aa r X i v : . [ h e p - t h ] J u l Quantum Spring from the Casimir Effect
Chao-Jun Feng ∗ and Xin-Zhou Li † Shanghai United Center for Astrophysics (SUCA),Shanghai Normal University, 100 Guilin Road, Shanghai 200234, P.R.China
The Casimir effect arises not only in the presence of material boundaries but also in space withnontrivial topology. In this paper, we choose a topology of the flat ( D + 1)-dimensional spacetime,which causes the helix boundary condition for a Hermitian massless scalar field. Especially, Casimireffect for a massless scalar field on the helix boundary condition is investigated in two and threedimensions by using the zeta function techniques. The Casimir force parallel to the axis of the helixbehaves very much like the force on a spring that obeys the Hooke’s law when the ratio r of the pitchto the circumference of the helix is small, but in this case, the force comes from a quantum effect,so we would like to call it quantum spring . When r is large, this force behaves like the Newton’slaw of universal gravitation in the leading order. On the other hand, the force perpendicular to theaxis decreases monotonously with the increasing of the ratio r . Both forces are attractive and theirbehaviors are the same in two and three dimensions. PACS numbers: 03.70.+k, 11.10.-z
I. INTRODUCTION
Since the first work on Casimir effect performed byCasimir [1], it has been extensively studied [2] for morethan 60 years. Essentially, the casimir effect is a polariza-tion of the vacuum of some quantized fields, and it may bethought of as the energy due to the distortion of the vac-uum. Such a distortion may be caused either by the pres-ence of boundaries in the space-time manifold or by somebackground field like the gravity. Early works on thegravity effect were performed by Utiyama and DeWitt,see ref.[3][4]. In history, Casimir firstly predicts the effectof the boundaries and he found that there is an attractiveforce acting on two conducting plan-parallel plates in vac-uum. Since the last decade, the Casimir effect has beenpaid more attention due to the development of precisemeasurements [5], and it has been applied to the fabrica-tion of microelectromechanical systems (MEMS)[6]. Re-cently, some new methods have developed for computingthe Casimir energy between a finite number of compactobjects [7].The nature of the Casimir force may depend on (i)the background field, (ii) the spacetime dimensionality,(iii) the type of boundary conditions, (iv) the topologyof spacetime, (v) the finite temperature. The most evi-dent example of the dependence on the geometry is givenby the Casimir effect inside a rectangular box [2, 8].The detailed calculation of the Casimir force inside a D-dimensional rectangular cavity was shown in [9], in whichthe sign of the Casimir energy depends on the length ofthe sides. The Casimir force arises not only in the pres-ence of material boundaries, but also in spaces with non-trivial topology. For example, we get the scalar field on aflat manifold with topology of a circle S . The topology ∗ Electronic address: [email protected] † Electronic address: [email protected] of S causes the periodicity condition φ ( t,
0) = φ ( t, C ),where C is the circumference of S , imposed on the wavefunction which is of the same kind as those due to bound-ary. Similarly, the antiperiodic conditions can be drawnon a M¨obius strip. The ζ -function regularization pro-cedure is a very powerful and elegant technique for theCasimir effect. Rigorous extension of the proof of Epstein ζ -function regularization has been discussed in [10]. Vac-uum polarization in the background of on string was firstconsidered in [11]. The generalized ζ -function has manyinteresting applications, e.g., in the piecewise string [12].Similar analysis has been applied to monopoles [13], p-branes [14] or pistons [15].As we have known, there are many things that looklike the spring, for instance, DNA has the helix structurein our cells. Thus, it is interesting to find the effect of thehelix configuration presenting in the space-time manifoldfor quantum fields and as far as we know, no one has con-sidered this configuration before. In this paper, we haveinvestigated the Casimir effect for a massless scalar fieldon the circular helix structure in two (2D) and three (3D)dimensions by using the zeta function techniques, whichis a very useful and elegant technique in regularizing thevacuum energy.In next section we have calculated the Casimir energyand force by imposing the helix boundary conditions andwe find that the behavior of the force parallel to the axisof the helix is very much like the force on a spring thatobeys the Hooke’s law in mechanics when the r ≪ h to the circumference a of the helix. However, in this case, the force comes froma quantum effect, and so we would like to call the helixstructure as a quantum spring . When r is large, this forcebehaves like the Newton’s law of universal gravitationin the leading order and vanishes when r goes to theinfinity. The magnitude of this force has a maximumvalues at r = 0 . r ≈ .
494 (3D). On theother hand, the force perpendicular to the axis decreasesmonotonously with the increasing of the ratio r . Bothforces are attractive and their behaviors are the same intwo and three dimensions. We will give some discussionsand conclusions in the last section. II. EVALUATION OF THE CASIMIR ENERGYA. Topology of the flat (D+1)-dimensionalspacetime
As mentioned in Section I, the Casimir effect arise notonly in the presence of material boundaries, but also inspaces with nontrivial topology. For example, we get thescalar field on a flat manifold with topology of a circle S . The topology of S causes the periodicity condition φ ( t,
0) = φ ( t, C ). Before we consider complicated casesin the flat spacetime, we have to discuss the lattices.A lattice Λ is defined as a set of points in a flat (D+1)-dimensional spacetime M D +1 , of the formΛ = ( D X i =0 n i e i | n i ∈ Z ) , (1)where { e i } is a set of basis vectors of M D +1 . In terms ofthe components v i of vectors V ∈ M D +1 , we define theinner products as V · W = ǫ ( a ) v i w j δ ij , (2)with ǫ ( a ) = 1 for i = 0, ǫ ( a ) = − x − x plane, the sublattice Λ ′′ ⊂ Λ ′ ⊂ Λ areΛ ′ = { n e + n e | n , ∈ Z } , (3)and Λ ′′ = { n ( e + e ) | n ∈ Z } . (4)The unit cylinder-cell is the set of points U c = (cid:26) X = D X i =0 x i e i | ≤ x < a, − h ≤ x < , −∞ < x < ∞ , − L ≤ x T ≤ L (cid:27) , (5)where T = 3 , · · · , D . When L → ∞ , it contains preciselyone lattice point (i.e. X = 0), and any vector V hasprecisely one ”image” in the unit cylinder-cell, obtainedby adding a sublattice vector to it.In this paper, we choose a topology of the flat (D+1)-dimensional sapcetime: U c ≡ U c + u , u ∈ Λ ′′ , see Fig.1.This topology causes the helix boundary condition for aHermitian massless scalar field φ ( t, x + a, x , x T ) = φ ( t, x , x + h, x T ) , (6)where, if a = 0 or h = 0, it returns to the periodicityboundary condition. In calculations on the Casimir effect, extensive useis made of eigenfunctions and eigenvalues of the corre-sponding field equation. A Hermitian massless scalarfield φ ( t, x α , x T ) defined in a (D+1)-dimensional flatspacetime satisfies the free Klein-Gordon equation: (cid:0) ∂ t − ∂ i (cid:1) φ ( t, x α , x T ) = 0 , (7)where i = 1 , · · · , D ; α = 1 , T = 3 , · · · , D . Under theboundary condition (6), the modes of the field are then φ n ( t, x α , x T ) = N e − iω n t + ik x x + ik z z + ik T x T , (8)where N is a normalization factor and x = x, x = z ,and we have w n = k T + k x + (cid:18) − πnh + k x h a (cid:19) = k T + k z + (cid:18) πna + k z a h (cid:19) . (9)Here, k x and k z satisfy ak x − hk z = 2 nπ , ( n = 0 , ± , ± , · · · ) . (10)In the ground state (vacuum), each of these modes con-tributes an energy of w n /
2. The energy density of thefield is thus given by E D +1 = 12 a Z d D − k (2 π ) D − ∞ X n = −∞ s k T + k z + (cid:18) πna + k z a h (cid:19) , (11)where we have assumed a = 0 without losing generalities. a 2a 3a 4a 5a2hhx T x x -h-2h-3h-4h-5h -a-2a-3a 3h FIG. 1: The helix boundary condition can be induced by thetopology of spacetime.
B. Massless scalar field in dimension
In the 2 + 1 dimensional spacetime, we have the fol-lowing boundary condition to mimic the helix structure: φ ( t, x + a, z ) = φ ( t, x, z + h ) , (12)where h is regarded as the pitch of the helix, and we callthis condition the helix boundary condition. One cansee from eq.(12) that it would return to the cylindricalboundary conditions when h vanishes and for h = 0, thewhole system(the spring) does not have the cylindricalsymmetry. Therefore, the vacuum energy density is givenby E ( a, h ) = 12 a Z ∞−∞ dk π ∞ X n = −∞ s k + (cid:18) πna + ka h (cid:19) , (13)which is divergent, so we should regularize it to get afinite result. There many regularization method couldbe used to deal with the divergence, but in this paperwe would like to use the zeta function techniques, whichis a very useful and elegant technique in regularizing thevacuum energy. To use the ζ -function regularization, wedefine E ( s ) as E ( a, h ; s ) = √ γπa ∞ X n =1 Z ∞ dk (cid:0) k + 1 (cid:1) − s/ (cid:18) πnaγ (cid:19) − s , (14)for Re ( s ) > k integration, and here we have defined γ ≡ r , r = ha . (15)We will see in the following that the analytic continuationto the complex s plane is well defined at s = −
1. Thus,the regularized Casimir energy density is E R ( a, h ) = E ( a, h ; − k in eq.(14), see AppendixA1, we get E ( a, h ; s ) = 12 a r γπ (cid:18) πaγ (cid:19) − s Γ (cid:0) s − (cid:1) Γ (cid:0) s (cid:1) ζ ( s − , (16)where ζ ( s ) is the Riemann zeta function. The value ofthe analytically continued zeta function can be obtainedfrom the reflection relationΓ (cid:16) s (cid:17) ζ ( s ) = π s − Γ (cid:18) − s (cid:19) ζ (1 − s ) . (17)Taking s = −
1, we getlim s →− Γ (cid:18) s − (cid:19) ζ ( s −
1) = ζ (3)2 π , (18)then we have E R ( a, h ) = − ζ (3)2 πa γ − / = − ζ (3)2 πa (cid:18) r (cid:19) − / , (19)where we have used Γ( − /
2) = − √ π and if r = 0, itcome back to the cylindrical case with periodical bound-ary, see eq.(12). The Casimir force on the x direction ofthe helix is F a = − ∂E R ( a, h ) ∂a = − ζ (3)2 πa (cid:18) r (cid:19) − / , (20) which is always an attractive force and the magnitude ofthe force monotonously decreases with the increasing ofthe ratio r . Once r becomes large enough, the force canbe neglected. While, the Casimir force on the z directionis F h = − ∂E R ( a, h ) ∂h = − ζ (3)2 πa r (1 + r ) / , . (21)which has a maximum magnitude at r = 0 .
5. When r < .
5, the magnitude of the force increases with theincreasing of r until r = 0 .
5, and the force is almostlinearly depending on r when r ≪
1. So, it is just likethe force on a spring complying with the Hooke’s law,but in this case, the force originates from the quantumeffect, namely, the Casimir effect. Once r > .
5, themagnitude of the force decreases with the increasing of r . To illustrate the behavior of the Casimir force in thiscase, we plot them for each direction in Fig.2. - - - - - F a - - - - - - F h FIG. 2: The Casimir force on the x (left) and z (right) direc-tion in the unit 3 ζ (3) / (2 πa ) vs. the ratio r in 2 + 1 dimen-sion. The point corresponds to the maximum magnitude ofthe force at r = 0 . It should be noticed that in Fig.2, the behavior of theforces are different with respect to the ratio r , but thisdose not conflict with the relation (9), which shows thatlabeling the axes is a matter of convention, namely thefinal result should have the the symmetry of a ↔ h .The reason is the following, eq.(19) could be rewrittenin terms of a and h : E R ( a, h ) = − ζ (3)2 π (cid:18) a + h (cid:19) − / , (22)which respects the symmetry of a ↔ h in deed. And, onecan easily see that eqs. (20) and (21) are also under thissymmetry, if one rewritten these equations as F a = − ζ (3)2 π a ( a + h ) / , (23) F h = − ζ (3)2 π h ( a + h ) / , (24)which are all consistent with the relation (9). C. Massless scalar field in dimension
As in the 2 + 1 dimension case, the vacuum energydensity in 3 + 1 dimention is given by E ( a, h ) = 12 a Z ∞−∞ dk y dk z (2 π ) ∞ X n = −∞ s k + (cid:18) πna + k z a h (cid:19) , (25)where k = k y + k z . Again, to use the ζ -function regu-larization, we define E ( s ) as E ( a, h ; s ) = 14 π a ∞ X n =1 Z π dθ p ˜ γ · Z ∞ kdk (cid:0) k + 1 (cid:1) − s/ (cid:18) πna ˜ γ (cid:19) − s , (26)and for Re ( s ) >
1, and we have defined˜ γ ≡ r cos θ . (27)where cos θ = k z /k and r is still the ratio of h to a defined in eq.(15). We will see in the following that theanalytic continuation to the complex s plane is also welldefined at s = − E R ( a, h ) = E ( a, h ; − k and θ in eq.(26), see Appendix B, we get E ( a, h ; s ) = − ζ ( s − π (2 − s ) a (cid:18) πa (cid:19) − s F (cid:18) − s, , − r (cid:19) , (28)Taking s = −
1, we get ζ ( −
3) = from (17), and then E R ( a, h ) = − π a F (cid:18) , , − r (cid:19) . (29)Therefore, the Casimir force on the x direction of thehelix is F a = − ∂E R ( a, h ) ∂a = − π a (cid:20) F (cid:18) , , − r (cid:19) − r F (cid:18) , , − r (cid:19)(cid:21) , (30) which is always attractive and its magnitudemonotonously decreases with the increasing of theratio r . By the definition of the hypergeometric func-tion, we can expand eq.(30) up to arbitrary orders of r ,thus for small r , we get F a | r ≪ = − π a (cid:20) − r + O ( r ) (cid:21) , (31)while for large r , we asymptotically expand eq.(30) as F a | r ≫ = − π a (cid:20) πr + 112 πr + O (cid:18) r (cid:19)(cid:21) , (32)up to O ( r − ). Then, it is clear to see that, the force willbe vanished when r goes to infinity. On the other hand,the Casimir force on the z direction is F h = − ∂E R ( a, h ) ∂h = − π r a F (cid:18) , , − r (cid:19) , (33)and for r ≪ r ≫
1, we respectively have F h | r ≪ = − π a (cid:20) r − r + O ( r ) (cid:21) , (34)and F h | r ≫ = − π a (cid:20) πr + 25 πr + O (cid:18) r (cid:19)(cid:21) . (35)Therefore, for small r , the force linearly depends on r ,namely, F h = − Kr , K = π a , ( r ≪ , (36)which is very much like a spring obeying the Hooke’slaw with spring constant K in classical mechanics, butin this case, the force comes from a quantum effect, andwe would like to call it quantum spring , see Fig.3. When r is large, the force behaves like the Newton’s law ofuniversal gravitation, i.e. F h ∼ − /r in the leadingorder. Furthermore, there exists a maximum magnitudeof the force | F h | max when r takes a critical value r ≈ . F (cid:0) / , / , − r (cid:1) − F (cid:0) / , / , − r (cid:1) r = 0 . (37)To illustrate the behavior of the forces on the helix, weplot them for each direction in Fig.4. III. CONCLUSION AND DISCUSSION
In conclusion, we have investigated the Casimir effectwith a helix configuration in two and three dimensions,and it can be easily generalized to high dimensions. Wefind that the force parallel to the axis of the helix has aparticular behaviors that the Casimir force in the usualcase do no possesses. It behaves very much like the force hF h F a FIG. 3: Illustration of the
Quantum spring . - - - - - F a - - - - F h FIG. 4: The Casimir force on the x (left) and z (right) direc-tion in the unit π /a vs. the ratio r in 3 + 1 dimension. Thepoint corresponds to the maximum magnitude of the force at r = r ≈ . on a spring that obeys the Hooke’s law in mechanics when r ≪
1, and like the Newton’s law of universal gravi-tation when r ≫
1. Furthermore, the The magnitudeof this force has a maximum values at r = 0 . r ≈ .
494 (3D). So, we would like to call this he-lix configuration as a quantum spring , see Fig.3. On theother hand, the force perpendicular to the axis decreasesmonotonously with the increasing of the ratio r . Bothforces are attractive and their behaviors are the same intwo and three dimensions.It should be noticed that, the critical value r , atwitch the magnitude of the force gets its maximum valuedepends on the space-time dimensions. On a general D + 1-dimensional ( D ≥
3) flat space-time manifold, theCasimir energy density on the helix is roughly given by E ( a, h ) ∼ − F (cid:18) d − , , − r (cid:19) a − ( d +1) , (38)up to some coefficient. Then, the Casimir force on the z direction is roughly F h ∼ − r F (cid:18) d + 12 , , − r (cid:19) a − ( d +2) , (39)thus the critical value r satisfies4 F (cid:0) d + 1 / , / , − r (cid:1) − d + 1) F (cid:0) d + 3 / , / , − r (cid:1) r = 0 , (40)which can not be exactly solved but one can numericallycalculate the critical point r . In Fig.5, we have shownthe dependence of r on the space dimension d from twoto ten dimensions. r d FIG. 5: The critical value r vs . the space dimension d . In this paper, we have considered the massless scalarfield, and one can easily generalize it to a massive scalarfield. As is known that the Casimir effect disappearsas the mass of the field goes to infinity since there areno more quantum fluctuation in this limit, but of course,how the Casimir force varies as the mass changes is worthstudying [17], and we will study it in our further work[18], in which we will also consider the Casimir effectof the electromagnetic field in the helix configuration.Since this quantum spring effect may be detected in thelaboratory and be applied to the microelectromechanicalsystem, we suggest to do the experiment to verify ourresults. It should be noticed that, in the experiment orthe real application, the spring like Fig.3 should be soft,which means the force coming from the classical mechan-ics could be small enough, and the quantum effect dom-inates the behavior of the spring.
Acknowledgments
This work is supported by National Education Foun-dation of China grant No. 2009312711004 and Shang-hai Natural Science Foundation, China grant No.10ZR1422000.
Appendix A: The k integration in dimension The integration in eq.(14) is given by I ( s ) = Z ∞ dk (cid:0) k + 1 (cid:1) − s/ = Z ∞ F (cid:16) s , b, b ; − k (cid:17) dk = 12 Z ∞ F (cid:16) s , b, b ; − z (cid:17) z − / dz , (A1)where we have used (1 + z ) α = F ( − α, b, b ; − z ), and p F q is hypergeometric functions. By using [16] Z ∞ F ( a, b, c ; − z ) z − t − dz = Γ( a + t )Γ( b + t )Γ( c )Γ( − t )Γ( a )Γ( b )Γ( c + t ) (A2)where Γ( a ) is Gamma functions, we get I ( s ) = √ π (cid:0) s − (cid:1) Γ (cid:0) s (cid:1) , (A3)where we have used Γ(1 /
2) = √ π . Appendix B: The k and θ integration in dimension The integration in eq.(26) is given by I ( s ) = Z π dθ ˜ γ s − / Z ∞ kdk (cid:0) k + 1 (cid:1) − s/ = 12 − s Z π dθ ˜ γ s − / ( k + 1) − s (cid:12)(cid:12)(cid:12)(cid:12) ∞ k =0 = − − s Z π/ dθ (cid:18) r cos θ (cid:19) s − / = − − s Z x − / (1 − x ) − / · F (cid:18) − s, ,
12 ; − r x (cid:19) dx = − π − s F (cid:18) − s, , − r (cid:19) , (B1)where we have defined x = cos θ and we have used [16] Z (1 − x ) µ − x ν − p F q ( a , · · · , a p ; ν, b , · · · , b q ; ax ) dx = Γ( µ )Γ( ν )Γ( µ + ν ) p F q ( a , · · · , a p ; µ + ν, b , · · · , b q ; a ) . (B2) [1] H. B. G. Casimir, Indag. Math. , 261 (1948) [Kon. Ned.Akad. Wetensch. Proc. , 793 (1948 FRPHA,65,342-344.1987 KNAWA,100N3-4,61-63.1997)].[2] M. Bordag, G. L. Klimchitskaya, U. Mohideen andV. M. Mostepanenko, Advances in the Casimir Effect ,Oxford University Press, 2009.[3] R. Utiyama and B. S. DeWitt, J. Math. Phys. , 608(1962).[4] B. S. DeWitt, Phys. Rept. , 295 (1975).[5] R. S. Decca, D. Lopez, E. Fischbach, G. L. Klimchit-skaya, D. E. Krause and V. M. Mostepanenko, Phys. Rev.D , 077101 (2007) [arXiv:hep-ph/0703290].[6] F. M. Serry, D. Walliser, and G. J. Maclay,J.Microelectromech.Syst. , 193 (1995),H. B. Chan, V. A. Aksyuk, R. N. Kleiman, D. J. Bishop,and F. Capasso, Science , 1941 (2001).[7] T. Emig, N. Graham, R. L. Jaffe and M. Kardar, Phys.Rev. Lett. , 170403 (2007) [arXiv:0707.1862 [cond-mat.stat-mech]].[8] W. Lukosz, Physica , 109(1971).[9] X. Z. Li, H. B. Cheng, J. M. Li and X. H. Zhai, Phys.Rev. D , 2155 (1997);X. Z. Li and X. H. Zhai, J. Phys. A :11053-11057, 2001.[arXiv:hep-th/0205225].[10] E. Elizalde, S. D. Odintsov, A. Romeo, A. A. Bytsenkoand S. Zerbini, Zeta Regularization Techniques with Ap- plications , World Scientific, Singapore, 1993.[11] T. M. Helliwell and D. A. Konkowski, Phys. Rev. D ,1918 (1986).[12] X. Z. Li, X. Shi and J. Z. Zhang, Phys. Rev. D , 560(1991);I. H. Brevik, H. B. Nielsen and S. D. Odintsov, Phys.Rev. D , 3224 (1996).[13] E. R. Bezerra de Mello, V. B. Bezerra andN. R. Khusnutdinov, Phys. Rev. D , 063506 (1999)[arXiv:gr-qc/9903006].[14] X. Shi and X. . Li, Class. Quant. Grav. , 75 (1991).[15] X. H. Zhai and X. Z. Li, Phys. Rev. D , 047704 (2007)[arXiv:hep-th/0612155];X. H. Zhai, Y. Y. Zhang and X. Z. Li, Mod. Phys. Lett.A , 393 (2009) [arXiv:0808.0062 [hep-th]];R. M. Cavalcanti, Phys. Rev. D , 065015 (2004)[arXiv:quant-ph/0310184];M. P. Hertzberg, R. L. Jaffe, M. Kardar andA. Scardicchio, Phys. Rev. Lett. , 250402 (2005)[arXiv:quant-ph/0509071].[16] I.S. Gradshteyn and I.M. Ryzhik ; Alan Jeffrey, DanielZwillinger, editors. Table of Integrals, Series, and Prod-ucts , seventh edition. Academic Press, 2007. ISBN 978-0-12-373637-4 .[17] M. Bordag, U. Mohideen and V. M. Mostepanenko, Phys.Rept. , 1 (2001) [arXiv:quant-ph/0106045]., 1 (2001) [arXiv:quant-ph/0106045].