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The Casimir Effect: Some Aspects

The Casimir Effect: Some Aspects

Brazilian Journal of , vol. 36, no. 4A, 2006 1137

The : Some Aspects

Carlos Farina Universidade Federal do Rio de Janeiro, Ilha do Fundao,˜ 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 . We present two methods of computing Casimir , namely: the global method introduced by Casimir, based on the idea of zero-point of the electromagnetic field, and a local one, which requires the computation of the energy- 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 and radiation reaction forces on non-relativistic moving boundaries. Keywords: Quantum field ; Casimir effect

I. INTRODUCTION spent by a signal to travel from one to the other is of the order (or greater) than atomic characteristic A. Some history (r/c ≥ 1/ωMan, where ωmn are atomic transition ). Though this conjecture seemed to be very plausible, a rigorous demonstration was in order. Further, the precise expression The standard Casimir effect was proposed theoretically of the van der Waals for large (retarded by the dutch and humanist Hendrik Brugt Gerhard regime) should be obtained. Casimir (1909-2000) in 1948 and consists, basically, in the attraction of two parallel and perfectly conducting plates lo- Motivated by the disagreement between experiments and cated in vacuum [1]. As we shall see, this effect has its origin theory described above, Casimir and Polder [4] considered in colloidal chemistry and is directly related to the dispersive for the first , in 1948, the influence of retardation ef- van der Waals interaction in the retarded regime. fects on the van der Waals forces between two as well The correct explanation for the non-retarded dispersive as on the between an atom and a perfectly conducting van der Walls interaction between two neutral but polariz- wall. These authors obtained their results after lengthy calcu- able atoms was possible only after quantum was lations in the context of perturbative quantum electrodynam- properly established. Using a perturbative approach, London ics (QED). Since Casimir and Polder’s paper, retarded forces showed in 1930 [2] for the first time that the above mentioned between atoms or and walls of any kind are usu- 2 6 interaction is given by VLon(r) ≈ −(3/4)(~ω0α )/r , where ally called Casimir-Polder forces. They showed that in the α is the static polarizability of the atom, ω0 is the dominant retarded regime the van der Waals interaction potential be- 7 transition and r is the between the atoms. tween two atoms is given by VRet (r) = −23~cαAαB/(4πr ). In the 40’s, various experiments with the purpose of studying In contrast to London’s result, it falls as 1/r7. The change equilibrium in colloidal suspensions were made by Verwey in the power law of the dispersive when and Overbeek [3]. Basically, two types of force used to be we go from the non-retarded regime to the retarded one 7 8 invoked to explain this equilibrium, namely: a repulsive elec- (FNR ∼ 1/r → FR ∼ 1/r ) was measured in an experiment trostatic force between layers of charged adsorbed by with sheets of mica by D. Tabor and R.H.S. Winterton [5] the colloidal particles and the attractive London-van der Waals only 20 years after Casimir and Polder’s paper. A change forces. was observed around 150A,˚ which is the order of magnitude However, the experiments performed by these authors of the of the dominant transition (they worked in showed that London’s interaction was not correct for large dis- the range 50A˚ − 300A,˚ with an accuracy of ±4A).˚ They also tances. Agreement between experimental data and theory was showed that the retarded van der Waals interaction potential possible only if they assumed that the van der Waals interac- between an atom and a perfectly conducting wall falls as 1/r4, tion fell with the distance between two atoms more rapidly in contrast to the result obtained in the short distance regime than 1/r6. They even conjectured that the reason for such a (non-retarded regime), which is proportional to 1/r3 (as can different behaviour for large distances was due to the retar- be seen by the image method). Casimir and Polder were very dation effects of the electromagnetic interaction (the informa- impressed with the fact that after such a lengthy and involved tion of any change or fluctuation occurred in one atom should QED calculation, the final results were extremely simple. This spend a finite time to reach the other one). Retardation ef- is very clear in a conversation with . In Casimir’s fects must be taken into account whenever the time interval own words 1138 Carlos Farina

In the summer or autumn 1947 (but I am not ab- A direct consequence of dispersive van der Waals forces solutely certain that it was not somewhat earlier between two atoms or molecules is that two neutral but polar- or later) I mentioned my results to Niels Bohr, izable macroscopic bodies may also interact with each other. during a walk.“That is nice”, he said, “That is However, due to the so called non-additivity of van der Waals something new.” I told him that I was puzzled forces, the total interaction potential between the two bodies by the extremely simple form of the expressions is not simply given by a pairwise integration, except for the for the interaction at very large distance and case where the bodies are made of a very rarefied medium. he mumbled something about zero-point energy. In , the Casimir method provides a way of obtaining That was all, but it put me on a new track. this kind of interaction potential in the retarded regime (large distances) without the necessity of dealing explicitly with the Following Bohr’s suggestion, Casimir re-derived the results non-additivity problem. Retarded van der Waals forces are obtained with Polder in a much simpler way, by computing the usually called Casimir forces. A simple example may be in shift in the electromagnetic zero-point energy caused by the order. Consider two semi-infinite slabs made of polarizable presence of the atoms and the walls. He presented his result material separated by a distance a, as shown in Figure 1. in the Colloque sur la theorie´ de la liaison chimique, that took place at Paris in April of 1948: a I found that calculating changes of zero-point en- B r ergy really leads to the same results as the calcu- AB f lations of Polder and myself... rˆAB A short paper containing this beautiful result was published A −f in a French journal only one year later [6]. Casimir, then, decided to test his method, based on the variation of zero-point energy of the electromagnetic field caused by the interacting bodies in other examples. He knew that the existence of zero- FIG. 1: Forces between molecules of the left slab and molecules of the right slab. point energy of an atomic (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 Suppose the force exerted by a A of the left slab isotopes of ? Again, in Casimir’s own on a molecule B of the right slab is given by words we have the answer [8]: C fAB = − γ rAB , if there were two isotopes of empty you rAB could really easy confirm the existence of the zero-point energy. Unfortunately, or perhaps for- where C and γ are positive constants, rAB the distance between tunately, there is only one copy of empty space the molecules and rˆAB the unit vector pointing from A to B. and if you cannot change the atomic distance Hence, by a direct integration it is straightforward to show then you might change the shape and that was that, for the case of dilute media, the force per unit be- the idea of the attracting plates. tween the slabs is attractive and with modulus given by F C 0 A month after the Colloque held at Paris, Casimir presented slabs = , (2) his seminal paper [1] on the attraction between two parallel Area aγ−4 conducting plates which gave rise to the famous effect that 0 since then bears his name: where C is a positive constant. Observe that for γ = 8, which corresponds to the Casimir and Polder force, we obtain a force 4 On 29 May, 1948, ‘I presented my paper on between the slabs per unit area which is proportional to 1/a . the attraction between two perfectly conducting Had we used the Casimir method based on zero-point energy plates at a meeting of the Royal to compute this force we would have obtained precisely this Academy of Arts and . It was published kind of dependence. Of course, the numerical coefficients in the course of the year... would be different, since here we made a pairwise integration, neglecting the non-additivity problem. A detailed discussion As we shall see explicitly in the next section, Casimir ob- on the identification of the Casimir energy with the sum of tained an attractive force between the plates whose modulus van der Waals interaction for a dilute sphere can be per unit area is given by found in Milton’s book [9] (see also references therein). In 1956, Lifshitz and collaborators developed a general the- F(a) 1 dyn ≈ 0,013 , (1) ory of van der Waals forces [10]. They derived a powerful L2 (a/µm)4 cm2 expression for the force at zero as well as at fi- nite temperature between two semi-infinite dispersive media where a is the separation between the plates, L2 the area of characterized by well defined dielectric constants and sepa- each plate (presumably very large, i.e., L À a). rated by a slab of any other dispersive medium. They were Brazilian Journal of Physics, vol. 36, no. 4A, 2006 1139 able to derive and predict several results, like the variation of B. The Casimir’s approach the thickness of thin superfluid films in a remarkable agreement with the experiments [11]. The Casimir result for metallic plates can be reobtained from Lifshitz formula in the The novelty of Casimir’s original paper was not the predic- appropriate limit. The Casimir and Polder force can also be tion of an attractive force between neutral objects, once Lon- inferred from this formula [9] if we consider one of the media don had already explained the existence of a force between sufficiently dilute such that the force between the slabs may neutral but polarizable atoms, but the method employed by be obtained by direct integration of a single atom-wall inter- Casimir, which was based on the zero-point energy ot the elec- [12]. tromagnetic field. Proceeding with the canonical of the electromagnetic field without sources in the Coulomb The first experimental attempt to verify the existence of gauge we write the hamiltonian for the free radiation the Casimir effect for two parallel metallic plates was made field as by Sparnaay [13] only ten years after Casimir’s theoretical · ¸ . However, due to a very poor accuracy achieved 2 1 Hˆ = ~ω aˆ† aˆ + , (3) in this experiment, only compatibility between experimental ∑ ∑ k kα kα 2 data and theory was established. One of the great difficulties α=1 k was to maintain a perfect parallelism between the plates. Four where aˆ† and aˆ are the creation and operators decades have passed, approximately, until new experiments kα kα were made directly with metals. In 1997, using a torsion of a with momentum k and α. The en- pendulum Lamoreaux [14] inaugurated the new era of experi- ergy of the field when it is in the vacuum state, or simply the ments concerning the Casimir effect. Avoiding the parallelism , is then given by problem, he measured the Casimir force between a plate and a 2 ˆ 1 spherical lens within the proximity force approximation [15]. E0:=h0|H|0i = ∑ ∑ ~ωk , (4) This experiment may be considered a landmark in the history k α=1 2 of the Casimir effect, since it provided the first reliable exper- imental confirmation of this effect. One year later, using an which is also referred to as zero-point energy of the electro- atomic force microscope , Mohideen and Roy [16] measured 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 the Casimir force between a plate and a sphere with a better 1 accuracy and established an agreement between experimen- contribute to the energy of the field with 2 ~ωkα and total vac- tal data and theoretical of less than a few percents uum energy is then a divergent quantity given by an infinite (depending on the range of distances considered). The two sum over all possible modes. precise experiments mentioned above have been followed by The presence of two parallel and perfectly conducting many others and an incomplete list of the modern series of ex- plates imposes on the electromagnetic field the following periments about the Casimir effect can be found in [17]-[26]. boundary conditions: For a detailed analysis comparing theory and experiments see E × nˆ| = 0 [27, 28] plates (5)

We finish this subsection emphasizing that Casimir’s orig- B · nˆ|plates = 0 , inal predictions were made for an extremely idealized situ- ation, namely: two perfectly conducting (flat) plates at zero which modify the possible frequencies of the field modes. The temperature. Since the experimental accuracy achieved nowa- Casimir energy is, then, defined as the difference between the days is very high, any attempt to compare theory and exper- vacuum energy with and without the material plates. How- imental data must take into account more realistic boundary ever, since in both situations the vacuum energy is a divergent conditions. The most relevant ones are those that consider the quantity, we need to adopt a prescription to give finite conductivity of real metals and roughness of the surfaces a physical meaning to such a difference. Therefore, a precise involved. These conditions become more important as the dis- definition for the Casimir energy is given by tance between the two bodies becomes smaller. Thermal ef- "Ã ! Ã ! # fects must also be considered. However, in principle, these 1 1 ECas := lim ~ωk − ~ωk , (6) effects become dominant compared with the vacuum contri- s→0 ∑ 2 ∑ 2 kα I kα II bution for large distances, where the forces are already very small. A great number of papers have been written on these where subscript I means a regularized sum and that the fre- topics since the analysis of most recent experiments require quencies are computed with the boundary conditions taken the consideration of real boundary conditions. For finite con- into account, subscript II means a regularized sum but with ductivity effects see Ref. [30]; the simultaneous consideration no boundary conditions at all and s stands for the regularizing of roughness and finite conductivity in the proximity for ap- parameter. This definition is well suited for plane geometries proximation can be found in Ref. [31] and beyond PFA in Ref. like that analyzed by Casimir in his original . In more [32] (see also references cited in the above ones). Concerning complex situations, like those involving spherical shells, there the present status of controversies about the thermal Casimir are some subtleties that are beyond the purposes of this intro- force see Ref. [29] ductory article (the self-energy of a spherical shell depends on 1140 Carlos Farina ³ ´ its while the self-energy of a pair of plates is indepen- where, as usual, G(x,x0) = ih0|T φ(x)φ(x0) |0i. dent of the distance between them). Since the above lagrangian density does not depend explic- Observe that, in the previous definition, we eliminate the itly on x, Noether’s Theorem leads naturally to the following regularization prescription only after the subtraction is made. µν energy-momentum tensor (∂µT = 0) Of course, there are many different regularization methods. A quite simple but very efficient one is achieved by introducing µν ∂L ν µν a high frequency cut off in the zero-point energy expression, T = ∂ φ + g L , (10) ∂(∂µφ) as we shall see explicitly in the next section. This procedure can be physically justified if we note that the metallic plates which, after symmetrization, can be written in the form become transparent in the high frequency limit so that the high ³ ´ frequency contributions are canceled out from (6). 1 T = ∂ φ∂ φ + ∂ φ∂ φ + g L . (11) Though the calculation of the Casimir for the case µν 2 µ ν ν µ µν of two parallel plates is very simple, its determination may For our purposes, it is convenient to write the vacuum expec- become very involved for other geometries, as is the case, tation value (VEV) of the energy-momentum tensor in terms for instance, of a perfectly conducting spherical shell. Af- of the above Green as ter a couple of years of hard work and a “nightmare in Bessel h i functions”, Boyer [33] computed for the first time the Casimir i 0 0 0 α 2 0 h0|Tµν(x)|0i = − lim (∂µ∂ν+∂ν∂µ)−gµν(∂α∂ +m ) G(x,x ). pressure inside a spherical shell. Surprisingly, he found a re- 2 x0→x pulsive pressure, contrary to what Casimir had conjectured (12) five years before when he proposed a very peculiar model for In this context, the Casimir energy density is defined as the stability of the [34]. Since then, Boyer’s result has been confirmed and improved numerically by many au- ρC(x) = h0|T00(x)|0iBC − h0|T00(x)|0iFree , (13) thors, as for instance, by Davies in 1972 [35], by Balian and Duplantier in 1978 [36] and also Milton in 1978 [37], just to where the subscript BC means that the VEV must be computed mention some old results. assuming that the field satisfies the appropriate boundary con- The Casimir effect is not a peculiarity of the electromag- dition. Analogously, considering for instance the case of two netic field. It can be shown that any relativistic field under parallel plates perpendicular to the OZ axis (the generaliza- boundary conditions caused by material bodies or by a com- tion for other configurations is straightforward) the Casimir pactification of space has its zero-point energy force per unit area on one plate is given by modified. Nowadays, we denominate by Casimir effect any + − change in the vacuum energy of a quantum field due to any ex- FC = h0|Tzz |0i − h0|Tzz |0i, (14) ternal agent, from classical backgrounds and non-trivial topol- where superscripts + and − mean that we must evaluate hTzzi ogy to external fields or neighboring bodies. Detailed reviews on both sides of the plate. In other words, the desired Casimir of the Casimir effect can be found in [9, 38–41]. pressure on the plate is given by the discontinuity of hTzzi at the plate. Using equation (12), hTzzi can be computed by µ ¶ C. A local approach 2 i ∂ ∂ ∂ 0 h0|Tzz|0i = − lim − G(x,x ). (15) 2 x0→x ∂z ∂z0 ∂z2

In this section we present an alternative way of comput- Local methods are richer than global ones, since they pro- ing the Casimir energy density or directly the Casimir pres- vide much more about the system. Depending sure which makes use of a local quantity, namely, the energy- on the problem we are interested in, they are indeed neces- momentum tensor. Recall that in classical sary, as for instance in the study of radiative properties of an the total force on a distribution of charges and currents can atom inside a cavity. However, with the purpose of comput- be computed integrating the Maxwell stress tensor through ing Casimir in simple situations, one may choose, an appropriate closed surface containing the distribution. For for convenience, global methods. Previously, we presented simplicity, let us illustrate the method in a scalar field. The only the global method introduced by Casimir, based on the lagrangian density for a free scalar field is given by zero-point energy of the quantized field, but there are many others, namely, the generalized function method [42] and ¡ ¢ 1 1 Schwinger’s method [43, 44], to mention just a few. L φ,∂ φ = − ∂ φ∂µφ − m2φ2 (7) µ 2 µ 2

The field equation and the corresponding Green function are II. EXPLICIT COMPUTATION OF THE CASIMIR FORCE given, respectively, by

2 2 (−∂ + m )φ(x) = 0 ; (8) In this section, we show explicitly two ways of comput- ing the Casimir force per unit area in simple situations where (∂2 − m2)G(x,x0) = −δ(x − x0), (9) plane surfaces are involved. We start with the global approach Brazilian Journal of Physics, vol. 36, no. 4A, 2006 1141 introduced by Casimir which is based on the zero-point en- Using that ergy. Then, we give a second example where we use a local " # · ¸ approach, based on the energy-momentum tensor. We finish ∂2 1 ∞ ∂2 1 1 e−εnπ/a = , this section by sketching some results concerning the Casimir ∂ε2 ε ∑ ∂ε2 ε eεπ/a − 1 effect for massive fields. n=1 as well as the definition of Bernoulli’s numbers,

A. The electromagnetic Casimir effect between two parallel 1 ∞ tn−1 plates t = ∑ Bn , e − 1 n=0 n! we obtain As our first example, let us consider the standard (QED) " ( ) # Casimir effect where the quantized electromagnetic field is 2 2 ∞ ³ ´n−1 L 1 ∂ 1 Bn επ 6a constrained by two perfectly parallel conducting plates sepa- E(a) = + − 2π ε3 ∂ε2 ε ∑ n! a πε4 rated by a distance a. For convenience, let us suppose that one " n=0 plate is located at z = 0, while the other is located at z = a. L2 a 1 1 B ³π´3 = 6(B −1) + (1 + 2B ) + 4 The quantum electromagnetic potential between the metallic 2π 0 π ε4 1 ε3 12 a plates in the Coulomb gauge (∇ · A = 0) which satisfies the # ∞ ³ ´ appropriate BC is given by [45] Bn π n−1 + ∑ (n − 2)(n − 3)εn−4 . n=5 n! a Z µ ¶ L2 ∞ 0 2π~ 1/2 A(ρ,z,t) = d2κ × Using the well known values B = 1, B = − 1 and B = − 1 , (2π)2 ∑ ckaL2 0 1 2 4 30 ( n=0 and taking ε → 0+, we obtain ³nπz´ × a(1)(κ,n)(κˆ × zˆ)sin + a E(a) π2 1 = −~c · ) L2 24 × 30 a3 h nπ ³nπz´ κ ³nπz´i + a(2)(κ,n) i κsin − zˆ cos × ka a k a As a consequence, the force per unit area acting on the plate at z = a is given by × ei(κ·ρ−ωt) + h.c., (16) 2 £ ¤ F(a) 1 ∂Ec(a) π ~c 1 dyn where ω(κ,n) = ck = c κ2 + n2π2/a2 1/2, with n being a = − = − ≈ −0,013 , L2 L2 ∂a 240a4 4 cm2 non-negative and the prime in Σ0 means that for n = 0 (a/µm) an extra 1/2 factor must be included in the normalization of where in the last step we substituted the numerical values the field modes. The non-regularized Casimir energy then of ~ and c in order to give an idea of the strength of the reads Casimir pressure. Observe that the Casimir force between " # Z µ ¶1/2 the (conducting) plates is always attractive. For plates with ~c d2κ ∞ n2π2 nr 2 2 1cm2 of area separated by 1µm the modulus of this attrac- Ec (a) = L 2 κ + 2 ∑ κ + 2 2 (2π) n=1 a tive force is 0,013dyn. For this same separation, we have Z Z q −8 2 +∞ PCas ≈ 10 Patm, where Patm is the atmospheric pressure at ~c 2 d κ adkz 2 2 − L 2 2 κ + kz . sea level. Hence, for the idealized situation of two perfectly 2 (2π) −∞ 2π conducting plates and assuming L2 = 1 cm2, the modulus of −7 Making the variable transformation κ2 +(nπ/a)2 =: λ and in- the Casimir force would be ≈ 10 N for typical separations troducing exponential cutoffs we get a regularized expression used in experiments. However, due to the finite conductivity (in 1948 Casimir used a generic cutoff function), of real metals, the Casimir forces measured in experiments are smaller than these values. " Z Z L2 1 ∞ ∞ ∞ E r(a,ε) = e−εκκ2 dκ + ∑ e−ελλ2 dλ − 2π 2 0 nπ n=#1 a B. The Casimir effect for a scalar field with Robin BC Z ∞ Z ∞ − dn e−ελλ2dλ nπ 0 a In order to illustrate the local method based on the energy- " Ã ! 2 ∞ 2 −εnπ/a momentum tensor, we shall discuss the Casimir effect of a L 1 ∂ e massless scalar field submitted to Robin BC at two parallel = 3 + ∑ 2 − 2π ε n=1 ∂ε ε plates, which are defined as Z Z # ∞ ∂2 ∞ − dn e−ελdλ . (17) ∂φ ∂ε2 nπ φ|bound. = β |bound. , (18) 0 a ∂n 1142 Carlos Farina

1+iβiλ 0 where, by assumption, β is a non-negative parameter. How- where γi = (i = 1,2). Outside the plates, with z,z > a, ever, before computing the desired Casimir pressure, a few 1−iβiλ we have: comments about Robin BC are in order. µ ¶ First, we note that Robin BC interpolate continuously ıλ(z<−a) RR 0 e 1 ıλ(z<−a) −ıλ(z<−a) Dirichlet and Neumann ones. For β → 0 we reobtain Dirichlet g (z,z ) = e − e . (24) 2ıλ γ2 BC while for β → ∞ we reobtain Neumann BC. Robin BC al- ready appear in classical electromagnetism, classical mechan- Defining tµν such that ics, , heat and Schrodinger¨ [46] and even in the Z d2k dω study of interpolating partition functions [47]. A nice realiza- hT µν(x)i = ⊥ htµν(x)i, (25) tion of these conditions in the context of (2π) 2π can be obtained if we study a vibrating with its extremes we have attached to elastic supports [48? ]. Robin BC can also be used µ ¶ in a phenomenological model for a penetrable surfaces [49]. 1 ∂ ∂ ht33(z)i = lim + λ2 g(z,z0). (26) In fact, in the context of the model, it can be shown 2i z0→z ∂z ∂z0 that for frequencies much smaller than the plasma frequency (ω ¿ ωP) the parameter β plays the role of the plasma wave- The Casimir force per unit area on the plate at x = a is given length [? ]. In the context of QFT this kind of BC appeared by the discontinuity in ht33i: more than two decades ago [50, 51]. Recently, they have been Z 2 h i discussed in a variety of contexts, as in the AdS/CFT corre- d k⊥ dω 33 33 F = ht i|z=a − ht i|z=a . spondence [52], in the discussion of upper bounds for the ratio (2π) 2π − + /energy in confined [53], in the static Casimir After a straightforward calculation, it can be shown that effect [54], in the heat kernel expansion [55–57] and in the 4 Z one-loop of the λφ theory [58, 59]. As we 1 ∞ dξ ξ3 ³ ´³ ´ will see, Robin BC can give rise to restoring forces in the sta- F (β1,β2;a)=− 2 4 . 32π a 0 1+β1ξ/2a 1+β2ξ/2a eξ− 1 tic Casimir effect [54]. 1−β1ξ/2a 1−β2ξ/2a Consider a massless scalar field submitted to Robin BC on (27) two parallel plates: Depending on the values of parameters β1 and β2, restoring Casimir forces may arise, as shown in Figure 2 by the dotted ∂φ ∂φ φ| = β | ; φ| = −β | , (19) line and the thin solid line. z=0 1 ∂z z=0 z=a 2 ∂z z=a where we assume β1(β2) ≥ 0. For convenience, we write Z 3 d k 0 α G(x,x0) = α eıkα(x−x ) g(z,z0;k ,ω), (20) (2π)3 ⊥ pa4 (β1 = 0 ,β2 → ∞) with α = 0,1,2, gµν = diag(−1,+1,+1,+1) and the reduced 0 Green function g(z,z ;k⊥,ω) satisfies µ ¶ (β = 0 , β = 1) ∂2 1 2 + λ2 g(z,z0;k ,ω) = −δ(z − z0), (21) ∂z2 ⊥

2 2 2 0 where λ = ω −k⊥ and g(z,z ;k⊥,ω) is submitted to the fol- O a lowing boundary conditions (for simplicity, we shall not write k⊥,ω in the argument of g): ∂ ∂ g(0,z0) = β g(0,z0) ; g(a,z0) = −β g(a,z0) β = β = β = β → ∞ 1 ∂z 2 ∂z ( 1 2 0 or 1 2 ) It is not difficult to see that g(z,z0) can be written as   A(z0)(sinλz + β λcosλz), z < z0 1 4 g(z,z0)= FIG. 2: Casimir pressure, conveniently multiplied by a , as a func-  0 0 tion of a for various values of parameters β1 and β2. B(z )(sinλ(z− a)−β2λcosλ(z − a)), z > z (22) For points inside the plates, 0 < z,z0 < a, the final expression The particular cases of Dirichlet-Dirichlet, Neumann- for the reduced Green function is given by Neumann and Dirichlet-Neumann BC can be reobtained if ³ ´³ ´ we take, respectively, β1 = β2 = 0, β1 = β2 → ∞ and β1 = 1 ıλz< −ıλz< 1 ıλ(z>−a) ıλ(z>−a) γ e − e γ e − e 0; β2 → ∞. For the first two cases, we obtain gRR(z,z0) = − 1 2 , 2ıλ( 1 eıλa − 1 e−ıλa) Z ∞ 3 γ1 γ2 DD NN 1 ξ F (a) = F (a) = − 2 4 dξ . (28) (23) 32π a 0 eξ− 1 Brazilian Journal of Physics, vol. 36, no. 4A, 2006 1143

s−1 R ∞ ξ geometry. Other geometries may give rise to power law de- Using the representation dξ = ζR(s)Γ(s), 0 eξ−1 cays when ma → ∞. However, in some cases, the behav- where ζR is the , we get half the elec- iour of the Casimir force with ma may be quite unexpected. tromagnetic result, namely, The Casimir force may increase with ma before it decreases π2~c 1 monotonically to zero as ma → ∞ (this happens, for instance, F DD(a) = F NN(a) = − . (29) when Robin BC are imposed on a massive scalar field at two 480 a4 parallel plates [61]). For the case of mixed BC, we get In the case of massive fermionic fields, an analogous be- haviour is found. However, some care must be taken when Z ∞ 3 DN 1 ξ computing the Casimir energy density for , regarding F (a) = + 2 4 dξ , (30) 32π a 0 eξ + 1 what kind of BC can be chosen for this field. The point is Using in the previous equation the integral representation that is a first order equation, so that if we want R ξs−1 non-trivial solutions, we can not impose that the field satis- ∞ dξ = (1 − 21−s)Γ(s)ζ (s) we get a repulsive pres- 0 eξ+1 R fies Dirichlet BC at two parallel plates, for instance. The most sure (equal to half of Boyer’s result [60] obtained for the elec- appropriate BC for fermions is borrowed from the so called tromagnetic field constrained by a perfectly conducting plate MIT bag model for hadrons [62], which basically states that parallel to an infinitely permeable one), there is no flux of fermions through the boundary (the normal component of the fermionic current must vanish at the bound- 7 π2~c 1 F DN(a) = × . (31) ary). The Casimir energy per unit area for a massive fermionic 8 480 a4 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]) C. The Casimir effect for massive particles

1 f Ec (a,m) = In this subsection, we sketch briefly some results concern- L2 ing massive fields, just to get some feeling about what kind · ¸ Z ∞ q of influence the of a field may have in the Casimir ef- 1 2 2 2 ξ − ma −2ξ − 2 3 dξξ ξ − m a log 1 + e . (35) fect. Firstly, let us consider a massive scalar field submit- π a ma ξ + ma ted to Dirichlet BC in two parallel plates, as before. In this case, the allowed frequencies for the field modes are given The small and large mass limits are given, respectively, by h i1/2 2 n2π2 2 2 by ωk = c κ + + m , which lead, after we use the 1 7π m a2 E f (a,m) ≈ − + ; Casimir method explained previously, to the following result L2 c 2880a3 24a2 for the Casimir energy per unit area (see, for instance, Ref. 1 3(ma)1/2 [38]) E f (a,m) ≈ − e−2ma . (36) L2 c 25π3/2a3 2 ∞ 1 m 1 Since the first correction to the zero mass result has an oppo- 2 Ec(a,m) = − 2 ∑ 2 K2(2amn) , (32) L 8π a n=1 n site sign, also for a fermionic field small diminish the Casimir effect. In the large mass limit, ma → ∞, we have a where Kν is a modified , m is the mass of the behaviour analogous to that of the scalar field, namely, an ex- field and a is the distance between the plates, as usual. The ponential decay with ma (again this happens due to the plane limit of small mass, am ¿ 1, is easily obtained and yields geometry). We finish this section with an important observa- 1 π2 m2 tion: even a as light as the electron has a completely E (a,m) ≈ − + . (33) negligible Casimir effect. L2 c 1440a3 96a As expected, the zero mass limit coincides with our previous result (29) (after the force per unit area is computed). Observ- III. MISCELLANY 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. In this section we shall briefly present a couple of topics On the other hand, in the limit of large mass, am À 1, it can which are in some way connected to the Casimir effect and be shown that that have been considered by our research in the last 2 ³ ´1/2 years. For obvious reasons, we will not be able to touch all 1 m π −2ma 2 Ec(a,m) ≈ − 2 e (34) the topics we have been interested in, so that we had to choose L 16π a ma only a few of them. We first discuss how the Casimir effect Note that the Casimir effect disappears for m → ∞, since in of a charged field can be influenced by an external magnetic this limit there are no quantum fluctuations for the field any- field. Then, we show how the constitutive equations asso- more. An exponential decay with ma is related to the plane ciated to the Dirac quantum vacuum can be affected by the 1144 Carlos Farina presence of material plates. Finally, we consider the so called large mass limit described previously, huge magnetic fields dynamical Casimir effect. 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 A. Casimir effect under an external magnetic field 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 - The Casimir effect which is observed experimentally is that ment at the laboratory. 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 B. Magnetic permeability of the constrained Dirac vacuum (and hope) of enhancing the Casimir effect of and 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- In contrast to the classical vacuum, the quantum vacuum pairs which are continuously created and destroyed is far from being an empty space, inert and insensible to any from the Dirac vacuum would respond in such a way that external influence. It behaves like a macroscopic medium, in the corresponding Casimir energy would be greatly amplified. the sense that it responds to external agents, as for example This problem was considered for the first time in 1998 [65] electromagnetic fields or the presence of material plates. As (see also Ref. [66]). The influence of an external magnetic previously discussed, recall that the (standard) Casimir effect field on the Casimir effect of a charged scalar field was con- is nothing but the energy shift of the vacuum state of the field sidered in Ref. [67] and recently, the influence of a magnetic caused by the presence of parallel plates. There are many field on the fermionic Casimir effect was considered with the other fascinating phenomena associated to the quantum vac- more appropriate MIT BC [68]. uum, namely, the particle creation produced by the application For simplicity, let us consider a massive field un- of an electric field [69], the birefringence of the QED vacuum der anti-periodic BC (in the OZ direction) in the presence under an external magnetic field [70, 71] and the Scharnhorst of a constant and uniform magnetic field in this same direc- effect [72, 73], to mention just a few. This last effect pre- tion. After a lengthy but straightforward calculation, it can be dicts that the velocity of light propagating perpendicularly to shown that the Casimir energy per unit area is given by [66] two perfectly conducting parallel plates which impose (by as- sumption) BC only on the radiation field is slightly altered by ∞ E(a,B) 2(am)2 (−1)n−1 the presence of the plates. The expected relative variation in 2 = − 2 3 ∑ 2 K2(amn) ` π a n=1 n the velocity of light for typical values of possible experiments −36 ∞ Z µ 2¶ is so tiny (∆c/c ≈ 10 ) that this effect has not been con- eB ∞ 2 2 eBa n−1 −(n/2) σ−(am) /σ firmed yet. Depending on the of the material plates the − 2 ∑(−1) dσ e L , 4π an=1 0 σ velocity of light propagating perpendicular to the plates is ex- pected to diminish [74]. The inside a cavity where we introduced the Langevin function: L(ξ) = cothξ − was considered in [75]. 1/ξ. In the strong field limit, we have The negligible change in the velocity of light predicted by E(a,B) eBm ∞ (−1)n−1 Scharnhorst may be connected with the fact that the effect that ≈ − K1(amn) . (37) bears his name is a two-loop QED effect, since the classical `2 π2 ∑ n n=1 field of a traveling light wave interacts with the radiation field In this limit we can still analyze two distinct situations, only through the fermionic loop. namely, the small mass limit (ma ¿ 1) and the large mass With the purpose of estimating a change in the constitutive limit (ma À 1). Considering distances between the plates typ- equations of the quantum vacuum at the one-loop level, we ical of Casimir experiments (a ≈ 1µm), we have in the former were led to consider the fermionic field submitted to some BC. case The magnetic permeability µ of the constrained Dirac vacuum was computed by the first time in Ref. [76], but with the non- ρ (a,B) B c ≈ 10−4 × ; ma ¿ 1 , (38) realistic anti-periodic BC. In this case, the result found for the ρc(a,0) Tesla relative change in the permeability, ∆µ := µ−1, was also neg- ligible (an analogous calculation has also been made in the while in the latter case, context of scalar QED [77]). However, when the more real- istic MIT BC are imposed on the Dirac field at two parallel ρc(a,B) B ≈ 10−10 × ; ma À 1 . (39) plates things change drastically. In this case, it can be shown ρ (a,0) Tesla c that the magnetic permeability of the constrained Dirac vac-

In the above equations ρc(a,B) and ρc(a,0) are the Casimir uum is given by [78] energy density under the influence of the magnetic field and without it, respectively. Observe that for the Casimir ef- 1 π − 2 e2 1 e2 = 1 − + H(ma), fect of electrons and positrons, which must be treated as the µ(ma) 12π2 ma 6π2 ma Brazilian Journal of Physics, vol. 36, no. 4A, 2006 1145 where throughout this paper the perturbative method introduced by Z ½ µ ¶ ¾ Ford and Vilenkin [84]. This method was also applied suc- ∞ x x − 1 H(ma) = dx ln 1 + e−2max . cessfully to the case of the electromagnetic field under the 1 (x2 − 1)3/2 x + 1 influence of one moving (perfectly) conducting plate [85] as (40) well as an oscillating cavity formed by two parallel (perfectly) For confining distances of the order of 0,1µm, we have conducting plates [91]. −9 Let us then consider a massless scalar field φ in 1 + 1 in ∆µ := µ − 1 ≈ 10 . (41) the presence of one moving boundary which imposes on the field a Robin BC at one moving boundary when observed from The previous value is comparable to the magnetic permeabil- a co-moving inertial frame. By assumption, the movement ity of and Nitrogen at room temperature and at- of the boundary is prescribed, non-relativistic and of small mospheric pressure. Hence, an experimental verification of amplitude (δq(t) is the position of the plate at a generic in- this result seems to be not unfeasible (we will come back to stant t). Last assumptions may be stated mathematically by this point in the final remarks). |δq˙(t)| << c and |δq(t)| << c/ω0 , where ω0 corresponds to the (main) mechanical frequency. Therefore, we must solve 2 C. The Dynamical Casimir effect the following equation: ∂ φ(x,t) = 0, with the field satisfying a Robin BC at the moving boundary given by · ¸ ∂ ∂ 1 + δq˙(t) φ(x,t)| = φ(x,t)| . (42) As our last topic, we shall briefly discuss the so called dy- ∂x ∂t x=δq(t) β x=δq(t) namical Casimir effect, which consists, as the name suggests, of the consideration of a quantum field in the presence of where β is a non-negative parameter and the previous condi- moving boundaries. Basically, the between vacuum tion was already written in the laboratory frame. We are ne- fluctuations and a moving boundary may give rise to dissi- glecting terms of the order O(δq˙2/c2). The particular cases of pative forces acting on the boundary as well as to a particle Dirichlet and Neumann BC are reobtained by making β = 0 creation phenomenon. In some sense, these phenomena were and β → ∞, respectively. The dissipative forces for these par- expected. Recall that the static Casimir force is a fluctuat- ticular cases were studied at zero temperature as well as at ing quantity [79] and hence, using general arguments related finite temperature and also with the field in a to the fluctuation-dissipation theorem [80] dissipative forces in [100] (dissipative forces on a perfectly conducting mov- on moving boundaries are expected. Further, using arguments ing plate caused by the vacuum fluctuations of the electro- of energy conservation we are led to creation of real parti- magnetic field at zero and non-zero temperature were studied cles (, if we are considering the electromagnetic field in [101]). The perturbative approach introduced by Ford and [81]). For the above reasons, this topic is sometimes referred Vilenkin [84] consists in writing to as radiation reaction force on moving boundaries. After Schwinger’s suggestion that the phenomenon of φ(x,t) = φ0(x,t) + δφ(x,t) , (43) could be explained by the dynamical where φ (x,t) is the field submitted to a Robin BC at a sta- Casimir effect [82] (name coined by himself), a lot of work 0 tic boundary fixed at the origin and δφ(x,t) is the first order has been done on this subject. However, it was shown a few contribution due to the movement of the boundary. The total years later that this was not the case (see [9] and references force on the moving boundary may be computed with the aid therein for more details). of the corresponding energy-momentum tensor, namely, The dynamical Casimir effect already shows up in the case ¡ ¢ ¡ ¢ of one (moving) mirror [83–85]. However, oscillating cavi- δF(t) = h0|T 11 t,δq+(t) − T 11 t,δq−(t) |0i , (44) ties whose walls perform in parametric with a certain unperturbed field eigenfrequency may greatly where superscripts + and − mean that we must compute the enhance the effect [86–89]. Recently, a one dimensional os- energy-momentum tensor on both sides of the moving bound- cillating cavity with walls of different nature was considered ary. It is convenient to work with time Fourier transforms. [90]. The dynamical Casimir effect has also been analyzed for The susceptibility χ(ω) is defined in the Fourier space by a variety of three-dimensional geometries, including parallel plane plates [91], cylindrical [92], and rectangular δF (ω) =: χ(ω) δQ(ω) (45) [93], cylindrical [94] and spherical cavities [95]. For a review where δF (ω) and δQ(ω) are the Fourier transformations of concerning classical and quantum phenomena in cavities with δF(t) and δq(t), respectively. It is illuminating to compute moving boundaries see Dodonov [96] and for a variety of top- the total work done by the vacuum fluctuations on the moving ics on non-stationary Casimir effect including perspectives of boundary. It is straightforward to show that its experimental verification see the special issue [97]. In this section, we shall discuss the force exerted by the Z +∞ Z ∞ 1 2 quantum fluctuations of a massless scalar field on one moving δF(t)δq˙(t)dt = − dωωI mχ(ω)|δQ(ω)| . (46) −∞ π 0 boundary as well as the particle creation phenomenon in a un- usual example in 1+1 dimensions, where the field satisfies a Note that only the imaginary part of χ(ω) appears in the pre- Robin BC at the moving boundary [98, 99]. We shall follow vious equation. It is responsible for the dissipative effects and 1146 Carlos Farina

hence it is closely related with the total energy converted into where, by assumption, ω0T À 1 (this is made in order to sin- real particles. On the other hand, part of χ(ω), when it gle out the effect of a given Fourier component of the ). exists, does not contribute to the total work and hence it is not For this case, we obtain the following spectral distribution related to particle creation, but to dispersive effects. For the dN particular cases of Dirichlet or Neumann BC it can be shown (ω)= (δq )2Tω(ω − ω)× dω 0 0 that the susceptibility is purely imaginary and given by 2 2 [1 − β ω(ω0 − ω)] 3 × 2 2 2 2 Θ(ω0 − ω). (51) D N ~ω (1 + β ω )(1 + β (ω0 − ω) ) χ (ω) = χ (ω) = i 2 , (47) 6πc The spectral distributions for the particular cases of Dirichlet which implies or Neumann BC can be easily reobtained by making simply β = 0 and β → ∞, respectively. The results coincide and are ~ d3 given by [102] (we are making c = 1) δF(t) = δq(t). (48) 6πc2 dt3 dN(D) dN(N) (ω) = (ω) = (δq )2Tω(ω − ω)Θ(ω − ω) . Since I mχ(ω) > 0, for these cases the vacuum fluctuations dω dω 0 0 0 are always dissipating energy from the moving boundary. (52) However, for Robin BC an interesting thing happens. It A simple inspection in (51) shows that, due to the presence of can be shown that χ(ω) acquires also a real part, which gives the Heaviside step function, only the field modes with eigen- rise to a dispersive force acting on the moving boundary. The frequencies smaller than the mechanical frequency are ex- explicit expressions of R eχ(ω) and I mχ(ω) can be found cited. Further, the number of particles created per unit fre- in [98], but the general behaviour of them as functions of quency when Robin BC are used is always smaller than the ω is shown in Figure 3. For convenience, we normalize number of particles created per unit frequency when Dirichlet these quantities dividing them by the value I mχD(ω), where (or Neumann) BC are employed. the subscript D means that I mχ(ω) must be computed with Dirichlet BC.

1,0

Im Im

D

0,5

Re Im

D

0,0

-0,5

0 10 20 30 40 50

FIG. 4: Spectral distributions of created particles for: Dirichlet and Neumann BC (dashed line) and for some interpolating values of the FIG. 3: Imaginary and real parts of χ(ω) with Robin BC appropri- parameter β (dotted and solid lines). ately normalized by the value of I mχ(ω) for the Dirichlet BC. Figure 4 shows the spectral distribution for different val- ues of the parameter β, including β = 0 (dashed line), which Now, let us discuss briefly the particle creation phenom- corresponds to Dirichlet or Neumann BC. Note that for inter- enon under Robin BC. Here, we shall consider a semi-infinite polating values of β particle creation is always smaller than slab extending from −∞ to δq(t) following as before a pre- for β = 0 (dotted line). Depending on the value of β, particle scribed non-relativistic motion which imposes on the field creation can be largely suppressed (solid line). Robin BC at δq(t). It can be shown that the corresponding spectral distribution is given by [99]

" #2 IV. FINAL REMARKS Z ∞ 0 0 2 dN 4ω dω [δQ(ω − ω )] 0 2 0 (ω) = 2 2 2 ω 1−β ωω dω 1 + β ω 0 2π 1 + β2ω0 (49) In the last decades there has been a substantial increase in As an explicit example, let us consider the particular motion the study of the Casimir effect and related topics. It is remark- able that this fascinating effect, considered nowadays as a fun- −|t|/T δq(t) = δq0 e cos(ω0t), (50) damental one in QFT, was born in connection with colloidal Brazilian Journal of Physics, vol. 36, no. 4A, 2006 1147 chemistry, an essentially experimental . As we men- the fields in nature are interacting fields, like those in QED, tioned previously, the novelty of Casimir’s seminal work [1] etc. Hence, we could ask what are the first corrections to the (see also [6]) was the technique employed by him to compute Casimir effect when we consider interacting fields. In princi- forces between neutral bodies as is emphasized by Itzykson ple, they are extremely small. In fact, for the case of QED, the and Zuber [103]: first radiative correction to the Casimir energy density (con- sidering that the conducting plates impose BC only on the ra- By considering various types of bodies influenc- diation field) was firstly computed by Bordag et al [104] and is ing the vacuum configuration we may give an (1) (0) 9 αλ given by (a,α) = (a) c , where λ is the Comp- interesting interpretation of the forces acting on EC EC 32 a c them. (0) ton wavelength of the electron and EC (a) is the zeroth order However, the attractive or repulsive character of the Casimir contribution to the Casimir energy density. As we see, at least force can not be anticipated. It depends on the specific bound- for QED, radiative corrections to the Casimir effect are exper- ary conditions, the number of space-time dimensions, the na- imentally irrelevant. However, they might be relevant in the ture of the field (bosonic or fermionic), etc. The “mystery” bag model, where for λc ≈ a and also the prop- of the Casimir effect has intrigued even proeminent agators must be considered submitted to the bag BC [9]. Be- such as , as can be seen in his own statement sides, the study of radiative corrections to the Casimir effect [37]: provide a good laboratory for testing the validity of idealized BC in higher order of . ... one of the least intuitive consequences of quan- tum electrodynamics. Concerning the dynamical Casimir effect, the big challenge is to conceive an experiment which will be able to detect real There is no doubt nowadays about the existence of the (sta- photons created by moving boundaries or by an equivalent tic) Casimir effect, thanks to the vast list of accurate exper- physical system that simulates rapid motion of a boundary. An iments that have been made during the last ten years. It is ingenious proposal of an experiment has been made recently worth emphasizing that a rigorous comparison between the- by the Padova’s group [105]. There are, of course, many other ory and experimental data can be achieved only if the effects interesting aspects of the dynamical Casimir effect that has of temperature and more realistic BC are considered. In prin- been studied, as [106], mass correction ciple, the former are important in comparison with the vac- of the moving mirrors [107], etc. (see also the reviews [108]). 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 As a final comment, we would like to mention that surpris- an idea of a typical plasma (the plasma wave- ing results have been obtained when a deformed quantum field length is closely related to the penetration depth), recall that theory is considered in connection with the Casimir effect. It seems that the simultaneous assumptions of deformation and for Au we have λP ≈ 136nm. The Casimir effect has become an extremely active area boundary conditions lead to a new mechanism of creation of of research from both theoretical and experimental points of real particles even in a static situation [109]. Of course, the view and its importance lies far beyond the context of QED. Casimir energy density is also modified by the deformation This is due to its interdisciplinary character, which makes this [110]. Quantum field with different space-time sym- effect find applications in quantum field theory (bag model, metries, other than those governed by the usual Poincare´ al- for instance), cavity QED, atomic and , gebra (as for example the κ-deformed Poincare´ [111]) mathematical methods in QFT (development of new regular- may give rise to a modified relation, a desirable fea- ization and renormalization schemes), fixing new constraints ture in some tentative models for solving recent astrophysical in hypothetical forces, (nanomachines oper- paradoxes. ated by Casimir forces), condensed physics, and , models with compactified extra-dimensions, Acknowledgments: I am indebted to B. Mintz, P.A. Maia etc. Neto and R. Rodrigues for a careful reading of the manuscript In this work, we considered quantum fields interacting only and many helpful suggestions. I would like also to thank to all with classical boundaries. Besides, these were members of the Casimir group of UFRJ for enlightening dis- described by highly idealized BC. Apart from this kind of in- cussions that we have maintained over all these years. Finally, teraction, there was no other interaction present. However, I thank to CNPq for a partial financial support.

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