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arXiv:hep-th/0612232v1 20 Dec 2006

Brazilian Journal of Physics, vol 34, no. 4A, 2006, 1137-1149

1

The Casimir effect: some aspects

Carlos Farina

(1) Universidade Federal do Rio de Janeiro, Ilha do Fund˜ ao,

Caixa Postal 68528, Rio de Janeiro, RJ, 21941-972, Brazil

(Received)

We start this paper with a historical survey of the Casimir effect, showing that its origin is related to ex-

periments on colloidal chemistry. We present two methods of computing Casimir forces, namely: the global

method introduced by Casimir, based on the idea of zero-point energy of the quantum electromagnetic field, and

a local one, which requires the computation of the energy-momentum stress tensor of the corresponding field.

As explicit examples, we calculate the (standard) Casimir forces between two parallel and perfectly conducting

plates and discuss the more involved problem of a scalar field submitted to Robin boundary conditions at two

parallel plates. A few comments are made about recent experiments that undoubtedly confirm the existence of

this effect. Finally, we briefly discuss a few topics which are either elaborations of the Casimir effect or topics

that are related in some way to this effect as, for example, the influence of a magnetic field on the Casimir effect

of charged fields, magnetic properties of a confined vacuum and radiation reaction forces on non-relativistic

moving boundaries.

I.INTRODUCTION

A.Some history

The standard Casimir effect was proposed theoretically

by the dutch physicist and humanist Hendrik Brugt Gerhard

Casimir (1909-2000) in 1948 and consists, basically, in the

attraction of two parallel and perfectly conducting plates lo-

cated in vacuum [1]. As we shall see, this effect has its origin

in colloidal chemistry and is directly related to the dispersive

van der Waals interaction in the retarded regime.

The correct explanation for the non-retarded dispersive

van der Walls interaction between two neutral but polariz-

able atoms was possible only after quantum mechanics was

properly established. Using a perturbative approach, London

showed in 1930 [2] for the first time that the above mentioned

interaction is given by VLon(r) ≈ −(3/4)(?ω0α2)/r6, where

α is the static polarizability of the atom, ω0is the dominant

transition frequency and r is the distance between the atoms.

In the 40’s, various experiments with the purpose of studying

equilibrium in colloidal suspensions were made by Verwey

and Overbeek [3]. Basically, two types of force used to be

invoked to explain this equilibrium, namely: a repulsive elec-

trostaticforcebetweenlayersofchargedparticlesadsorbedby

thecolloidalparticlesandtheattractiveLondon-vanderWaals

forces.

However, the experiments performed by these authors

showedthatLondon’sinteractionwas notcorrectforlargedis-

tances. Agreement between experimentaldata and theory was

possible only if they assumed that the van der Waals interac-

tion fell with the distance between two atoms more rapidly

than 1/r6. They even conjectured that the reason for such a

different behaviour for large distances was due to the retar-

dation effects of the electromagnetic interaction (the informa-

tion of any change or fluctuation occurred in one atom should

spend a finite time to reach the other one). Retardation ef-

fects must be taken into account whenever the time interval

spent by a light signal to travel from one atom to the other

is of the order (or greater) than atomic characteristic times

(r/c≥1/ωMan, where ωmnare atomic transition frequencies).

Thoughthisconjectureseemedtobeveryplausible,arigorous

demonstration was in order. Further, the precise expression

of the van der Waals interaction for large distances (retarded

regime) should be obtained.

Motivated by the disagreement between experiments and

theory described above, Casimir and Polder [4] considered

for the first time, in 1948, the influence of retardation ef-

fects on the van der Waals forces between two atoms as well

as on the force between an atom and a perfectly conducting

wall. These authors obtained their results after lengthy calcu-

lations in the context of perturbative quantum electrodynam-

ics (QED). Since Casimir and Polder’s paper, retarded forces

between atoms or molecules and walls of any kind are usu-

ally called Casimir-Polder forces. They showed that in the

retarded regime the van der Waals interaction potential be-

tween two atoms is given by VRet(r) = −23?cαAαB/(4πr7).

In contrast to London’s result, it falls as 1/r7. The change

in the power law of the dispersive van der Waals force when

we go from the non-retarded regime to the retarded one

(FNR∼ 1/r7→ FR∼ 1/r8) was measured in an experiment

with sheets of mica by D. Tabor and R.H.S. Winterton [5]

only 20 years after Casimir and Polder’s paper. A change

was observed around 150˚ A, which is the order of magnitude

of the wavelength of the dominant transition (they worked in

the range 50˚ A−300˚ A, with an accuracy of ±4˚ A). They also

showed that the retarded van der Waals interaction potential

betweenan atomanda perfectlyconductingwall falls as 1/r4,

in contrast to the result obtained in the short distance regime

(non-retarded regime), which is proportional to 1/r3(as can

be seen by the image method). Casimir and Polder were very

impressed with the fact that after such a lengthy and involved

QEDcalculation,thefinalresultswereextremelysimple. This

is very clear in a conversation with Niels Bohr. In Casimir’s

own words

In the summer or autumn 1947 (but I am not ab-

solutely certain that it was not somewhat earlier

or later) I mentioned my results to Niels Bohr,

during a walk.“That is nice”, he said, “That is

something new.” I told him that I was puzzled

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et al.

by the extremely simple form of the expressions

for the interaction at very large distance and

he mumbled something about zero-point energy.

That was all, but it put me on a new track.

FollowingBohr’s suggestion,Casimir re-derivedthe results

obtainedwithPolderina muchsimplerway,bycomputingthe

shift in the electromagnetic zero-point energy caused by the

presence of the atoms and the walls. He presented his result

in the Colloque sur la th´ eorie de la liaison chimique, that took

place at Paris in April of 1948:

I foundthat calculatingchanges of zero-point en-

ergy really leads to the same results as the calcu-

lations of Polder and myself...

A short paper containingthis beautiful result was published

in a French journal only one year later [6]. Casimir, then,

decidedtotesthismethod,basedonthevariationofzero-point

energy of the electromagnetic field caused by the interacting

bodies in other examples. He knew that the existence of zero-

point energy of an atomic system (a hot stuff during the years

that followed its introduction by Planck [7]) could be inferred

by comparing energy levels of isotopes. But how to produce

isotopes of the quantum vacuum? Again, in Casimir’s own

words we have the answer [8]:

if there were two isotopes of empty space you

could really easy confirm the existence of the

zero-point energy. Unfortunately, or perhaps for-

tunately, there is only one copy of empty space

and if you cannot change the atomic distance

then you might change the shape and that was

the idea of the attracting plates.

A monthafter the Colloqueheld at Paris, Casimir presented

his seminal paper [1] on the attraction between two parallel

conducting plates which gave rise to the famous effect that

since then bears his name:

On 29 May, 1948, ‘I presented my paper on

the attraction between two perfectly conducting

plates at a meeting of the Royal Netherlands

Academy of Arts and Sciences. It was published

in the course of the year...

As we shall see explicitly in the next section, Casimir ob-

tained an attractive force between the plates whose modulus

per unit area is given by

F(a)

L2

≈ 0,013

1

(a/µm)4

dyn

cm2,

(1)

where a is the separation between the plates, L2the area of

each plate (presumably very large, i.e., L ≫ a).

A direct consequence of dispersive van der Waals forces

between two atoms or molecules is that two neutral but polar-

izable macroscopic bodies may also interact with each other.

However, due to the so called non-additivity of van der Waals

forces, the total interaction potential between the two bodies

is not simply given by a pairwise integration, except for the

case where the bodies are made of a very rarefied medium.

In principle, the Casimir method provides a way of obtaining

this kind of interaction potential in the retarded regime (large

distances) without the necessity of dealing explicitly with the

non-additivity problem. Retarded van der Waals forces are

usually called Casimir forces. A simple example may be in

order. Consider two semi-infinite slabs made of polarizable

material separated by a distance a, as shown in Figure 1.

a

A

B

ˆ rAB

−f

f

rAB

FIG. 1: Forces between molecules of the left slab and molecules of

the right slab.

Suppose the force exerted by a molecule A of the left slab

on a molecule B of the right slab is given by

fAB= −C

rγ

AB

rAB,

whereC andγ are positiveconstants, rABthe distance between

the molecules and ˆ rABthe unit vector pointing from A to B.

Hence, by a direct integration it is straightforward to show

that, for the case of dilute media, the force per unit area be-

tween the slabs is attractive and with modulus given by

Fslabs

Area=

C′

aγ−4,

(2)

whereC′is a positive constant. Observe that for γ = 8, which

correspondsto theCasimir andPolderforce,we obtaina force

between the slabs per unit area which is proportional to 1/a4.

Had we used the Casimir method based on zero-point energy

to compute this force we would have obtained precisely this

kind of dependence. Of course, the numerical coefficients

would be different, since here we made a pairwise integration,

neglecting the non-additivity problem. A detailed discussion

on the identification of the Casimir energy with the sum of

van der Waals interaction for a dilute dielectric sphere can be

found in Milton’s book [9] (see also references therein).

In 1956,Lifshitz and collaboratorsdevelopeda generalthe-

ory of van der Waals forces [10]. They derived a powerful

expression for the force at zero temperature as well as at fi-

nite temperature between two semi-infinite dispersive media

characterized by well defined dielectric constants and sepa-

rated by a slab of any other dispersive medium. They were

able to derive and predict several results, like the variation of

the thickness of thin superfluid helium films in a remarkable

agreement with the experiments [11]. The Casimir result for

metallic plates can be reobtained from Lifshitz formula in the

appropriate limit. The Casimir and Polder force can also be

inferred from this formula [9] if we consider one of the media

sufficiently dilute such that the force between the slabs may

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Brazilian Journal of Physics, vol 34, no. 4A, 2006, 1137-1149

3

be obtained by direct integration of a single atom-wall inter-

action [12].

The first experimental attempt to verify the existence of

the Casimir effect for two parallel metallic plates was made

by Sparnaay [13] only ten years after Casimir’s theoretical

prediction. However, due to a very poor accuracy achieved

in this experiment, only compatibility between experimental

data and theory was established. One of the great difficulties

was to maintain a perfect parallelism between the plates. Four

decades have passed, approximately, until new experiments

were made directly with metals. In 1997, using a torsion

pendulum Lamoreaux [14] inaugurated the new era of experi-

mentsconcerningthe Casimir effect. Avoidingtheparallelism

problem,he measuredthe Casimir force between a plate and a

spherical lens within the proximity force approximation [15].

This experiment may be considered a landmark in the history

of the Casimir effect, since it provided the first reliable exper-

imental confirmation of this effect. One year later, using an

atomic force microscope , Mohideen and Roy [16] measured

the Casimir force between a plate and a sphere with a better

accuracy and established an agreement between experimen-

tal data and theoretical predictions of less than a few percents

(depending on the range of distances considered). The two

precise experiments mentioned above have been followed by

manyothers and an incompletelist of the modernseries of ex-

periments about the Casimir effect can be found in [17]-[26].

For a detailed analysis comparing theory and experiments see

[27, 28]

We finish this subsection emphasizing that Casimir’s orig-

inal predictions were made for an extremely idealized situ-

ation, namely: two perfectly conducting (flat) plates at zero

temperature. Since the experimentalaccuracy achievednowa-

days is very high, any attempt to compare theory and exper-

imental data must take into account more realistic boundary

conditions. The most relevant ones are those that consider the

finiteconductivityofrealmetalsandroughnessofthesurfaces

involved. Theseconditionsbecomemoreimportantas the dis-

tance between the two bodies becomes smaller. Thermal ef-

fects must also be considered. However, in principle, these

effects become dominant compared with the vacuum contri-

bution for large distances, where the forces are already very

small. A great number of papers have been written on these

topics since the analysis of most recent experiments require

the consideration of real boundary conditions. For finite con-

ductivityeffects see Ref. [30]; the simultaneousconsideration

of roughness and finite conductivity in the proximity for ap-

proximationcanbe foundinRef. [31]andbeyondPFAin Ref.

[32] (see also references cited in the above ones). Concerning

the present status of controversies about the thermal Casimir

force see Ref. [29]

B.The Casimir’s approach

The novelty of Casimir’s original paper was not the predic-

tion of an attractive force between neutral objects, once Lon-

don had already explained the existence of a force between

neutral but polarizable atoms, but the method employed by

Casimir,whichwasbasedonthezero-pointenergyottheelec-

tromagnetic field. Proceeding with the canonical quantization

of the electromagnetic field without sources in the Coulomb

gauge we write the hamiltonian operator for the free radiation

field as

ˆ H =

2

∑

α=1∑

k

?ωk

?

ˆ a†

kαˆ akα+1

2

?

,

(3)

where ˆ a†

of a photon with momentum k and polarization α. The en-

ergy of the field when it is in the vacuum state, or simply the

vacuum energy, is then given by

kαand ˆ akαare the creation and annihilation operators

E0:=?0|ˆ H|0? =∑

k

2

∑

α=1

1

2?ωk,

(4)

which is also referred to as zero-point energy of the electro-

magnetic field in free space. Hence, we see that even if we do

not have any real photon in a given mode, this mode will still

contribute to the energy of the field with1

uum energy is then a divergent quantity given by an infinite

sum over all possible modes.

The presence of two parallel and perfectly conducting

plates imposes on the electromagnetic field the following

boundary conditions:

2?ωkαand total vac-

E× ˆ n|plates = 0

(5)

B· ˆ n|plates = 0 ,

whichmodifythepossiblefrequenciesofthe fieldmodes. The

Casimir energy is, then, defined as the difference between the

vacuum energy with and without the material plates. How-

ever, since in both situations the vacuum energy is a divergent

quantity,we needto adopta regularizationprescriptiontogive

a physical meaning to such a difference. Therefore, a precise

definition for the Casimir energy is given by

ECas:= lim

s→0

??

∑

kα

1

2?ωk

?

I

−

?

∑

kα

1

2?ωk

?

II

?

,

(6)

where subscript I means a regularized sum and that the fre-

quencies are computed with the boundary conditions taken

into account, subscript II means a regularized sum but with

no boundaryconditions at all and s stands for the regularizing

parameter. This definition is well suited for plane geometries

like that analyzed by Casimir in his original work. In more

complexsituations, like those involvingspherical shells, there

are some subtleties that are beyond the purposes of this intro-

ductoryarticle (the self-energyof a spherical shell dependson

its radius while the self-energy of a pair of plates is indepen-

dent of the distance between them).

Observe that, in the previous definition, we eliminate the

regularization prescription only after the subtraction is made.

Of course, there are many different regularizationmethods. A

quite simple but very efficient one is achieved by introducing

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et al.

a high frequency cut off in the zero-point energy expression,

as we shall see explicitly in the next section. This procedure

can be physically justified if we note that the metallic plates

becometransparentin thehighfrequencylimit sothat thehigh

frequency contributions are canceled out from equation (6).

Though the calculation of the Casimir pressure for the case

of two parallel plates is very simple, its determination may

become very involved for other geometries, as is the case,

for instance, of a perfectly conducting spherical shell. Af-

ter a couple of years of hard work and a “nightmare in Bessel

functions”, Boyer [33] computed for the first time the Casimir

pressure inside a spherical shell. Surprisingly, he found a re-

pulsive pressure, contrary to what Casimir had conjectured

five years before when he proposed a very peculiar model for

the stability of the electron [34]. Since then, Boyer’s result

has been confirmed and improved numerically by many au-

thors, as for instance, by Davies in 1972 [35], by Balian and

Duplantier in 1978 [36] and also Milton in 1978 [37], just to

mention some old results.

The Casimir effect is not a peculiarity of the electromag-

netic field. It can be shown that any relativistic field under

boundary conditions caused by material bodies or by a com-

pactification of space dimensions has its zero-point energy

modified. Nowadays, we denominate by Casimir effect any

changein thevacuumenergyofa quantumfielddueto anyex-

ternalagent,fromclassical backgroundsandnon-trivialtopol-

ogy to external fields or neighboringbodies. Detailed reviews

of the Casimir effect can be found in [9, 38, 39, 40, 41].

C. A local approach

In this section we present an alternative way of comput-

ing the Casimir energy density or directly the Casimir pres-

sure which makes use of a local quantity, namely, the energy-

momentum tensor. Recall that in classical electromagnetism

the total force on a distribution of charges and currents can

be computed integrating the Maxwell stress tensor through

an appropriate closed surface containing the distribution. For

simplicity, let us illustrate the method in a scalar field. The

lagrangian density for a free scalar field is given by

L?φ,∂µφ?= −1

The field equation and the corresponding Green function are

given, respectively, by

2∂µφ∂µφ−1

2m2φ2

(7)

(−∂2+m2)φ(x) = 0 ;(8)

(∂2−m2)G(x,x′) = −δ(x−x′),

(9)

where, as usual, G(x,x′) = i?0|T

Since the above lagrangian density does not depend explic-

itly on x, Noether’s Theorem leads naturally to the following

energy-momentumtensor (∂µTµν= 0)

∂L

∂(∂µφ)∂νφ+gµνL ,

?

φ(x)φ(x′)

?

|0?.

Tµν=

(10)

which, after symmetrization, can be written in the form

Tµν=1

2

?

∂µφ∂νφ+∂νφ∂µφ

?

+gµνL .

(11)

For our purposes, it is convenient to write the vacuum expec-

tation value (VEV) of the energy-momentum tensor in terms

of the above Green function as

?0|Tµν(x)|0?=−i

2lim

x′→x

?

(∂′

µ∂ν+∂′

ν∂µ)−gµν(∂′

α∂α+m2)

?

G(x,x′).

(12)

In this context, the Casimir energy density is defined as

ρC(x) = ?0|T00(x)|0?BC−?0|T00(x)|0?Free,

(13)

wherethesubscriptBC meansthattheVEVmustbecomputed

assuming that the field satisfies the appropriateboundarycon-

dition. Analogously, considering for instance the case of two

parallel plates perpendicular to the OZ axis (the generaliza-

tion for other configurations is straightforward) the Casimir

force per unit area on one plate is given by

FC= ?0|T+

zz|0?−?0|T−

zz|0?,

(14)

where superscripts + and − mean that we must evaluate ?Tzz?

on both sides of the plate. In other words, the desired Casimir

pressure on the plate is given by the discontinuity of ?Tzz? at

the plate. Using equation (12), ?Tzz? can be computed by

?0|Tzz|0? = −i

2lim

x′→x

?∂

∂z

∂

∂z′−∂2

∂z2

?

G(x,x′).

(15)

Local methods are richer than global ones, since they pro-

vide much more information about the system. Depending

on the problem we are interested in, they are indeed neces-

sary, as for instance in the study of radiative properties of an

atom inside a cavity. However, with the purpose of comput-

ing Casimir energies in simple situations, one may choose,

for convenience, global methods. Previously, we presented

only the global method introduced by Casimir, based on the

zero-point energy of the quantized field, but there are many

others, namely, the generalized zeta function method [42] and

Schwinger’s method [43, 44], to mention just a few.

II.EXPLICIT COMPUTATION OF THE CASIMIR FORCE

In this section, we show explicitly two ways of comput-

ing the Casimir force per unit area in simple situations where

plane surfaces are involved. We start with the global approach

introduced by Casimir which is based on the zero-point en-

ergy. Then, we give a second example where we use a local

approach, based on the energy-momentum tensor. We finish

this section by sketching some results concerning the Casimir

effect for massive fields.

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A. The electromagnetic Casimir effect between two parallel

plates

As our first example, let us consider the standard (QED)

Casimir effect where the quantized electromagnetic field is

constrained by two perfectly parallel conducting plates sepa-

rated by a distance a. For convenience,let us suppose that one

plate is located at z = 0, while the other is located at z = a.

The quantum electromagnetic potential between the metallic

plates in the Coulomb gauge (∇·A = 0) which satisfies the

appropriate BC is given by [45]

A(ρ,z,t) =

L2

(2π)2

?

∞ ′

∑

n=0

Z

d2κ

?2π?

ckaL2

?1/2

×

×

a(1)(κ,n)(ˆ κ×ˆ z)sin

?nπz

?

a

?

+

+ a(2)(κ,n)

?

inπ

kaκsin

?nπz

a

−ˆ zκ

kcos

?nπz

a

???

×

× ei(κ·ρ−ωt)+ h.c. ,

where ω(κ,n) = ck = c?κ2+n2π2/a2?1/2, with n being a

an extra 1/2 factor must be included in the normalization of

the field modes. The non-regularized Casimir energy then

reads

(16)

non-negativeinteger and the prime in Σ′means that for n = 0

Enr

c(a) =

?c

2

Z

L2d2κ

(2π)2

−?c

2

?

κ+2

∞

∑

n=1

Z+∞

−∞

?

κ2+n2π2

a2

?1/2?

κ2+k2z.

Z

L2d2κ

(2π)2

adkz

2π

2

?

Making the variable transformationκ2+(nπ/a)2=:λ and in-

troducing exponential cutoffs we get a regularized expression

(in 1948 Casimir used a generic cutoff function),

Er(a,ε) =L2

2π

?

1

2

Z∞

0

e−εκκ2dκ+

∞

∑

n=1

?

Z∞

nπ

a

e−ελλ2dλ −

−

Z∞

0

dn

Z∞

nπ

a

e−ελλ2dλ

=L2

2π

?

1

ε3+

∞

∑

n=1

dn∂2

∂2

∂ε2

?

e−εnπ/a

ε

?

?

−

−

Z∞

0

∂ε2

Z∞

nπ

a

e−ελdλ

.

(17)

Using that

∂2

∂ε2

?

1

ε

∞

∑

n=1

e−εnπ/a

?

=∂2

∂ε2

?1

ε

1

eεπ/a−1

?

,

as well as the definition of Bernoulli’s numbers,

1

et−1=

∞

∑

n=0

Bntn−1

n!

,

we obtain

E (a) =L2

2π

?

?

Bn

n!

1

ε3+∂2

∂ε2

?

ε4+(1+2B1)1

1

ε

∞

∑

n=0

Bn

n!

?επ

a

?n−1?

ε3+B4

?

−6a

πε4

?

=L2

2π

6(B0−1)a

π

1

12

?π

a

?3

+

∞

∑

n=5

?π

a

?n−1

(n−2)(n−3)εn−4

.

Usingthewell knownvaluesB0=1, B1=−1

and taking ε → 0+, we obtain

2andB4=−1

30,

E (a)

L2

= −?c

π2

24×30·1

a3

As a consequence, the force per unit area acting on the plate

at z = a is given by

F(a)

L2

= −1

L2

∂Ec(a)

∂a

= −π2?c

240a4≈ −0,013

1

(a/µm)4

dyn

cm2,

where in the last step we substituted the numerical values

of ? and c in order to give an idea of the strength of the

Casimir pressure. Observe that the Casimir force between

the (conducting) plates is always attractive. For plates with

1cm2of area separated by 1µm the modulus of this attrac-

tive force is 0,013dyn. For this same separation, we have

PCas≈ 10−8Patm, where Patmis the atmospheric pressure at

sea level. Hence, for the idealized situation of two perfectly

conducting plates and assuming L2= 1 cm2, the modulus of

the Casimir force would be ≈ 10−7N for typical separations

used in experiments. However, due to the finite conductivity

ofreal metals, the Casimir forcesmeasuredin experimentsare

smaller than these values.

B.The Casimir effect for a scalar field with Robin BC

In order to illustrate the local method based on the energy-

momentum tensor, we shall discuss the Casimir effect of a

massless scalar field submitted to Robin BC at two parallel

plates, which are defined as

φ|bound.= β∂φ

∂n|bound.,

(18)

where, by assumption, β is a non-negative parameter. How-

ever, before computing the desired Casimir pressure, a few

comments about Robin BC are in order.

First, we note that Robin BC interpolate continuously

Dirichlet and Neumannones. For β→0 we reobtainDirichlet

BC while for β → ∞ we reobtain Neumann BC. Robin BC al-

ready appear in classical electromagnetism,classical mechan-

ics, wave, heat and Schr¨ odingerequations[46] andevenin the

study of interpolating partition functions [47]. A nice realiza-

tion of these conditions in the context of classical mechanics