the story was given by Candelas and Weinberg [29] : for example, for a scalar field, for N 1 the Casimir energy is negative; then for N 3, 5, . . . , 19, the Casimir energy

is positive, and for N 2 2 1 the energy becomes increasingly negative. [See

Table 1 0. 1 and Fig. 10.2.] For even N the result is divergent; if a cutoff is

introduced, the coefficient of the logarithmic divergence is negative for all

N [25 ]. [See Table 1 0.2 and Fig. 10.5 . ] These calculations will be treated

in detail in Chapter 10.

Until recently, the balance of our knowledge of the dimensional de­

pendence of the Casimir effect referred to the force computed in paral­

lelepiped geometries, where only interior modes are computed. Calculations

of the Casimir energies inside rectangular cavities were first given by Lukosz

[77] (see also [78, 79] ) and later by Ruggiero, Zimerman, and Villani [80,

81] and by Ambj¢rn and Wolfram [82] . In general these results are highly

questionable, because no exterior contributions can be included, because it

is impossible to separate the Klein-Gordon equation, for example, in the

exterior of a rectangular cavity. (Thus we will defer the discussion of these

calculations until Sees. 6 . 1 and 7.1.2.) The exception is the case of infinite

parallel plates embedded in a D-dimensional space and separated by a dis­

tance 2a; that is, the.re is one longitudinal dimension and D - 1 transverse

dimensions. The result for the force per unit area for a scalar field satisfy­

ing Dirichlet or Neumann boundary conditions as found by Ambj0rn and

= ( ; ) Wolfram [82] is [see (2.35)] :F -a-D-1T2D-27f-(D+l)/2Dr D 1 ((D + 1 ) ,

(1 .34)

which we have plotted in Fig. 1 . 2 . Note that :F has a simple pole (due to the

10

Introduction to the Casimir Effect

<EFBFBD>C+l t::!

0.0

<EFBFBD>

-25. 0-5' .0 -3.0 -1.0 - 1.0 3.0 5.-<2D>"'0
D
Fig. 1 . 2 A plot of the Casimir force per unit area Fin (1.34) for -5 < D < 5 for the case of a slab geometry (two parallel plates) .
gamma function) at D = -1 . However, F is not infinite at the other poles
of the gamma function, which are located at all the negative odd integral values of D, because the Riemann zeta function vanishes at all negative even values of its argument. One interesting and well-known special case
of (1 .34) is D = 1 [83]:
( 1 .35)
where the negative sign indicates that the force is attractive. (This was originally calculated in the context of the string theory for the potential
between heavy quarks.) For the case of D = 3 we recover precisely one­ half Casimir's result ( 1 .12) (of course, with a -->- 2a), suggesting, perhaps
misleadingly, that each electrodynamic mode contributes one half the total. But, all these cases are essentially one-dimensional.
However, more recently we have examined the Casimir force of a scalar field satisfying Dirichlet boundary conditions on a spherical shell in D
dimensions [32] . The details of this calculation will be given in Chap­
ter 9. The numerical results are shown in Fig. 9.1. We find that the

Applications

11

Casimir stress vanishes as D --> oo (largely because the surface area of
a D-dimensional sphere tends to zero as D --> ) oo , and also vanishes
when D is a negative even integer. Remarkably, the stress has simple
poles at positive even integer values of D. These results for scalar, or
in waveguide terminology, TE modes, are generalized to include the TM
modes as well, which behave qualitatively similarly [33]. Does this mean
that we cannot make sense of the Casimir effect in two space dimen­
sions, an arena of tremendous interest in condensed matter physics [30, 31]? Some hints of a resolution of this serious difficulty are suggested in Sec. 9.3.

1.4 Applications
It might seem to the reader that the Casimir effect is an esoteric aspect of quantum mechanics of interest only to specialists. That this is not the case should be apparent from the duality of this effect with van der Waals forces between molecules. The structure of gross matter is therefore intimately tied to the Casimir effect. But we can be more specific in citing true field­ theoretic application of these phenomena.
Perhaps the first extensive reference to the Casimir effect in particle
physics occurred with the development of the bag model of hadrons [84, 85, 86, 87, 88, 89] . There a hadron was modeled as a quark and an an­
tiquark, or three quarks, confined to the interior of a (spherical) cavity. Asymptotic freedom was implemented by positing that the interior of the
cavity was a chromomagnetic vacuum (p, = 1), while the exterior was a
perfect chromomagnetic conductor (p, = oo) . Predictions could readily be made for masses and magnetic moments, in terms of a few parame­ ters, principally, the bag constant B, the strong coupling constant a8, the
radius of the nucleon RN, and the "zero-point energy" parameter Z, as
well as quark masses. These parameters were, in fact, determined from fits to the data. In particular, the parameter Z, which occurred in the
bag-model Hamiltonian as a term -Zja, where a is the bag radius, was assumed to be positive. In fact, Boyer's result [13] already suggested that
was in error, since the Casimir effect for a spherical shell is repulsive. How­ ever, in this situation, there are no exterior modes, so the result is less clear. Nevertheless, it has been argued that a repulsive result is expected
theoretically, and that the model must be modified correspondingly [18,

12

Introduction to the Casimir Effect

1 9] . These issues, and other hadronic applications of the Casimir effect will be treated in Chapter 6. See also Sec. 1 1 .4.
We have already referred to higher-dimensional theories, such as Kaluza­
Klein models, in which extra dimensions are curled up into finite topolog­
ical structures, where the scale is set by the Planck length. This gives
rise to large observable vacuum energy effects [28, 29, 25] . The hope is that quantum fluctuations of the gravitational (and other) fields can sta­
bilize the geometry and explain why all dimensions are not of the scale
LP!anck <20> 10-33 em. Although there is now quite an extensive literature on
this subject, progress has been slow because of the technical difficulties as­
sociated with implementing the required Vilkovisky-DeWitt formalism [90, 91]. The status of this important interface between "unified" theories and
cosmology will be related in Chapter 10.
We will come back to earth with a recounting of the Maxwell-Chern­
Simons Casimir effect in Chapter 8. In two space dimensions it is possible
to introduce a mass term for the photon without spoiling gauge invariance.
That is, in place of the Maxwell Lagrangian we can write

.c

-

- 41 FJ.LV FJ.LV

+

4 1

J.L01{3
JLf

Faf3

A J.L"

( 1 . 36)

The mass term has the form of the Chern-Simons topological Lagrangian
which occur for the anyon fields perhaps relevant for the fractional quantum
Hall effect [92, 93, 94, 95] and for high temperature superconductivity [96, 97, 98] . Here, however, we are regarding AJ.L to be a physical photon field
somehow trapped in two dimensions. We will attempt to make sense of
the Casimir effect in two dimensions by use of the procedure described in
Chapter 9, and contrast scalar and vector fields. Salient features include [30, 31]

• For parallel lines vector and scalar fields give identical attractive
Casimir forces. This again illustrates the universal character of
one-dimensional geometries.
• For a circular boundary vector and scalar fields give completely
different results for the Casimir effect; in the leading approximation, the scalar force is repulsive, while the vector is attractive. • The following dimensional reduction theorem holds true for the
massless theory (Jl = 0): The Casimir effect for a right circular cylinder in (3+1 ) dimensions for a vector field [24] , as described in Chapter 7, reduces, as the component of momentum along the

- ) Local Effects

13

longitudinal axis goes to zero, kz 0, to the sum of the Casimir

effects for a scalar and for a vector field in (2+1) dimensions:

(3 + 1)v--)- (2 + 1)v + (2 + 1)s, (kz---)0). (1.37)

We will discuss the temperature dependence and possibilities of observing
this phenomenon in condensed matter systems in Chapter 8.

1.5 Local Effects

To this point we have considered the Casimir effect as a global phenomenon. The observable Casimir force on a macroscopic bounding surface is a collec­ tive effect, and localization of the phenomenon would seem to be nonunique.
[The nonlocalization of physical phenomena should already be familiar at
the classical level in connection with radiation. For example, the question
of whether a uniformly accelerated charge radiates or not is only answerable if the beginning and end of the acceleration process is specified; and even then it is only a manner of speaking to say that the radiation is associated
with the beginning and end of the process. See, for example, Ref. [99] .]
Nevertheless, we have already indicated that construction of the energy­
momentum tensor T<>"" is an important route toward calculation of the
Casimir effect. Further if one imposes conservation and tracelessness of that tensor in electromagnetism,

(1.38)

one can infer [100] a_unique vacuum expectation value ofthe energy-momen­

tum tensor for the case of parallel conducting plates located at z = 0 and

z = a:

- - (<28> )<29> <20> <20> 7[2

(

T <20>-'"

)

-

-

720a4

1 0 0 -1 0

(1.39)

00 03

The energy density and the force per area given in ( 1 . 12) are contained
in this result. Besides having great interest in their own right, these local
effects could have important gravitational consequences [101]. Local effects
also must be understood if one is to correctly interpret the divergences
inherent in the theory. The nature of boundary divergences, a subject to

14

Introduction to the Casimir Effect

which we will return later, was first studied systematically by Deutsch and Candelas [102 ] . These issues and consequences, particularly in hadronic physics, will be discussed in Chapter 1 1 .

1.6 Sonoluminescence

One of the most intriguing phenomena in physics today is sonolumines­

cence [34] . In the experiment, a small (radius rv 10-3 em) bubble of air or

other gas is injected into water, and subjected to an intense acoustic field

(overpressure rv 1 atm, frequency rv 2 X 104 Hz) . If the parameters are

carefully chosen, the repetitively collapsing bubble emits an intense flash

of light at minimum radius (something like a million optical photons are

emitted per flash) , yet the process is sufficiently non-catastrophic that a

single bubble may continue to undergo collapse and emission 20,000 times

a second for many minutes, if not months. Many curious properties have

been observed, such as sensitivity to small impurities, strong temperature

dependence, necessity of small amounts of noble gases, possible strong iso­

tope effect, etc. [34] .

No convincing theoretical explanation of the light-emission process has

yet been put forward. This is certainly not for want of interesting the­

oretical ideas. One of the most intriguing suggestions was advocated by

Schwinger [35, 36] , based on a reanalysis of the Casimir effect. Specifically,

he proposed that the Casimir effect be generalized to the spherical volume

defined by the bubble (as we will discuss in Chapter 5), and with the static

boundary conditions replaced by time-varying ones. He called this idea

the dynamical Casimir effect. Unfortunately, although Schwinger began

the general reformulation of the static problem in Refs. [103, 104 ] [most of

which had been, unbeknownst to him, given earlier [16] (see Chapter 5), he

did not live to complete the program. Instead, he proposed a rather naive

approximation of subtracting the zero-point energy ! 2: fiw of the medium

from that of the vacuum, leading, for a spherical bubble of radius a in a

medium with index of refraction n, to a Casimir energy proportional to the

volume of the bubble:

J (dk) (1- _!_) Ebulk

=

47ra3 3

(27r)3 <20>2 k

n ·

(1 . 40)

Of course, this is quartically divergent. If one puts in a suitable ultraviolet

Radiative Corrections

15

cutoff, one can indeed obtain the needed 10 MeV per flash. On the other
hand, one might have serious reservations about the physical meaning of
such a divergent result.
In Chapter 5 we will carefully study the basis for this model for sono­ luminescence. We will argue there that the leading term ( 1 .40) is to be
removed by subtracting the contribution the formalism would give if either medium filled all space. Doing so still leaves us with a cubically divergent
Casimir energy; but we will argue further that this cubic divergence can
plausibly be removed as a contribution to the surface energy. The remain­
ing finite energy has been determined by a number of authors [39, 40, 41, 42] to be positive and small:

E c

rv

23(n - 1)2 384na

'

In - 11 « 1,

(1.41)

is at least ten orders of magnitude too small, and of the wrong sign, to be relevant to sonoluminescence. This result is also equivalent to the finite van
der Waals self-interaction of a spherical bubble [38], as shown in Sec. 5.9.
It remains to be confirmed whether this adiabatic approximation is valid in the extreme situation present in the sonoluminescing environment. A dynamical calculation is called for, and first steps toward that theory will be sketched. That, and a discussion of the contradictory literature on this
evolving subject, will be detailed in Chapter 12.

1.7 Radiative Corrections
All of the effects so far described are at the one-loop quantum level. Two­
loop effects have been considered by a few authors [105, 106, 107] . Results
have been given both for parallel conducting plates and a conducting spher­
ical shell, and will be described in Chapter 13. These effects are certainly
negligible in QED-the typical correction is down not merely by a factor of the fine structure constant a, but by the ratio of the (small) Compton wavelength Ac of the electron to the geometrical size a of the macroscopic system. However, such corrections could be important in hadronic systems,
where O:s "' 1 and Ac rv a; but there the relevant calculations have not been done. For the status of this important topic, see Chapter 13.

16

Introduction to the Casimir Effect

1.8 Other Topics

The above summary does not do justice to all the work carried out over many years on the theory and applications of quantum vacuum energy. Many of these topics will come up in the appropriate places in the text. For example, the Casimir effect could be relevant to the physics of cosmic strings, and a brief discussion of some of the literature on this subject will
appear in Chapters 7 and 1 1 . The Casimir energy of a closed string itself
will be discussed in Appendix B.

1.9 Conclusion
It is the aim of this monograph to provide a unified, yet comprehensive, treatment of the Casimir effect in a wide variety of domains. Although from textbooks one might conclude that the Casimir effect is an esoteric subject with little practical consequence, I hope this introduction has convinced the reader of the pervasive nature of the zero-point fluctuation phenomena. These phenomena lie at the very heart of quantum mechanics, and, as noted above, what we discuss here are just the first quantum corrections to classical configurations. The subtleties and difficulties encountered in all but the simplest of the Casimir effect calculations demonstrate that we are only beginning to understand the quantum nature of the universe.

1.10 General References
Mathematical references used freely throughout this book include Whit-­
taker and Watson [108] , Gradshteyn and Rhyzik [109] , Prudnikov, Brychkov, and Marichev [1 10] , and Abramowitz and Stegun [11 1].
Finally, reference should be made to review articles on the Casimir
effect and its applications, by Plunien, Muller, and Greiner [112] and by Mostepanenko and 'Irunov [1 13] . The latter authors have also written a book-length review of the subject [1 14] . Marginally related are books by Levin and Micha [1 15] and by Krech [1 16] . An excellent book is that of Milonni [1 1 7] , but the orientation of that treatise is quite different.
After completion of this manuscript, a long review article has appeared
by Bordag, Mohideen, and Mostepanenko [118] , which is quite complemen-

General References

17

tary to the present volume, being in the end primarily concerned with the experimental situation.

Tilis page is intentionally left blank

Chapter 2
Casimir Force Between Parallel Plates
2.1 Introduction
It is often stated that zero-point energy in quantum field theory is not observable, and for this reason the theory should be defined by normal ordering. That such a conclusion is incorrect was recognized by Casimir in
1948 when he showed that zero-point fluctuations in electromagnetic fields
gave rise to an attractive force between parallel, perfectly conducting plates
[1] . His result, at zero temperature, for the force per unit area between such
plates separated by a distance a is
(2.1) Experiments have confirmed this Casimir Effect. Our aim in this chapter is
to rederive Casimir's result using careful Green's function techniques which should lay to rest any uneasiness concerning control of infinities in the prob­ lem. The formalism developed here will be applied in subsequent chapters to derive Casimir forces in more complicated geometries and topologies, and make application to fundamental physics issues from hadrons to cosmology.
In this chapter, we will first, in Sec. 2.2, provide a simple, unphysical,
derivation of the Casimir effect between two idealized plates using dimen­ sional regularization. The Green's function approach in the case of a scalar field satisfying Dirichlet boundary conditions will then be given in Sec.
2.3. Here we calculate the force on one of the plates by looking both at
the normal-normal component of the stress tensor, and by computing the
Casimir energy. In Sec. 2.4 we consider a massive scalar field. The nonzero temperature Casimir effect is examined in Sec. 2.5, with specific attention
19

20

Casimir Force Between Parallel Plates

to the high- and low-temperature limits. The full electromagnetic case is
treated at last in Sec. 2.6, where we introduce a Green's dyadic formula­ tion. The work of the chapter is concluded with Sec. 2. 7, where the Casimir
force on parallel surfaces due to fluctuations in a massless fermionic field
satisfying bag-model boundary conditions is treated.
The Casimir effect evolved out of the earlier work, published the same
year, 1948, by Casimir and Polder, who considered the retarded dispersive forces between polarizable atoms, the constituents of dielectric media [50] .
(As mentioned, this result was verified by a cavity calculation involving
zero-point energy [51].) In particular, Casimir and later, and more explic­
itly, Lifshitz recognized that the "Casimir" forces between bodies having different dielectric constants can be interpreted, in the limit of tenuous me­
dia, to arise from the retarded (1 /r7) and the short-range ( 1/r6) van der
Waals potentials between the molecules which make up the bodies, and that
these van der Waals forces are a result of quantum fluctuations. We will
discuss these questions and their experimental consequences in Chapter 3.

2.2 Dimensional Regularization

We begin by presenting a simple, "modern," derivation of the Casimir ef­ fect in its original context, the electromagnetic force between parallel, un­ charged, perfectly conducting plates. No attempt at rigor will be given, for the same formulae will be derived by a consistent Green's function tech­ nique in the following section. Nevertheless, the procedure illustrated here correctly produces the finite, observable force starting from a divergent for­ mal expression, without any explicit subtractions, and is therefore of great utility in practice.
For simplicity we consider a massless scalar field cjJ confined between
two parallel plates separated by a distance a. (See Fig. 2 . 1 . ) Assume that
the field satisfies Dirichlet boundary conditions on the plates, that is

cjJ(z = 0) = cjJ(z =a) = 0.

(2.2)

The Casimir force between the plates results from the zero-point energy per unit transverse area

(2.3)

Dimensional Regularization

21

z=O

z=a

Fig. 2. 1 Geometry of parallel, infinitesimal plates.

where we have set h = c = 1, and introduced normal modes labeled by the positive integer n and the transverse momentum k.
To evaluate (2.3) we employ dimensional regularization. That is, we
let the transverse dimension be d, which we will subsequently treat as a
continuous, complex variable. It is also convenient to employ the Schwinger
proper-time representation for the square root:

where we have used the Euler representation for the gamma function. We
next carry out the Gaussian integration over k:

Finally, we again use the Euler representation, and carry out the sum over

( ) ( ) n by use of the definition of the Riemann zeta function:

E

=

-

1 4ft

1 (47r)d/2

7r -;:;

d+l r

- d-+2-1

((-d- 1).

(2.6)

When d is an odd integer, this expression is indeterminate, but we can use

22

Casimir Force Between Parallel Plates

G) C ; ) the reflection property r ((z)7r-z/2 = r

z ((1 - z)7r(z-l)/2

(2.7)

to rewrite (2.6) as

(2.8)

We note that analytic continuation in d is involved here: (2.5) is only valid

if Re d < - 1 and the subsequent definition of the zeta function is only valid

if Re d < -2. In the physical applications, d is a positive integer.

We evaluate this general result (2.8) at d = 2. This gives for the energy

- per unit area in the transverse direction

E

=

-

1

71"2
440

-a1 3

'

(2.9)

where we have recalled that ((4) = 1r4/90. The force per unit area between

the plates is obtained by taking the negative derivative of u with respect

to a:

= - Fs

-

- &[)a

[

=

-

11"2 480

-a1 4

.

(2.10)

The above result (2.10) represents the Casimir force due to a scalar field.
It is tempting (and, in this case, is correct) to suppose that to obtain the force due to electromagnetic field fluctuations between parallel conducting
plates, we simply multiply by a factor of 2 to account for the two polariza­
tion states of the photon. Doing so reproduces the classic result of Casimir
(2.1):

Fern

=

-

71"2
240

1 a4

.

(2.11)

A correct derivation of this result will be given in Sec. 2.6.

2.3 Scalar Green's Function
We now rederive the result o f Sec. 2.2 by a physical and rigorous Green's
function approach. The equation of motion of a massless scalar field </>

Scalar Green's Function

23

produced by a source K is

(2.12)

from which we deduce the equation satisfied by the corresponding Green's function

-82G(x, x') = b(x - x') .

(2.13)

For the geometry shown in Fig. 2.1, we introduce a reduced Green's function g(z, z') according to the Fourier transformation

J J G(x, x') =

(

ddk 2 7r ) d

e

' k

·

(

x-

x

'

)

dw 27r

e

-t· w

(t -t'

)

g

(

z

,

,
z

)

,

(2.14)

where we have suppressed the dependence of g on k and w, and have allowed
z on the right hand side to represent the coordinate perpendicular to the plates. The reduced Green's function satisfies

(- ::2 ) - A2 g(z, z') = b(z - z'),

(2.15)

where >.2 = w2 - k2 . Equation (2. 15) is to be solved subject to the boundary conditions (2.2) , or

g(O, z') = g(a, z') = 0.

( 2 . 1 6)

We solve (2.15) by the standard discontinuity method. The form of the

solution is

{ g(z' z') =

A sin >.z,

O < z < z' < a,

B sin >.(z - a) , a > z > z' > O,

(2.17)

= which makes use of the boundary condition on the plates (2.16) . According
to (2.15), g is continuous at z z', but its derivative has a discontinuity:

A sin >.z' - B sin >.(z' - a) = 0, A>. cos >.z' - B>. cos >.(z' - a) = 1 .

The solution t o this system o f equations is

A

=

-

<EFBFBD> >.

sin

>.(z' sin >.a

a) '

B

=

-

<EFBFBD> >.

sin Az' sin >.a

'

( 2 . 1 8 a) ( 2 . 18b)
( 2 . 1 9 a) (2.19b)

24

Casimir Force Between Parallel Plates

which implies that the reduced Green's function is

g(z,

z')

=

-

1
).. s1.n )..a

sin .Xz<

sin

.X (z>

-

a),

(2.20)

where z> (z< ) is the greater (lesser) of z and z'. From knowledge of the Green's function we can calculate the force on
the bounding surfaces from the energy-momentum or stress tensor. For a scalar field, the stress tensor* is

(2.21)

where the Lagrap.ge density is

(2.22)

What we require is the vacuum expectation value of T11v which can be obtained from the Green's function according to

(¢(x)¢(x')) = <20>G(x, x') , 2

(2.23)

a time-ordered product being understood in the vacuum expectation value.

By virtue of the boundary condition (2.2) we compute the normal-normal

component of the stress tensor for a given w and k (denoted by a lowercase letter) on the boundaries to be

(tzz) = ;/]z8z,g(z, z')lz--+z'=O,a = <20>.X cot .Xa.

(2.24)

We now must integrate on the transverse momentum and the frequency to get the force per unit area. The latter integral is best done by performing a complex frequency rotation,

W ---+ i(, A ---+ i )k2 + (2 = i<>.

(2.25)

Thus, the force per unit area is given by

J J :F

=

-

1
2

dd k (2n)d

d( 2n

<EFBFBD>

coth

<EFBFBD>a.

This integral does not exist.

(2.26)

*The ambiguity in defining the stress tensor has no effect. We can add to Tl"v an arbitrary multiple of (81"8v - 9!"v82)cp2 [119, 120] . But the zz component of this tensor on the surface vanishes by virtue of (2.2). Locally, however, there is an effect. See
Chapter 11.

Scalar Green's Function

25

What we do now is regard the right boundary at z = a, for example,
to be a perfect conductor of infinitesimal thickness, and consider the flux
of momentum to the right of that surface. To do this, we find the Green's
function which vanishes at z = a, and has outgoing boundary conditions as
e'kz z -->- oo, ,.., A calculation just like that which led to (2.20) yields for z, z' > a,

(2.27)

The corresponding normal-normal component of the stress tensor at z = a lS

(2.28)

So, from the discontinuity in tzz, that is, the difference between (2.24) and (2.28), we find the force per unit area on the conducting surface:

J J F

=

1 2 - -

dd k - (211- ')d

- 2d1(1' <20><>:(coth <20><>:a - 1 ) .

(2.29)

We evaluate this integral using polar coordinates:

F

=

-

(

Ad+ I 211')d+

l

Jro =

<EFBFBD><EFBFBD>:d

d<EFBFBD><EFBFBD>:

e

<EFBFBD><EFBFBD>:
2<EFBFBD><a

-

1

.

(2.30)

Here An is the surface area of a unit sphere in n dimensions, which is most
easily found by integrating the multiple Gaussian integral

(2.31)

in polar coordinates. The result is

An

=

21l'n/2
f(n/2)

'

When we substitute this into (2.30) and use the identity

(2.32)

(2.33)

as well as one of the defining equations for the Riemann zeta function,

Jro= dy eyYs--11 = r(s)((s),

(2.34)

26

Casimir Force Between Parallel Plates

we find for the force per unit transverse area

(2.35)

Evidently, (2.35) is the negative derivative of the Casimir energy (2.8) with respect to the separation between the plates:

:F

=

-

a£ oa

. '

(2.36)

this result has now been obtained by a completely well-defined approach.
The force per unit area, (2.35), is plotted in Fig. 1 .2, where a __,_ 2a and
d = D - 1.
We can also derive the same result by computing the energy from the energy-momentum tensort . The relevant component is+

( 2 . 37)

so when the vacuum expectation value is taken, we find from (2.20)

(too)

=

-

-2-v4,

<EFBFBD>
sm Aa

[(w2

+

k2

)

sin

>..z

sin

>..

(

z

- a)

+ >..2 cos >..z cos >..(z - a)]

-

22').

1 sm.

>.. a

[w2

cos

>..a

-

k2

cos

>..(2z-

a)]

.

(2.38)

We now must integrate this over z to find the energy per area between the plates. Integration of the second term in (2.38) gives a constant, indepen­ dent of a, which will not contribute to the force. The first term gives

1a0 dz (too) = - w<> 222/a1 cot >..a.

(2.39)

As above, we now integrate over w and k, after we perform the complex
frequency rotation. We obtain

( 2 . 40 )
tAgain, the ambiguity in the stress tensor is without effect, because the extra term here is '172cf,2 , which upon integration over space becomes a vanishing surface integral. +As noted after (1.22) , we would get the same integrated energy if we dropped the second,
Lagrangian, term in Too there, that is, used Too = &oc/J&oc/J.

Scalar Green's Function

27

If we introduce polar coordinates so that ( = "' cos B, we see that this differs from (2.26) by the factor of a(cos2 B) . Here

(cos2 B)

=

f07r

cos2 B sind- 1 B dB fo1r sind- 1 B dB

=

d

1 +

1

'

(2.41)

which uses the integral

(2.42)

Thus, we again recover (2.8).
For the sake of completeness, we note that it is also possible to use the eigenfunction expansion for the reduced Green's function. That expansion is

g(z,

z')

_ -

<EFBFBD>
a

<EFBFBD>L..J
n=1

sin(mrz/a) sin(mrz'fa) n21r2;a2 - A' 2

·

(2.43)

When we insert this into the stress tensor we encounter

(2.44)

We subtract and add .\2 to the numerator of this divergent sum, and omit

the divergent part, which is simply a constant in .\. As we will discuss more

fully later, such terms correspond to 8 functions in space and time (contact

terms) , and should be omitted, since we are considering the limit as the

space-time points coincide. We evaluate the resulting finite sum by use of

the following expression for the cotangent:

00 cot

1rx

=

- 7r1X

+

-21xr

k"L=".'1.J

1 x2 - k2 --- -= -=-=-

(2.45)

So in place of (2.24) we obtain

which agrees with (2.24) apart from an additional contact term.

(2.46)

28

Casimir Force Between Parallel Plates

In passing, we note that in the case of periodic boundary conditions,

(2.43) is replaced by

<EFBFBD> g (z ,

z')

=

<EFBFBD> 1

n

00
oo

ein2 1r ( z - z ' ) j a
(2mr/a)2 - ,\2

'

(2.47)

so we see immediately that

(2.48)

for the forces referring to periodic and Dirichlet boundary conditions. The corresponding energies are related by

£p(a) = 2£D(a/2).

(2.49)

See Ref. [82] .

Yet another method was proposed by Schwinger in Ref. [103] , based on

the proper-time representation of the effective action,

1oo -<2D> W =

2

ds e-isH '
sa ---+0 S

( 2 . 50 )

see Ref. [121]. He used it there in attempting to construct the Casimir energy of a dielectric sphere, which we shall discuss in Chapter 5, but
it may be easily applied to the calculation of the Casimir effect between
parallel plates<65>see Refs. [122, 123, 124, 125, 126] .
--' Bordag, Hennig, and Robaschik [127] consider the Casimir effect be­
tween plates described by 8-function potentials. If g is the strength of the potential, as ag oo we recover the Casimir energy for Dirichlet plates, (2.9).

2.4 Massive Scalar

It is easy to modify the discussion of Sec. 2.3 to include a mass f-1 for the

( ::2 - ) scalar field. The reduced Green's function now satisfies the equation

-

,\2 g(z, z') = o(z - z'),

(2.51)

where

(2.52)

Massive Scalar

29

0.020

;?" <:l:
<EFBFBD>':<;:;- 0.010

Fig. 2.2 Scalar Casimir force per unit area, :F, between parallel plates a.s a function of
ma.ss for d = 2.
instead of (2. 15) , so the reduced Green's function between the plates has just the form (2.20). The calculation proceeds just as in Sec. 2.3, and we find, in place of (2.30)
(2.53)
When we substitute the value of Ad+l given by (2.32) , and introduce a
dimensionless integration variable, we find for the force per unit area
For d = 2 this function is plotted in Fig. 2.2. Ambj0rn and Wolfram [82] considered the case of massive fields in D dimensions, which had been first treated by Hays in the two-dimensional case [128] . They expressed
the result in terms of the corresponding energy, which can be presented in
alternative forms (there is a misprint in (2.18) of Ref. [82] for the first form

30

Casimir Force Between Parallel Plates

here)

1
ad+ l

1
2d+ln(d+l)/2r

( d!l )

J{o 'XJ

d t

td ln

( 1

-

e-2yft2+fJ.2a2 )

( 2 . 5 5 a)

(2.55b)

where the last form is a rapidly convergent sum of modified Bessel functions.

Because the Casimir energy a massive scalar field between parallel plates

vanishes exponentially as the mass goes to infinity, we anticipate that the

Casimir energy, nonrelativistically, is zero. One can see this directly using

a simple zeta-function regularization technique. Write w = p2/2t-t, and

evaluate the corresponding zero-point energy by writing (s --+ 1 )

<EFBFBD> J ( ) 1
2 Lw

=

1
4t-t

(X)

ddk
(2n)d

k2

+

n27r2
<EFBFBD>

s

2d+2n(

1
l+d)/2t-tad+2s

r(1

/2

+

s

+ d/2)(
r(-s)

(

1

+

2s

+

d)

.

(2.56)

---+ [see (2.4)] . Notice that this reduces to 1/2t-t times (2.8) for s = 1/2. Evi­
dently, the s 1 limit vanishes for all d > -2.

2 . 5 Finite Temperature
We next turn to a consideration of the Casimir effect at nonzero tempera­ ture. In this case, fluctuations arise not only from quantum mechanics but from thermal effects. In fact, as we will shortly see, the high-temperature limit is a purely classical phenomenon. Finite temperature effects were first
discussed by Lifshitz [7] , but then considered more fully by Fierz, Sauer, and Mehra [129, 3, 4] . (Fi<46>rz's early calculation referred only to the energy, and not the free energy or force.) Hargreaves [10] analyzed the discrepancy between the results of Lifshitz [7] and Sauer [3] , which turned out to be the result of transcription errors in the former paper [130] . An excellent treat­
ment of the parallel plate problem using the stress-tensor approach and the
method of images was given by Brown and Maclay [100], for both zero and
finite temperatures. A multiple-scattering formulation was presented by
Balian and Duplantier [13 1 , 14] .

Finite Temperature

31

Formally, we can easily obtain the expression for the Casimir force be­ tween parallel plates at nonzero temperature. In (2.29) we replace the imaginary frequency by

(2.57)

where f3 = 1/kT, T being the absolute temperature. Correspondingly, the frequency integral is replaced by a sum on the integer n:

!00 - oo

d( -27r

---t

1 -(3

n

I0:0
= - oo

.

(2.58)

Thus, (2.29) is replaced by

(2.59)

where "'n = Jk2 + (2nn/(3)2. We first consider the high-temperature limit. When T ---> oo ((3 -->- 0),
apart from exponentially small corrections (considered in Sec. 3.2), the contribution comes from the n = 0 term in the sum in (2.59). That integral is easily worked out in polar coordinates using (2.32) and (2.34). The result is

(2.60)

In particular, for two and three dimensions, d = 1 and d = 2, respectively,

d=1: d=2;

:FT-->oo rv -kT<6B>
82(n4(a3a2)3'. :FT-->oo rv -kT

(2.61a) (2.61b)

Notethatifwe applythesame procedure to the energyexpression (2.40) we find that the energy vanishes in the high temperature limit, because (0 = 0. Accordingly, the entropy approaches a constant. This is as expected from thermodynamics, as discussed by Ref. [132].
This high-temperature limit should be classical. Indeed, we can de­ rive this same result from the classical limit of statistical mechanics. The

32

Casimir Force Between Parallel Plates

Helmholtz free energy for massless bosons is F = -kTln Z, ln Z = - 2:)n(1 - e-f3Pi), (2.62)

from which the pressure on the plates can be obtained by differentiation:

We

make

the

momentum-space

p=
sum

- 8avF· explicit

for

our

d

+

1

spatial

(2.63)
geometry:

(2.64)

Now, for high the first order
F = Vk

TtteeJ2rma_mp!d!e:is_nraJ/t3u. (rdW2e7d,rke)fd3can-n-=f+- -w0o,roiw!t2ee/3te2hsxep(arnenas2du72rl2tth+aes

exponential, k2)s ls=O'

a

n

d
(

keep
2.65)

where we have used the identity

ln<EFBFBD> = :s<>t=O·

(2.66)

This trick allows us to proceed done, the s derivative acts only

as on

1in/rS(e-cs. )2:

.

2.

After

the

k

integration

is

1r(<28>s) ls=O = -1,

(2.67)

( so we easily find the result from (2.7)

F

=

-V

kT
(2a.jif) d+ I

(

(d

+

1

)

r

1 d) -+2- .

(2.68)

The pressure, the force per unit area on the plates, is obtained by applying the following differential operator to the free energy:

(2.69)

where V = Aa; A being the d-dimensional area ofthe plates. The result of this operation coincides with (2.60).
The low-temperature limit (T ---+ 0 or f3 ---+ ) oo is more complicated because P is not analytic at T = 0. The most convenient way to proceed

Finite Temperature

33

is to resum the series in (2.59) by means ofthe Poisson sum formula, which says that if c(a) is the Fourier transform of b(x),

c(a)

=

1 - 21!"

100 - 00

b ( x ) e- u.:>x

dx,

( 2 . 70 )

then the following identity holds:

00

00

L b(n) = 21r L c(21rn).

n = - cx:>

n= - oo

(2. 71)

Here, we take

(2.72)

Introducing polar coordinates for k, changing from k to the dimensionless integration variable z = 2a"'x' and interchanging the order of x and z integration, we find for the Fourier transform§

= c(a)

Ad 2 2d1rd+ 1 ad+ 1

ioo
0

dz z2 z - 1 - e --

1f3z/4d-rxra 0

cos ax

( z 2

-

(- 41l("3ax ) 2)

(d-2)/2

(2.73)

The x integral in (2.73) is easily expressed in terms of a Bessel function: It

lS

cos ) - (J4z1d- l"a-1

1
0

1

du

( -a41(l-J"az u (1 - u2) (d-2)/2

2 -

a( l - d) /2z (d- l) / 2

( - 81t"a ) (3

(

d

-

3)

/

2
yc1l"

r

( d )
-

J(d- 1)/2

( a(Jz ) -41-l"a

·

( 2 . 74 )

We thus encounter the z integral

I(s)

=

1oo
0

dz z(d+3)/2 ez - 1

J(d- 1)j2 (sz) ,

(2.75)

where s = a(J/41l"a. The zero-temperature limit comes entirely from a = 0:

(2. 76)

§ It is obvious that d = 2 is an especially simple case. The calculation then is described
in Sec. 3.2.

34

Casimir Force Between Parallel Plates

So we require the small-x behavior of the Bessel function,

(X) X --+- J(d-1)/2(x) rv

2

(d-1)/2 1 r ( d!1 ) '

0,

( 2 . 77 )

whence ( ) ( ) I(s) "'

-s 2

(d-

1)/2

- 2d+ l y?r

- d +2 1

r

-d 2

+

1

((d + 2)'

s --+- 0.

(2.78)

Inserting this into (2.73) we immediately recover the zero-temperature re­

sult (2.35).

We now seek the leading correction to this. We rewrite I(s) as

I(s)

=

1 s(d+5)/2

r=
fo

dy y 1-

(ed+- 3y)f/s2

e-

yfs

J(d-

1

)j2

(Y)

100 00 1 , 8 (d+5) /2

0

dY Y(d+3)/2e-yfs n" L =O e-nyfsJ(d- 1)/2 (Y)

(2 ·79)

where we have employed the geometric series. The Bessel-function integral

has an elementary form:

r=
J 0

d y

J q

( y )

e

-

by

=

(y'ij2"TI - b)q y'ij2"TI

( 2. 80)

) p (!!_)p is the fundamental integral, and the form we want can be written as

r=
}0

dy 1q (y

y

e-by

=

(-1)P

db

(Jb2+1 - b)q Jb2 + 1 '

(2.81)

provided p is a nonnegative integer. (For the application here, this means

d is odd, but we will be able to analytically continue the final result to arbitrary d.) Then we can write I(s) in terms ofthe series

00 I(s)

=

( - 1 ) (d+3) /2 s(d+S)/2

L
1=1

f(l),

l = n + 1,

(2.82)

where

( ) [J ] f(l) =

- d (d+3)/2 dljs

(

<EFBFBD> )

2

+

1

-

<EFBFBD>

(d- 1)/2

V(<28>)2 + 1

(2.83)

Finite Temperature

35

We evaluate the sum in (2.82) by means ofthe Euler-Maclaurin summation formula, which has the following formal expression:
<EFBFBD>f(l) = 100dlf(l) + <20>[f(oo) + f(1)]

+

f k=1

(2<>) ! B2k [f(2k- 1) (oo)

-

j(2k-1)

(1)].

(2.84)

Here, Bn represents the nth Bernoulli number. Since we are considering

a =/= 0 (the a = 0 term was dealt with in the previous paragraph), the low-temperature limit corresponds to the limit s -->- oo. It is easy then to see that f(oo) and all its derivatives there vanish. The function f at 1 has

( the general form

j(1) -

!_ dE

)

(d+3)

/2

[v'f2+1 - E] (d-
v'f2+1

1)

/2

'

E s = 1 / 0. __,_

(2.85)

By examining the various possibilities for odd d, d = 1, d = 3, d =5, and

so on, we find the result

Because it is easily seen that before the limit E ---> 0 is taken, f(1) is an even function of E, it follows that the odd derivatives of f evaluated at 1 that appear in (2.84) vanish. Finally, the integral in (2.84) is that ofa total

( derivative:

J1

oo

d

l

f

(

l

)

=

-

1
E -

d
- dE

)

(d+1)/2

[ E] ( v'f2+v1'f_ 2+1d-1)/2

=

-j(l).

(2.87)

Thus,

the

final

expression I(s) rv s(

for
1
d+S)

I ( /2

s) 2(

is d-

1

)

/

2

7f

-

1

/

2

r

(

<EFBFBD>
2

+

1

)

.

(2.88)

Note that a choice of analytic continuation has been made so as to avoid

oscillatory behavior in d.

( ( We return to (2.73). It may be written as

c(a) -_ r_ 2(<28>"d_ d//2_ 2) 22d1fd+11ad+la(1-d)/2

8 1ra
73

)

(d-

3)

/2

Vif

r

(d

/

2

)

!

)a(3 41fa

36

Casimir Force Between Parallel Plates

(<28> =

2-d- 17r-d/2- 1

( 4na ) d+1 f3

_1 _ ad+ 1

r

2

+

) 1

_1 _ ad+2

.

(2.89)

The correction to the zero-temperature result (2.35) is obtained from

L :J:T-->0 corr

-

- <20> 2

CXl

c(2nn)

J3 n=1

-n-d/2-1 r(d/2 + 1 )((d + 2)/3-d-2 .

(2.90)

[ Thus, the force per unit area in the low-temperature limit has the form

1

+

_ 1 d+ 1

( 2a) d+2] f3

,

(2.91)

ofwhich the d = 1 and d = 2 cases are familiar:

d= 1 :

(2.92a)

d=2 :

(2.92b)

These equations are incomplete in that they omit exponentially small terms; for example, in the last square bracket, we should add the term

_

240
1f

<EFBFBD>
f3

e

-

1r

f3/

a

.

(2.93)

We will discuss such corrections in Sec. 3.2. Mitter and Robaschik [133] considered the Casimir effect between two
plates where the temperature between the plates T is different from the external temperature T'. If T' < T the difference in thermal pressure can balance the Casimir attraction.

2.6 Electromagnetic Casimir Force
We now turn to the situation originally treated by Casimir: the force be­ tween parallel conducting plates due to quantum fluctuations in the electro­ magnetic field. An elegant procedure,, which can be applied to much more
'II One advantage of this scheme is its explicit gauge invariance, as contrasted with meth­ ods making use of the Green's functions for the potentials.

Electromagnetic Casimir Force

37

complicated geometries, involves the introduction of the Green's dyadic, de­ fined as the response function between the (classical) electromagnetic field and the polarization source [ 1 1 ] :

J E(x) = d4x' r(x, x') · P (x') .

(2.94)

In the following we will use the Fourier transform of r in frequency:

J <20><> r(x, x') =

e-iw(t-t') r(r, r'; w ) ,

(2.95)

which satisfies the Maxwell equations

V X r = iw<I> , -V x <I> - iwr = iwlJ(r - r').

(2.96a) (2.96b)

The second Green's dyadic appearing here is solenoidal,

v . <I> = 0,

(2.96c)

as is r if a multiple of a S function is subtracted:

V · r' = 0, r' = r + H(r - r') .

(2.96d)

The system of first-order equations (2.96a) , (2.96b) can be easily converted to second-order form:

(\72 + w2)r' = -V x (V x l ) cS(r - r') , (\72 + w2) <I> = iwV x H(r - r').

(2.97a) (2.97b)

The system of equations (2.97a) , (2.97b) is quite general. We specialize to the case of parallel plates by introducing the transverse Fourier trans­ form:

J k r'(r'

r' · '

w)

=

(

d2 27f

)

2

eik

(

r

j_

-

r't_

l

g

(

z

'

z'

·
'

k

'

w

)

.

(2.98)

The equations satisfied by the various Cartesian components of r may be easily worked out once it is recognized that

(2.99)

38

Casimir Force Between Parallel Plates

In terms of the Fourier transforms, these equations are

(::2 ) - k2 + w2

a2
( ) az2

-

k 2

+

w 2

a2
( ) az2

-

k 2

+

w 2

a2
( ) ( ) az2

-

k 2

+

W 2

a2
( ) ( ) az2

-

k 2

+

w 2

a2
( ) az2

-

k 2

+

W 2

9zz = -k2J(z - z1) ,

9zx

=

-

z. kx

a az

J

(z

- zI) ,

9zy

=

-z.ky

a az

J

( z

-

zI)

,

9xx =

-ky2

+

a2 az2

il(z - z1) ,

9yy =

- kx2

+

a2 az2

J(z - z1),

9xy = kx kyb(z - Z1) .

( 2 . 1 00a) (2 . 1 00b) (2.1 00c) (2.100d) (2.1 00e) (2. 100f)

We solve these equations subject to the boundary condition that the transverse components of the electric field vanish on the conducting sur­ faces, that is,

ll X r1 l z=O,a = 0,

(2.101)

where n is the normal to the surface. That means any x or y components
vanish at z = 0 or at z = a. Therefore, 9xy is particularly simple. By the
standard discontinuity method, we immediately find [cf. (2.20)]

9xy = 9yx = .Aksxm. ky.Aa (ss) ,

(2.102)

where

(ss) = sin .Az< sin .A(z> - a) .

(2.103)

To find 9xx we simply subtract a J function: 9<>x = 9xx - J(z - Z1).
Then, we again find at once

(2.104)

(2. 105a)

Electromagnetic Casimir Force

39

and similarly

9zz , To determine the boundary condition on

( 2. 105b) we recall the solenoidal con­

dition on r', (2 .96d) , which implies that
:Z9zzl = 0.

( 2 . 1 06)

z=O,a

9zz k2 This then leads straightforwardly to the conclusion = - A sm. Aa ( )cc ,

(2 . 107)

where

( )cc = cos Az< cos A(z> - a) .

(2.108)

The remaining components have the property that the functions are dis­

continuous, while, apart from a o function, their derivatives are continuous:

9zx ikx = --s1.- n /,\-a (cs) ,

(2 .1 09a)

9zy iky = - -s1.-n /,\-a (cs),

(2. 109b)

where

{ (cs) =

cos Az sin A(z' - a) , z < z', sin Az' cos A(z - a) , z > z'.

(2 . 1 1 0)

Similarly, where

9xz ikx = s-1.- n A,-a (sc),

9yz iky a s1n ( )c = -. -A,- s

,

{ (sc) =

sin AZ cos .A(z' - a) , z < z', cos Az1 sin A(z - a) , z > z',

(2 . 1 1 1 a) ( 2 . 1 1 1b)
(2.112)

40

Casimir Force Between Parallel Plates

which just reflects the symmetry
r'(r, r') =r'(r', r).

(2.113)

The normal-normal component of the electromagnetic stress tensor is

(2.114)

The vacuum expectation value is obtained by the replacements

(E(x)E

(

x'

)

)

=

<EFBFBD>
2

r

(x

,

x'

)

,

(H(x)H(x'))

=

-

1 -2;-

1 2 W

v

X

r(x,

x')

X

io;;o l v

.

In terms of the Fourier transforms, we have

(2. 1 1 5a) (2.115b)

(2.116)

When the appropriate Green's functions are inserted into the above, enor­ mous simplification occurs on the surface, and we are left with

(tzz ) lz=O,a = i.\ COt Aa,

(2. 117)

which indeed is twice the scalar result (2.24) , as claimed at the end of Sec. 2.2.

2.6. 1 Variations
The force between a perfectly conducting plate and a perfectly permeable one was worked out by Boyer [134] and studied more recently in Refs. [135,
1 36] . A repulsive result, -<2D> times that for a scalar field with Dirichlet
boundary conditions, (2.9), is found at zero temperature. It is extremely interesting that this answer differs only by a sign from the fermionic Casimir force we will derive in the next section. Kenneth and Nussinov [137] derive the Casimir effect between plates which conduct in single, different, direc­ tions. As expected, when the conductivities are parallel, the energy is 1 /2 that for ordinary conductors.

Fermionic Casimir Force

41

Fluctuations in the Casimir stress have been considered by Barton [138] .
If T is the observation time, then for T » a,

(2.118)
far beyond experimental reach. The stress correlation function was ana­ lyzed by Barton in Ref. [139] . See also Refs. [140, 141, 142] .
How real is Casimir energy? Just as real as any other form, as demon­ strated by Jaekel and Reynaud [143] , who consider the mechanical and inertial properties of Casimir energy and conclude that "vacuum fluctua­ tions result in mechanical effects which conform with general principles of mechanics . "
Mention should also be made of the Scharnhorst effect, in which light
speeds greater than the vacuum speed of light are possible in a parallel plate capacitor, as an induced consequence of the Casimir effect [144, 145, 146] . It is interesting that Schwinger in 1990 wrote a manuscript, which may never have been submitted to a journal, that claimed, in contradiction with the above referenced results, that the effect was nonuniform, dispersive, and persisted if only a single plate was present [147] .

2.7 Fermionic Casimir Force

We conclude this Chapter with a discussion of the force on parallel surfaces due to fluctuations in a massless Dirac fermionic field. For this simple ge­ ometry, the primary distinction between this case and what has gone before lies in the boundary conditions. The boundary conditions appropriate to the Dirac equation are the so-called bag-model boundary conditions. That
is, if n<>-' represents an outward normal at a boundary surface, the condition
on the Dirac field 'lj; there is

(1 + in · ')')'l/; = 0.

(2.119)

For the situation of parallel plates at z = 0 and z = a, this means

(2 . 1 20)

at z = 0 and z = a, respectively. In the following, we will choose a rep­
resentation of the Dirac matrices in which iT's is diagonal, in 2 X 2 block

42
form,

Casimir Force Between Parallel Plates

(2.121a)

while

I 0

=

(

0
i

-i
0

)

'

(2.121b)

from which the explicit form of all the other Dirac matrices follow from
I = ir015a.
The effect of fermionic fluctuations was first investigated by Johnson [87], and quoted in Ref. [86]. (The bag model and its boundary conditions were introduced in [84].)

2 . 7 . 1 Summing Modes
It is easiest, but not rigorous, to sum modes as in Sec. 2.2. We introduce a Fourier transform in time and the transverse spatial directions,

(2.122)

so that the Dirac equation for a massless fermion -ir81/J = 0 becomes, in

the

coordinate

system

in (

which
- w =F

ik:zli)esUa±lo±ngktvh±e

x
=

axi 0,

s

,

(2.123a)

( :z) 0, ±ku± + -w ± i V± =

(2.123b)

where the subscripts indicate the eigenvalues of if5 and and u and v are eigenvectors of 0"3 with eigenvalue +1 or -1, respectively. This system of equations is to be solved to the boundary conditions (2.120), or

u+ + u- lz=O = 0,

(2.124a)

0, v+ - v- lz=O =

(2.124b)

0, u+ - u- lz=a =

(2.124c)

0. V+ + V- lz=a =

(2.124d)

Fermionic Casimir Force

43

The solution is straightforward. Each component satisfies

= where >.2 w2 - k2, so each component is expressed as follows:

= A , u+ + u_ = v+ - v_

sin .Xz B sin .Xz ,

= u+ - u_ = C sin .A(z - a) ,
v+ + v- D sin .X(z - a) .

(2.125)
(2.126a) (2.126b) (2.126c) (2.126d)

Inserting these into the Dirac equation (2.123a) and (2. 123b) , we find, first, a condition on .\:

cos .\a = 0,

(2.127)

or

.\a = (n + 21 )1r,

(2.128)

where n is an integer. We then find two independent solutions for the

coefficients:

A
B
C
D

= = # =

0,
0ii<EFBFBD><EFBFBD>, ((--11t)nAA,,

(2.129a) (2.129b) (2 . 1 29c)
( 2 . 1 29d)

and

A = 0,

#B 0,

= iC

}5__ .\

(

-

1

)

n

B

'

= iD .::.'\_ (-1)nB.

( 2 . 1 30a) (2.130b) (2.130c)
(2 . 1 30d)

Thus, when we compute the zero-point energy, we must sum over odd inte­ gers, noting that there are two modes, and remembering the characteristic

44

Casimir Force Between Parallel Plates

minus sign for fermions: Instead of (2.3), the Casimir energy is

(2.131)
which is <20> x 2 times the scalar result (2.9) because ((-3) = -B4j4 = 1 /1 20.
(The factor of 2 refers to the two spin modes of the fermion.)

2 . 7 . 2 Green's Function Method

Again, a more controlled calculation starts from the equation satisfied by the Dirac Green's function,

,.,.;.aG(x, x') = b(x - x'), z

(2.132)

subject to the boundary condition

(1 + in · r)GJz=O,a = 0.

(2.133)

We introduce a reduced, Fourier-transformed, Green's function,

J dw J d2k w) G(x' x') =

e-iw(t-t' ) 2n

(2n

)

2

eik·

(x-x'

)

9

(

z

'

z'

' ·

k '

'

(--lw 13;. <20>) which satisfies

+ 1 · k + z oz 9(z, z') = b(z - z').

(2.134) (2.135)

Introducing the representation for the gamma matrices given above, we find

( ) that the components of 9 corresponding to the + 1 or - 1 eigenvalues of i")'5,

9 =

9++ 9-+

9+9--

)

(2. 1 36)

Fermionic Casimir Force

45

satisfy the coupled set of equations
( : ) -w ± rr · k =f ia-3 z 9±± = 0, ( : ) -w ± rr · k =f ia-3 z 9±'f = =t=i5(z - z') .

(2.137a) (2.137b)

( ) We then resolve each of these components into eigenvectors of a-3 :

9±± =

u±( +±) v±(+±)

u(±-±) v±( -±)

(2.138)

and similarly for 9±'f . These components satisfy the coupled equations

( ) -w

=f

2 .

a az

u±C±±l ± kv±C±±J = 0,

( ) ±ku±c±±J +

-w

±

2 .

a az

v±c±±J = 0,

( ) -w

=f

2 .

a az

uC±+'Jf ± kv±C+'fJ = =f2.u, (z - z') '

(-w ) =f

2 .

a az

uc±-'fJ ± kv±c-'fJ = 0 ,

(2.139a) ( 2 . 1 39b) (2.139c) (2.139d)

(-w ) ±ku±(+'f) +

±

i

.!.!._ az

v±(+'f) = 0,

( ) ±ku(±-'f) + -w ± i.a!.!z._ v±(-'f) = =t=i5(z - z'),

(2.139e) ( 2 . 1 39f)

which aside from the inhomogeneous terms are replicas of (2. 1 23a) and (2.123b) . These equations are to be solved subject to the boundary condi­ tions

U<EFBFBD>±<EFBFBD> - U<>±<EFBFBD> lz=a = 0,

u<EFBFBD>±<EFBFBD> + u<>±<EFBFBD> l z=O = 0,

± U<><55>

-

± U<><55> lz=a

=

0,

u<EFBFBD>±<EFBFBD> + u<>±<EFBFBD> l z=O = 0,

v+(±+) + v_(±+) \ z=a = 0 ,

- V<><56> v<><76> lz=O = 0,

( 2 . 1 40a) (2.140b) ( 2 . 140c) (2.140d) ( 2 . 1 40e) ( 2 . 1 40f)

46

Casimir Force Between Parallel Plates

vv+<2B><±±-<2D>l

v<±l l + __ z=a - v<>±<EFBFBD> lz=O

=
=

0, 0.

( 2 . 1 40g) ( 2 . 1 40h)

The solution is straightforward. We find

[ ] u+(++) = u(-+-)* = v+<-+)* = v-<--)

= 1 2 cos >.a

cos

>. ( z

+

z I

-

a)

+

iW T

sm.

.X ( z

+

z I

-

a)

, (2 . 141 a)

-- v+<++l = v-<+-l = u+<-+l* = u-<--l*

=

- 2i k>.

1 cos >.a

sm.

.X

(z

+

z

1

-

a)

'

u<EFBFBD>+<2B> = u<>+<2B>* = -v<>-J* = -v<>-<2D>

(2.141b)

<EFBFBD> ] 2

1 cos Aa

[E(z

-

z1)

cos

>. (z>

-

z<

-

a)

- i sin .X (z> - z< - a) ,

( 2 . 1 4 1 c)

-- v<>+J = v<>+<2B> = u<>-<2D> = u<>-<2D> -ik 1 sm. .X(z> - z< - a) , 2>. cos >.a

( 2 . 1 41 d)

where

_ { E(z

z1) -_

1 if z > z1,
-1 if z < z1•

(2.142)

We now insert these Green's functions into the vacuum expectation

Tflv (l'fl<66>av l'v<>8fl) gflv value of the energy-momentum tensor. The latter is

= <20>'l/J!'o<>

+

'lj; + £ ,

( 2 . 1 43 )

£

=

1 1
- 2 'l/Jl'-; 8'l/J .

( 2 . 1 44)

We take the vacuum expectation value by the replacement

( 2 . 1 45)

where G is the fermionic Green's function computed above. Because we are interested in the limit as x1 __._ x we can ignore the Lagrangian term in the

Fermionic Casimir Force

47

energy-momentum tensor, leaving us with

so for a given frequency and transverse momentum,

-2 i

a ;u:z:;-

tr

3 a

(

9-

+

+

9+ - ) Iz' _,z

i 2

0 EJz

[u-(++)

+

u+(+-)

-

(v-(-+)

+

v+(--) )] 1z, ->z ·

(2.146) ( 2 . 1 47)

When we insert the solution found above (2. 141c) , we obtain

(t33)

=

2 i <20> Re EJz

u-(++) ·

(2.148)

Carrying out the differentiation and setting z = z' we find instead of (2.24),

(2.149)

where again we ignore the 5-function term. We now follow the same procedure given in Sec. 2 .3: The force per unit
area is

(2.150)

As in (2.26) we omit the 1 in the last square bracket: The same term is present in the vacuum stress outside the plates, so cancels out when we compute the discontinuity across the plates. We are left with, then,

100 'L
.r

_ -

-

1 1 6n2a4

0

dx x3
eX + 1 ·

(2.151)

But

(2.152)

48
so here we find

Casimir Force Between Parallel Plates

which is, indeed, i times the scalar force given in (2.10).

(2.153)

Chapter 3
Casimir Force Between Parallel Dielectrics

3.1 The Lifshitz Theory

The formalism given in Sec. 2.6 can be readily extended to dielectric bodies [1 1]. The starting point is the effective action in the presence of an external
polarization source P:

J W = (dx) [P · (-A - V¢) + EE · (-A - V¢)

-H . (V X A) + 21 (H2 - EE2)J ,

(3.1)

which, upon variation with respect to H , E, A, and ¢ , yields the appro­ priate Maxwell's equations. Thus, because W is stationary with respect to these field variations, its response to a change in dielectric constant is explicit:

(3.2)
Comparison of iJ. W with the second iteration of the source term in the
vacuum persistence amplitude,

(3.3)

allows us to identify the effective product of polarization sources,

iP(x)P(x') jeff = HE b(x - x') .

(3.4)

49

50

Casimir Force Between Parallel Dielectrics

Thus, the numerical value of the action according to (2.94) ,
<EFBFBD> J <20> J W = (dx)P(x) · E(x) = (dx)(dx')P(x) · r(x, x') · P (x' ) , (3.5)
implies the following change in the action when the dielectric constant is varied slightly,

(3.6)

where the repeated indices on the dyadic indicate a trace. In view of (3.2) , this is equivalent to the vacuum-expectation-value replacement ( 2 . 1 1 5a) .
For the geometry of parallel dielectric slabs, shown in Fig. 1 . 1 , where the dielectric constants in the three regions are

z < 0,

0 < z < a,

(3.7)

a < z,

the components of the Green's dyadics may be expressed in terms of the

TE (transverse electric or H) modes and the TM (transverse magnetic or

E) modes,* given by the reduced scalar Green's functions satisfying

( ::2 ) - + k2 - w2E gH(z, z') = 8(z - z'),

a
( ) az - -

1
E -

a
- az'

+

k2 E -

-

w 2

gE(z, z') = 5(z - z'),

(3.8a) (3.8b)

where, quite generally, E = E(z) , E1 = E(z'). The nonzero components of the
Fourier transform g given by (2.98) are easily found to be (in the coordinate
system where k lies along the +x axis)

-

- -1E 8(z

- z1)

+

1
E -

a
- az

1
- E1

a
- az'

g

E

'

= w2gH,

=

-

1
-E

8 (

z

-

z ')

+

-EkE21 gE ,

= z. -EkE1 -aaz gE '

(3.9a) (3.9b) (3.9c) (3.9d)

*We have changed the notation from that of the original reference [ 1 1 ] . The TE modes are denoted by H, the TM modes are denoted by E, to be consistent with the notation used later in the book.

The Lifshitz Theory

51

9zx = -z. -EkE1 - aaz' gs

( 3 . 9e)

The trace required in the change of the action (3.6) is obtained by taking

( ) I the limit z' --> z, and consequently omitting delta functions:

9kk =

w 2 g H

+

k2 -EE'

g s

+

1
-E

a -az

1
-E1

a - az'

g s

.
z=z'

(3. 10)

This appears in the change of the energy when the second interface is
displaced by an amount 8a,

&(z) = -8a(Ez - E3)8(z - a),

namely (A is the transverse area)

J - JE
A

=

i 2 -

dw
-
27r

(dk) - ( 27r ) 2

dz

8E(z)gkk (z,

z;

k,

w)

=

-8a

:F,

where the force per unit area is

(3.11) (3.12)

(3. 13)

Because gH , gE and <20> %z <20> 8<>, gE are all continuous, while EE1 is not, we

interpret interface

the trace of g in (3.12 from opposite sides,

) symmetrically; we let z and

so the term

€k€2, gE

-->

gE _]£_
EI E2

z' approach the . Subsequently,

we may evaluate the Green's function on a single side of the interface. In

terms of the notation (for E = 1 , <20> = -i>-. in the notation used in the

previous chapter)

(3.14)

which is positive after a complex frequency rotation is performed (it is automatically positive for finite temperature) , the magnetic (TE) Green's
function is in the region z, z' > a,

(3.15)

where the reflection coefficient is

(3.16)

52

Casimir Force Between Parallel Dielectrics

with the denominator here being

d

=

K3 K3

+
-

"'I KI

K3 K3

+
-

K2 e2<65><:3a K2

_

1_

(3.17)

The electric (TM) Green's function gE has the same form but with the

replacement

(3.18)

except in the exponentials; the corresponding denominator is denoted by
d' . [It is easy to see that gH reduces to (2.27) when r = - 1 ; the results in
Sec. 2.6 follow from (3.9a)-(3.9e) in the coordinate system adopted here.]
Evaluating these Green's functions just outside the interface, we find
for the force on the surface per unit area

where the first bracket comes from the TE modes, and the second from the TM modes. The first term in each bracket, which does not make reference to the separation a between the surfaces, is seen to be a change in the volume energy of the system. These terms correspond to the electromagnetic energy required to replace medium 2 by medium 3 in the displacement volume. They constitute the so-called bulk energy contribution. (It will be discussed further in Chapter 12.) The remaining terms are the Casimir force. We rewrite the latter by making a complex rotation in the frequency,

(3.20)

This gives for the force per unit area at zero temperature

:F.C_ra=soimi. r

=

-

-811!-"2

Jf0'x:;

d<EFBFBD>'>"

r= j 0

dk2

2K3

(d-1

+

d'- 1 )

.

(3.21)

From this, we can obtain the finite temperature expression immediately by the substitution

(3.22)

(3.23)

Applications

53

the prime being a reminder to count the n 0 term with half weight. These results agree with those of Lifshitz et al. [7, 67, 8] . The connection between Casimir's ideas of zero-point energy and Lifshitz' theory ofretarded dispersion forces appears in Boyer's paper [148] .t
Note that the same result (3.21 ) may be easily rederived by computing the normal-normal component of the stress tensor on the surface, Tzz , pro­ vided two constant stresses are removed, terms which would be present if either medium filled all space. The difference between these two constant stresses,

J rzvzol

=

.
-z

(dk) (27r ) 2

dw 27r

( 1'£ 2

-

1'£ 3 )

'

(3.24)

precisely corresponds to the deleted volume energy in the previous calculation.

3.2 Applications

Various applications can be made from this general formula (3.21) . In

particular, if we set the intermediate material to be vacuum, E3 = 1 , and set El = E2 = oo, so that 1'£1 = 1'£2 = oo, <20>/'£ = ;/'£ = 0, we recover the Casimir force (2. 1 1) between parallel, perfectly conducting plates. More generally,

we can let the intermediate material have a dispersive permittivity, so that

we obtain

F

=
=

-

_18r1712_r2}r0Jr=o=dd((e2)(vrtVf;E2=£(ea(d)-!3'£21e

2M41'-£ = -2

1 4

7r2 0y't

a4

'

(3.25)

which, until the last step, still admits of dispersion. This last expression is an obvious generalization of Casimir's result to a dielectric-filled capacitor. Note that the corresponding energy per unit area is

(3.26)

tThe nonretarded part of the Lifshitz formula, for d « .>.., the "principal absorption wavelength of the material," was rederived in 1968 by van Kampen, Nijboer, and Schram [149] , using the "argument principle" described in Appendix A to evaluate the zero­ point energy.

54

Casimir Force Between Parallel Dielectrics

which is a considerably simpler expression than, but equivalent to, that given in Ref. [150] . (See their note added in proof.) [Cf. (2.55a) .]

3.2 . 1 Temperature Dependence for Conducting Plates
As for the temperature dependence, we note that we must take this limit with special care for the static situation w = 0. In order to enforce correctly the electrostatic boundary conditions, we adopt the prescription that we take the limit E ---+ oo before we set w = 0. Doing so for the temperature­ dependent version of (3.21 ) gives

(3.27)

where ;,,.. = in (2.59) for

dk2=+

(2nn/{3)2
2. Notice

,

This is exactly twice that if we had simply

the scalar let E1 ,2 ---+

result given oo, the first

denominator structure in (3.21) for n = 0 would not contribute, which,

among other consequences, would imply an incorrect T ---+ 0 limit.+ Defining

(3.28)

100 -- we find the Casimir force for arbitrary temperature to be

-r-T

1 <20>I

.r -- - -4nf-3a3 <20>

n=O

nt

y 2

d y
eY

1
-

1

.

(3.29)

As noted in the previous chapter, the high-temperature, t » 1, limit is

particularly easy to obtain, for then the n = 0 term is the only one which

is not exponentially small. Including the first of these exponentially small

-- -- ( ) corrections (from n = 1 ) we find for large T

F T

cv

-

1
4nf3a3

(

(3

)

-

1
2nf3a3

1 + t + -t22

-t e,

f3 « 4na.

(3.30)

+ Remarkably, this seemingly obvious prescription has become controversial. Bostrom and Sernelius [151 , 152] claim that the n = 0 mode of the TE mode should be omit­ ted. This would give rise to a significant temperature correction in the experimen­ tally accessible region, while none is seen-see below. That this claim is incorrect has been demonstrated by Lamoreaux [ 153] . See also Svetovoy and Lokhanin [154, 155] , who obtain a large linear temperature correction, again in contradiction with ex­ periment. A sensible explanation of the ambiguities which led to these erroneous results appears in Bordag et a!. [156] .

Applications

55

[The term which is not exponentially small is twice that given in (2.61b).] This coincides with the result first found by Sauer [3] and Mehra [5, 4, 6]. See also Levin and Rytov [157]. Our form (3.21) is especially suited to obtain the high temperature limit, in contradistinction to the forms obtained by Sauer and Mehra and by Brown and Maclay [100].
The low-temperature limit was also worked out in the previous chapter, but for a general value of d, the transverse dimension. There are significant simplifications when d = 2. We use the Poisson sum formula (2.71) for functions related by a Fourier transformation (2.70). Here we take

100 dy-1-, b(n) =

y2

1 lnlt

eY -

(3.31)

which has the Fourier transform for a =1- 0

1 100 dx cos ax 100 1 1 o c(a) = - 7r

xt y2 dy - eY - -

[<5B> coth1rz - . !._] I -__!__<5F>

2 2z 1ra dz2

z=a/t

[4 (1 (1 !._.] =

_ _I_
1ra

7r3 e-2rraft
_

+ e-2rraft) e- 2rra f t ) 3

_

a3

·

(3.32)

Here we have interchanged the order of integration and used the fact that

1
0

00

-eYd-y-1

sm.

z

y

=

7r
2 -

coth

1r

z

-

-21z

'

(3.33)

which may be easily derived from (2.45). The evaluation of c(O) is easily accomplished directly, or by expanding coth1rz in the above, yielding

15t . c(O) = - 7r3

(3.34)

We therefore find an alternative form for the sum in (3.29)

which, apart from a factor, expresses the general temperature dependence of the Casimir force. From this form, it is very easy to obtain the low-

56

Casimir Force Between Parallel Dielectrics

temperature limit

{3 » 47ra.

(3.36)

again in agreement with Sauer and Mehra [3, 5, 4, 6] , and with that found by

Lifshitz [7, 8] when the transcription error there is corrected. The result is,

of course, twice that given in (2.92b) , including the exponential correction

(2.93). We recognize the second term here as the blackbody radiation

pressure arising from thermal fluctuations above the plate, z > a, (external

7!"2 !__ (F) fluctuations),§ so we write

FT

=

-
45

( kT) 4

-

oa

A

'

(3.37)

where the corresponding (internal) free energy per unit area is (<28> = akT)

where the constant X is undetermined by (3.36). Under the inversion
symmetry discovered by Ravndal and Tollefsen [158] , a generalization of
that found by Brown and Maclay [100] , this low temperature result can be
extended to the high-temperature limit by the inversion formula

(3 . 39)

so here

7!"2 ( ) f(<28>) = -45 e + -41 X<> + <20> -417r + <20> e47fE' <20> » 1.

(3.40)

2 The corresponding force is, from (3.37) ,

F T

"'

-

X
2a3

kT

-

kT 41ra3

(t

+ 2t + 2)e-t,

t » 1,

(3.41)

where t = 47rakT. We see that the Stefan's law contribution cancelled be­ tween the interior and exterior modes, and that the first term expresses the
correct linear behavior shown in (3.30) with X = ((3)/27r. The exponen­
tially small term in (3.30) is reproduced, which shows the (limited) efficacy
of this inversion symmetry.

§ See footnote 7 of Ref. [100] .

Applications

57

Recently, these results have been rederived by semiclassical orbit theory by Schaden and Spruch [159] .

3 . 2 . 2 Finite Conductivity

Another interesting result, important for the recent experiments [71 , 72] , is the correction for an imperfect conductor, where for frequencies above the infrared, an adequate representation for the dielectric constant is [99]

E(w)

=

1

-

w--24
w

,

(3.42)

where the plasma frequency is,

Wp2 = <20> 4ne2N ,

(3.43)

where e and m are the charge and mass of the electron, and N is the number

density of free electrons in the conductor. A simple calculation shows, at

[ (.!!._) ] zero temperature [10, 11],

n2
:F ;::; - 240a4

1

-

_8
3ft

_.!._
ea

N

1/2

.

(3.44)

If we define a penetration parameter, or skin depth, by fJ = 1/wp, we can

( ) write the result out to second order as [160, 1 1 4]

:F ;::; - - 2410f- 2a4

1

-

-136

fJ
-a

+

24-aJ22

.

(3.45)

Recently, Lambrecht, Jaekel, and Reynaud [161] analyzed the Casimir force between mirrors with arbitrary frequency-dependent reflectivity, and found that it is always smaller than that between perfect reflectors.

3.2.3 van der Waals Forces
Now suppose the central slab consists of a tenuous medium and the sur­ rounding medium is vacuum, so that the dielectric constant in the slab
'<27><>Although we have used rationalized Heaviside-Lorentz units in our electromagnetic ac­ tion formalism, that is without effect, in that the one-loop Casimir effect is independent of electromagnetic units. For considerations where the electric charge and polarizability appear, it seems more convenient to use unrationalized Gaussian units.

58

Casimir Force Between Parallel Dielectrics

differs only slightly from unity, E - 1 « 1.
Then, with a simple change of variable, K = (p,
we can recast the Lifshitz formula (3.21) into the form

(3.46) (3.47)

If the separation of the surfaces is large compared to the characteristic wavelength characterizing E, a(c » 1 , we can disregard the frequency de­ pendence of the dielectric constant, and we find

For short distances, a(c « 1, the approximation is

(3.49)

(3.50)

These formulas are identical with the well-known forces found for the com­ plementary geometry in Ref. [1 1].
Now we wish to obtain these results from the sum of van der Waals forces, derivable from a potential of the form

V = - - B .
r"<22>

(3.51)

We do this by computing the energy (N = density of molecules)

f E

=

-

<EFBFBD>
2

BN2

Jro

dz

Jro

dz'

(drj_ ) (d<J

[ ( r _1_

-

) 2 rI_1_

1 +

( z

-

z')2] /'Y 2

'

(3.52)

If we disregard the infinite self-interaction terms (analogous to dropping

the volume energy terms in the Casimir calculation) , we get [ 1 1 , 38]

(3.53)

Applications

59

So then, upon comparison with (3.49) , we set 'Y = 7 and in terms of the polarizability,

we find

a

=

E-1 47rN

'

(3.54)

(3.55)

or, equivalently, we recover the retarded dispersion potential of Casimir and Polder [50] ,

V = - - 4213r - ar72 '

(3.56)

whereas for short distances we recover from (3.50) the London potential [49] ,

(3.57)

which are the quantitative forms of (1.6) and (1 .9), given in (1 . 30) and ( 1 .29), respectively.

3.2.4

Force between Polarizable Molecule and a Dielectric Plate

As a final application of these ideas, we will calculate the energy of inter­ action between a molecule of polarizability a(w) and a dielectric slab. This energy is given by (3.6) with

bE(r, w) = 47ra(w)J(r - R) ,

(3.58)

which expresses the change in the dielectric constant when a molecule is inserted in the vacuum at R. We will suppose that the dielectric slab
occupies the region of space z < 0 with vacuum above it. The appropriate Green's functions here, referring to a single interface, are trivially obtained
from those discussed in Sec. 3 . 1 . In region 2, gH has the form of (3.15) with
the reflection coefficient r given by

2K 1 + r = + K Kl --- ,

(3.59)

60

Casimir Force Between Parallel Dielectrics

where

(3.60)

which is obtained from The energy is then (R3

(3.16)
= z)

by

taking

the

limits

a

--+

oo ,

E2

=

1,

EJ

=

E1 .

J E

=

i 2

dw 27!"

(dk) (2n)2

4na(w)gkk

(z,

z;

k,

w)

,

(3.61)

where, from (3. 10) ,

(3.62)
The necessary contact term here is easily deduced from the physical require­ ment that the energy of interaction go to zero as the separation gets large,
z --> oo, which effectively removes the w2j"' term in gkk . Therefore, the
interaction energy between the molecule and the dielectric slab separated
by a distance z is

(3.63)

One application of this result refers to the attraction of a molecule by a perfectly conducting plate. We merely take E1 --> oo and then easily find

E

=

-

<EFBFBD>
8nz4

'

(3.64)

a result first calculated by Casimir and Polder [50] . This result was exper­ imentally verified by Sukenik, Boshier, Cho, Sandoghar, and Hinds [162] . (Actually, they measured the force on an atom between two plates, the general theory of which was given by Barton [163] .) A second, particu­ larly interesting possibility occurs when the molecule is of the same type as those composing the dielectric slab. When the common dielectric constant

Experimental Verification of the Casimir Effect

61

is close to unity, the energy of interaction, to lowest order in E - 1 , is

where we have expressed the polarizability of the molecule in terms of the dielectric constant according to (3.54) . Lifshitz et al. [7, 8, 9] have considered the limiting behavior of large separations (small () where E can be considered to be constant.
A recent proposal by Ford and Svaiter [164] suggests focusing the vac­ uum modes of a quantized field by a parabolic mirror, thereby enhanc­ ing the Casimir-Polder force on an atom, which would be drawn into the focus of the mirror. The approach used in that paper is a semiclassi­ cal approximation, based on geometrical optics. It is related to the cal­ culations of Schaden and Spruch [165, 166] who used a semiclassical ap­ proximation and geometrical optics to calculate Casimir energies between pairs of conductors, plates, a plate and a sphere, spheres, and concen­ tric spheres, in the approximation that the separations of the objects are small compared to all radii of the objects. They also provide a rigorous derivation of the proximity theorem result of Derjaguin [167, 168, 54, 55, 56] , which is discussed in the next section.
3.3 Experimental Verification of the Casimir Effect
Attempts to measure the Casimir effect between solid bodies date back to the middle 1950s. The early measurements were, not surprisingly, some­ what inconclusive [55 , 57, 58, 59, 60, 61, 62, 63, 64, 65] . The Lifshitz theory (3.2 1 ) , for zero temperature, was, however, confirmed accurately in the experiment of Sabisky and Anderson in 1973 [66] . So there could be no serious doubt of the reality of zero-point fluctuation forces. For a review of the earlier experiments, see Ref. [169] .
New technological developments allowed for dramatic improvements in experimental techniques in recent years, and thereby permitted nearly di­ rect confirmation of the Casimir force between parallel conductors. First, in 1996 Lamoreaux used a electromechanical system based on a torsion pen­ dulum to measure the force between a conducting plate and a sphere [68, 69] . The force per unit area is, of course, no longer given by ( 1 . 12) or (1.13), but may be obtained from that result by the proximity force theorem [170]

62

Casimir Force Between Parallel Dielectrics

which here says that the attractive force F between a sphere of radius R and a flat surface is simply the circumference of the sphere times the energy per unit area for parallel plates, or, from ( 1 .12),

F

=

271"R E(d)

=

-

1r3 360

R d

he d2 '

R » d,

(3.66)

where d is the distance between the plate and the sphere at the point of

closest approach, and R is the radius of curvature of the sphere at that

point. The proof of (3.66) is quite simple. If R » d, each element of the

sphere may be regarded as parallel to the plane, so the potential energy of

1 the sphere is

V (d) = J(o" 21rR sin BR dB E(d + R(1 - cos B)) = 21rR

R dx E(d + R - x).
-R

(3.67)

To obtain the force between the sphere and the plate, we differentiate with

respect to d:

jR av ad -R ax F = -- = 21rR

dx -a E(d + R - x)

= 21rR[E(d) - E (d + 2R)] <20> 21rR E (d) , d « R,

(3.68)

provided that E (a) falls off with a. This result was already given in Refs. [54, 55, 57] . The proximity theorem itself dates back to a paper by Derjaguin in 1934 [167, 168].
Lamoreaux in 1997 [68, 69] claimed an agreement with this theoreti­ cal value at the 5% level, although it seems that finite conductivity was not included correctly, nor were roughness corrections incorporated [171]. Further, Lambrecht and Reynaud [172] analyzed the effect of conductivity and found discrepancies with Lamoreaux [70] , and therefore stated that it is too early to claim agreement between theory and experiment. See also Refs. [173, 1 74] .
An improved experimental measurement was reported in 1998 by Mo­ hideen and Roy [71], based on the use of an atomic force microscope. They included finite conductivity, roughness, and temperature corrections, al­ though the latter remains beyond experimental reach. II Spectacular agree­ ment with theory at the 1% level was attained. Improvements were subse­ quently reported [72, 73] . (The nontrivial effects of corrugations in the sur-

li The low temperature correction for the force between a perfectly conducting sphere and

Experimental Verification of the Casimir Effect

63

face were examined in Ref. [175] . ) Erdeth [74] used template-stripped sur­ faces, and measured the Casimir forces with similar devices at separations of 20-100 nm. Rather complete analyses of the roughness, conductivity, and temperature correction to this experment have now been published [176, 156, 177] .
Very recently, a new measurement of the Casimir force (3.66) has been announced by a group at Bell Labs [75] , using a micromachined torsional device, by which they measure the attraction between a polysilicon plate and a spherical metallic surface. Both surfaces are plated with a 200 nm film of gold. The authors include finite conductivity [172, 1 78] and sur­ face roughness corrections [179, 1 80] , and obtain agreement with theory at better than 0.5% at the smallest separations of about 75 nm. However, potential corrections of greater than 1 % exist, so that limits the level of verification of the theory. Their experiment suggests novel nanoelectrome­ chanical applications.
The recent intense experimental activity is very encouraging to the de­ velopment of the field. Coming years, therefore, promise ever increasing experimental input into a field that has been dominated by theory for five decades.

a perfectly conducting plate is [156, 68, 69]

[1 ] pT = - <20> R + 360((3) (Td)3 - 16(Td)4 .

360 d3

11"3

(3.69)

For the closest separations yet measured, d <20> 100 nm, this correction is only <20> 10-5 at room temperature.

Tilis page is intentionally left blank

Chapter 4
Casimir Effect with Perfect Spherical Boundaries

4.1 Electromagnetic Casimir Self-Stress on a Spherical Shell

The zero-point fluctuations due to parallel plates, either conducting or in­ sulating, give rise to an attractive force, which seems intuitively under­ standable in view of the close connection with the attractive van der Waals interactions. However, one's intuition fails when more complicated geome­ tries are considered.
In 1956 Casimir proposed that the zero-point force could be the Poincare stress stabilizing a semiclassical model of an electron [12] . For definiteness, take a naive model of an electron as a perfectly conducting shell of radius a carrying a total charge e. The Coulomb repulsion must be balanced by some attractive force; Casimir proposed that that could be provided by the vacuum fluctuation energy, so that the effective energy of the configuration would be

E

--

- e2 2a

-

-Z
a

nc '

(4 . 1 )

where the Casimir energy i s characterized by a pure number Z . The would open the way for a semiclassical calculation of the fine-structure constant, for stability results if E = 0 or

a = -eh2e = 2Z.

(4.2)

Unfortunately as Tim Boyer was to discover a decade later after a heroic calculation [13] , the Casimir force in this case is repulsive, Z = -0.04618. The sign results from delicate cancellations between interior and exterior

65

66

Casimir Effect with Perfect Spherical Boundaries

modes, and between TE and TM modes, so it appears impossible to predict the sign a priori.
Boyer's calculation was rather complicated, involving finding the zeroes of Bessel functions. Boyer's expression was subsequently evaluated with greater precision by Davies [76] . In the late 1970s two independent calcu­ lations appeared confirming this surprising result. Balian and Duplantier [14] used a multiple scattering formalism to obtain a quite tractable form for the Casimir energy for both zero and finite temperature, while Mil­ ton, DeRaad, and Schwinger [15] exploited the Green's function technique earlier developed for the parallel plate geometry. We will describe the lat­ ter approach here. In particular the Green's dyadic formalism of Sec. 2.6 may be used, except now the modes must be described by vector spherical harmonics, defined by [99, 1 8 1 , 182, 183, 1 84]

(4.3)

where L is the orbital angular momentum operator,

V'. L

=

1
-:-r

x

(4.4)

I

Notice that we may take l ;:: 1, because spherically symmetric solutions to

Maxwell's equations do not exist for w -=/=- 0. The vector spherical harmonics

satisfy the orthonormality condition

J dO. XT'm' · Xzm = 5ll' 5mm' 1

(4.5)

as well as the sum rule

+ l
"L'
m=-l

IXzm(B, ¢)12

=

2l 47r

1.

(4.6)

The divergenceless dyadics r' and <P may be expanded in terms of vector

( <20>V' ) spherical harmonics as r' = L tzXzm +

X gzXzm '

( 4 . 7a)

( <20>V' ) lm
<P = L gzXzm -

X hXzm ,

( 4 . 7b )

lm

where the second suppressed tensor index is carried by the coefficient func­
tions fz , gz , h, gz .

Electromagnetic Casimir Self-Stress on a Spherical Shell

67

Inserting this expansion into the first-order equations (2.96a) , (2.96b) ,
and using the properties of the vector spherical harmonics, we straightfor­
wardly find [15] that the Green's dyadic may be expressed in terms of two
scalar Green's functions, the electric and the magnetic:

lm

- v X [Gz (r, r')Xzm (D)X!m(D')] X v'}

+ 8-function terms,

(4.8)

where the expression "8-function terms" refers to terms proportional to

spatial delta functions. These terms may be omitted, as we are interested

in the limit in which the two spatial points approach coincidence. These

( ) { } scalar Green's functions satisfy the differential equation

<EFBFBD> r

<EFBFBD> dr2 r

_

l(l + r2

1)

+

w2

Fz (r, r') Gz(r, r')

= _2r_2 8(r _ r') ,

(4.9)

subject to the boundary conditions that they be finite at the origin (the center of the sphere), which picks out the spherical Bessel function of the
first kind, jz , there, and correspond to outgoing spherical waves at infinity,*
which selects out the spherical Hankel function of the first kind, hjll . On
the surface of the sphere, we must have

: l Fz (a, r') = 0, r rGz (r, r') r=a = 0,

(4.10)

so that F is the TE (H) , and G is the TM (E) , Green's function. The result
is that

(4.1 1 )

where Gf i s the vacuum Green's function (k = lwl),
(4.12)
*The terminology refers to the associated Helmholtz equation, so the behavior at spatial infinity is eikr jr. The time dependence is e-iwt , where k = l w l . Thus, in field-theoretic terms, we are using the usual causal or Feynman Green's function.

68

Casimir Effect with Perfect Spherical Boundaries

and in the interior and the exterior of the sphere respectively,
r,r' < a : { ;: } = -Ac,pikjz(kr)jz(kr'), (4.13a) r,r' > a : { ;: } = -Bc,pikhi1)(kr)hjll(kr'), (4.13b)

where the coefficients are

A F

--

BF-

1

--

hz(1) jz(

(ka) ka)

'

(4. 14a)

A G

_
-

BG_

1

-_

[kah?\ka)] [kajz(ka)]'

'

'

(4.14b)

ka. the prime signifying differentiation with respect to the argument From
the electromagnetic energy density we may derive the following formula for the energy of the system

00 E = J(dr)<29>2<EFBFBD> J_oo d2wne-iw(t-t')

x

L00
1=1

L l m=-

l

{

k

2

[

Pz

(r

,

r

'

)

+

Gz

(

r

,

r'

)

]

X

1

m

(

n

)

·

Xim

(

O'

)

- v X Xzm(n) . [Fz(r,r') + Gz(r,r')] · Xim(n') X v'}lr=r'.

(4.15)

Note here that the vacuum term in the Green's functions has been removed,

since that corresponds to the zero-point energy that would be present in this

formalism if no bounding surface were present. Here we are putting the two

spatial points coincident, while we leave a temporal separation, T = t - t',

which is only to be set equal to zero at the end of the calculation, and

therefore serves as a kind of regulator. The integration over the solid angle

and the sum on m may be easily carried out, with the result

;i <20>(2l + 00r2dr(2k2[P1 + Gz](r,r) E =

1) £: <20><>e-iwr 1

+

1 r2

:rr

{

d<EFBFBD>1

r'

[Pl(r,r') + Gl(r,r')J }r'=r).

(4.16)

Electromagnetic Casimir Self-Stress on a Spherical Shell

69

The integral of the derivative term here is equal to zero, which can be seen

by explicit calculation. The radial integral over Bessel functions is simply

done using recurrence relations. (For further details, see Chapter 5.) The

z ht (z) result is, in Minkowski spacetime, with

<EFBFBD>

J { (zzjjt1)' ((zzjjtz))"' (zzhhtt)' ((zzhhtz))"' } . E

=

..!:._ 2a

<EFBFBD> (2l
1=1

+

1)

d

(wa)
2n

e

-

iwT

z

= ka, and

+

+

= h<>1\z),
+

(4. 1 7)

Now it may be verified that the integrand in (4. 17) has the following

analytic properties in the complex variable ( = k:

• The singularities lie in the lower half plane or on the real axis.t Consequently, the integration contour C in w lies j ust above the real axis for w > 0, and just below the real axis for w < 0.
• For Im ( > 0, the integrand goes to zero as 1 / \ (\ 2 . (This is a weaker condition than specified in Ref. [15] .) This convergent behavior is
the result of including both interior and exterior contributions.

Then we may write the energy of the sphere as

1 E =

c

dw - 2n

e

-

'. w-r

g

(

j

w

j

)

,

(4.18)

where the integrand satisfies the dispersion relation

100 1
g(jwl) = -; 'IT<49>

_ 00

d( (2

-

( W 2

-

. g((),
<EFBFBD>E

(4.19)

) because the singularities of g(( occur only for Im ( ::; 0. Now we can carry
out the w integral in (4. 18) to obtain

(4.20)

Finally, we rewrite the result in Euclidean space by making the Euclidean

transformation ijTj ---+ j T4 j > 0, so that we have the representation

1oo _l_
2j(j

e

-

I

C:

II

T41

=

dk4 eik4T4
_00 2n k<> + (2 ·

(4.21)

t H a large external sphere i s added, as Hagen [185] advocates, the singularities arising from modes in the annulus become real. This is because there is then no energy radiated to infinity. However, this has no effect on the stress on the inner sphere [186].

70

Casimir Effect with Perfect Spherical Boundaries

Thus the Euclidean transform of the energy is

(4.22)

In effect, then, the Euclidean transformation is given by the recipe w ---+- ik4,

lwl ----+ i j k4 j , T ---+- iT4. In particular, the energy (4. 17) is transformed into

the expression

- 00 100 ( ) EE =

1 - 21ra

2_ 1=1

)

2

1

+

1

)

-2 1

dy ei8Yx
_ 00

<EFBFBD> + <20> s'
_l

+

s"

+

e'
_l

e"

s1 s1 e1 e1

= 00 1 100 -- 211ra 21=1 )21 + 1) -2 _00 dy ei8Yx-ddx ln(1 - A2d,

(4.23)

where

(4.24)

is written in terms of Ricatti-Bessel functions of imaginary argument,

(4.25)

In the above we have used the value of the Wronskian,

(4.26)

Here, as a result of the Euclidean rotation,

X = IYI,

y 1 = k -:- a 1.s real, as 1.s 0 = -1:- -T ---+- 0.

1

1 a

(4.27)

The same formula may be derived by computing the stress on the surface

through use of the stress tensor [15] , the force per unit area being given by

the discontinuity

Trrlrr==aa+- F =

- -1 =

2

f) -;:;-

E,

4na ua

(4.28)

E being given by (4.17) , or by (4.23) after the Euclidean rotation.

Electromagnetic Casimir Self-Stress on a Spherical Shell

71

A very rapidly convergent evaluation of this formula can be achieved by using the uniform asymptotic expansions for the Bessel functions:

(4.29a)
l __._ oo , (4.29b)

where

X = VZ,

ry

=

t -

1

+

ln

1

z
+ t-

1

,

(4.30)

and the uk (t) are polynomials in t of definite parity and of order 3k [1 1 1 ] ,
the first few of which are

u1 (t)

=

1 24

(3t

-

5t3 ) ,

(4.31a)

uz(t)

=

1

1 152

(81t2

-

462t4 +

285t6) ,

(4.31b)

u3 (t)

=

1 414720

(30375t3

-

369603t5

+

765765t1

-

425425t9)

,

(4.31c)

U4(t)

=

1 398131

20

(4465125t4

-

941

21

676t6

+ 349922430t8

- 446185740t10 + 185910725t12).

(4.31d)

If we now write

E

=

-

1 2a

L00
1=1

J(

l

,

o)

,

(4.32)

we easily find from the leading approximation,

(2v)2

ln(1

-

>.[2)

rv

-

(1

1 + z2)3

,

(4.33)

so that

l J(l,

0)

rv

3 32

,

___. 00.

(4.34)

o -=/= In order to obtain a finite sum, therefore, we must keep 0 until the end

of the calculation. By adding and subtracting the leading approximation

72

Casimir Effect with Perfect Spherical Boundaries

to the logarithm, we can write

+ J(l, o) = Rt St (8),

(4.35)

where

1CXldz [( + + + z2)3] Rt

=

-

1 - 271"

0

2l

1 ) 2 ln(1 - >.t2)

1 ( 1

= J(l, 0) --332 , (4.36)

and

loo z z ddz + z2)3 . 4171" Sl(o) = -

d i8vz

-=

e

1 (1

(4.37)

By use of the Euler-Maclaurin sum formula (2.84) , we can work out the
sum

( 4.38)

precisely the negative ofthe value of a single term at E = O!t The sum of the
remainder, L;1 Rt , is easily evaluated numerically, and changes this result
by less than 2%. Thus the result for the Casimir energy for a spherical
conducting shell is found to be

E

=

0.092353 2a

.

(4.40)

This agrees with the result found in 1968 by Boyer [13], evaluated more precisely by Davies [76] , and confirmed by a completely different method by Balian and Duplantier in 1978 [14] . Recently, this result has been re­ confirmed, using a zeta function method, by Leseduarte and Romeo [187, 188] . Reconsiderations using direct mode summation have also appeared [189, 190, 191].
It is, of course, possible to derive the result using potentials and ghost
fields. Unlike in our manifestly gauge-invariant approach, gauge invariance
must then be verified. See Ref. [192, 193] .

*This result may be formally obtained by zeta-function regularization:

L00 V8 = (T8 - 1)((-s) - T 8 ,
l=l

s < -1,

(4.39)

so if we formally extrapolate to s = 0, the angular momentum sum of unity becomes -1.

Electromagnetic Casimir Self-Stress on a Spherical Shell

73

Eberlein [194] considered the fluctuations of the force on a sphere. If
the observation time T is large compared to the radius of the sphere, T » a,

(4.41)

which is two orders of magnitude smaller than for parallel plates, as seen in (2. 1 1 8) .
Bordag, Elizalde, Kirsten, and Leseduarte [195] examined a scalar field with mass /-l subject to a spherical boundary. Divergences were encountered, which were removed by renormalizing constants in a classical Hamiltonian,

= Hclassical pV + CYS + Fa + k + -ah ,

(4.42)

= where V = 47ra3/3, S 47ra2• Although this would seem to make it

impossible to determine the Casimir energy, which is of the form of hja,

a renormalization prescription was imposed that only the contributions

corresponding to /-l <20> oo were to removed. Doing so left mass corrections

which did not decrease exponentially, as they did for parallel plates, as

discussed in Sec. 2 .4. Clearly there are issues here yet to be resolved. The

completely finite result for a massless scalar will be derived in Chapter 9.

4.1 . 1 Temperature Dependence

Balian and Duplantier [14] also considered the temperature dependence of the electromagnetic Casimir effect for a sphere. They computed the free energy in both the low and high temperature limits, with the results

F

"'

0.04618
a

-

(1!'a)

3

(kT)4
-1 - 5

'

kT « 1Ia'

F

"'

- 4 kT

(ln kTa +

0.769)

-

1
3840kTa2

,

kT » 1/a.

(4.43a) (4.43b)

In view of the relation between the force and the energy [which follows from applying the substitution (2.58) on the zero-temperature expression (4.28) for the force] the corresponding expressions for the energy are

(4.44)
Note that, unlike the situation for parallel plates, discussed after (2.61b),
ET does not vanish in the T _, oo limit. See Ref. [132, 196, 1 97] .

74

Casimir Effect with Perfect Spherical Boundaries

We sketch the derivation of these results in the leading uniform asymp­
totic approximation, where the approximation (4.33) holds. Then we may
write the approximation for the energy at finite temperature as a double
sum:§

(4.45)

As (3 -+ 0 we can approximate the sum over l by an integral,

(4.46)

where in the last step we adopted a zeta-function evaluation. Alternatively, we could keep the ei21rnaJjf3 point-splitting factor in the n sum, which then
evaluates as

<EFBFBD> <20>
1=1

e21fina8If3

=

- <20>2

-

_.!:_ 2i

cot

nao
(3

'

(4.47)

the real part of which is correctly -1/2. For low temperature, (3 » a, we instead replace the sum on n in (4.45)
by an integral,

=

- - 2n3a

00
"<22>"'"
1=1

- 179r2 [6

+

6ov

-

2(ov)3] e-v8

3

64a '

(4.48)

where the l sum may be carried out directly, or as in Ref. [15] . There are
no power of T corrections in this approximation.

§This resembles the double-sum representation found by Brown and Maclay for parallel plates [100) .

Fermion Fluctuations

75

To obtain the latter, it is necessary to use the exact expression (4.23) ,
which for finite T becomes

E T

=

-

1
27fa

<EFBFBD> L.). 2l l=l

+

1

)

12n3a

<EFBFBD>
<EFBFBD>
n=O

fn,l

,

(4.49)

where

<EFBFBD> fn,l = Xn d n ln (1 - Af(Xn)) ,

Xn

=

2nan T

,

(4.50)

where we note that fo,l = 0. We may evaluate this by use of the Euler­

Maclaurin sum formula (2.84) . Now the correction to the zero temperature

<EFBFBD> l I ( ) result (4.40) comes from the neighborhood of n = 0, where

d fn,l n=O = 0,

ds dns

fn,l

n=O

=

-6

12n3a

3 8ll ,

(4.51)

so that

kT « 1ja. (4.52)

4.2 Fermion Fluctuations

The corresponding calculation for a massless spin-1/2 particle subject to bag model boundary conditions (2. 1 19) on a spherical surface,

,)cis (1 + in .

=

o,

(4.53)

was carried out by Ken Johnson [198] and by Milton [23] . The result is also

a repulsive stress, of less than one-half the magnitude of the electromagnetic

result. (Recall that for parallel plates, the reduction factor was 7/8.)

(f'<27>O) In this case we wish to solve the Green's function equation G(x, x') = 8(x - x')

(4.54)

subject to the boundary condition (4.53). In the same representation for the gamma matrices used before in Sec. 2.7, this may be easily achieved in

76

Casimir Effect with Perfect Spherical Boundaries

terms of the total angular momentum eigenstates (J = L + (1/2)u):

( ) zJl=MJ±1/2 (0) =

l + 1/2 =t= M 2z + 1

1/2 y;lM- 1/2 (O) I +)

( ) =f

l + 1/2 ± M 2[ + 1

112
YlM+I/2 (0) 1 -) .

(4.55)

These may be interchanged by the radial spin operator U . r<>zJl=MJ± 1/2 = zJl=MJ=f l/2 .

(4.56)

These harmonics satisfy the addition theorem, the analog of (4.6)

J "<22>"

tr zJHM1/2 (o)zJJM±1/2 (0)*

M=- J

_
-

21 +
47r

1 .

(4.57)

From this point, it is straightforward to derive the fermionic Green's func­

tion (the function

details are
G(O) by

given

in

Ref.

[19]) .

It

differs

from

the

free

Dirac

Green's

c = ccoJ + c,

(4.58)

where, using a matrix notation for the two-dimensional spin space spanned by zJHM1/2 '

G- ±'f

=

- z. k

"" <20> J

hJ+Ij2(ka)jJ+lj2 (ka) [JJ+ 1/2 ( ka) J 2

-

hJ-1j2(ka)JJ-1j2 (ka) [JJ-1j2 (ka)J2

( ) =t=iwjJ+1;2 (kr)jJ+ 1;2 (kr') kjJ+lj2 (kr)]J- 1j2 (kr')
x -k]J_ 1;2 (kr)JJ+Ij2 (kr') =t=iWJJ- 1j2 (kr)JJ- 1j2 (kr') '

G- ±±

=

-ik L
J

1/k2a2 [JJ+Ij2 (ka)J2 - [JJ-1j2 (ka)]2

(4.59a)

( ) -ikwjj+1;2 (kr)JJ+Ij2 (kr') =fWJJ+Ij2 (kr)]J- 1j2 (kr')
x =fWJJ-1;2 (kr)]J+1;2 (kr') ik]J- 1j2 (kr)JJ- 1j2 (kr') · (4.59b)

Here, as in Sec. 2.7.2, the subscripts denote the eigenvalues of i')'5 .
Once the Green's function is found, it can be used in the usual way to
compute the vacuum expectation value of the stress tensor, which in the

Fermion Fluctuations

77

Dirac case is given by (2 . 143) , which leads directly to (unlike in Sec. 2 .7, a

= ! l factor of 2 is included for the charge trace) (Trr) tr 1 · rG(x, x')

.

( 4.60)

x' <20>x

The discontinuity of the stress tensor across the surface of the sphere gives

the energy according to

(4.61)
A quite straightforward calculation (the details are given in Refs. [19, 23])
gives the result for the sum of exterior and exterior modes, again, in terms of modified spherical Bessel functions:

(4.62)

The argument of the logarithm may also be written in an alternative form

(4.63)

where

>.z

=

.!!:._
dx

[

(

z

+

<EFBFBD>
2

)

-

<EFBFBD>
2

.!.:_!
dx

]

<EFBFBD> x

.

(4.64)

This expression may again be numerically evaluated through use of the uniform asymptotic approximants, with the result

E = 0.0204.

(4.65)

a

Somewhat less precision was obtained because, in this case, the leading uniform asymptotic approximation vanished. This result has been verified, to perhaps one more significant figure,

E

- _

0.02037

a

'

(4.66)

by Elizalde, Bordag, and Kirsten [199] . See also Ref. [200] .

Tilis page is intentionally left blank

Chapter 5
The Casimir Effect of a Dielectric Ball: The Equivalence of the Casimir
Effect and van der Waals Forces
A natural generalization of the considerations of the previous Chapter is to allow the spherical shell to be replaced by a dielectric ball, with permit­ tivity E. The Casimir energy, or self-stress, for such a situation was first considered by me in 1 980 [16] . This is a rather more subtle situation than the situation considered above, because when the speed of light is differ­ ent on the two sides of the boundary, the zero-point energy is not finite. However, as we shall see, it is possible to extract an unambiguous finite part, at least in the dilute approximation, by regulating the divergences, and renormalizing physical parameters. In this Chapter we will consider the most general situation, in which a ball of radius a, composed of a ma­ terial having permittivity E1 and permeability p,', is embedded in a uniform medium having permittivity E and permeability p,. Dispersion is included by allowing these electromagnetic parameters to depend on the frequency w. This configuration allows us to apply the results to the situation of sono­ luminescence, for example, where a bubble of air (E1 <20> 1 , p,' = 1 ) is inserted into a standing acoustic wave in water (E > 1 , p, = 1 ) . This application will be discussed in Chapter 12.
5 . 1 Green's Dyadic Formulation
We use the Green's dyadic formulation of Chapter 4, as modified for dielec­ tric materials. In terms of Green's dyadics, Maxwell's equations become in a region where E and p, are constant and there are no free charges or
79

80

The Casimir Effect of a Dielectric Ball

currents [cf. (2.96a)<29>(2.96d)]

V X r = iw<l> , v . <I> = 0,

-J11 V X <l> = -I.WEr1 '

v . r' = o,

(5.1)

in which r ' = r + 1/E, where 1 includes a spatial delta function. The
two solenoidal Green's dyadics given here satisfy the following second-order
equations:

(\72

+

w 2 EJ1) r '

=

1
E V - <20>

X

(V

X

1),

(\72 + w2EJ1)<l> = iwJlV X 1 .

(5.2a) (5.2b)

These can be expanded in terms of vector spherical harmonics (4.3) as

follows

( ) L r'(r, r') = lm

tz (r, r')Xzm (D) + W-E2- · Jl v X gz (r, r')Xzm (n)

' (5.3a)

<EFBFBD> ( <20> ) <I>(r, r') = .iiz (r, r')Xzm (D) - V x .fz (r, r')Xzm(D) . (5.3b)

When these are substituted in Maxwell's equations (5. 1 ) we obtain, first,

(5.4)

and then the second-order equations

j (Dz + w2 w)gz (r, r') = iwJl d!1" Xim(!1") · V" x 1 ,

(5.5a)

-<2D> j (Dz + w2J1E)fz (r, r') =

d!l" Xim (D") · V" x (V" x 1)

=

<EFBFBD>E

Dz

1 r 2

J(r

- r')Xim(D'),

where the spherical Bessel operator is

(5.5b)

(5.6) These equations can be solved in terms of Green's functions satisfying
(5.7)

Green's Dyadic Formulation

81

Let us specialize to the case of a sphere of radius a centered on the origin,
with properties E1, JJ1 in the interior and E, /J outside. Then the solutions
to (5.7) have the form

r, r' < a, r, r' > a,

(5.8)

where

k' = lwl <20>,

(5.9)

and h1 = h}1l is the spherical Hankel function of the first kind. Specifically,
we have

]z (r, r') = w2JJFz (r, r')X!m(r!') , gz(r, r1) = -iw!J"V1 x Gz (r, r1)Xim(r!1) ,

( 5 . 1 0 a) (5. 10b)

where Fz and Gz are Green's functions of the form (5.8) with the constants

A and B determined by the boundary conditions given below. Given Fz ,

L { Gz , the fundamental Green's dyadic is given by the generalization of (4.8) , r1(r, r1) = lm W2JJFz (r, r1)Xzm (r!)Xim(r!1)

} -

1 -E

V

X

Gz (r,

rI ) Xzm ( r ! ) Xi'm (r!I )

X

<EFBFBD>I V

+<2B>E r12 J(r - T1)Xzm(r!)Xim(r!1) .

(5 . 1 1 )

Because of the boundary conditions that

-1 B.L
/J

(5.12)

be continuous at r = a, we find for the constants A and B in the two Green's functions in (5. 1 1)

JE/1ez (x1)ef(x) - y'?/lez (x)ef (x1)

<EFBFBD><EFBFBD>
JE/1s1(x1)s[ (x) - y'?/ls1 (x)s[ (x1)

<EFBFBD><EFBFBD>

Ac

=

y'?/lez (x')ef (x) - JE/1ez (x)ef (x1) flz

( 5 . 1 3a) (5 . 13b) (5 . 1 3c)

82

The Casimir Effect of a Dielectric Ball

Be

=

JE'Jls1

(x')

s;

(x)

<EFBFBD>
l:ll

y'EJ?s1

(x)sf

(x')

.

( 5 . 1 3d)

Here we have introduced x = ka, x' = k'a, the Riccati-Bessel functions

el (x) = xhl (x) , s1 (x) = xj1 (x),

(5.14)

and the denominators

l:l1 = .fi/7sl (x')ef (x) - <20>sf(x')el (x),

A1 = <20>s1 (x')e; (x) - .fi/7s; (x')e1 (x),

(5.15)

and have denoted differentiation with respect to the argument by a prime.

5 . 2 Stress o n the Sphere

We can calculate the stress (force per unit area) on the sphere by computing the discontinuity of the radial-radial component of the stress tensor:

F = (Trr) (a-) - (Trr) (a+) ,

(5.16)

where

(5 . 1 7)

The vacuum expectation values of the product of field strengths are given di­ rectly by the Green's dyadics computed in Section 5 . 1 ; according to (2.1 15a) and (2.115b),

i (E(r)E(r')) = r(r, r'),

w i(B (r)B(r'))

=

-

1
2

v

X

r (r, r')

X

fo;;o. l v '

( 5 . 1 8a) ( 5 . 1 8b)

where here and in the following we ignore o functions because we are in­

terested in the limit as r' ---+ r. It is then rather immediate to find for the

stress on the sphere (the limit t' ---+ t is assumed)

Joo dw f F

=

_1_ 2ia2

e-iw(t-t') 21 + 1

-oo 27f

1=1 47f

{ [ ( ) J I x

(E' - E)

k2 -

a2

F1

+

E

l(l

+
,
E

1)

+

-1
E

a
ar

r

a
fir!

r '

G1

r=r'=a+