The Casimir Effect: Some Aspects

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London-van attractive the and particles be adsorb colloidal particles to the charged el of used repulsive layers force between a force of namely: trostatic equilibrium, types this two explain Verwey Basically, to by invoked made [3]. were Overbeek suspensions and studyin of colloidal purpose in the with equilibrium experiments various 40’s, the In α hwdi 90[]frtefis ieta h bv mentioned above by the that given time is first the interaction for [2] Lond was 1930 approach, in mechanics perturbative showed a quantum polariz- Using after but established. only neutral properly possible two was between atoms interaction able Walls der van regime. retarded disper the the in to origi interaction its related Waals has der directly effect van is this and see, chemistry shall colloidal we in plates As conducting [1]. the vacuum perfectly in in and cated basically, parallel consists, two and Gerhard of Brugt attraction 1948 Hendrik in humanist (1909-2000) and Casimir physicist dutch the by stesai oaiaiiyo h atom, the of polarizability static the is oee,teeprmnspromdb hs authors these by performed experiments the However, h orc xlnto o h o-eadddispersive non-retarded the for explanation correct The theoretically proposed was effect Casimir standard The / r 6 hyee ojcue htterao o uha such for reason the that conjectured even They . fcagdfils antcpoete facnndvcu an vacuum confined a of th boundaries. properties example, moving for magnetic as, effect fields, this ar charged to which of way topics some few in a related experi are discuss that recent briefly about we Finally, made field are effect. scalar comments this a few of A problem involved plates. more parallel the f discuss Casimir and (standard) energy-mo the plates the calculate of we computation examples, the explicit As requires which zero-poin of one, idea of local the a methods on based two Casimir, present by We introduced method chemistry. colloidal on periments esatti ae ihahsoia uvyo h aii ef Casimir the of survey historical a with paper this start We .INTRODUCTION I. .Sm history Some A. 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[5] Polder’s Winterton 150 and R.H.S. around Casimir and observed Tabor after was D. years by 20 mica only of sheets with eg rmtennrtre eiet h eaddone retarded the to regime non-retarded when force the Waals ( der from van dispersive 1 go the as we of falls law it power the result, in London’s to contrast In ewe nao n efcl odcigwl al s1 as falls wall potentia conducting interaction perfectly a Waals and der atom an van between retarded the that showed eaddrgm h a e al neato oeta be- by potential given interaction is Waals atoms der two van tween the regime retarded ( nnrtre eie,wihi rprinlt 1 regi to distance proportional short is the which in obtained regime), result (non-retarded the to contrast in ewe tm rmlclsadwlso n idaeusu- are kind any of walls called and ally molecules force retarded or paper, atoms Polder’s between and electrodyna Casimir quantum Since perturbative (QED). of ics conducting ca context lengthy perfectly the after a in results and lations their ef- atom obtained well authors retardation an as These atoms between of wall. two force influence between the forces the on Waals as der 1948, van in the considered on [4] time, fects Polder first and the Casimir for above, described theory (retard distances large obtained. for be expressio interaction should Waals precise regime) der the van Further, the of order. in rigoro a was plausible, very demonstration be to seemed conjecture this Though svr la nacnesto ihNesBh.I Casimir’s In Bohr. Niels with conversation words a own in clear Th very involve simple. extremely and is were lengthy results a final very the such calculation, were after QED Polder that and fact the Casimir with method). impressed image the by seen be re ewe w aalladpretyconducting perfectly and parallel two between orces F r nuneo antcfil nteCsmreffect Casimir the on field magnetic a of influence e ihreaoain fteCsmrefc rtopics or effect Casimir the of elaborations either e / NR oiae ytedsgemn ewe xeiet and experiments between disagreement the by Motivated nryo h unu lcrmgei ed and field, electromagnetic quantum the of energy t c etmsrs esro h orsodn field. corresponding the of tensor stress mentum et htudutdycnr h xsec of existence the confirm undoubtedly that ments ≥ umte oRbnbudr odtosa two at conditions boundary Robin to submitted ∼ optn aii ocs aey h global the namely: forces, Casimir computing aito ecinfre nnon-relativistic on forces reaction radiation d oehn e. odhmta a puzzled was I that him is “That told said, I Bohr, he Niels new.” nice”, to something is results walk.“That my a mentioned during earlier I somewhat ab- later) not not or was am it I that (but certain 1947 solutely autumn or summer the In et hwn htisoii srltdt ex- to related is origin its that showing fect, 1 1 / / ω r Man 7 aii-odrforces Casimir-Polder A ˚ → − where , 300 F R ∼ ,wt nacrc of accuracy an with A, ˚ ω 1 / mn r ,wihi h re fmagnitude of order the is which A, ˚ 8 r tmctasto frequencies). transition atomic are a esrdi nexperiment an in measured was ) V Ret hysoe hti the in that showed They . ( r = ) − 23 / ± ~ r c 4 7 h change The . α ) hyalso They A). ˚ A α / r B 3 / ( a can (as 4 π / lcu- r me r m- 7 ed us 4 is ) 1 n d s . l , 2 et al.

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 principle, 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 B I found that calculating changes of zero-point en- rAB f ergy really leads to the same results as the calcu- lations of Polder and myself... rÂB A A short paper containing this beautiful result was published −f in a French journal only one year later [6]. Casimir, then, decided to test his method, based on the variation of zero-point FIG. 1: Forces between molecules of the left slab and molecules of the right slab. 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 Suppose the force exerted by a molecule A of the left slab that followed its introduction by Planck [7]) could be inferred on a molecule B of the right slab is given by by comparing energy levels of isotopes. But how to produce C isotopes of the quantum vacuum? Again, in Casimir’s own fAB = − γ rAB , words we have the answer [8]: rAB γ if there were two isotopes of empty space you where C and are positive constants, rAB the distance between could really easy confirm the existence of the the molecules and rÂB the unit vector pointing from A to B. zero-point energy. Unfortunately, or perhaps for- Hence, by a direct integration it is straightforward to show tunately, there is only one copy of empty space that, for the case of dilute media, the force per unit area be- and if you cannot change the atomic distance tween the slabs is attractive and with modulus given by then you might change the shape and that was F C ′ the idea of the attracting plates. slabs = , (2) Area aγ−4 A month after the Colloque held at Paris, Casimir presented ′ γ his seminal paper [1] on the attraction between two parallel where C is a positive constant. Observe that for = 8, which correspondsto the Casimir and Polder force, we obtain a force conducting plates which gave rise to the famous effect that 4 since then bears his name: between the slabs per unit area which is proportional to 1/a . Had we used the Casimir method based on zero-point energy On 29 May, 1948, ‘I presented my paper on to compute this force we would have obtained precisely this the attraction between two perfectly conducting kind of dependence. Of course, the numerical coefficients plates at a meeting of the Royal Netherlands would be different, since here we made a pairwise integration, Academy of Arts and Sciences. It was published neglecting the non-additivity problem. A detailed discussion in the course of the year... on the identification of the Casimir energy with the sum of van der Waals interaction for a dilute dielectric sphere can be As we shall see explicitly in the next section, Casimir ob- found in Milton’s book [9] (see also references therein). tained an attractive force between the plates whose modulus In 1956, Lifshitz and collaborators developed a general the- per unit area is given by ory of van der Waals forces [10]. They derived a powerful expression for the force at zero temperature as well as at fi- F(a) 1 dyn ≈ 0,013 , (1) nite temperature between two semi-infinite dispersive media L2 (a/µm)4 cm2 characterized by well defined dielectric constants and separated by a slab of any other dispersive medium. They were where a is the separation between the plates, L2 the area of able to derive and predict several results, like the variation of each plate (presumably very large, i.e., L ≫ a). the thickness of thin superfluid helium films in a remarkable A direct consequence of dispersive van der Waals forces agreement with the experiments [11]. The Casimir result for between two atoms or molecules is that two neutral but polar- metallic plates can be reobtained from Lifshitz formula in the izable macroscopic bodies may also interact with each other. appropriate limit. The Casimir and Polder force can also be However, due to the so called non-additivity of van der Waals inferred from this formula [9] if we consider one of the media forces, the total interaction potential between the two bodies sufficiently dilute such that the force between the slabs may Brazilian Journal of Physics, vol 34, no. 4A, 2006, 1137-1149 3 be obtained by direct integration of a single atom-wall inter- neutral but polarizable atoms, but the method employed by action [12]. Casimir, which was based on the zero-pointenergy ot the elec- The first experimental attempt to verify the existence of tromagnetic field. Proceeding with the canonical quantization the Casimir effect for two parallel metallic plates was made of the electromagnetic field without sources in the Coulomb by Sparnaay [13] only ten years after Casimir’s theoretical gauge we write the hamiltonian operator for the free radiation prediction. However, due to a very poor accuracy achieved field as in this experiment, only compatibility between experimental 2 data and theory was established. One of the great difficulties † 1 Hˆ = ∑ ∑~ωk aˆ αaˆkα + , (3) was to maintain a perfect parallelism between the plates. Four k 2 α=1 k decades have passed, approximately, until new experiments were made directly with metals. In 1997, using a torsion † wherea ˆkα anda ˆkα are the creation and annihilation operators pendulum Lamoreaux [14] inaugurated the new era of experi- of a photon with momentum k and polarization α. The en- ments concerning the Casimir effect. Avoiding the parallelism ergy of the field when it is in the vacuum state, or simply the problem, he measured the Casimir force between a plate and a vacuum energy, is then given by spherical lens within the proximity force approximation [15]. This experiment may be considered a landmark in the history 2 ˆ 1 ~ω of the Casimir effect, since it provided the first reliable exper- E0:=h0|H|0i = ∑ ∑ k , (4) α 2 imental confirmation of this effect. One year later, using an k =1 atomic force microscope , Mohideen and Roy [16] measured which is also referred to as zero-point energy of the electro- the Casimir force between a plate and a sphere with a better magnetic field in free space. Hence, we see that even if we do accuracy and established an agreement between experimen- not have any real photon in a given mode, this mode will still tal data and theoretical predictions of less than a few percents 1 ~ω contribute to the energy of the field with 2 kα and total vac- (depending on the range of distances considered). The two uum energy is then a divergent quantity given by an infinite precise experiments mentioned above have been followed by sum over all possible modes. many others and an incomplete list of the modern series of ex- The presence of two parallel and perfectly conducting periments about the Casimir effect can be found in [17]-[26]. plates imposes on the electromagnetic field the following For a detailed analysis comparing theory and experiments see boundary conditions: [27, 28] We finish this subsection emphasizing that Casimir’s orig- E × nˆ|plates = 0 inal predictions were made for an extremely idealized situ- (5) ation, namely: two perfectly conducting (flat) plates at zero B · nˆ|plates = 0 , temperature. Since the experimental accuracy achieved nowadays is very high, any attempt to compare theory and exper- which modify the possible frequencies of the field modes. The imental data must take into account more realistic boundary Casimir energy is, then, defined as the difference between the conditions. The most relevant ones are those that consider the vacuum energy with and without the material plates. How- finite conductivity of real metals and roughnessof the surfaces ever, since in both situations the vacuum energy is a divergent involved. These conditions become more important as the dis- quantity, we need to adopt a regularization prescription to give tance between the two bodies becomes smaller. Thermal ef- a physical meaning to such a difference. Therefore, a precise fects must also be considered. However, in principle, these definition for the Casimir energy is given by effects become dominant compared with the vacuum contribution for large distances, where the forces are already very 1 ~ω 1 ~ω small. A great number of papers have been written on these ECas := lim ∑ 2 k − ∑ 2 k , (6) s→0 α α topics since the analysis of most recent experiments require " k !I k !II # the consideration of real boundary conditions. For finite con- where subscript I means a regularized sum and that the fre- ductivity effects see Ref. [30]; the simultaneous consideration quencies are computed with the boundary conditions taken of roughness and finite conductivity in the proximity for ap- into account, subscript II means a regularized sum but with proximationcan be foundin Ref. [31] and beyondPFA in Ref. no boundary conditions at all and s stands for the regularizing [32] (see also references cited in the above ones). Concerning parameter. This definition is well suited for plane geometries the present status of controversies about the thermal Casimir like that analyzed by Casimir in his original work. In more force see Ref. [29] 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 dependson B. The Casimir’s approach 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 The novelty of Casimir’s original paper was not the predic- regularization prescription only after the subtraction is made. tion of an attractive force between neutral objects, once Lon- Of course, there are many different regularization methods.A don had already explained the existence of a force between quite simple but very efficient one is achieved by introducing 4 et al. a high frequency cut off in the zero-point energy expression, which, after symmetrization, can be written in the form as we shall see explicitly in the next section. This procedure can be physically justified if we note that the metallic plates 1 T ν = ∂ φ∂νφ + ∂νφ∂ φ + g νL . (11) become transparentin the high frequencylimit so that the high µ 2 µ µ µ frequency contributions are canceled out from equation (6). Though the calculation of the Casimir pressure for the case For our purposes, it is convenient to write the vacuum expec- of two parallel plates is very simple, its determination may tation value (VEV) of the energy-momentum tensor in terms become very involved for other geometries, as is the case, of the above Green function as for instance, of a perfectly conducting spherical shell. Af- ter a couple of years of hard work and a “nightmare in Bessel i ∂′ ∂ ∂′ ∂ ∂′ ∂α 2 ′ h0|Tµν(x)|0i = − lim ( µ ν+ ν µ)−gµν( α +m ) G(x,x ). functions”, Boyer [33] computed for the first time the Casimir 2 x′→x pressure inside a spherical shell. Surprisingly, he found a re- h i (12) pulsive pressure, contrary to what Casimir had conjectured In this context, the Casimir energy density is defined as five years before when he proposed a very peculiar model for ρ the stability of the electron [34]. Since then, Boyer’s result C(x)= h0|T00(x)|0iBC − h0|T00(x)|0iFree , (13) has been confirmed and improved numerically by many authors, as for instance, by Davies in 1972 [35], by Balian and where the subscript BC means that the VEV must be computed Duplantier in 1978 [36] and also Milton in 1978 [37], just to assuming that the field satisfies the appropriate boundary con- mention some old results. dition. Analogously, considering for instance the case of two The Casimir effect is not a peculiarity of the electromag- parallel plates perpendicular to the OZ axis (the generaliza- netic field. It can be shown that any relativistic field under tion for other configurations is straightforward) the Casimir boundary conditions caused by material bodies or by a com- force per unit area on one plate is given by pactification of space dimensions has its zero-point energy + − modified. Nowadays, we denominate by Casimir effect any FC = h0|Tzz |0i − h0|Tzz |0i, (14) changein the vacuumenergyof a quantumfield due to any external agent, from classical backgroundsand 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, 39, 40, 41]. pressure on the plate is given by the discontinuity of hTzzi at the plate. Using equation (12), hTzzi can be computed by

C. A local approach ∂ ∂ ∂2 i ′ h0|Tzz|0i = − lim − G(x,x ). (15) 2 x′→x ∂z ∂z′ ∂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 information 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 electromagnetism 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 energies 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 1 1 others, namely, the generalized zeta function method [42] and L φ,∂ φ = − ∂ φ∂µφ − m2φ2 (7) µ 2 µ 2 Schwinger’s method [43, 44], to mention just a few. The field equation and the corresponding Green function are given, respectively, by (−∂2 + m2)φ(x) = 0 ; (8) II. EXPLICIT COMPUTATION OF THE CASIMIR FORCE

(∂2 − m2)G(x,x′) = −δ(x − x′), (9) In this section, we show explicitly two ways of comput- where, as usual, G(x,x′)= ih0|T φ(x)φ(x′) |0i. ing the Casimir force per unit area in simple situations where plane surfaces are involved. We start with the global approach Since the above lagrangian density does not depend explicitly on x, Noether’s Theorem leads naturally to the following introduced by Casimir which is based on the zero-point en- µν ergy. Then, we give a second example where we use a local energy-momentum tensor (∂µT = 0) approach, based on the energy-momentum tensor. We ﬁnish ν ∂L ν ν this section by sketching some results concerning the Casimir T µ = ∂ φ + gµ L , (10) ∂(∂µφ) effect for massive ﬁelds. Brazilian Journal of Physics, vol 34, no. 4A, 2006, 1137-1149 5

A. The electromagnetic Casimir effect between two parallel we obtain plates L2 1 ∂2 1 ∞ B επ n−1 6a E(a)= + ∑ n − π ε3 ∂ε2 ε πε4 2 " ( n=0 n! a ) # As our first example, let us consider the standard (QED) L2 a 1 1 B π 3 Casimir effect where the quantized electromagnetic field is = 6(B −1) + (1 + 2B ) + 4 π 0 π ε4 1 ε3 constrained by two perfectly parallel conducting plates sepa- 2 " 12 a rated by a distance a. For convenience,let us suppose that one ∞ Bn π n−1 plate is located at z = 0, while the other is located at z = a. + ∑ (n − 2)(n − 3)εn−4 . n=5 n! a # The quantum electromagnetic potential between the metallic plates in the Coulomb gauge (∇ · A = 0) which satisfies the 1 1 appropriate BC is given by [45] Using the well known values B0 = 1, B1 = − 2 and B4 = − 30 , and taking ε → 0+, we obtain 2 ∞ ′ 1/2 L 2π~ 2 A(ρ,z,t) = ∑ d2κ × E(a) π 1 (2π)2 Z ckaL2 = −~c · n=0 L2 24 × 30 a3 nπz × a(1)(κ,n)(κˆ × zˆ)sin + As a consequence, the force per unit area acting on the plate ( a at z = a is given by π π κ π (2) n n z n z + a (κ,n) i κsin − zˆ cos × F(a) 1 ∂E (a) π2~c 1 dyn ka a k a = − c = − ≈−0,013 , ) 2 2 ∂ 4 4 2 κ ρ ω h i L L a 240a (a/µm) cm × ei( · − t) + h.c., (16) where in the last step we substituted the numerical values ω κ κ2 2π2 2 1/2 where ( ,n) = ck = c + n /a , with n being a of ~ and c in order to give an idea of the strength of the non-negative integer and the prime in Σ′ means that for n = 0 Casimir pressure. Observe that the Casimir force between an extra 1/2 factor must be included in the normalization of the (conducting) plates is always attractive. For plates with the field modes. The non-regularized Casimir energy then 1cm2 of area separated by 1µm the modulus of this attrac- reads tive force is 0,013dyn. For this same separation, we have ∞ 1/2 −8 ~ 2κ 2π2 PCas ≈ 10 Patm, where Patm is the atmospheric pressure at nr c 2 d 2 n E (a) = L κ + 2 ∑ κ + sea level. Hence, for the idealized situation of two perfectly c 2 Z (2π)2 a2 " n=1 # conducting plates and assuming L2 = 1 cm2, the modulus of ~c d2κ +∞ adk −7 2 z κ2 2 the Casimir force would be ≈ 10 N for typical separations − L 2 2 + kz . 2 Z (2π) Z−∞ 2π used in experiments. However, due to the finite conductivity q κ2 π 2 λ of real metals, the Casimir forces measured in experiments are Making the variable transformation +(n /a) =: and in- smaller than these values. troducing exponential cutoffs we get a regularized expression (in 1948 Casimir used a generic cutoff function),

2 ∞ ∞ ∞ B. The Casimir effect for a scalar field with Robin BC L 1 εκ ελ E r(a,ε)= e− κ2 dκ + ∑ e− λ2 dλ − 2π 2 Z Z nπ " 0 n=1 a ∞ ∞ ελ In order to illustrate the local method based on the energy- − dn e− λ2dλ Z Z nπ momentum tensor, we shall discuss the Casimir effect of a 0 a # massless scalar field submitted to Robin BC at two parallel plates, which are defined as L2 1 ∞ ∂2 e−εnπ/a = + ∑ − π ε3 ∂ε2 ε ∂φ 2 " n=1 ! φ β |bound. = ∂ |bound. , (18) ∞ 2 ∞ n ∂ ελ − dn e− dλ . (17) Z ∂ε2 Z nπ where, by assumption, β is a non-negative parameter. How- 0 a # ever, before computing the desired Casimir pressure, a few Using that comments about Robin BC are in order. 2 ∞ 2 First, we note that Robin BC interpolate continuously ∂ 1 ε π ∂ 1 1 ∑ e− n /a = , Dirichlet and Neumann ones. For β → 0 we reobtain Dirichlet ∂ε2 ε ∂ε2 ε eεπ/a − 1 " n=1 # BC while for β → ∞ we reobtain Neumann BC. Robin BC al- as well as the definition of Bernoulli’s numbers, ready appear in classical electromagnetism, classical mechanics, wave, heat and Schrödinger equations [46] and even in the ∞ n−1 1 t study of interpolating partition functions [47]. A nice realiza- t = ∑ Bn , e − 1 n=0 n! tion of these conditions in the context of classical mechanics 6 et al. can be obtained if we study a vibrating string with its extremes we have attached to elastic supports [48? ]. Robin BC can also be used 1 ∂ ∂ in a phenomenological model for a penetrable surfaces [49]. ht33(z)i = lim + λ2 g(z,z′). (25) 2i z′→z ∂z ∂z′ In fact, in the context of the plasma model, it can be shown that for frequencies much smaller than the plasma frequency ω ω β The Casimir force per unit area on the plate at x = a is given ( ≪ P) the parameter plays the role of the plasma wave- 33 length [? ]. In the context of QFT this kind of BC appeared by the discontinuity in ht i: more than two decades ago [50, 51]. Recently, they have been d2k dω discussed in a variety of contexts, as in the AdS/CFT corre- F ⊥ t33 t33 = π π h i|z=a− − h i|z=a+ . spondence [52], in the discussion of upper boundsfor the ratio Z (2 ) 2 h i entropy/energy in confined systems [53], in the static Casimir After a straightforward calculation, it can be shown that effect [54], in the heat kernel expansion [55, 56, 57] and in λφ4 the one-loop renormalization of the theory [58, 59]. As 1 ∞ dξξ3 F (β ,β ;a)=− . we will see, Robin BC can give rise to restoring forces in the 1 2 π2 4 β ξ β ξ 32 a Z0 1+ 1 /2a 1+ 2 /2a eξ− 1 static Casimir effect [54]. 1−β1ξ/2a 1−β2ξ/2a Consider a massless scalar field submitted to Robin BC on (26) two parallel plates: Depending on the values of parameters β1 and β2, restoring ∂φ ∂φ Casimir forces may arise, as shown in Figure 2 by the dotted 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 4 3 α pa ′ d kα ıkα(x−x′) ′ G(x,x )= e g(z,z ;k ,ω), (20) (β1 = 0 ,β2 → ∞) Z (2π)3 ⊥ with α = 0,1,2, gµν = diag(−1,+1,+1,+1) and the reduced ′ (β1 = 0 , β2 = 1) Green function g(z,z ;k⊥,ω) satisfies ∂2 + λ2 g(z,z′;k ,ω)= −δ(z − z′), (21) ∂z2 ⊥ O a λ2 ω2 2 ′ ω where = −k⊥ and g(z,z ;k⊥, ) is submitted to the following boundary conditions (for simplicity, we shall not write k⊥,ω in the argument of g): (β β 0 or β β ∞) ∂ ∂ 1 = 2 = 1 = 2 → g(0,z′)= β g(0,z′) ; g(a,z′)= −β g(a,z′) 1 ∂z 2 ∂z FIG. 2: Casimir pressure, conveniently multiplied by a4, as a func- It is not difficult to see that g(z,z′) can be written as tion of a for various values of parameters β1 and β2. ′ ′ A(z )(sinλz + β1λcosλz), z < z g(z,z′)= The particular cases of Dirichlet-Dirichlet, Neumann-  ′ λ β λ λ ′  B(z )(sin (z− a)− 2 cos (z − a)), z > z Neumann and Dirichlet-Neumann BC can be reobtained if we take, respectively, β1 = β2 = 0, β1 = β2 → ∞ and β1 = For points inside the plates, 0 z z′ a, the final expression  < , < 0; β2 → ∞. For the first two cases, we obtain for the reduced Green function is given by ∞ ξ3 DD NN 1 1 ıλz< −ıλz< 1 ıλ(z>−a) ıλ(z>−a) F a F a dξ (27) γ e − e γ e − e ( )= ( )= − 2 4 ξ . 1 2 32π a Z0 e − 1 gRR(z,z′)= − , 2ıλ( 1eıλa − 1 e−ıλa) γ1 γ2 ξs−1 ∞ ξ ζ Γ (22) Using the integral representation 0 d ξ = R(s) (s), β λ e −1 1+i i ′ ζ R where γi = (i = 1,2). Outside the plates, with z,z > a, where R is the Riemann zeta function, we get half the elec- 1−iβiλ we have: tromagnetic result, namely, ıλ(z −a) 2 e < 1 λ λ π ~c 1 gRR(z,z′)= eı (z<−a) − e−ı (z<−a) . (23) F DD(a)= F NN (a)= − . (28) 2ıλ γ 480 a4 2 Defining tµν such that For the case of mixed BC, we get

2 ∞ ν d k dω ν 1 ξ3 hT µ (x)i = ⊥ htµ (x)i, (24) F DN ξ π π (a)=+ 2 4 d ξ , (29) Z (2 ) 2 32π a Z0 e + 1 Brazilian Journal of Physics, vol 34, no. 4A, 2006, 1137-1149 7

Using in the previous equation the integral representation what kind of BC can be chosen for this field. The point is ∞ ξs−1 that Dirac equation is a first order equation, so that if we want dξ = (1 − 21−s)Γ(s)ζ (s) we get a repulsive pres- 0 eξ+1 R R non-trivial solutions, we can not impose that the field satis- sure (equal to half of Boyer’s result [60] obtained for the elec- fies Dirichlet BC at two parallel plates, for instance. The most tromagnetic field constrained by a perfectly conducting plate appropriate BC for fermions is borrowed from the so called parallel to an infinitely permeable one), MIT bag model for hadrons [62], which basically states that there is no flux of fermions through the boundary (the normal 7 π2~c 1 F DN (a)= × . (30) component of the fermionic current must vanish at the bound- 8 480 a4 ary). The Casimir energyper unit area for a massive fermionic field submitted to MIT BC at two parallel plates was first computed by Mamayev and Trunov [63] (the massless fermionic C. The Casimir effect for massive particles field was first computed by Johnson in 1975 [64]) 1 1 ∞ ξ − ma E f ξξ ξ2 2 2 −2ξ 2 c (a,m)= − 2 3 d − m a log 1 + e . In this subsection, we sketch briefly some results concern- L π a Zma ξ + ma ing massive fields, just to get some feeling about what kind q of influence the mass of a field may have in the Casimir ef- The small and large mass limits are given, respectively, by fect. Firstly, let us consider a massive scalar field submit- π2 ted to Dirichlet BC in two parallel plates, as before. In this 1 f 7 m Ec (a,m) ≈ − + ; case, the allowed frequencies for the field modes are given L2 2880a3 24a2 2 2 1/2 1/2 ω κ2 n π 2 1 f 3(ma) −2ma by k = c + 2 + m , which lead, after we use the E (a,m) ≈ − e . (34) a L2 c 25π3/2a3 Casimir methodh explainedi previously, to the following result for the Casimir energy per unit area (see, for instance, Ref. Since the first correction to the zero mass result has an oppo- [38]) site sign, also for a fermionic field small masses diminish the Casimir effect. In the large mass limit, ma → ∞, we have a ∞ 1 m2 1 behaviour analogous to that of the scalar field, namely, an ex- E (a,m)= − ∑ K (2amn) , (31) 2 c π2 2 2 ponential decay with ma (again this happens due to the plane L 8 a n=1 n geometry). We finish this section with an important observa- where Kν is a modified Bessel function, m is the mass of the tion: even a particle as light as the electron has a completely field and a is the distance between the plates, as usual. The negligible Casimir effect. limit of small mass, am ≪ 1, is easily obtained and yields

1 π2 m2 III. MISCELLANY E (a,m) ≈− + . (32) L2 c 1440a3 96a

As expected, the zero mass limit coincides with our previous In this section we shall briefly present a couple of topics result (28) (after the force per unit area is computed). Observ- which are in some way connected to the Casimir effect and ing the sign of the first correction on the right hand side of last that have been considered by our research group in the last equation we conclude that for small masses the Casimir effect years. For obvious reasons, we will not be able to touch all is weakened. the topics we have been interested in, so that we had to choose On the other hand, in the limit of large mass, am ≫ 1, it can only a few of them. We first discuss how the Casimir effect be shown that of a charged field can be influenced by an external magnetic 1 m2 π 1/2 field. Then, we show how the constitutive equations asso- E (a,m) ≈− e−2ma (33) ciated to the Dirac quantum vacuum can be affected by the L2 c 16π2 a ma presence of material plates. Finally, we consider the so called Note that the Casimir effect disappears for m → ∞, since in dynamical Casimir effect. this limit there are no quantum fluctuations for the field any- more. An exponential decay with ma is related to the plane geometry. Other geometries may give rise to power law de- A. Casimir effect under an external magnetic field cays when ma → ∞. However, in some cases, the behaviour of the Casimir force with ma may be quite unexpected. The Casimir force may increase with ma before it decreases mono- The Casimir effect which is observed experimentally is that tonically to zero as ma → ∞ (this happens, for instance, when associated to the photon field, which is a massless field. As Robin BC are imposed on a massive scalar field at two parallel we mentioned previously, even the electron field already ex- plates [61]). hibits a completely unmeasurable effect. With the purpose In the case of massive fermionic fields, an analogous be- (and hope) of enhancing the Casimir effect of electrons and haviour is found. However, some care must be taken when positrons we considered the influence of an external elec- computing the Casimir energy density for fermions, regarding tromagnetic field on their Casimir effect. Since in this case 8 et al. we have charged fields, we wondered if the virtual electron- B. Magnetic permeability of the constrained Dirac vacuum 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. In contrast to the classical vacuum, the quantum vacuum This problem was considered for the first time in 1998 [65] is far from being an empty space, inert and insensible to any (see also Ref. [66]). The influence of an external magnetic external influence. It behaves like a macroscopic medium, in field on the Casimir effect of a charged scalar field was con- the sense that it responds to external agents, as for example sidered in Ref. [67] and recently, the influence of a magnetic electromagnetic fields or the presence of material plates. As field on the fermionic Casimir effect was considered with the previously discussed, recall that the (standard) Casimir effect more appropriate MIT BC [68]. is nothing but the energy shift of the vacuum state of the field For simplicity, let us consider a massive fermion field un- caused by the presence of parallel plates. There are many der anti-periodic BC (in the OZ direction) in the presence other fascinating phenomena associated to the quantum vac- of a constant and uniform magnetic field in this same direc- uum, namely, the particle creation produced by the application tion. After a lengthy but straightforward calculation, it can be of an electric field [69], the birefringence of the QED vacuum shown that the Casimir energy per unit area is given by [66] 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 E(a,B) 2(am)2 (−1)n−1 = − ∑ K (amn) two perfectly conducting parallel plates which impose (by as- 2 π2 3 2 2 ℓ a n=1 n sumption) BC only on the radiation field is slightly altered by ∞ ∞ 2 the presence of the plates. The expected relative variation in eB 2σ 2 σ eBa − ∑(−1)n−1 dσ e−(n/2) −(am) / L , the velocity of light for typical values of possible experiments 4π2a Z σ n=1 0 is so tiny (∆c/c ≈ 10−36) that this effect has not been confirmed yet. Depending on the nature of the material plates the where we introduced the Langevin function: L(ξ)= cothξ − velocity of light propagating perpendicular to the plates is ex- 1/ξ. In the strong field limit, we have pected to diminish [74]. The Scharnhorst effect inside a cavity was considered in [75]. ∞ The negligible change in the velocity of light predicted by E(a,B) eBm (−1)n−1 ≈− ∑ K (amn) . (35) Scharnhorst may be connected with the fact that the effect that 2 π2 1 ℓ n=1 n 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. In this limit we can still analyze two distinct situations, With the purpose of estimating a change in the constitutive namely, the small mass limit (ma ≪ 1) and the large mass equations of the quantum vacuum at the one-loop level, we limit (ma ≫ 1). Considering distances between the plates typ- were led to consider the fermionic field submitted to some BC. ical of Casimir experiments (a ≈ 1µm), we have in the former The magnetic permeability µ of the constrained Dirac vacuum case 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 ρc(a,B) B ∆ ≈ 10−4 × ; ma ≪ 1 , (36) relative change in the permeability, µ := µ−1, was also neg- ρc(a,0) Tesla ligible (an analogous calculation has also been made in the context of scalar QED [77]). However, when the more real- while in the latter case, 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- ρ (a,B) B c ≈ 10−10 × ; ma ≫ 1 . (37) uum is given by [78] ρc(a,0) Tesla 1 π − 2 e2 1 e2 = 1 − + H(ma), µ(ma) 12π2 ma 6π2 ma In the above equations ρc(a,B) and ρc(a,0) are the Casimir energy density under the influence of the magnetic field and where without it, respectively. Observe that for the Casimir ef- ∞ x x − 1 fect of electrons and positrons, which must be treated as the H(ma)= dx ln 1 + e−2max . Z1 (x2 − 1)3/2 x + 1 large mass limit described previously, huge magnetic fields are needed in order to enhance the effect (far beyond accessi- (38) ble fields in the laboratory). In other words, we have shown For confining distances of the order of 0,1µm, we have that though the Casimir effect of a charged fermionic field can ∆µ := µ − 1 ≈ 10−9 . (39) indeed be altered by an external magnetic field, the universal constants conspired in such a way that this influence turns out The previous value is comparable to the magnetic permeabil- to be negligible and without any chance of a direct measure- ity of Hydrogen and Nitrogen at room temperature and atmo- ment at the laboratory. spheric pressure. Hence, an experimental verification of this Brazilian Journal of Physics, vol 34, no. 4A, 2006, 1137-1149 9 result seems to be not unfeasible (we will come back to this of the boundary is prescribed, non-relativistic and of small point in the final remarks). amplitude (δq(t) is the position of the plate at a generic in- stant t). Last assumptions may be stated mathematically by |δq˙(t)| << c and |δq(t)| << c/ω0 , where ω0 corresponds C. The Dynamical Casimir effect 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

As our last topic, we shall briefly discuss the so called dy- ∂ ∂ 1 namical Casimir effect, which consists, as the name suggests, + δq˙(t) φ(x,t)| δ = φ(x,t)| δ . (40) ∂x ∂t x= q(t) β x= q(t) of the consideration of a quantum field in the presence of moving boundaries. Basically, the coupling between vacuum where β is a non-negative parameter and the previous condi- fluctuations and a moving boundary may give rise to dissi- tion was already written in the laboratory frame. We are ne- pative forces acting on the boundary as well as to a particle glecting terms of the order O(δq˙2/c2). The particular cases of creation phenomenon. In some sense, these phenomena were Dirichlet and Neumann BC are reobtained by making β = 0 expected. Recall that the static Casimir force is a fluctuat- and β → ∞, respectively. The dissipative forces for these par- ing quantity [79] and hence, using general arguments related ticular cases were studied at zero temperature as well as at to the fluctuation-dissipation theorem [80] dissipative forces finite temperature and also with the field in a coherent state on moving boundaries are expected. Further, using arguments in [100] (dissipative forces on a perfectly conducting mov- of energy conservation we are led to creation of real parti- ing plate caused by the vacuum fluctuations of the electro- cles (photons, if we are considering the electromagnetic field magnetic field at zero and non-zero temperature were studied [81]). For the above reasons, this topic is sometimes referred in [101]). The perturbative approach introduced by Ford and to as radiation reaction force on moving boundaries. Vilenkin [84] consists in writing After Schwinger’s suggestion that the phenomenon of sonoluminescence could be explained by the dynamical φ(x,t)= φ0(x,t)+ δφ(x,t) , (41) Casimir effect [82] (name coined by himself), a lot of work has been done on this subject. However, it was shown a few where φ0(x,t) is the field submitted to a Robin BC at a static years later that this was not the case (see [9] and references boundary fixed at the origin and δφ(x,t) is the first order con- therein for more details). tribution due to the movementof the boundary. The total force The dynamical Casimir effect already shows up in the case on the moving boundary may be computed with the aid of the of one (moving) mirror [83, 84, 85]. However, oscillating corresponding energy-momentum tensor, namely, cavities whose walls perform vibrations in parametric reso- nance with a certain unperturbed field eigenfrequency may δF(t)= h0|T 11 t,δq+(t) − T 11 t,δq−(t) |0i , (42) greatly enhance the effect [86, 87, 88, 89]. Recently, a one dimensional oscillating cavity with walls of different nature where superscripts + and − mean that we must compute the was considered [90]. The dynamical Casimir effect has also energy-momentumtensor on both sides of the moving bound- been analyzed for a variety of three-dimensional geometries, ary. It is convenient to work with time Fourier transforms. including parallel plane plates [91], cylindrical waveguides The susceptibility χ(ω) is defined in the Fourier space by [92], and rectangular [93], cylindrical [94] and spherical cavi- δ ω χ ω δ ω ties [95]. For a review concerning classical and quantum phe- F ( )=: ( ) Q( ) (43) nomena in cavities with moving boundaries see Dodonov [96] where δF (ω) and δQ(ω) are the Fourier transformations of and for a variety of topics on non-stationary Casimir effect δ δ including perspectives of its experimental verification see the F(t) and q(t), respectively. It is illuminating to compute special issue [97]. the total work done by the vacuum fluctuations on the moving In this section, we shall discuss the force exerted by the boundary. It is straightforward to show that quantum fluctuations of a massless scalar field on one moving +∞ 1 ∞ boundary as well as the particle creation phenomenonin a un- δF(t)δq˙(t)dt = − dωωI mχ(ω)|δQ(ω)|2 . (44) Z ∞ π Z usual example in 1+1 dimensions, where the field satisfies a − 0 Robin BC at the moving boundary [98, 99]. We shall follow Note that only the imaginary part of χ(ω) appears in the pre- throughout this paper the perturbative method introduced by vious equation. It is responsible for the dissipative effects and Ford and Vilenkin [84]. This method was also applied suc- hence it is closely related with the total energy converted into cessfully to the case of the electromagnetic field under the real particles. On the other hand, the real part of χ(ω), when it influence of one moving (perfectly) conducting plate [85] as exists, does not contribute to the total work and hence it is not well as an oscillating cavity formed by two parallel (perfectly) related to particle creation, but to dispersive effects. For the conducting plates [91]. particular cases of Dirichlet or Neumann BC it can be shown φ Let us then consider a massless scalar field in 1 + 1 in that the susceptibility is purely imaginary and given by the presence of one moving boundary which imposes on the field a Robin BC at one movingboundarywhen observedfrom ~ω3 χD(ω)= χN (ω)= i , (45) a co-moving inertial frame. By assumption, the movement 6πc2 10 et al. which implies β = 0 and β → ∞, respectively. The results coincide and are given by [102] (we are making c = 1) ~ d3 δF(t)= δq(t). (46) 6πc2 dt3 (D) (N) dN dN 2 (ω)= (ω) = (δq0) T ω(ω0 − ω)Θ(ω0 − ω) . Since I mχ(ω) > 0, for these cases the vacuum fluctuations dω dω are always dissipating energy from the moving boundary. (50) However, for Robin BC an interesting thing happens. It A simple inspection in (49) 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 (or Neumann) BC are employed. these quantities dividing them by the value I mχD(ω), where the subscript D means that I mχ(ω) must be computed with Dirichlet BC.

1,0

Im Im

0,5

Re Im

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

2 dN 4ω ∞ dω′ [δQ(ω − ω′)]2 IV. FINAL REMARKS (ω)= ω′ 1−β2ωω′ ω β2ω2 π 2 2 d 1 + Z0 2 1 + β ω′ " # (47) As an explicit example, let us consider the particular motion In the last decades there has been a substantial increase in the study of the Casimir effect and related topics. It is remark- −|t|/T δq(t)= δq0 e cos(ω0t), (48) able that this fascinating effect, considered nowadays as a fun- damental one in QFT, was born in connection with colloidal where, by assumption, ω T 1 (this is made in order to sin- 0 ≫ chemistry, an essentially experimental science. As we men- gle out the effect of a given Fourier componentof the motion). tioned previously, the novelty of Casimir’s seminal work [1] For this case, we obtain the following spectral distribution (see also [6]) was the technique employed by him to compute dN forces between neutral bodies as is emphasized by Itzykson (ω)= (δq )2T ω(ω − ω)× dω 0 0 and Zuber [103]: 2 2 [1 − β ω(ω0 − ω)] × Θ(ω0 − ω). (49) By considering various types of bodies inﬂuenc- (1 + β2ω2)(1 + β2(ω − ω)2) 0 ing the vacuum conﬁguration we may give an The spectral distributions for the particular cases of Dirichlet interesting interpretation of the forces acting on or Neumann BC can be easily reobtained by making simply them. Brazilian Journal of Physics, vol 34, no. 4A, 2006, 1137-1149 11

However, the attractive or repulsive character of the Casimir first radiative correction to the Casimir energy density (con- force can not be anticipated. It depends on the specific bound- sidering that the conducting plates impose BC only on the ra- ary conditions, the number of space-time dimensions, the na- diation field) was firstly computed by Bordag et al [104] and is ture of the field (bosonic or fermionic), etc. The “mystery” (1) (0) 9 αλ given by E (a,α)= E (a) c , where λ is the Comp- of the Casimir effect has intrigued even proeminent physicists C C 32 a c (0) such as Julian Schwinger, as can be seen in his own statement ton wavelength of the electron and EC (a) is the zeroth order [37]: contribution to the Casimir energy density. As we see, at least for QED, radiative corrections to the Casimir effect are exper- ... one of the least intuitive consequencesof quan- imentally irrelevant. However, they might be relevant in the tum electrodynamics. bag model, where for quarks λc ≈ a and also the quark prop- There is no doubt nowadays about the existence of the agators must be considered submitted to the bag BC [9]. Be- (static) Casimir effect, thanks to the vast list of accurate ex- sides, the study of radiative corrections to the Casimir effect periments that have been made during the last ten years. It is provide a good laboratory for testing the validity of idealized worth emphasizing that a rigorous comparison between theory BC in higher order of perturbation theory. and experimental data can be achieved only if the effects of Concerning the dynamical Casimir effect, the big challenge temperature and more realistic BC are considered. In princi- is to conceive an experiment which will be able to detect real ple, the former are important in comparison with the vacuum photons created by moving boundaries or by an equivalent contribution for large distances, while the latter can not be physicalsystem that simulates rapid motion of a boundary. An neglected for short distances. Typical ranges investigated in ingenious proposal of an experiment has been made recently Casimir experiments are form 0.1µm to 1.0µm and, to have an by the Padova’s group [105]. There are, of course, many other idea of a typical plasma wavelengths (the plasma wavelength interesting aspects of the dynamical Casimir effect that has is closely related to the penetration depth), recall that for Au been studied, as quantum decoherence [106], mass correction we have λP ≈ 136nm. of the moving mirrors [107], etc. (see also the reviews [108]). The Casimir effect has become an extremely active area As a final comment, we would like to mention that surpris- of research from both theoretical and experimental points of ing results have been obtained when a deformed quantum field view and its importance lies far beyond the context of QED. theory is considered in connection with the Casimir effect. It This is due to its interdisciplinary character, which makes this seems that the simultaneous assumptions of deformation and effect find applications in quantum field theory (bag model, boundary conditions lead to a new mechanism of creation of for instance), cavity QED, atomic and molecular physics, real particles even in a static situation [109]. Of course, the mathematical methods in QFT (development of new regular- Casimir energy density is also modified by the deformation ization and renormalization schemes), fixing new constraints [110]. Quantum field theories with different space-time sym- in hypothetical forces, nanotechnology (nanomachines oper- metries, other than those governed by the usual Poincaréal- ated by Casimir forces), condensed matter physics, gravitation gebra (as for example the κ-deformed Poincaréalgebra [111]) and cosmology, models with compactified extra-dimensions, may give rise to a modified dispersion relation, a desirable fea- etc. ture in some tentative models for solving recent astrophysical In this work, we considered quantum fields interacting only paradoxes. with classical boundaries. Besides, these interactions were described by highly idealized BC. Apart from this kind of in- Acknowledgments: I am indebted to B. Mintz, P.A. Maia teraction, there was no other interaction present. However, Neto and R. Rodrigues for a careful reading of the manuscript the fields in nature are interacting fields, like those in QED, and many helpful suggestions. I would like also to thank to all etc. Hence, we could ask what are the first corrections to the members of the Casimir group of UFRJ for enlightening dis- Casimir effect when we consider interacting fields. In princi- cussions that we have maintained over all these years. Finally, ple, they are extremely small. In fact, for the case of QED, the I thank to CNPq for a partial financial support.

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