I. INTROUCTION The dependence of physical quantities on the number of dimensions is of considerable interest [,]. In particular, by expanding in the n

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1 OKHEP{96{7 Vector Casimir eect for a -dimensional sphere Kimball A. Milton epartment of Physics and Astronomy, University of Oklahoma, Norman, OK 739 (November 3, 996) Abstract The Casimir energy or stress due to modes in a -dimensional volume subject to TM (mixed) boundary conditions on a bounding spherical surface is calculated. Both interior and exterior modes are included. Together with earlier results found for scalar modes (TE modes), this gives the Casimir eect for uctuating \electromagnetic" (vector) elds inside and outside a spherical shell. Known results for three dimensions, rst found by Boyer, are reproduced. Qualitatively, the results for TM modes are similar to those for scalar modes: Poles occur in the stress at positive even dimensions, and cusps (logarithmic singularities) occur for integer dimensions. Particular attention is given the interesting case of =...Kk,.39.Ba,..Jj,.5.Kc Typeset using REVTEX milton@phyast.nhn.uoknor.edu

2 I. INTROUCTION The dependence of physical quantities on the number of dimensions is of considerable interest [,]. In particular, by expanding in the number of dimensions one can obtain nonperturbative information about the coupling constant [3{6]. Useful expansions have also been obtained in inverse powers of the dimension [7]. In a previous paper we investigated the dimensional dependence of the Casimir stress on a spherical shell of radius a in space dimensions [8]. Specically, we studied the Casimir stress (the stress on the sphere is equal to the Casimir force per unit area multiplied by the area of the sphere) that is due to quantum uctuations of a free massless scalar eld satisfying irichlet boundary conditions on the shell. That is, the Green's functions satisfy the boundary conditions G(x;t;x ;t ) =: (.) jxj=a Here, following the suggestion of [9], we calculate the TM modes (for which H r =)in the same situation. The TM modes are modes which satisfy mixed boundary conditions on the r G(x;t;x ;t ) =; (.) jxj=r=a as opposed to the TE modes (for which E r = ), which satisfy irichlet boundary conditions (.) on the surface, and are equivalent to the scalar modes found in [8]. The organization of this paper is straightforward. In Sec. II we construct the Green's functions in -dimensional space by direct solution of the dierential equation, subject to the boundary condition (.). Then, the Casimir stress is computed from the vacuum expectation value of the energy-momentum-stress tensor, expressed as derivatives of the Green's functions. The resulting expression for the Casimir stress on a -dimensional spherical shell takes the form of an innite sum of integrals over modied Bessel functions; the dimension appears explicitly as well as in the orders of the Bessel functions. By combining the results

3 for the TM modes found here with those for the TE modes found earlier [8], we obtain a general expression for vector modes subject to perfect conductor boundary conditions on the spherical shell, which expression agrees with that found long ago for three dimensions []. Asacheck, in Sec. III we rederive the same result from the vacuum expectation value of the energy density of the eld. In Sec. IV we examine this expression for the TM Casimir stress in detail. We show that for < a constant can be added to the series without eect; a suitable constant is chosen so that each term in the series exists (each of the integrals converges). We show how to evaluate the sum of the series numerically for all real, using two methods: one involving Riemann zeta functions, and the second involving continuation in dimension. Both methods give the same numerical results. When > the TM Casimir stress is real, and nite except when is an even integer. The well-known = 3 result [,3,4,] is reproduced when the n = mode is removed. When the Casimir stress is complex; there are logarithmic singularities in the complex- plane at = ; ; ; ; ; :::. In the Appendix, the important case of = is discussed [5{7]. II. STRESS TENSOR FORMALISM The calculation given in this paper for the Casimir stress on a spherical shell follows very closely the Green's function technique described in [8], and we will therefore be concise, and emphasize the signicant dierences. A. Green's function The two-point Green's function G(x;t;x ;t ) satises the inhomogeneous r G(x;t;x ;t )= () (x x )(t t ); (.) 3

4 subject to the boundary condition (.) on the surface jxj = a. We solve this equation by the standard discontinuity method. In particular, we divide space into two regions, region I, the interior of a sphere of radius a and region II, the exterior of the sphere. In addition, in region I we will require that G be nite at the origin x =and in region II we will require that G satisfy outgoing-wave boundary conditions at jxj =. We begin by taking the time Fourier transform of G: G! (x; x )= Z The dierential equation satised by G! is dt e i!(t t) G(x;t;x ;t ): (.)! + r G! (x; x )= () (x x ): (.3) To solve this equation we introduce polar coordinates and seek a solution that has cylindrical symmetry; i.e., we seek a solution that is a function only of the three variables r = jxj, r = jx j, and, the angle between x and x so that x x = rr cos. In terms of these polar variables (.3) + @ G! (r;r ;)= (r r )() ( )= r sin : (.4) We solve (.4) using the method of separation of variables. The angular dependence is given in terms of the ultraspherical (Gegenbauer) polynomial [8] C ( +=) n (z) (n =;;;3;:::); (.5) where z = cos. The general solution to (.4) is an arbitrary linear combination of separatedvariable solutions; in region I the Green's function has the form (with = n += and k = j!j) G! (r;r ;)= a n r = J (kr)c ( +=) n (z) (r <r <a) (.6a) and 4

5 G! (r;r ;)= r = [b n J (kr)+c n J (kr)] C ( +=) n (z) (r <r<a): (.6b) [Note that J (x) and J (x) are linearly independent so long as is not an integer. Thus, in writing (.6b), we assume explicitly that is not an even integer. We also assumed > in writing down (.6a), so that J is excluded because it is singular at r =.] The general solution to (.4) in region II has the form and G! (r;r ;)= G! (r;r ;)= d n r = H () (kr)c ( +=) n (z) (r >r >a) (.7a) r = e n H () (kr)+f n H () (kr) C ( +=) n (z) (r >r>a): (.7b) The arbitrary coecients a n, b n, c n, d n, e n, and f n are uniquely determined by six conditions; namely, the mixed boundary condition (.) at r = a, [b n J (ka)+c n J (ka)] + ka[b n J(ka)+c n J (ka)]= (.8a) and [e n H () (ka)+f n H () (ka)] + ka[e n H () (ka)+f n H () (ka)] = ; (.8b) the condition of continuity at r = r, a n J (kr )=b n J (kr )+c n J (kr ) (.8c) and d n H () (kr )=e n H () (kr )+f n H () (kr ); (.8d) and the jump condition in the rst derivative of the Green's function at r = r, b n J(kr )+c n J (kr ) a n J(kr )= 4(r ) k (.8e) 5

6 and e n H () (kr )+f n H () (kr ) d n H () (kr )= 4(r ) : k (.8f) Solving these equations for the coecients, we easily nd the Green's function to be, in region I, G! (r;r ;)= where ) 8(rr ) = C(= n (cos )[J (kr < )J (kr > ) J (kr)j (kr )] ; sin (.9a) and, in Region II, G! (r;r ;)= = i J (ka)+kaj (ka) J (ka)+kaj(ka) ; (.9b) 6(rr ) n (cos ) = C(= ) H () (kr)h () (kr ) i h H () (kr < )H () (kr > ) ; (.a) where = H () (ka)+kah () (ka) H () (ka)+kah () (ka) : (.b) B. Stress Tensor For a scalar eld, we can calculate the induced force per unit area on the sphere from the stress-energy tensor T (x;t), dened by T '(x;t)@ '(x;t) '(x;t)@ '(x;t): (.) The radial scalar Casimir force per unit area f on the sphere is obtained from the radialradial component of the vacuum expectation value of the stress-energy tensor []: 6

7 f = hjt rr in T rr outji j r=a : (.) To calculate f we exploit the connection between the vacuum expectation value of the elds and the Green's function, hjt(x;t)(x ;t )ji=ig(x;t;x ;t ); (.3) so that the force density is given by the derivative of the Green's function at equal times, G(x;t;x ;t): f G(x;t;x G(x;t;x ;t) out x=x ; jxj=a : (.4) It is a bit more subtle to calculate the force density for the TM modes. For a given frequency, we write ht rr i = i rr r r +! r? r? G! ; (.5) where, if we average over all directions, we can integrate by parts on the transverse derivatives, r? r?!r?! n(n+ ) r ; (.6) where the last replacement, involving the eigenvalue of the Gegenbauer polynomial, is appropriate for a given mode n [see (.4) of [8]]. As for the radial derivatives, they are r r = r r ; r r = r r ; (.7) which, by virtue of (.), implies that the r r r r term does not contribute to the stress on the sphere. In this way, we easily nd the following formula for the contribution to the force per unit area for interior modes, In the TM mode, the radial derivatives correspond to tangential components of E, which must vanish on the surface. See []. 7

8 f TM in = and for exterior modes, where f TM out = Z i dx (+)= a + ( ) x Z i dx (+)= a + ( ) x w(n; )(x n(n + )) s n(x) s n(x) ; (.8) w(n; )(x n(n + )) e n(x) e n(x) ; (.9) w(n; ) = (n + ) (n + ) ; (.) n! x = ka, and the generalized Ricatti-Bessel functions are s n (x) =x = J (x); e n (x) =x = H () (x): (.) It is a small check to observe that for =werecover the known result [5]! f TM m (x) = = Z i 8 a 3 dx x m= m x J m (x) Jm(x) + H () H () m (x) ; (.) where the half-weight atn= is a result of the limit!. In two dimensions, the vector Casimir eect consists of only the TM mode contribution. In general, we can combine the TE mode contribution, given in [8], and the TM mode contribution, found here, into the following simple formula : f TM+TE i w(n; ) = (+)= a + ( ) Z s n (x) dx x s n (x) + e n(x) e n (x) + s n(x) s n(x) + e n(x) e n(x) : (.3) It will be noted that, for =3,this result agrees with that found for the usual electrodynamic Casimir force/area, when the n = mode is properly excluded. [See (4.7) of [] with the cuto =.] Of course, this only coincides with electrodynamics in three dimensions. The number of electrodynamic modes changes discontinuously with dimension, there being We will not concern ourselves with a constant term in the integrand, which we will deal with in Sec. IV. 8

9 only one in =, the TM mode, and none in =,in general there being modes. Equation (.3) is of interest in a mathematical sense, because signicant cancellations do occur between TE and TM modes in general. The integrals in (.8) and (.9) are oscillatory and therefore very dicult to evaluate numerically. Thus, it is advantageous to perform a rotation of 9 degrees in the complex-! plane. The resulting expression for f TM is f TM = + w(n; ) a + Z dx x d dx ln h x (3 ) x = K (x) x = I (x) i : (.4) III. ENERGY ERIVATION As a check of internal consistency, it would be reassuring to derive the same result by integrating the energy density due to the eld uctuations. The latter is computable from the vacuum expectation value of the stress tensor, which in turn is directly related to the Green's function, G! : ht i = i Z d! (! + r r )G! r=r : (3.) Again, because we are going to integrate this over all space, we can integrate by parts, replacing, in eect, r r! r!! ; (3.) which uses the Green's function equation (.3). [Point splitting is always implicitly assumed, so that delta functions may be omitted.] Then, using the area of a unit sphere in dimensions, we nd the Casimir energy to be given by A = = (=) ; (3.3) 9

10 Z Z E = i= d! (=)! r dr G! (r;r): (3.4) So, from the form for the Green's function given in (.9a) and (.a), we see that we need to evaluate integrals such as Z a which are given in terms of the indenite integral Z dx xz (x)z (x) = x rdrj (kr)j (kr); (3.5) x Z (x)z (x)+z(x)z (x) ; (3.6) valid for any two Bessel functions Z, Z of order. Thus we nd for the Casimir energy of the TM modes the formula E TM = i w(n; ) ( )a Z dx (x n(n + )) x sn (x) s n(x) + e n(x) e +( )x : (3.7) n(x) We obtain the stress on the spherical shell by dierentiating this expression with respect to a (which agrees with (.8) and (.9), apart from the constant in the integrand), followed again doing the complex frequency rotation, which yields where the integrals are F TM = a ( ) w(n; )Q n ; (3.8) Q n = Z dx x d ln q(x); (3.9) dx where q(x) = I (x)+ x (I +(x)+i K (x) (x)) x (K +(x)+k (x)) : (3.) This agrees with the form found directly from the force density, (.4), again, apart from a constant in the x integrand.

11 IV. NUMERICAL EVALUATION OF THE STRESS We now need to evaluate the formal expression (3.8) for arbitrary dimension. We implicitly assumed in its derivation that > and that was not an even integer, but we will argue that (3.8) can be continued to all. A. Convergent reformulation of (3.8) First of all, it is apparent that as it sits, the integral Q n in (3.9) does not exist. (The form in (.4) does exist for the special case of = 3.) As in the scalar case [8], we argue that since w(n; ) = for <; (4.) we can add an arbitrary term, independent of n, to Q n in (3.8) without eect as long as <. In eect then, we can multiply the quantity in the logarithm in (3.9) by an arbitrary power of x without changing the value for the force for <. We choose that multiplicative factor to be =x because then a simple asymptotic analysis shows that the integrals converge. Then, we analytically continue the resulting expression to all. The constant is, of course, without eect in (3.9), but allows us to integrate by parts, ignoring the boundary terms. The result of this process is that the expression for the Casimir force is still given by (3.8), but with Q n replaced by Z Q n = dx ln q(x) being given by (3.). x q(x) ; (4.) Now the individual integrals in (3.8) converge, but the sum still does not. We can see this by making the uniform asymptotic approximations for the Bessel functions in (3.) [9], which leads to Q n ::: (n!): (4.3)

12 (Note that the coecients in this expansion depend on the dimension, unlike the scalar case, given in (3.7) of [8].) Because of this behavior, it is apparent that the series diverges for all positive, except for =, where the series truncates. There were actually two procedures which were used to turn the corresponding sum in the scalar case into a convergent series, and to extract numerical results, although only one of those procedures was described in the paper [8]. In that procedure, we subtract from the summand the leading terms in the =n expansion, derived from (4.3), identifying those summed subtractions with Riemann zeta functions: F TM a + Q + ( ) ( ) " " NX n= ( )+ w(n; )Q n n + K+ X k= KX k= b k (k + ) b K+ N X n=! b k n k n K : (4.4) Here b k are the coecients in the asymptotic expansion of the summand in (3.8), of which the rst two are ( )( ) b = ; (4.5a) b = : 9 (4.5b) In (4.4) we keep K terms in the asymptotic expansion, and, after N terms in the sum, we approximate the subtracted integrand by the next term in the large n expansion. The series converges for <K+, so more and more terms in the asymptotic expansion are required as increases. There is a second method which gives identical results, and is, in fact, more convergent. The results given in [8] were, in fact, rst computed by this procedure, which is based on analytic continuation in dimension. Here, we simply subtract from Q n the rst two terms in the asymptotic expansion (4.3), and then argue, as a generalization of (4.), that (n + ) n! = for <; (n + ) = for <: (4.6) n! Therefore, by continuing from negative dimension, we argue that we can make the subtraction without introducing any additional terms. Thus, if we dene

13 ^Q n = Q n ; (4.7) we have F TM = a ( ) a ( ) w(n; ) ^Qn NX w(n; ) ^Qn + g() n=n+! (n + ) n! ; (4.8) where g() is the coecient of 4 in (4.3). The last sum in (4.8) can be evaluated according to (n + ) n!(n+=) = (=) ( =) sin = : (4.9) The approximation given in (4.8) converges for <4. B. Casimir stress for integer The case of integers is of special note, because, for those cases, the series truncates. For example, for = only the n =, terms appear, where the integrals cancel by virtue of the symmetry of Bessel functions, K (x) =K (x); I n (x)=i n (x); (4.) for n an integer. However, using the rst procedure (4.4), we have a residual zeta function contribution: F TM = = a (Q Q + ) = a; (4.) because both ( ) and ( ) have simple poles, with residue, at =. This result for = is the negative of the result found in the scalar case, (3.) of [8], which is a direct consequence of the fact that the n term in the asymptotic expansion cancels when the TE and TM modes are combined [compare (4.5a) with the corresponding term in (3.3) of [8].] The continuation in method gives the same result, because then 3

14 F TM = = a ( ^Q + ^Q ^Q )= a 64 + = 64 a ; (4.) where a limiting procedure,!, is employed to deal with the singularity which occurs for n =,where!. For the negative even integers we achieve a similar cancellation between pairs of integers, with no zeta function residual because the functions no longer have poles there. For example, for = we have F TM = = a (Q Q +Q 3 Q 4 )= (4.3) because Q = Q 4 and Q = Q 3. Again, the other method of regularization gives the same result when a careful limit is taken. For odd integer, trucation occurs without cancellation, because I 6= I. For example, for =, F TM = = (Q + Q )= :6 + :63i: (4.4) C. Numerical results We have used both methods described above to extract numerical results for the stress on a sphere due to TM uctuations in the interior and exterior. Results are plotted in Fig.. Salient features are the following: As in the scalar case, poles occur for positive even dimension. The integrals become complex for < because the function q(x), (3.), occurring in the logarithm develops zeros. (This phenomenon started at = for the scalar case.) Correspondingly, there are logarithmic singularities, and cusps, occurring at,,,,,..., rather than just at the nonpositive even integers. The sign of the Casimir force changes dramatically with dimension. Here this is even more striking than in the scalar case, where the sign was constant between the poles 4

15 for >. For the TM modes, the Casimir force vanishes for =:6, being repulsive for <<:6 and attractive for :6 <<4. Also in Fig., the results found here are compared with those found in the scalar or TE case, [8]. The correspondence is quite remarkable. In particular, for < the qualitative structure of the curves are very similar when the scale of the dimensions in the TE case is reduced by a factor of ; that is, the interval << in the TE case corresponds to the interval <<inthetm, << for TE corresponds to << for TM, etc. Physically, the most interesting result is at =3. The TM mode calculated here has the value F TM =3 = :4. However, if we wish to compare this to the electrodynamic result [], we must subtract o the n = mode, which is given in terms of the integral Q = :433 = =4, which displays the accuracy of our numerical integration. Similarly removing the n = mode (= =4) from the result quoted in (3.4) of [8], F TM =3 = :868, gives agreement with the familiar result [,3,4,,9]: F TM+TE n> =3 = :46 a ; (4.5) To conclude, this paper adds one more example to our collection of known results concerning the dimensional and boundary dependence of the Casimir eect. Unfortunately, we are no closer to understanding intuitively the sign of the phenomenon. ACKNOWLEGEMENTS I thank the US epartment of Energy for nancial support. I am grateful to A. Romeo, Y. J. Ng, C. M. Bender, and M. Bordag for useful conversations and correspondence. 5

16 APPENIX: TOWAR A FINITE =CASIMIR EFFECT The truly disturbing aspect of our results here and in [8] are the pole in even dimensions. In particular many very interesting condensed matter systems are well-approximated by being two dimensional. Are we to conclude that the Casimir eect does not exist in two dimensions? One trivial way to extract a nite answer from our expressions, which have simple poles at = (I will set aside the logarithmic singularity there in the TM mode, because that only occurs in one integral, Q ), is to average over the singularity. If we do so for the scalar result in [8], we obtain while for the TM result here, we nd F TE = = :34 a ; (A) F TM = = :34 a ; (A) which numbers, incidentally, are remarkably close to the leading Q term, as stated 3 in [5], which are :4 and :54. But, there seems to be no reason to have any belief in these numbers. However, something remarkable does happen in the scalar case. If we use the rst procedure, (4.4), we note that the poles can arise both from the integrals and from the explicit zeta functions. For the latter, let the dependence on be given by r()=( ) which has a pole at =. When we average over the pole, we obtain r( + ) r( ) lim = r (); (A3)! where the prime denotes dierentiation. For the scalar modes it is easy to verify that r () = : Thus, there is no contribution from those subtracted terms. In other words, they 3 The integral in (A) of [5] was not evaluated very accurately there. The value, good to 6 gures, should be in our notation, Q scalar =:8837. Similarly Q TM =:599. 6

17 might just as well be omitted, which is what we would do if we inserted a cuto and simply dropped the divergent terms. (This procedure does give the correct = 3 results.) This provides some evidence for the validity of the procedure which yields (A). Unfortunately, the same eect does not occur for the TM modes, r () 6=. Nor does it occur for higher dimensions, =4,6,..., even for scalars. And, even for scalars, it is not clear how the divergences of a massive (+) theory can be removed. So we are no closer to solving the divergence problem in even dimensions. 4 It is clear there is much more work to do on Casimir phenomena. 4 For a discussion of the inadequacies of the dubious procedure of attempting to extract a nite result in [5] see [9]. 7

18 REFERENCES [] C. M. Bender, S. Boettcher, and L. Lipatov, Phys. Rev. Lett. 68, 3674 (99). [] C. M. Bender, S. Boettcher, and L. Lipatov, Phys. Rev. 46, 5557 (99). [3] C. M. Bender and S. Boettcher, Phys. Rev. 48, 499 (993). [4] C. M. Bender and S. Boettcher, Phys. Rev. 5, 875 (995). [5] C. M. Bender, S. Boettcher, and L. R. Mead, J. Math. Phys. 35, 368 (994). [6] C. M. Bender, S. Boettcher, and M. Moshe, J. Math. Phys, 35, 494 (994). [7] C. M. Bender, L.. Mlodinow, and N. Papanicolaou, Phys. Rev. A 5, 35 (98); M. unn, T. C. Germann,. Z. Goodson, C. A. Traynor, J.. Morgan III,. K. Watson, and. R. Herschbach, J. Chem. Phys., 5987 (994). [8] C. M. Bender and K. A. Milton, Phys. Rev. 5, 6547 (994). [9] S. Leseduarte and A. Romeo, Ann. Phys. (N.Y.) 5, 448 (996). [] T. H. Boyer, Phys. Rev. 74, 764 (968). [] J. C. Slater and N. H. Frank, Electromagnetism (McGraw-Hill, New York, 947), p. 53. [] K. A. Milton, L. L. eraad, Jr., and J. Schwinger, Ann. Phys. (N.Y.) 5, 388 (978). [3] B. avis, J. Math. Phys. 3, 34 (97). [4] R. Balian and R. uplantier, Ann. Phys. (N.Y.), 65 (978). [5] K. A. Milton and Y. J. Ng, Phys. Rev. 46, 84 (99). [6] S. Sen, J. Math. Phys., 968 (98). [7] L. L. eraad, Jr. and K. A. Milton, Ann. Phys. (N.Y.) 36, 9 (98). [8] Handbook of Mathematical Functions, ed. by M. Abramowitz and I. A. Stegun (National Bureau of Standards, Washington,.C., 964), Chap.. 8

19 [9] See Chap. 9 of [8]. 9

20 FIGURES. F TM = solid. F TE = dashed Fa FIG.. A plot of the TM Casimir stress F TM for <<4 on a spherical shell, compared with F TE, taken from [8]. For < ( < ) the stress F TM (F TE ) is complex and we have plotted Re F.