Electromagnetic Field Correlators, Maxwell Stress Tensor, and the Casimir Effect for Parallel Walls

Brazilian Journal of Physics, vol. 35, no. 3A, September, 2005 657

F. C. Santos, J. J. Passos Sobrinho, and A. C. Tort Instituto de F´ısica, Universidade Federal do Rio de Janeiro, Cidade Universitaria´ - Ilha do Fundao˜ - Caixa Postal 68528, 21945-970 Rio de Janeiro RJ, Brazil Received on 23 May, 2005 We evaluate the quantum electromagnetic ﬁeld correlators associated with the electromagnetic vacuum distorted by the presence of two plane parallel conducting walls and in the presence of a conducting wall parallel to a perfectly magnetically permeable one. Regularization is performed through the generalized zeta funtion technique. Results are applied to rederive the atractive and repulsive Casimir effect through Maxwell stress tensor. Surface divergences are shown to cancel out when stresses on both sides of the material surface are taken into account.

I. INTRODUCTION the vacuum distortions caused by the presence of these walls through the quantized version of the Maxwell stress tensor. According to Casimir [1], the (macroscopically) observable Though the approach chosen here has many points in common vacuum energy of a quantum field is the regularized difference with Ref. [7] cited above and Green functions techniques [8], between the zero point energies with and without the external it is a different alternative in the sense that it relies on objects conditions demanded by the particular physical situation at known as correlators, which are regularized vacuum expec- hand. In the case of the quantized electromagnetic field con- tation values of products of the electromagnetic field compo- fined between two infinite parallel conducting walls separated nents taken at the same point of space and for equal times. by a distance a, Casimir’s conception of the vacuum energy These correlators contain all the information we need about leads to a force per unit area between the walls given by the local behavior of the vacuum expectation values of ele- ments of Maxwell stress tensor, in particular, their behavior F π~c near both sides of the material surface in question, a feature = − . (1) A 240a4 crucial to the obtention of the correct result. The local behavior of the Maxwell tensor, or of the relativistic symmetrical Until 1997 only one experiment involving Casimir’s original stress-energy tensor, is extremely important because as shown setup had been performed [2]. Lately, however, the exper- by, for example, Deutsch and Candelas [9] with the help of imental observation of this tiny force with metallic surfaces Green functions techniques, as we approach the boundaries was significantly improved by a series of new experiments we find strong divergencies that cannot be simply removed by due to Lamoreaux, Mohideen and Roy, and Mohideen [3]. usual renormalization procedures. Besides reviewing the ob- The concept of an observable vacuum energy can be extended tention of the electromagnetic Casimir for the standard case of to all quantum fields and several types of boundary condi- two perfectly conducting parallel walls, we will also consider tions and/or applied external fields. An updated review of all a pair of parallel walls, one of them perfectly permeable. This these Casimir effects can be found in the monograph by Bor- setup was first proposed by Boyer [10] who analyzed them dag, Mohideen, and Mostepananko [4], but see also, Mostepa- from the viewpoint of random electrodymanics and it is the nenko and Trunov, and Plunien et al. [5]. simplest example of a repulsive Casimir force. For both cases, The local approach to the electromagnetic Casimir effect the conducting plate and Boyer’s setup we also construct the was initiated by Brown and Maclay who calculated the renor- symmetrical stress tensor. We will employ Gaussian units and malized stress energy tensor between two parallel perfectly set c = ~ = 1. conducting plates by means of Green functions techniques [6]. An interesting approach to the standard Casimir effect is the one due to Gonzales [7]. Analyzing the vacuum oscillations of the electromagnetic field confined by two parallel II. MAXWELL STRESS TENSOR AND THE slabs separated by a gap of size a, Gonzales pointed out that ELECTROMAGNETIC FIELD CORRELATORS the apparently non-objectionable definition of the vacuum energy given above could easily lead to conceptual errors and Our aim in this section is to obtain an expression for the stressed the fact that in any Casimir interaction calculation, quantum version of the electromagnetic force per unit area contributions from both sides of the material surfaces involved that acts on plane material wall. Material wall here means a must be taken into account. This is so because the vacuum perfectly conducting square surface(ε → ∞) or a perfectly per- pressure always pushes the material surfaces involved, there- meable one (µ → ∞) whose linear dimension L is much larger fore, the repulsiveness or atractiveness of the Casimir force than others relevant dimensions envolved such as the distance depends on the discontinuity of the relevant component of the between two of those walls. The physical interaction between quantized Maxwell stress tensor at the location of the surface. any of the two types of walls considered here and the vac- The purpose of this paper is to pursue this line of reason- uum electromagnetic field is mimicked by the imposition of ing by analyzing the stresses on parallel plane walls due to appropriate boundary conditions on the electromagnetic field 658 F. C. Santos et al. on the location of the walls. The Cartesian components of the though not equal, to the ones employed by Lutken¨ e Ravn- Maxwell stress tensor (in Gaussian units) are given by [11] dal [16]. We will also show how to obtain the corresponding µ ¶ results for another unusual but intersting setup [13]. 1 1 1 T = E E − δ E2 + B B − δ B2 (2) i j 4π i j 2 i j i j 2 i j III. CORRELATORS FOR CASIMIR’S SETUP where i, j = x,y,z. Suppose that the material surface is placed perpendicularly to the OZ axis. Upon quantizing the electro- Consider an experimental setup consisting in two infinite magnetic field we can write the quantum version of (2). For perfectly conducting parallel plates (ε → ∞) kept at a fixed instance, distance a from each other. We will choose the coordinates ® h ® D E ® D E i axis in such a way that the OZ direction is perpendicular to ˆ 1 ˆ 2 ˆ 2 ˆ2 ˆ2 Tzz = Ez − Ek + Bz − Bk , (3) the plates. One of the plates will be placed at z = 0 and the 0 8π 0 0 0 0 other one at z = a. The field must satisfy the following bound- ® ˆ 2 ˆ 2 ˆ 2 ˆ2 ˆ2 ˆ2 ˆ ˆ ary conditions on the plates: the tangential components Ex e where Ek = Ex + Ey and Bk = Bx + By; O 0 ≡ h0|O|0i de- notes a vacuum expectation value. The other Cartesian com- Ey of the electric field and the normal component Bz of the magnetic field must be zero on the plates. Since there are ponents of this tensor can be obtained ® in an analogous way. The quantum macroscopic force Fˆ on the wall can be eval- no real charges or currents it will be convenient to work in 0 ∇ Φ uated by integrating the quantum version of the classical result the Coulomb gauge in which · A(r,t) = 0, and = 0, thus ∂ ∂ ∇ [11] E(r,t) = − A(r,t)/ t and B(r,t) = ×A(r,t). These physical boundary conditions combined with the choice of gauge Z allow us to rewrite the boundary conditions in terms of the F = T˜ · nˆ da, (4) ∂R components of the vector potential A(r,t) in the following way: at z = 0 we will have, were nˆ is the outward normal at the boundary ∂R and R is any region containing all or part of the wall. Classically, (4) can be obtained by integrating the Lorentz force per unit volume Ax(x,y,0,t) = 0; Ay(x,y,0,t) = 0; acting on charge and current distributions and eliminating the sources in favor of the fields. From a quantum point of view, ∂ we see that the problem of evaluating the pressure the material Az(x,y,0,t) = 0, (5) surface due to the distorted zero point oscillations of the elec- ∂z tromagnetic field is reduced to the evaluation of the vacuum and at z = a , expectation value of the quantum operators Eî (r,t)Eˆ j (r,t), Bî (r,t)Bˆ j (r,t), and Eî (r,t)Bˆ j (r,t). The evaluation of these correlators depends on the specific choice of the boundary Ax(x,y,a,t) = 0; Ay(x,y,a,t) = 0; conditions. A regularization recipe will also be necessary, for these objects are mathematically ill-defined. For the setup en- volving conducting plates these correlators were evaluated by ∂ A (x,y,a,t) = 0 . (6) Lutken¨ and Ravndal [16], see also [12]. They can be also ob- ∂z z tained from the coincidence limit of the photon propagator between conducting plates evaluated by Bordag et al [17]. In the The vector potential operator Aˆ (r,t) that satisfies the wave next section we will evaluate and regularize these correlators equation, the Coulmb gauge and the boundary conditions can by means of analitycal continuation techniques [13] similar, be written as

³ ´ 1 ∞ 00 Z 2 n ³ ´ 1 π 2 d κ nπz ˆ ( , ) = √ (1)(κ, )κˆ × ˆ A r t π ∑ ω aˆ n zsin a n=0 a · ³ ´ ³ ´¸¾ in nπz κ nπz κ ρ ω + aˆ(2)(κ,n) κˆ sin − zˆ cos ei( · − t) + h.c, (7) ωa a ω a where κ = (kx,ky) and ρ is the position arrow on the XY plane. The normal frquencies are given by r π2 ω = ω(κ,n) = κ2 + n2 , (8) a2 ” with kx,ky ∈ R e n ∈ N−1. The symbol ∑ indicates that the term corresponding to n = 0 for normaliztion reasons must be multiplied by 1/2 . The Fourier coefﬁcients aˆ(λ)(κ,n) where λ = 1,2 is the polarization index, are operators in the photon Brazilian Journal of Physics, vol. 35, no. 3A, September, 2005 659 ocupation number space and satisfy h i (λ) †(λ0) 0 0 ¡ 0¢ aˆ (κ,n),aˆ (κ ,n ) = δλλ0 δnn0 δ κ − κ . (9)

It is convenient to write the vector potential in the general form

∞ 00 Z 2 2 (λ) (λ) −iω(κ,n)t Aˆ (r,t) = ∑ d κ ∑ aˆ (κ,n)Aκn (r)e + h.c, (10) n=0 λ=1

(λ) where Aκn (r) are the modal functions. The modal functions for each polarization state must obey Helmholtz equation and the boundary conditions given above. In our case the modal functions are ³π´ 1 ³ π ´ (1) 1 2 1 n z −iκ·ρ Aκ (r) = √ sin e κˆ × zˆ , (11) n π a ω a and · ¸ ³π´ 1 π ³ π ´ κ ³ π ´ (2) 1 2 1 in n z n z −iκ·ρ Aκ (r) = √ κˆ sin − zˆ cos e . (12) n π a ω aω a ω a

The next step is to evaluate the electric field operator Eˆ (r,t). Recalling that aˆ(λ)(κ,n)|0i = 0, we first write the correlators < Eî(r,t)Eˆ j(r,t)i0 in the general form ˆ ˆ ∗ < Ei(r,t)E j(r,t)i0 = ∑Eiα(r)E jα(r), (13) α where we have introduced the modal functions Eiα(r) for the electric field. In our case (11) and (12) yield

µ ¶ 1 i ω(κ,n)π 2 ³nπz´ E(1) (r) = sin e−iκ·ρ (κˆ × zˆ) , (14) iκn π a a i and,

µ ¶ 1 · ¸ i ω(κ,n)π 2 inπ ³nπz´ κ ³nπz´ E(2) (r) = κ sin − zˆ cos e−iκ·ρ , (15) iκn π a i aω(κ,n) a i ω(κ,n) a respectively. Taking (14) and (15) into (13), we write κˆ i = cosφδix +sinφδiy, zˆi = δiz e (zˆ ×κˆ)i = sinφδix −cosφδiy, where φ is the azimuthal angle on the XY plane and we have performed all angular integrals. In this way we end up with

µ ¶ k ³π´ δ ∞ 00 ³ π ´Z ∞ ˆ ˆ 2 i j 2 n z κκω κ hEi(r,t)E j(r,t)i0 = π ∑ sin d ( ,n) a 2 n=0 a 0 µ ¶ k ³π´³π´2 δ ∞ 00 ³ π ´Z ∞ 2 i j 2 2 n z κκω−1 κ + π ∑ n sin d ( ,n) a a 2 n=0 a 0 µ ¶ ³π´ ∞ 00 ³ π ´Z ∞ 2 δ⊥ 2 n z κκ3 ω−1 κ + π i j ∑ cos d ( ,n), (16) a n=0 a 0

δk δ δ δ δ δ⊥ δ δ where i j := ix jx + iy jy e i j := iz jz. Equation (16) is a formal expression for the correlator hEi(r,t)E j(r,t)i0, since it is mathematically ill-defined unless a regularization recipe is prescribed. We will regularize the integrals in (16) with the help of analytical continuation methods. Consider, for instance, the first integral on the r.h.s. of (16) and let us rewrite it as follows µ ¶ µ ¶ Z ∞ 2π2 1/2 Z ∞ 2π2 1/2−s κκ κ2 n κκ κ2 n d + 2 → d + 2 . 0 a 0 a The first term in (16) can be rewritten as

µ ¶ k Z µ ¶ 1 −s 2 ³π´ δ ∞” ³nπz´ ∞ n2π2 2 T = i j ∑ sin2 dκκ κ2 + , (17) 1 π 2 a 2 n=0 a 0 a 660 F. C. Santos et al. the second as

µ ¶ k Z µ ¶− 1 −s 2 ³π´³π´2 δ ∞” ³nπz´ ∞ n2π2 2 T = i j ∑ n2 sin2 dκκ κ2 + , (18) 2 π 2 a a 2 n=0 a 0 a and the third one as

µ ¶ Z µ ¶− 1 −s 2 ³π´ ∞” ³nπz´ ∞ n2π2 2 T = δ⊥ ∑ cos2 dκκ3 κ2 + . (19) 3 π i j 2 a n=0 a 0 a Let us assume that ℜs is large enough to give precise mathematical meaning to these integrals. After evaluating them and make use of the analytical continuation of the results we will take the limit s → 0. Let us see, for instance, what happens with T1. Making use of the following representation of Euler beta function [19] Z ∞ ³ ´ ¡ ¢ν−1 B µ µ ν dxxµ−1 x2 + c2 = ,1 − ν − cµ+2 −2 , (20) 0 2 2 2 ¡ ¢ Γ Γ Γ ℜ ν µ ℜ where B(x,y) = (x) (y)/ (x + y), and that holds for + 2 < 1 and µ > 0, we obtain

µ ¶ / − Z ∞ 2π2 1 2 s ³ π´3−2s Γ ³ π´3−2s κκ κ2 n 1 n (s − 3/2) 1 n d + 2 = = . (21) 0 a 2 a Γ(s − 1/2) (2s − 3) a

Taking this result into T1 we obtain

µ ¶ k " µ ¶# ³π´3−2s δ ∞ π 1 i j 3−2s 2n z T1 = ζR(2s − 3) − ∑ n cos , (22) 2s − 3 a 2a n=0 a where ζR(z) is the well-known Riemann zeta function. Taking the limit s → 0, we have

k " # 1 ³π´4 δ 1 1 ∞ d3 T = − i j + ∑ sin(2nξ) , (23) 1 π ξ3 3 a 2 120 8 n=1 d where we made use of the fact that ζR(−3) = 1/120, deﬁned ξ := πz/a, and wrote

1 d3 n3 cos(2nξ) = − × sin(2nξ). (24) 8 dξ3 The sum on the R.H.S. of (23) can be regularized in many ways. A quick though non-rigorous way is to write

k " # 1 ³π´4 δ 1 1 d3 ∞ T = − i j + ∑ sin(2nξ) , (25) 1 π ξ3 3 a 2 120 8 d n=1 and express the summand in terms of exponential functions of imaginary argument thereby transforming each one of the sums into the euclidean space. In this way we obtain Ã ! ∞ 1 ∞ ∞ ∑ sin(2nξ) = ∑ exp(i2nξ) − ∑ exp(−i2nξ) n=1 2i n=1 n=1 Ã ! ∞ ∞ ¡ ¢ 1 ξ ξ0 = ∑ exp(2n E ) − ∑ exp −2n E (26) 2i n=1 n=1

where we have made the substitution iξ → −ξE in the ﬁrst can be easily performed and the result is ξ ξ0 sum and i → E in the second. Each one of the sums above ∞ 1 ∑ sin(2nξ) = cot(ξ) (27) n=1 2 Brazilian Journal of Physics, vol. 35, no. 3A, September, 2005 661

It follows that Notice that some care must be taken when we aply this procedure to the third term. This is so because the term corre- ³ ´ δk · 3 ¸ 1 π 4 1 1 d 1 sponding to n = 0 in T3 is not zero. In fact its contribution T = − i j + cot(ξ) . (28) 1 3π a 2 120 8 dξ3 2 is:

Treating the two other terms in (16) in a similar manner we obtain µ ¶ Z 2 ³π´ ∞ T (n = 0) = δ⊥ dκκ −2−2s, (31) k · ¸ 3 π i j 1 ³π´4 δ 1 1 d3 1 a 0 T = − i j + cot(ξ) , (29) 2 π a 2 120 8 dξ3 2 and which diverges when the regularization is removed. However · ¸ 4 ³π´4 δ⊥ 1 1 d3 1 this term is non-physical and can be safely ignored. Finally, T = i j − cot(ξ) . (30) collecting all partial results we have 3 3π a 2 120 8 dξ3 2

hEi(r,t)E j(r,t)i0 = T1 + T2 + T3 · ¸ ³π´4 ³ ´ 2 k ⊥ 1 = −δ + δ + δi jF(ξ) . (32) a 3π i j 120

The function F (ξ) is deﬁned by

1 d3 1 F (ξ) := − cot(ξ), (33) 8 dξ3 2 and its expansion about ξ = 0 is given by

3 1 ¡ ¢ F (ξ) ≈ ξ−4 + + O ξ2 . (34) 8 120 Near ξ = π (which corresponds to z = a) we make the replacement ξ → ξ − π. Notice that due to the behavior of F (ξ) near ξ = 0,a, strong divergences predominate in the behavior of the correlators near the plates. By applying the exactly the same procedure we obtain the magnetic ﬁeld correlators · ¸ ³π´4 ³ ´ 2 k ⊥ 1 hBi(r,t)B j(r,t)i0 = −δ + δ − δi jF(ξ) . (35) a 3π i j 120

A direct evaluation also shows that the correlators < namic [10] and it is the simplest case of a repulsive Casimir Ei(r,t)B j(r,t)i0 are zero. effect that can be found in the literature. The boundary conditions now are: (a) the tangential components Ex and Ey of the electric field as well as the normal component Bz of the mag- IV. CORRELATORS FOR BOYER’S SETUP netic field must vanish on the surface of the plate at z = 0; (b) the tangential components of Bx e By of the magnetic field as well as normal conponent Ez of the electric field must vanish The other setup we are interested in is that one in which a on the surface of the plate at z = a. These boundary conditions perfectly conducting plate is placed at z = 0 and perfectly per- translated in terms of the components of the vector potential meable plate is placed at z = a. This setup was analyzed for read the first time by Boyer in the contetxt of stochastic electrody-

∂ A (x,y,0,t) = 0; A (x,y,0,t) = 0; A (x,y,0,t) = 0, (36) x y ∂z z 662 F. C. Santos et al. at z = 0, and at z = a ∂ ∂ A (x,y,a,t) = 0; A (x,y,a,t) = 0; A (x,y,a,t) = 0. (37) ∂z x ∂z y z

The appropriate vector potential operator Aˆ (r,t) is given by [13]

³ ´ 1 ∞ Z 2 ½ ·µ ¶ ¸ 1 π 2 d κ 1 πz ˆ ( , ) = √ (1)(κ, )κˆ × ˆ + A r t π ∑ ω aˆ n zsin n a n=0 2 a " ·µ ¶ ¸ ·µ ¶ ¸#) i(n + 1 ) 1 πz κ 1 πz + aˆ(2)(κ,n) κˆ 2 sin n + − zˆ cos n + ωa 2 a ω 2 a ×ei(κ·ρ−ωt) + h.c., (38) where as before κ = (kx,ky) and ρ is the position vector on the XY plane. The normal frequencies are given s µ ¶ 1 2 π2 ω(κ,n) = κ2 + n + , (39) 2 a2 with kx,ky ∈ R and n ∈ N−1. Notice that contrary to the case of two conducting plates normalization does not require the we multiply the term corresponding to n = 0 by 1/2. The electric and magnetic ﬁeld correlators for Boyer’s setup can be evaluated with the same technique employed before [13]. In fact, it is not hard to convince ourselves that it is sufﬁcient to perform the substitution n → n + 1/2 and follow the same steps as before to obtain

hEi(r,t)E j(r,t)i0 = T1 + T2 + T3, (40) where, for example

¡ ¢ k ( µ ¶ ) Γ s − 3 ³π´3−2s δ ∞ 1 3−2s 1 h ³ πz´i T = ¡ 2 ¢ i j ∑ n + 1 − cos 2(n + 1/2) , (41) 1 Γ 1 a 2a 2 2 a s − 2 n=0

The terms T2 e T1 show a similar structure. The main difference with respect to the two conducting plate case is that now we have to deal with the Hurwitz zeta function ζH (z,q) which has a series representation given by [19] ∞ ζ 1 H (z,q) = ∑ z , (42) n=0 (n + q) with ℜz > 1, and q 6= 0,−1,−2,..... In our case we must set µ ¶ 1 ∞ 1 3−2s ζH (2s − 3, ) = ∑ n + . (43) 2 n=0 2 It follows that in the limit s → 0 we have

k ( µ ¶ µ ¶ µ µ ¶ ¶) ³π´4 δ ∞ 3 π 1 i j ζ 1 1 1 z T1 = − π H −3, + ∑ n + cos 2 n + . (44) a 3 2 2 n=0 2 2 a

In the same way we obtain for T2 e T1 the results

k ( µ ¶ µ ¶ µ µ ¶ ¶) ³π´4 δ ∞ 3 π 1 i j ζ 1 1 1 z T2 = − π H −3, − ∑ n + cos 2 n + . (45) a 2 2 n=0 2 2 a and ( µ ¶ µ ¶ µ µ ¶ ¶) ³π´4 δ⊥ ∞ 3 π 2 i j ζ 1 1 1 z T3 = π H −3, + ∑ n + cos 2 n + . (46) a 3 2 2 n=0 2 2 a Brazilian Journal of Physics, vol. 35, no. 3A, September, 2005 663

Notice that this time we do not have the divergent contribution corresponding to n = 0 as in the case of the conducting plates. Adding the three terms we have ½ µ ¶ ® ³π´4 2 1 Eˆ (r,t)Eˆ (r,t) = (−δk + δ⊥)ζ −3, i j 0 a 3π i j i j H 2 µ ¶ µ µ ¶ ¶) ∞ 1 3 1 πz + ∑ n + cos 2 n + . (47) n=0 2 2 a

¡ ¢ ζ 1 ξ π The numerical value H −3, 2 can be obtained from [19] where as before := z /a. We can write B (q) ζ (−n,q) = − n+1 , (48) H n + 1 d3 (2n + 1)3 cos[(2n + 1)ξ] = − sin[(2n + 1)ξ] (52) dξ3 where n ∈ N and Bn+1(q) is a Bernoulli polynomial deﬁned by [19]

n and formally we have n! n−k Bn(x) = ∑ Bpx , (49) p=0 p!(n − p)! 1 d3 ∞ G(ξ) = − ∑ sin[(2n + 1)ξ]. (53) where Bp is a Bernoulli number. The relevant polynomial here ξ3 8 d n=0 is:

4 3 2 1 B4(x) = x − 2x + x − . (50) Writing sin[(2n + 1)ξ] in terms of exponentials of imaginary 30 argument and passing to the euclidean space we obtain after ¡ ¢ ζ 1 some simple manipulations With B4(1/2) = (7/8) × (1/30), it follows that H −3, 2 = −(7/8)(1/120). The sum can be regularized with the same technique employed before. In fact, we can deﬁne the function ξ 1 d3 1 G( ) by G(ξ) = − . (54) 8 dξ3 2sin(ξ) µ ¶ · µ ¶ ¸ ∞ 1 3 1 G(ξ) : = ∑ n + cos 2 n + ξ n=0 2 2 ∞ Collecting all® partial results we ﬁnally obtain for 1 3 ξ ˆ ˆ = ∑ (2n + 1) cos[(2n + 1) ], (51) Ei (r,t)E j (r,t) 0 the result 8 n=0

¡ ¢ "µ ¶ k ⊥ # ® ³π´4 2 7 −δ + δ Eˆ (r,t)Eˆ (r,t) = − i j + δ G(ξ) . (55) i j 0 a 3π 8 120 i j ® ˆ ˆ Proceeding in the same way in the evaluation of Bi (r,t)B j (r,t) 0 we obtain ¡ ¢ "µ ¶ k ⊥ # ® ³π´4 2 7 −δ + δ Bˆ (r,t)Bˆ (r,t) = − i j − δ G(ξ) . (56) i j 0 a 3π 8 120 i j

Observe that near ξ = 0 the function G(ξ) behaves as But near ξ = π its behavior is slightly different

3 7 1 h i G(ξ) = − (ξ − π)−4 + + O (ξ − π)2 . (58) 8 8 120 ® 3 7 1 ¡ ¢ Again, a direct calculation shows that Eˆi (r,t)Bˆ j (r,t) = 0 G(ξ) = ξ−4 − + O ξ2 , (57) 0 8 8 120 for this case. 664 F. C. Santos et al.

ε ε,µ ε,µ As before the divergent behavior of the correlators near the ^ ^ ^ ^n'=-z n=z plates we are interested in is an effect of the distortions of the electromagnetic oscillations with respect to a situation where the plates are not present. This fact has received the attention Z Z' of several authors, see for example [9, 16]. z=0 z=a z=l

V. THE CASIMIR EFFECT FOR CONDUCTING PLATES

FIG. 1: Three-plate setup for the obtention of the Casimir force per In order to apply the above results to Casimir’s original unit area. The plate at z = ` is auxiliary. setup we consider for convenience three parallel perfectly conducting plates perpendicular to the OZ axis at z = 0, z = a and z = `. The quantum version of Maxwell tensor is

® ® 1 ® ® 1 ® 4π Tˆ = Eˆ Eˆ − δ Eˆ 2 + Bˆ Bˆ − δ Bˆ 2 (59) i j 0 i j 0 2 i j 0 i j 0 2 i j 0 Making use of the correlator given by (32) we obtain the following partial results · ¸ ® ® ³π´4 2 1 Eˆ 2(z,t) = Eˆ 2(z,t = − + F (ξ) , (60) x 0 y 0 a 3π 120

· ¸ ® ³π´4 2 1 Eˆ 2(z,t = + F (ξ) , (61) z 0 a 3π 120 ® ® ® ˆ ˆ ˆ ˆ ˆ ˆ also Ex(z,t)Ey(z,t) 0 = Ex(z,t)Ez(z,t) 0 = Ey(z,t)Ez(z,t) 0 = 0. In the same way, making use of (35) we obtain · ¸ ® ® ³π´4 2 1 Bˆ2(z,t) = Bˆ2(z,t = − − F (ξ) , (62) x 0 y 0 a 3π 120

· ¸ ® ³π´4 2 1 Bˆ2(z,t = − F (ξ) , (63) z 0 a 3π 120 ® ® ® ˆ ˆ ˆ ˆ ˆ ˆ and also Bx(z,t)By(z,t) 0 = Bx(z,t)Bz(z,t) 0 = By(z,t)Bz(z,t) 0 = 0. The components of the quantum version of Maxwell tensor can be easily evaluated. For instance ® ® ® ® π ˆ ˆ 2 ˆ 2 ˆ 2 8 Tzz(z,t) = Ez (z,t) − Ex (z,t) − Ey (z,t) 0 0® 0® 0® + Bˆ2(z,t) − Bˆ2(z,t) − Bˆ2(z,t) . (64) z 0 x 0 y 0 performing the necessary substitutions we have ® π2 Tˆ = . (65) zz 0 240a4 In the same way ® ® 1 ¡ ® ® ¢ Tˆ = Tˆ = − Eˆ 2(z,t) + Bˆ2(z,t) xx 0 yy 0 8π z 0 z 0 π2 = − (66) 720a4

Notice how conveniently the divergent parts near the plates cancel out yielding ﬁnite results. Brazilian Journal of Physics, vol. 35, no. 3A, September, 2005 665

Let us obtain now the Casimir force per unit area between and Ez = 0. Making use of the correlator given by (55) the fol- the conducting plates. Consider Figure (V) and the plate at lowing partial results: z = a. The Casimir force per unit area on this plate is · ¸ ® ® ³π´4 2 7 1 Eˆ 2(z,t) = Eˆ 2(z,t = × + G(ξ) , Fz L R x 0 y 0 a 3π 8 120 = −Tzz(z → a ) + Tzz(z → a ) A (72) π2 π2 ·µ ¶ ¸ = − + , (67) ® ³π´4 2 7 1 240a4 240(` − a)4 Eˆ 2(z,t = − × + G(ξ) , (73) z 0 a 3π 8 120 where z → aL,R means that z tends to a from the left/right. ® ® and Eˆ (z,t)Eˆ (z,t) = Eˆ (z,t)Eˆ (z,t) = Taking the limit ` → ∞ we obtain the expected result for the x ® y 0 x z 0 ˆ ˆ Casimir force per unit area Ey(z,t)Ez(z,t) 0 = 0. By the same token making use of the correlator given by (56) we obtain F π2 z ³ ´ · ¸ = − 4 . (68) ® ® π 4 2 7 1 A 240a Bˆ2(z,t) = Bˆ2(z,t = × − G(ξ) , x 0 y 0 a 3π 8 120 The minus sign shows that the resulting pressure pushes to- (74) wards the region between the plates. If simultaneously we ·µ ¶ ¸ take the limits `,a → ∞ keeping the distance ` − a constant. ® ³π´4 2 7 1 Bˆ2(z,t = − × − G(ξ) , (75) The pressure changes its sign but it still pushes the plate at z 0 a 3π 8 120 z = a towards the one at z = `. In order to calculate the renormalized symmetrical stress- and also Θˆ µν ren energy tensor h (z)i0 we evaluate the energy density ® ® ® ρ Θˆ 00 ren Bˆ (z,t)Bˆ (z,t) = Bˆ (z,t)Bˆ (z,t) = Bˆ (z,t)Bˆ (z,t) = 0 (z) ≡ h (z)i0 in the region between the plates as x y 0 x z 0 y z 0 Θˆ xx ren Θˆ yy ren (76) well as the saptial components h (z)i0 , h (z)i0 , and hΘˆ zz (z)iren. The energy density is given by Proceeding as in the case of the conducting plates we obtain 0 the following results for the componenets of the quantum ver- 1 ¡ ® ® ¢ sion of Maxwell tensor ρ(r,t) = Eˆ 2 (r,t) + Bˆ 2 (r,t) (69) π 0 0 8 ® ® 7 π2 Tˆxx = Tˆyy = × , (77) making use of the correlators given by (32) and (35) we obtain 0 0 8 720a4

π2 and ρ(a) = − . (70) µ ¶ 720a4 ® 7 π2 Tˆzz = − × . (78) This result is due to the fact that the divergent pieces in (32) 0 8 240a4 and (35) cancel out yielding a finite result for the vacuum en- i j Notice that it is sufficient to multiply the results obtained for ergy density. Recalling that Θ (z) = −Ti j(z), see [11], with help of (2), (32) and (35) the remanescent components of the Casimir’s setup by the factor (−7/8) in order to obtain the symmetrical stress-energy tensor are easily obtained. The fi- results corresponding to Boyer’s setup. nal result is In order to obtain the Casimir force per unit area for this setup it is convenient to place a third conducting plate at z = `. π2 Then the Casimir force per unit area on the plate at z = a will hΘˆ µν(z)iren = diag (−1,1,1,−3), (71) 0 720a4 be given by which is in perfect agreement with Brown and Maclay’s re- Fz L R µ ren µν ren = −Tzz(z → a ) + Tzz(z → a ) sults [6]. Notice also that hΘˆ µ(z)i = gµνhΘˆ (z)i = 0, A 0 0 µ ¶ µ ¶ with gµν = diag(1,−1,−1,−1). 7 π2 7 π2 = − − × + − × .(79) 8 240a4 8 240(` − a)4

VI. THE CASIMIR EFFECT FOR ONE CONDUCTING Taking the limit ` → ∞ we obtain a Casimir force which PLATE AND AN INFINITELY PERMEABLE ONE pushes the plate at z = a towards the region z > a given by

Let us consider now the setup proposed by Boyer [10] F 7 π2 z = × (80) which consists of a perfectly conducting plate placed perpen- A 8 240a4 dicularly to the OZ axis at z = 0 and another infinitely permeable one parallel to the first placed at z = a. The boundary This is the result obtained by Boyer [10] for this setup using conditions on the conducting plate are as before Ex = Ey = 0 stochastic electrodynamic methods and it is one of the sim- and Bz = 0, and for the infinitely permeable plate: Bx = By = 0 plest example of a repulsive Casimir force. 666 F. C. Santos et al.

In order to evaluate the symmetrical stress-energy tensor we the Casimir energy and pressure and the symmetrical trace- first evaluate the Casimir energy density. Making use of (55) less stress energy tensor. We have shown that for the cases e (56) we have we had in mind here finite results are obtained only when we consider what happens on both sides of the surface boundary. 7 π2 This consideration provided the mechanism by which precise ρ = × . (81) 8 240a4 cancellations occurred and finite results were obtained. These cancellations occur only for the simple geometry considered As in the case of the conducting plates a simple calculation in this paper. This is in agreement with, for example, Ref. [9] shows that the stress energy tensor for Boyer’s setup is given and should be considered as a concrete example of the behav- by ior of quantized fields near and on boundary surfaces. This behavior depends on the geometry of the boundaries and can 7 π2 Θˆ µν ren become quite complicated. It can depend on the local curva- h (z)i0 = × 4 diag (1,−1,−1,3). (82) 8 720a ture of the boundary, for instance. Finally, it must be men- µ ν tioned that the equal time and space electromagnetic field cor- As before hΘˆ (z)iren = g νhΘˆ µ (z)iren = 0. µ 0 µ 0 relators calculated here were can be also employed to rederive VII. CONCLUSIONS the atractive and the repulsive Casimir effect by means of a quantum version of the Lorentz force [20]. They can be also In this paper we have shown how to employ the equal applied to the Casimir-Polder interaction between an atom and time and space electromagnetic field correlators evaluated be- material surfaces [21] tween parallel material surfaces to rederive results concernig

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