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Brazilian Journal of Physics, vol. 36, no. 4A, 2006 1137
The Casimir Effect: Some Aspects
Carlos Farina
Universidade Federal do Rio de Janeiro, Ilha do Fund˜ ao,
Caixa Postal 68528, Rio de Janeiro, RJ, 21941-972, Brazil
Received on 10 August, 2006
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.
Keywords: Quantum field theory; Casimir effect
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-
trostatic force between layers of charged particles adsorbed by
thecolloidalparticlesandtheattractiveLondon-vanderWaals
forces.
However, the experiments performed by these authors
showedthatLondon’sinteractionwasnotcorrectforlargedis-
tances. Agreement between experimental data 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
between an atom and a perfectly conducting wall 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
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1138Carlos Farina
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
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.
Following Bohr’s suggestion, Casimir re-derived the results
obtainedwithPolderinamuchsimplerway, 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 found that calculating changes of zero-point en-
ergy really leads to the same results as the calcu-
lations of Polder and myself...
A short paper containing this 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 month after the Colloque held 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 and γ are positive constants, 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
corresponds to the Casimir and Polder force, we obtain a 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 collaborators developed a general the-
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
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Brazilian Journal of Physics, vol. 36, no. 4A, 20061139
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
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-
ments concerning the Casimir effect. Avoiding the parallelism
problem, he measured the 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
many others and an incomplete list of the modern series 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 experimental accuracy achieved nowa-
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
finite conductivity of real metals and roughness of the surfaces
involved. These conditions become more important as 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-
ductivity effects see Ref. [30]; the simultaneous consideration
of roughness and finite conductivity in the proximity for ap-
proximation can be found in Ref. [31] and beyond PFA in 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 ,
which modify the possible frequencies of the field modes. 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 need to adopt a regularization prescription to give
a physical meaning to such a difference. Therefore, a precise
definition for the Casimir energy is given by
??
kα
ECas:= lim
s→0
∑
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 boundary conditions 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
complex situations, like those involving spherical shells, there
are some subtleties that are beyond the purposes of this intro-
ductory article (the self-energy of a spherical shell depends on
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1140Carlos Farina
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 regularization methods. A
quite simple but very efficient one is achieved by introducing
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
become transparent in the high frequency limit so that the high
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
change in the vacuum energy of a quantum field due to any ex-
ternal agent, from classical backgrounds and non-trivial topol-
ogy to external fields or neighboring bodies. Detailed reviews
of the Casimir effect can be found in [9, 38–41].
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
2∂µφ∂µφ−1
2m2φ2
(7)
The field equation and the corresponding Green function are
given, respectively, by
(−∂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-momentum tensor (∂µTµν= 0)
?
φ(x)φ(x?)
?
|0?.
Tµν=
∂L
∂(∂µφ)∂νφ+gµνL,
(10)
which, after symmetrization, can be written in the form
?
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
?
In this context, the Casimir energy density is defined as
Tµν=1
2
∂µφ∂νφ+∂νφ∂µφ
?
+gµνL.
(11)
?0|Tµν(x)|0?=−i
2lim
x?→x
(∂?
µ∂ν+∂?
ν∂µ)−gµν(∂?
α∂α+m2)
?
G(x,x?).
(12)
ρC(x) = ?0|T00(x)|0?BC−?0|T00(x)|0?Free,
wherethesubscriptBC meansthattheVEVmustbecomputed
assuming that the field satisfies the appropriate boundary con-
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
(13)
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
x?→x
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.
2lim
?∂
∂z
∂
∂z?−∂2
∂z2
?
G(x,x?).
(15)
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
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Brazilian Journal of Physics, vol. 36, no. 4A, 20061141
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.
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
?
d2κ
?2π?
ckaL2
?1/2
×
×
a(1)(κ,n)(ˆ κ×ˆ z)sin
?
× 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
?
−?c
2
(2π)2
?nπz
?
a
?
+
+ a(2)(κ,n)
inπ
kaκsin
?nπz
a
−ˆ zκ
kcos
?nπz
a
???
×
(16)
non-negative integer and the prime in Σ?means that for n = 0
Enr
c(a) =
?c
2
?
L2d2κ
(2π)2
?
κ+2
∞
∑
n=1
?+∞
?
κ2+n2π2
a2
?
?1/2?
κ2+k2z.
L2d2κ
−∞
adkz
2π
2
Making the variable transformation κ2+(nπ/a)2=: λ and in-
troducing exponential cutoffs we get a regularized expression
(in 1948 Casimir used a generic cutoff function),
?
2
?∞
?
n=1
?∞
Er(a,ε) =L2
2π
1
?∞
0
e−εκκ2dκ+
∞
∑
n=1
?
?∞
nπ
a
e−ελλ2dλ −
−
0
dn
?∞
nπ
a
e−ελλ2dλ
=L2
2π
1
ε3+
∞
∑
∂2
∂ε2
?
e−εnπ/a
ε
?
?
−
−
0
dn∂2
∂ε2
?∞
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
?n−1(n−2)(n−3)εn−4
1
ε
∞
∑
n=0
Bn
n!
?επ
a
?n−1?
ε3+B4
?
−6a
πε4
?π
?
=L2
2π
6(B0−1)a
?π
π
1
12a
?3
+
∞
∑
n=5
a
.
Using the well known values B0=1, B1=−1
and taking ε → 0+, we obtain
E(a)
L2
2and B4=−1
30,
= −?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
of real metals, the Casimir forces measured in experiments are
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)
Page 6
1142Carlos Farina
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 Neumann ones. For β→0 we reobtain Dirichlet
BC while for β → ∞ we reobtain Neumann BC. Robin BC al-
ready appear in classical electromagnetism, classical mechan-
ics, wave, heat and Schr¨ odinger equations [46] and even in the
study of interpolating partition functions [47]. A nice realiza-
tion of these conditions in the context of classical mechanics
can be obtained if we study a vibrating string with its extremes
attached to elastic supports [48? ]. Robin BC can also be used
in a phenomenological model for a penetrable surfaces [49].
In fact, in the context of the plasma model, it can be shown
that for frequencies much smaller than the plasma frequency
(ω ? ωP) the parameter β plays the role of the plasma wave-
length [? ]. In the context of QFT this kind of BC appeared
more than two decades ago [50, 51]. Recently, they have been
discussed in a variety of contexts, as in the AdS/CFT corre-
spondence [52], in the discussion of upper bounds for the ratio
entropy/energy in confined systems [53], in the static Casimir
effect [54], in the heat kernel expansion [55–57] and in the
one-loop renormalization of the λφ4theory [58, 59]. As we
will see, Robin BC can give rise to restoring forces in the sta-
tic Casimir effect [54].
Consider a massless scalar field submitted to Robin BC on
two parallel plates:
φ|z=0= β1∂φ
where we assume β1(β2) ≥ 0. For convenience, we write
?
with α = 0,1,2, gµν= diag(−1,+1,+1,+1) and the reduced
Green function g(z,z?;k⊥,ω) satisfies
?∂2
where λ2=ω2−k2
lowing boundary conditions (for simplicity, we shall not write
k⊥,ω in the argument of g):
∂
∂zg(0,z?) ;
It is not difficult to see that g(z,z?) can be written as
For points inside the plates, 0 < z,z?< a, the final expression
for the reduced Green function is given by
?
2ıλ(1
∂z|z=0 ; φ|z=a= −β2∂φ
∂z|z=a,
(19)
G(x,x?) =
d3kα
(2π)3eıkα(x−x?)αg(z,z?;k⊥,ω),
(20)
∂z2+λ2
?
g(z,z?;k⊥,ω) = −δ(z−z?),
(21)
⊥and g(z,z?;k⊥,ω) is submitted to the fol-
g(0,z?) = β1
g(a,z?) = −β2
∂
∂zg(a,z?)
g(z,z?)=
A(z?)(sinλz+β1λcosλz),
z < z?
B(z?)(sinλ(z−a)−β2λcosλ(z−a)), z > z?
(22)
gRR(z,z?) = −
1
γ1eıλz<−e−ıλz<??
1
γ2eıλ(z>−a)−eıλ(z>−a)?
γ1eıλa−1
γ2e−ıλa)
,
(23)
where γi=1+iβiλ
we have:
gRR(z,z?) =eıλ(z<−a)
1−iβiλ(i=1,2). Outside the plates, with z,z?>a,
?1
Defining tµνsuch that
2ıλγ2eıλ(z<−a)−e−ıλ(z<−a)
?
.
(24)
?Tµν(x)? =
?d2k⊥
(2π)
dω
2π?tµν(x)?,
(25)
we have
?t33(z)? =1
2ilim
z?→z
?∂
∂z
∂
∂z?+λ2
?
g(z,z?).
(26)
The Casimir force per unit area on the plate at x = a is given
by the discontinuity in ?t33?:
?d2k⊥
After a straightforward calculation, it can be shown that
F =
(2π)
dω
2π
?
?t33?|z=a−−?t33?|z=a+
?
.
F (β1,β2;a)=−
1
32π2a4
?∞
0
dξ ξ3
??1+β2ξ/2a
?1+β1ξ/2a
1−β1ξ/2a1−β2ξ/2a
?
eξ−1
.
(27)
Depending on the values of parameters β1and β2, restoring
Casimir forces may arise, as shown in Figure 2 by the dotted
line and the thin solid line.
pa4
a
O
(β1= 0 ,β2→ ∞)
(β1= β2= 0 or β1= β2→ ∞)
(β1= 0 , β2= 1)
FIG. 2: Casimir pressure, conveniently multiplied by a4, as a func-
tion of a for various values of parameters β1and β2.
The particular cases of Dirichlet-Dirichlet, Neumann-
Neumann and Dirichlet-Neumann BC can be reobtained if
we take, respectively, β1= β2= 0, β1= β2→ ∞ and β1=
0; β2→ ∞. For the first two cases, we obtain
FDD(a) =FNN(a) = −
1
32π2a4
?∞
0
dξ
ξ3
eξ−1.
(28)
Page 7
Brazilian Journal of Physics, vol. 36, no. 4A, 2006 1143
Using the integral representation
where ζRis the Riemann zeta function, we get half the elec-
tromagnetic result, namely,
?∞
0dξ
ξs−1
eξ−1= ζR(s)Γ(s),
FDD(a) =FNN(a) = −π2?c
For the case of mixed BC, we get
480
1
a4.
(29)
FDN(a) = +
1
32π2a4
?∞
0
dξ
ξ3
eξ+1,
(30)
Using in the previous equation the integral representation
?∞
sure (equal to half of Boyer’s result [60] obtained for the elec-
tromagnetic field constrained by a perfectly conducting plate
parallel to an infinitely permeable one),
0dξξs−1
eξ+1= (1−21−s)Γ(s)ζR(s) we get a repulsive pres-
FDN(a) =7
8×π2?c
480
1
a4.
(31)
C. The Casimir effect for massive particles
In this subsection, we sketch briefly some results concern-
ing massive fields, just to get some feeling about what kind
of influence the mass of a field may have in the Casimir ef-
fect. Firstly, let us consider a massive scalar field submit-
ted to Dirichlet BC in two parallel plates, as before. In this
case, the allowed frequencies for the field modes are given
by ωk= c
Casimir method explained previously, to the following result
for the Casimir energy per unit area (see, for instance, Ref.
[38])
?
κ2+n2π2
a2 +m2?1/2, which lead, after we use the
1
L2Ec(a,m) = −m2
8π2a
∞
∑
n=1
1
n2K2(2amn) ,
(32)
where Kνis a modified Bessel function, m is the mass of the
field and a is the distance between the plates, as usual. The
limit of small mass, am ? 1, is easily obtained and yields
1
L2Ec(a,m) ≈ −
As expected, the zero mass limit coincides with our previous
result (29) (after the force per unit area is computed). Observ-
ing the sign of the first correction on the right hand side of last
equation we conclude that for small masses the Casimir effect
is weakened.
On the other hand, in the limit of large mass, am?1, it can
be shown that
m2
16π2a
Note that the Casimir effect disappears for m → ∞, since in
this limit there are no quantum fluctuations for the field any-
more. An exponential decay with ma is related to the plane
π2
1440a3+m2
96a.
(33)
1
L2Ec(a,m) ≈ −
?π
ma
?1/2
e−2ma
(34)
geometry. Other geometries may give rise to power law de-
cays when ma → ∞. However, in some cases, the behav-
iour of the Casimir force with ma may be quite unexpected.
The Casimir force may increase with ma before it decreases
monotonically to zero as ma → ∞ (this happens, for instance,
when Robin BC are imposed on a massive scalar field at two
parallel plates [61]).
In the case of massive fermionic fields, an analogous be-
haviour is found. However, some care must be taken when
computing the Casimir energy density for fermions, regarding
what kind of BC can be chosen for this field. The point is
that Dirac equation is a first order equation, so that if we want
non-trivial solutions, we can not impose that the field satis-
fies Dirichlet BC at two parallel plates, for instance. The most
appropriate BC for fermions is borrowed from the so called
MIT bag model for hadrons [62], which basically states that
there is no flux of fermions through the boundary (the normal
component of the fermionic current must vanish at the bound-
ary). The Casimir energy per unit area for a massive fermionic
field submitted to MIT BC at two parallel plates was first com-
puted by Mamayev and Trunov [63] (the massless fermionic
field was first computed by Johnson in 1975 [64])
1
L2Ef
c(a,m) =
−
1
π2a3
?∞
madξξ
?
ξ2−m2a2log
?
1+ξ−ma
ξ+mae−2ξ
?
.
(35)
The small and large mass limits are given, respectively, by
1
L2Ef
1
L2Ef
c(a,m) ≈ −
c(a,m) ≈ −3(ma)1/2
7π2
2880a3+
m
24a2;
25π3/2a3e−2ma.
(36)
Since the first correction to the zero mass result has an oppo-
site sign, also for a fermionic field small masses diminish the
Casimir effect. In the large mass limit, ma → ∞, we have a
behaviour analogous to that of the scalar field, namely, an ex-
ponential decay with ma (again this happens due to the plane
geometry). We finish this section with an important observa-
tion: even a particle as light as the electron has a completely
negligible Casimir effect.
III.MISCELLANY
In this section we shall briefly present a couple of topics
which are in some way connected to the Casimir effect and
that have been considered by our research group in the last
years. For obvious reasons, we will not be able to touch all
the topics we have been interested in, so that we had to choose
only a few of them. We first discuss how the Casimir effect
of a charged field can be influenced by an external magnetic
field. Then, we show how the constitutive equations asso-
ciated to the Dirac quantum vacuum can be affected by the
Page 8
1144Carlos Farina
presence of material plates. Finally, we consider the so called
dynamical Casimir effect.
A.Casimir effect under an external magnetic field
The Casimir effect which is observed experimentally is that
associated to the photon field, which is a massless field. As
we mentioned previously, even the electron field already ex-
hibits a completely unmeasurable effect. With the purpose
(and hope) of enhancing the Casimir effect of electrons and
positrons we considered the influence of an external elec-
tromagnetic field on their Casimir effect. Since in this case
we have charged fields, we wondered if the virtual electron-
positron pairs which are continuously created and destroyed
from the Dirac vacuum would respond in such a way that
the corresponding Casimir energy would be greatly amplified.
This problem was considered for the first time in 1998 [65]
(see also Ref. [66]). The influence of an external magnetic
field on the Casimir effect of a charged scalar field was con-
sidered in Ref. [67] and recently, the influence of a magnetic
field on the fermionic Casimir effect was considered with the
more appropriate MIT BC [68].
For simplicity, let us consider a massive fermion field un-
der anti-periodic BC (in the OZ direction) in the presence
of a constant and uniform magnetic field in this same direc-
tion. After a lengthy but straightforward calculation, it can be
shown that the Casimir energy per unit area is given by [66]
E(a,B)
?2
= −2(am)2
π2a3
∞
∑
n=1
∞
∑
n=1
(−1)n−1
n2
?∞
K2(amn)
−
eB
4π2a
(−1)n−1
0dσ e−(n/2)2σ−(am)2/σL
?eBa2
σ
?
,
where we introduced the Langevin function: L(ξ) = cothξ−
1/ξ. In the strong field limit, we have
E(a,B)
?2
≈ −eBm
π2
∞
∑
n=1
(−1)n−1
n
K1(amn) .
(37)
In this limit we can still analyze two distinct situations,
namely, the small mass limit (ma ? 1) and the large mass
limit (ma?1). Considering distances between the plates typ-
ical of Casimir experiments (a ≈ 1µm), we have in the former
case
ρc(a,B)
ρc(a,0)≈ 10−4×
B
Tesla;
ma ? 1 ,
(38)
while in the latter case,
ρc(a,B)
ρc(a,0)≈ 10−10×
B
Tesla;
ma ? 1 .
(39)
In the above equations ρc(a,B) and ρc(a,0) are the Casimir
energy density under the influence of the magnetic field and
without it, respectively.Observe that for the Casimir ef-
fect of electrons and positrons, which must be treated as the
large mass limit described previously, huge magnetic fields
are needed in order to enhance the effect (far beyond accessi-
ble fields in the laboratory). In other words, we have shown
that though the Casimir effect of a charged fermionic field can
indeed be altered by an external magnetic field, the universal
constants conspired in such a way that this influence turns out
to be negligible and without any chance of a direct measure-
ment at the laboratory.
B.Magnetic permeability of the constrained Dirac vacuum
In contrast to the classical vacuum, the quantum vacuum
is far from being an empty space, inert and insensible to any
external influence. It behaves like a macroscopic medium, in
the sense that it responds to external agents, as for example
electromagnetic fields or the presence of material plates. As
previously discussed, recall that the (standard) Casimir effect
is nothing but the energy shift of the vacuum state of the field
caused by the presence of parallel plates. There are many
other fascinating phenomena associated to the quantum vac-
uum, namely, the particlecreationproducedby theapplication
of an electric field [69], the birefringence of the QED vacuum
under an external magnetic field [70, 71] and the Scharnhorst
effect [72, 73], to mention just a few. This last effect pre-
dicts that the velocity of light propagating perpendicularly to
two perfectly conducting parallel plates which impose (by as-
sumption) BC only on the radiation field is slightly altered by
the presence of the plates. The expected relative variation in
the velocity of light for typical values of possible experiments
is so tiny (∆c/c ≈ 10−36) that this effect has not been con-
firmed yet. Depending on the nature of the material plates the
velocity of light propagating perpendicular to the plates is ex-
pected to diminish [74]. The Scharnhorst effect inside a cavity
was considered in [75].
The negligible change in the velocity of light predicted by
Scharnhorst may be connected with the fact that the effect that
bears his name is a two-loop QED effect, since the classical
field of a traveling light wave interacts with the radiation field
only through the fermionic loop.
With the purpose of estimating a change in the constitutive
equations of the quantum vacuum at the one-loop level, we
wereledtoconsiderthefermionicfieldsubmittedtosomeBC.
The magnetic permeability µ of the constrained Dirac vacuum
was computed by the first time in Ref. [76], but with the non-
realistic anti-periodic BC. In this case, the result found for the
relative change in the permeability, ∆µ :=µ−1, was also neg-
ligible (an analogous calculation has also been made in the
context of scalar QED [77]). However, when the more real-
istic MIT BC are imposed on the Dirac field at two parallel
plates things change drastically. In this case, it can be shown
that the magnetic permeability of the constrained Dirac vac-
uum is given by [78]
1
µ(ma)= 1−π−2
12π2
e2
ma+
1
6π2
e2
maH(ma),
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Brazilian Journal of Physics, vol. 36, no. 4A, 20061145
where
H(ma) =
?∞
1
dx
x
(x2−1)3/2ln
?
1+
?x−1
x+1
?
e−2max
?
.
(40)
For confining distances of the order of 0,1µm, we have
∆µ := µ−1 ≈ 10−9.
(41)
The previous value is comparable to the magnetic permeabil-
ity of Hydrogen and Nitrogen at room temperature and at-
mospheric pressure. Hence, an experimental verification of
this result seems to be not unfeasible (we will come back to
this point in the final remarks).
C.The Dynamical Casimir effect
As our last topic, we shall briefly discuss the so called dy-
namical Casimir effect, which consists, as the name suggests,
of the consideration of a quantum field in the presence of
moving boundaries. Basically, the coupling between vacuum
fluctuations and a moving boundary may give rise to dissi-
pative forces acting on the boundary as well as to a particle
creation phenomenon. In some sense, these phenomena were
expected. Recall that the static Casimir force is a fluctuat-
ing quantity [79] and hence, using general arguments related
to the fluctuation-dissipation theorem [80] dissipative forces
on moving boundaries are expected. Further, using arguments
of energy conservation we are led to creation of real parti-
cles (photons, if we are considering the electromagnetic field
[81]). For the above reasons, this topic is sometimes referred
to as radiation reaction force on moving boundaries.
After Schwinger’s suggestion that the phenomenon of
sonoluminescence could be explained by the dynamical
Casimir effect [82] (name coined by himself), a lot of work
has been done on this subject. However, it was shown a few
years later that this was not the case (see [9] and references
therein for more details).
The dynamical Casimir effect already shows up in the case
of one (moving) mirror [83–85]. However, oscillating cavi-
ties whose walls perform vibrations in parametric resonance
with a certain unperturbed field eigenfrequency may greatly
enhance the effect [86–89]. Recently, a one dimensional os-
cillating cavity with walls of different nature was considered
[90]. The dynamical Casimir effect has also been analyzed for
a variety of three-dimensional geometries, including parallel
planeplates[91], cylindricalwaveguides[92], andrectangular
[93], cylindrical [94] and spherical cavities [95]. For a review
concerning classical and quantum phenomena in cavities with
moving boundaries see Dodonov [96] and for a variety of top-
ics on non-stationary Casimir effect including perspectives of
its experimental verification see the special issue [97].
In this section, we shall discuss the force exerted by the
quantum fluctuations of a massless scalar field on one moving
boundary as well as the particle creation phenomenon in a un-
usual example in 1+1 dimensions, where the field satisfies a
Robin BC at the moving boundary [98, 99]. We shall follow
throughout this paper the perturbative method introduced by
Ford and Vilenkin [84]. This method was also applied suc-
cessfully to the case of the electromagnetic field under the
influence of one moving (perfectly) conducting plate [85] as
well as an oscillating cavity formed by two parallel (perfectly)
conducting plates [91].
Let us then consider a massless scalar field φ in 1+1 in
the presence of one moving boundary which imposes on the
fieldaRobinBCatonemovingboundarywhenobservedfrom
a co-moving inertial frame. By assumption, the movement
of the boundary is prescribed, non-relativistic and of small
amplitude (δq(t) is the position of the plate at a generic in-
stant t). Last assumptions may be stated mathematically by
|δ˙ q(t)| << c
to the (main) mechanical frequency. Therefore, we must solve
the following equation: ∂2φ(x,t) = 0, with the field satisfying
a Robin BC at the moving boundary given by
?∂
where β is a non-negative parameter and the previous condi-
tion was already written in the laboratory frame. We are ne-
glecting terms of the order O(δ˙ q2/c2). The particular cases of
Dirichlet and Neumann BC are reobtained by making β = 0
and β → ∞, respectively. The dissipative forces for these par-
ticular cases were studied at zero temperature as well as at
finite temperature and also with the field in a coherent state
in [100] (dissipative forces on a perfectly conducting mov-
ing plate caused by the vacuum fluctuations of the electro-
magnetic field at zero and non-zero temperature were studied
in [101]). The perturbative approach introduced by Ford and
Vilenkin [84] consists in writing
and |δq(t)| << c/ω0, where ω0corresponds
∂x+δ˙ q(t)∂
∂t
?
φ(x,t)|x=δq(t)=1
βφ(x,t)|x=δq(t).
(42)
φ(x,t) = φ0(x,t)+δφ(x,t) ,
(43)
where φ0(x,t) is the field submitted to a Robin BC at a sta-
tic boundary fixed at the origin and δφ(x,t) is the first order
contribution due to the movement of the boundary. The total
force on the moving boundary may be computed with the aid
of the corresponding energy-momentum tensor, namely,
δF(t) = ?0|T11?t,δq+(t)?−T11?t,δq−(t)?|0? ,
where superscripts + and − mean that we must compute the
energy-momentum tensor on both sides of the moving bound-
ary. It is convenient to work with time Fourier transforms.
The susceptibility χ(ω) is defined in the Fourier space by
(44)
δF (ω) =: χ(ω) δQ(ω)
(45)
where δF (ω) and δQ(ω) are the Fourier transformations of
δF(t) and δq(t), respectively. It is illuminating to compute
the total work done by the vacuum fluctuations on the moving
boundary. It is straightforward to show that
?+∞
Note that only the imaginary part of χ(ω) appears in the pre-
vious equation. It is responsible for the dissipative effects and
−∞
δF(t)δ˙ q(t)dt = −1
π
?∞
0
dωωImχ(ω)|δQ(ω)|2. (46)
Page 10
1146Carlos Farina
hence it is closely related with the total energy converted into
real particles. On the other hand, the real part of χ(ω), when it
exists, does not contribute to the total work and hence it is not
related to particle creation, but to dispersive effects. For the
particular cases of Dirichlet or Neumann BC it can be shown
that the susceptibility is purely imaginary and given by
χD(ω) = χN(ω) = i?ω3
6πc2,
(47)
which implies
δF(t) =
?
6πc2
d3
dt3δq(t).
(48)
Since Imχ(ω) > 0, for these cases the vacuum fluctuations
are always dissipating energy from the moving boundary.
However, for Robin BC an interesting thing happens. It
can be shown that χ(ω) acquires also a real part, which gives
rise to a dispersive force acting on the moving boundary. The
explicit expressions of R eχ(ω) and Imχ(ω) can be found
in [98], but the general behaviour of them as functions of
ω is shown in Figure 3. For convenience, we normalize
these quantities dividing them by the value ImχD(ω), where
the subscript D means that Imχ(ω) must be computed with
Dirichlet BC.
??
?
FIG. 3: Imaginary and real parts of χ(ω) with Robin BC appropri-
ately normalized by the value of Imχ(ω) for the Dirichlet BC.
Now, let us discuss briefly the particle creation phenom-
enon under Robin BC. Here, we shall consider a semi-infinite
slab extending from −∞ to δq(t) following as before a pre-
scribed non-relativistic motion which imposes on the field
Robin BC at δq(t). It can be shown that the corresponding
spectral distribution is given by [99]
dN
dω(ω) =
4ω
1+β2ω2
?∞
0
dω?
2π
[δQ(ω−ω?)]2
1+β2ω?2
ω?
?
1−β2ωω?
?2
(49)
As an explicit example, let us consider the particular motion
δq(t) = δq0e−|t|/Tcos(ω0t),
(50)
where, by assumption, ω0T ? 1 (this is made in order to sin-
gle out the effect of a given Fourier component of the motion).
For this case, we obtain the following spectral distribution
dN
dω(ω)= (δq0)2Tω(ω0−ω)×
[1−β2ω(ω0−ω)]2
(1+β2ω2)(1+β2(ω0−ω)2)Θ(ω0−ω). (51)
The spectral distributions for the particular cases of Dirichlet
or Neumann BC can be easily reobtained by making simply
β = 0 and β → ∞, respectively. The results coincide and are
given by [102] (we are making c = 1)
×
dN(D)
dω
(ω) =dN(N)
dω
(ω) = (δq0)2Tω(ω0−ω)Θ(ω0−ω) .
(52)
A simple inspection in (51) shows that, due to the presence of
the Heaviside step function, only the field modes with eigen-
frequencies smaller than the mechanical frequency are ex-
cited. Further, the number of particles created per unit fre-
quency when Robin BC are used is always smaller than the
number of particles created per unit frequency when Dirichlet
(or Neumann) BC are employed.
FIG. 4: Spectral distributions of created particles for: Dirichlet and
Neumann BC (dashed line) and for some interpolating values of the
parameter β (dotted and solid lines).
Figure 4 shows the spectral distribution for different val-
ues of the parameter β, including β = 0 (dashed line), which
corresponds to Dirichlet or Neumann BC. Note that for inter-
polating values of β particle creation is always smaller than
for β = 0 (dotted line). Depending on the value of β, particle
creation can be largely suppressed (solid line).
IV. FINAL REMARKS
In the last decades there has been a substantial increase in
the study of the Casimir effect and related topics. It is remark-
able that this fascinating effect, considered nowadays as a fun-
damental one in QFT, was born in connection with colloidal
Page 11
Brazilian Journal of Physics, vol. 36, no. 4A, 20061147
chemistry, an essentially experimental science. As we men-
tioned previously, the novelty of Casimir’s seminal work [1]
(see also [6]) was the technique employed by him to compute
forces between neutral bodies as is emphasized by Itzykson
and Zuber [103]:
By considering various types of bodies influenc-
ing the vacuum configuration we may give an
interesting interpretation of the forces acting on
them.
However, theattractiveorrepulsivecharacteroftheCasimir
force can not be anticipated. It depends on the specific bound-
ary conditions, the number of space-time dimensions, the na-
ture of the field (bosonic or fermionic), etc. The “mystery”
of the Casimir effect has intrigued even proeminent physicists
such as Julian Schwinger, as can be seen in his own statement
[37]:
... oneoftheleastintuitiveconsequencesofquan-
tum electrodynamics.
There is no doubt nowadays about the existence of the (sta-
tic) Casimir effect, thanks to the vast list of accurate exper-
iments that have been made during the last ten years. It is
worth emphasizing that a rigorous comparison between the-
ory and experimental data can be achieved only if the effects
of temperature and more realistic BC are considered. In prin-
ciple, the former are important in comparison with the vac-
uum contribution for large distances, while the latter can not
be neglected for short distances. Typical ranges investigated
in Casimir experiments are form 0.1µm to 1.0µm and, to have
an idea of a typical plasma wavelengths (the plasma wave-
length is closely related to the penetration depth), recall that
for Au we have λP≈ 136nm.
The Casimir effect has become an extremely active area
of research from both theoretical and experimental points of
view and its importance lies far beyond the context of QED.
This is due to its interdisciplinary character, which makes this
effect find applications in quantum field theory (bag model,
for instance), cavity QED, atomic and molecular physics,
mathematical methods in QFT (development of new regular-
ization and renormalization schemes), fixing new constraints
in hypothetical forces, nanotechnology (nanomachines oper-
atedbyCasimirforces), condensedmatterphysics, gravitation
and cosmology, models with compactified extra-dimensions,
etc.
In this work, we considered quantum fields interacting only
with classical boundaries. Besides, these interactions were
described by highly idealized BC. Apart from this kind of in-
teraction, there was no other interaction present. However,
the fields in nature are interacting fields, like those in QED,
etc. Hence, we could ask what are the first corrections to the
Casimir effect when we consider interacting fields. In princi-
ple, they are extremely small. In fact, for the case of QED, the
first radiative correction to the Casimir energy density (con-
sidering that the conducting plates impose BC only on the ra-
diationfield)wasfirstlycomputedbyBordagetal[104]andis
given byE(1)
32
ton wavelength of the electron and E(0)
contribution to the Casimir energy density. As we see, at least
for QED, radiative corrections to the Casimir effect are exper-
imentally irrelevant. However, they might be relevant in the
bag model, where for quarks λc≈ a and also the quark prop-
agators must be considered submitted to the bag BC [9]. Be-
sides, the study of radiative corrections to the Casimir effect
provide a good laboratory for testing the validity of idealized
BC in higher order of perturbation theory.
C(a,α)=E(0)
C(a)9
αλc
a, where λcis the Comp-
C(a) is the zeroth order
Concerning the dynamical Casimir effect, the big challenge
is to conceive an experiment which will be able to detect real
photons created by moving boundaries or by an equivalent
physicalsystemthatsimulatesrapidmotionofaboundary. An
ingenious proposal of an experiment has been made recently
by the Padova’s group [105]. There are, of course, many other
interesting aspects of the dynamical Casimir effect that has
been studied, as quantum decoherence [106], mass correction
of the moving mirrors [107], etc. (see also the reviews [108]).
As a final comment, we would like to mention that surpris-
ing results have been obtained when a deformed quantum field
theory is considered in connection with the Casimir effect. It
seems that the simultaneous assumptions of deformation and
boundary conditions lead to a new mechanism of creation of
real particles even in a static situation [109]. Of course, the
Casimir energy density is also modified by the deformation
[110]. Quantum field theories with different space-time sym-
metries, other than those governed by the usual Poincar´ e al-
gebra (as for example the κ-deformed Poincar´ e algebra [111])
maygiverisetoamodifieddispersionrelation, adesirablefea-
ture in some tentative models for solving recent astrophysical
paradoxes.
Acknowledgments: I am indebted to B. Mintz, P.A. Maia
Neto and R. Rodrigues for a careful reading of the manuscript
and many helpful suggestions. I would like also to thank to all
members of the Casimir group of UFRJ for enlightening dis-
cussions that we have maintained over all these years. Finally,
I thank to CNPq for a partial financial support.
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