Casimir-Polder potential for parallel metallic plates in background of a conical defect

Transcription

1 Amenian Jounal of Physics, 7, vol., issue, pp. 8-9 Casimi-Polde potential fo paallel metallic plates in backgound of a conical defect A.Kh. Gigoyan Institute of Applied Poblems in Physics NAS RA, 5 Nesessian Steet, Yeevan, Amenia ashot.g@gmail.com Received 6 August 7 Abstact. A closed expession is povided fo the Casimi-Polde potential fo the inteaction of a polaziable micopaticle and two conducting plates in the geomety of a conical defect with an abitay plana angle deficit. The plates ae pependicula to the axis of the defect. The behavio of the potential is investigated in vaious asymptotic egions fo the values of the paametes in the poblem. The Casimi-Polde foces ae epulsive with espect to the defect and attactive with espect to the close plate. Keywods: Casimi-Polde foces, conical defects, vacuum polaization. Intoduction The conical defects ae among the most popula classes of topological defects that appea in field theoy and condensed matte physics as a esult of symmety baking phase tansitions [, ]. A wellknown example of this type of defects is the cosmic stings, which may have been ceated by phase tansitions in the ealy Univese. In the simplest model, the conical defect outside the coe is descibed by flat space-time with a plana angle deficit. Though the local geomety is the same as that fo the Minkowski space-time, the global popeties ae diffeent. The nontivial topology of a conical defect gives ise to a numbe of inteesting physical effects. In paticula, the polaization of the vacuum aound the defect has been lagely investigated in the liteatue fo scala, femionic and vecto fields (see, fo instance, efeences given in []). This effect is a consequence of the modification of vacuum fluctuations of quantum fields by the nontivial topology. The change in the vacuum fluctuations induced by conical defects leads to Casimi-Polde foces acting on a polaizable micopaticle (fo a geneal intoduction see []). Fo a staight defect, these foces have been investigated in [5, 6]. It has been shown that fo an isotopic polaizability tenso the foce is always epulsive. The foce in the pesence of an additional conducting plate pependicula to the defect is studied in [7]. The case of a conducting cylinde coaxial with the sting has been discussed in [8]. The coesponding esults show that the conical defect may be used to contol the stength and the sign of the Casimi-Polde foce in both these configuations. In this pape, we investigate the Casimi-Polde foces in the geomety of a conical defect with two paallel conducting plates pependicula to the defect axis. The vacuum expectation value (VEV) of the enegy-momentum tenso fo the electomagnetic field in the same geomety has been discussed in [9]. In paticula, the Casimi foces acting on the plates wee investigated and it has been shown that those foces ae always attactive. The poblem with two boundaies in conical spacetime fo a femionic field has been consideed ecently in [].

2 Gigoyan ǀǀ Amenian Jounal ov Physics, 7, vol., issue The pape is oganized as follows. In the next section we descibe the geomety and pesent the electomagnetic field modes fo the electic field. The Casimi-Polde potential is detemined by the VEV of the electic field squaed and, in section, we evaluate this VEV fo the geomety unde consideation. The asymptotic behavio of the Casimi-Polde potential in vaious limiting egions fo the paametes is investigated in section. The main esults ae summaized in section 5.. Poblem setup and the electic field modes Fo a conical defect along the -axis of cylindical coodinates (, φ, z), the line element is witten as, with a plana angle deficit ds = dt d dφ dz, () whee φ φ and <. We assume the pesence of two paallel metallic plates pependicula to the axis and placed at z = and z= a. Because of the bounday conditions imposed on the vacuum fluctuations of the electic and magnetic fields by the pesence of the plates, the spectum of these fluctuations is changed. This gives ise the Casimi-Polde foces acting on a polaizable paticle. Hee we conside a simple case of an isotopic polaizability neglecting the dispesion effects. In this appoximation, the Casimi-Polde inteaction enegy is expessed as U, = α E () whee is the static polaizability of the micopaticle and E is the VEV of the electic field squaed. This static limit is a good appoximation fo sepaations z, a z,, whee ω j is the ω j eigenfequency in the dispesion law fo the polaizability. Fo example, in the oscillato model one has g j α( ω) = j ω ω, whee g ae the oscillato stengths. j j Fo the evaluation of the VEV we use the mode sum fomula E E = β β β E E, () whee {, β β} E E is a complete set of modes fo the electic field, specified by a collective set of quantum numbes β. The mode functions obey the standad pefect conducto bounday conditions on the plates, namely, n E = at z = and z= a, with n beings the nomal to the plates. β 8

3 Casimi-Polde Potential fo Paallel Metallic Plates ǀǀ Amenian Jounal of Physics, 7, vol., issue Fist we will conside the egion between the plates, z a. The coesponding modes fo the electomagnetic field have been aleady pesented in [9]. In that pape the VEV of the electic field squaed has been consideed as well. Hee we deive an altenative epesentation, moe convenient fo both theoetical and numeical analysis. The electomagnetic field modes ae divided into two classes coesponding to tansvese magnetic (TM) and tansvese electic (TE) waves. We will distinguish them by the supescipt λ : λ = fo TM modes and λ = fo TE waves. The coesponding mode functions ead i ( φ ωt) βl = β βl, () ( λ ) (, ) E x C E z e π whee q =, m =, ±, ±,..., and,, φ modes one has l = coespond to the (, φ, z) -components. Fo TM ( ) ' (, ) = γ ( γ ) sin E z k J kz β ( ) (, ) = ( γ ) sin ( ) (, ) = γ ( γ ) cos β Eβ z ik J kz E z J kz (5) whee γ = ω k, γ <, J with espect to the agument of the function. Fo TE modes v x is the Bessel function and the pime means the deivative Eβ z J kz () (, ) =ω ( γ ) sin () ' (, ) =ωγ ( γ ) sin E z i J kz β (6) () ( z) and Eβ, =. The eigenvalues fo k ae quantized by the bounday conditions on the plates: π n k = kn =, whee n =,,,... fo TM modes and n =,,... fo TE modes. The nomalization constant a is given by C β δ n q =, ωγ a (7) fo both TM and TE modes. 8

4 Gigoyan ǀǀ Amenian Jounal ov Physics, 7, vol., issue. Casimi-Polde potential Substituting the mode functions into the mode-sum fomula (), fo the VEV of the electic field squaed we find ' ' q γ q γ E = d k z J + d k + k z G a γ cos ( n ) ( γ ) γ ( n γ ) sin ( n ) ( γ ), ω mn, = mn, = a ω (8) with the notation ' G x = J x + x J x. (9) In (8), the pime on the sign of the summations means that the tems m= and n= should be taken with the coefficient ½. Fo the futhe tansfomation we use the integal epesentation ds s e ω = ω π. s () Plugging this into (8) and intoducing new integation vaiables ae evaluated by using the fomulae x = and y = γ, the integals ove s p p+ y x x dyy e J y = x x xe I x, p = ( ) p+ y x x dyy e G y x x xe xi x, () whee. I x is the modified Bessel function of the fist kind. The fist integal in () is given, fo example, in []. The emaining integals ae obtained fom the fomulas given in []. Fo the VEV one obtains p =, and ' 8q kn x x E = dx xe cos ( knz) x( xe ) + π a mn, = { x x ( k z) ( k x) e ( xe ) } I ( x) + sin + n n x x x () 8

5 Casimi-Polde Potential fo Paallel Metallic Plates ǀǀ Amenian Jounal of Physics, 7, vol., issue Next we wite the functions cos ( kz ) and in tems of the function cos n kz and use the elation n ' x + kn x x( zan) e cos( knz) = e n= π n= a () and simila elation fo z =. The fomula () diectly follows fom the Poisson esummation fomula (see also []). As a esult, the VEV of the electic field squaed is decomposed into two pats E = E + E, () whee = xan E dxxe x an π n= + x x + + x x + Iq x x (5) and = (( ) ) ( x + ) + x( ( zan) ) E dxxe x z an π n= x I x x q ( x) (6) Hee we have intoduced the function q ' x ( x) qe I ( x) I =. m= (7) Fo this function we have the following epesentation [] ( π ) [ ] cosh qsin q e I q ( x) = e dy l= π cosh q x y xsi ( qy) cos( qπ ) (8) with the notation =sin(/) and [ ] q is the intege pat of q. The pime on the summation sign means that the tems l = and l = q (if q is and even intege) should be taken with coefficients /. 85

6 Gigoyan ǀǀ Amenian Jounal ov Physics, 7, vol., issue Substituting (8) into (5), afte integating ove, we find [ q ] π qsin E = + f ( s, ) l a dy πa 9 l= π cosh qy cos qπ ( qπ ) f cosh y, ( a), (9) whee the fist tem in the squae backets comes fom the l = tem in (8) and f ( x y) ( ) n x + xy +, = +. xy xy n ( + ) n= () The pime on the summation sign in (9) means that in the case of even intege values of q the tem with l = q should be taken with an additional coefficient ½. The contibution of the fist tem in the ight-hand side of () to E is expessed in tems of the function n [ q ] q c q = s q dy cosh n n l sin ( π ) l n = cosh ( qy) cos( qπ ), y () with n = that is given by c ( q) ( q )( q ) = + 9. This contibution coincides with the enomalized VEV of the field squaed in the bounday-fee cosmic sting spacetime [] (fo the VEV of the enegy-momentum tenso see [5, 6]): E ( s) ( q )( q + ) =. 8π () Fo the adial component of the coesponding Casimi-Polde foce one finds (moe geneal esults fo an anisotopic polaizability tenso with dispesion ae given in [6]) F ( q )( q + s ) = α 9π 5 () This foce is epulsive. 86

7 Casimi-Polde Potential fo Paallel Metallic Plates ǀǀ Amenian Jounal of Physics, 7, vol., issue In a simila way, by using the esult ( n z a) π + cos ( π za) ( π za) + =, n= sin () fo the contibution (6) we find ( π za) ( π ) [ q ] π + cos E = + f ( s,, ) l a z a a sin z a πa l= qsin π ( qπ ) f cosh y, ( a), z a dy cosh ( qy) cos( qπ) (5) with the function f ( x, y, u) ( ) + x nu xy =. ( n u) + xy n= (6) By taking into account (9) and (5), the Casimi-Polde potential is pesented as [ q ] α U( ) = UM ( ) + f ( sl,( a), z a) π a l= qsin π ( qπ ) f cosh y, ( a), z a dy cosh ( qy) cos( qπ) (7) with the function 87

8 Gigoyan ǀǀ Amenian Jounal ov Physics, 7, vol., issue (,, ) (, ) (,, ) f x y u = f x y f x y u = + = n= ( xy + n ) ( n u) + xy n x + xy x nu xy. (8) In (7) U M ( ) ( π za) ( π z a) πα+ cos =. 8a sin 5 (9) At the absence of the cosmic sting, one has q = and this tem suvives only. Hence, it pesents the potential in the egion between the plates on the Minkowski bulk. The emaining contibution in (7) is f xy,, u = f xyu,,. Fom hee it follows that induced by the conical defect. It is easily seen that the Casimi-Polde potential is symmetic with espect to the plane z = a. Fo intege values of the paamete q, the integal tem in (7) vanishes. In this special case, the expession we have deived above fo the VEV of the electic field squaed coincides with the esult pesented in [9]. The Casimi-Polde potential in the geomety of a single plate at z = is obtained fom the esult pesented above taking the limit a fo fixed values of and z. In this limit, the nonzeo contibution in (7) comes fom the tem n = only and fo the Casimi-Polde potential one finds [ q ] () α α () U ( ) = + c ( q) + f ( sl, z ) 8πz π l= qsin π ( qπ ) () ( cosh, ) f y z dy cosh ( qy) cos( qπ), () with the notation f () ( x y) ( ) ( x+ y) x y+ x, =. () The fist tem in the ight-hand side is the potential fo a plate in Minkowski bulk. The pat with the fist tem in the figue baces coesponds to the potential when the plate is absent. In the egion the Casimi-Polde potential is given by (). The coesponding fomula in the egion z> a is obtained by the eplacement z z a. Hence, we have found the Casimi-Polde potential in the entie egion fo the poblem at hand. 88

9 Casimi-Polde Potential fo Paallel Metallic Plates ǀǀ Amenian Jounal of Physics, 7, vol., issue. Asymptotic analysis and numeical esults Fist of all we note that the poblem is symmetic with espect to the plane z = a and the z - component of the Casimi-Polde foce, Fz = zu( ), vanishes on that plane, F z z = =. Let us a conside the behavio of the Casimi-Polde potential in the asymptotic egions of the paametes. Fo z, a z, the leading contibution comes fom the fist tem in the ight-hand side of () and it coincides with the coesponding quantity in a conical space in the absence of boundaies, given by (). Fo points close to the plate, z a,, the dominant contibution to (7) comes fom the fist tem in the squae backets of (9). Expanding it fo small za, to the leading ode one gets U α 8π z ( ) () The leading tem in the ight-hand side coincides with the VEV of the field squaed induced by a single conducting plate in Minkowski space-time. Fo lage values of a f xyu,, fo y. In this ange the dominant contibution to the seies in (8) comes fom lage values of y and we can we need the asymptotic expession fo the function f x, y, u π xy. Fo eplace the summation by the integation. To the leading ode one finds the VEV of the field squaed we get U ( ) U ( ) ( q) αc M + 8a () This shows that the adial component of the Casimi-Polde foce behaves as F U ( ) α c 8a ( q). () The foce is epulsive with espect to the sting. Recall that nea the sting and fo points not too close 5 to the plates, the adial foce is given by () and behaves as. In figue we have plotted the Casimi-Polde potential in the egion between the plates, ( ) au α, as a function of the escaled coodinates a and za fo a conical defect with plana angle deficit coesponding to.5 q =. 89

10 Gigoyan ǀǀ Amenian Jounal ov Physics, 7, vol., issue Figue : Casimi-Polde potential in the egion between the plates, as a function of the escaled distances fom the defect and fom the plate at z =. 5. Conclusion We have investigated the Casimi-Polde inteaction of a polaizable micopaticle in the geomety of two paallel conducting plates pependicula to the axis of a staight conical defect. In the static limit, when the dispesion of the polaizability can be neglected, the Casimi-Polde potential is expessed in tems of the VEV of the electic field squaed. Fo the evaluation of the latte we have used the method of diect summation ove a complete set of the electomagnetic field modes in the poblem unde consideation. We have povided a closed expession fo the Casimi-Polde potential fo an abitay value of the plana angle deficit, given by (7) in the egion between the plates. In the emaining egions, the potential descibes the inteaction between a polaizable micopaticle and a single plate. The coesponding expession fo the egion z < is pesented as (). Nea the plates and not too close to the defect, the leading tem in asymptotic expansion coincides with the Casimi-Polde potential fo a single plate in Minkowski bulk and is given by (). Fo points close to the defect, the leading contibution comes fom the pat coesponding to the geomety in the absence of the plates with the 5 adial foce given by (). The latte decays as. At lage distances fom the defect, the adial foce is pesented as () and the decay is slowe, like. 9

11 Casimi-Polde Potential fo Paallel Metallic Plates ǀǀ Amenian Jounal of Physics, 7, vol., issue Refeences [] A. Vilenkin and E. P. S. Shellad, Cosmic Stings and Othe Topological Defects (Cambidge Univesity Pess, Cambidge, England, 99); M. B. Hindmash and T. W. B. Kibble, Rep. Pog. Phys. 58 (995) 77. [] D. R. Nelson, Defects and Geomety in Condensed Matte Physics (Cambidge Univesity Pess, Cambidge, ). [] A. Mohammadi, E. R. Bezea de Mello, A. A. Sahaian, Class. Quantum Gav. (5) 5. [] V.A. Pasegian, Van de Waals foces: A Handbook fo Biologists, Chemists, Enginees, and Physicists (Cambidge Univesity Pess, Cambidge, 5). [5] V. M. Badeghyan, A. A. Sahaian, J. Contemp. Phys. 5 (). [6] A. A. Sahaian, A. S. Kotanjyan, Eu. Phys. J. 7 () 765. [7] E. R. Bezea de Mello, V. B. Bezea, H. F. Mota, A. A. Sahaian, Phys. Rev. D 86 () 65. [8] A. A. Sahaian, A. S. Kotanjyan, Phys. Lett. B 7 (). [9] E. R. Bezea de Mello, A. A. Sahaian, A. Kh. Gigoyan, J. Phys. A: Math. Theo. 5 () 7. [] A. Kh. Gigoyan, A. R. Mktchyan, A. A. Sahaian, Int. J. Mod. Phys. D 6 (7) 756. [] P. Pudnikov, Yu. A. Bychkov, O. I. Maichev, Integals and Seies (Godon and Beach, New Yok, 986), Vol.. [] A. A. Sahaian, V. F. Manukyan, N. A. Sahayan, axiv:76.79 (to appea in Eu. Phys. J. C). [] E. R. Bezea de Mello, A. A. Sahaian, Class. Quantum Gav. 9 () 56. [] E. R. Bezea de Mello, V. B. Bezea, A. A. Sahaian, Phys. Lett. B 65 (7) 5. [5] V. P. Folov, E. M. Seebiany, Phys. Rev. D 5 (987) 779. [6] J. S. Dowke, Phys. Rev. D 6 (987) 7. 9