The Casimir effect is an interesting phenomenon in the sense that it provides
us with one of the primitive means of extracting the energy out of the vacuum.
Since the original work of Casimir a number of works have appeared in extending
the result to the case of more general topological and dynamical configurations
of the boundary condition and to the circumstances at finite temperature and
gravity. In the studies of the Casimir effects it is common to assume the free
electromagnetic field in the bounded region. It may be interesting to extend
our arguments for fields other than the electromagnetic field. The Casimir
effect due to the free fermionic fields has been investigated by several
authors and has been found to result in an attractive force under the suitable
physical boundary conditions.Comment: 12 pages, 6 figures, REVTe

AGUS PURWANTO

Casimir H. B. G.

Chodos A.

HIROYUKI ABE

Igarashi Y.

Johnson K.

JUNYA HASHIDA

TAIZO MUTA

English

arXiv

Crossref

PATH-INTEGRAL FORMULATION OF CASIMIR EFFECTS IN SUPERSYMMETRIC QUANTUM ELECTRODYNAMICS

The path-integral method of calculating the Casimir energy between two parallel conducting plates is developed within the framework of supersymmetric quantum electrodynamics at vanishing temperature as well as at finite temperature. The choice of the suitable boundary condition for the photino on the plates is argued and the physically acceptable condition is adopted which eventually breaks the supersymmetry. The photino mass term is introduced in the Lagrangian and the photino mass dependence of the Casimir energy and pressure is fully investigated

Abe, Hiroyuki

Hashida, Junya

Muta, Taizo

Purwanto, Agus

Hiroshima University Institutional Repository

arXiv:hep-ph/9905286 v1   10 May 1999HUPD-9906Path-Integral Formulation of Casimir Effectsin Supersymmetric Quantum ElectrodynamicsHiroyuki Abe, Junya Hashida, Taizo Muta and Agus PurwantoDepartment of Physics, Hiroshima UniversityHigashi-Hiroshima, Hiroshima 739-8526, Japan(March 6, 2006)AbstractThe path-integral method of calculating the Casimir energy between two par-allel conducting plates is developed within the framework of supersymmetricquantum electrodynamics at vanishing temperature as well as at finite tem-perature. The choice of the suitable boundary condition for the photino onthe plates is argued and the physically acceptable condition is adopted whicheventually breaks the supersymmetry. The photino mass term is introducedin the Lagrangian and the photino mass dependence of the Casimir energyand pressure is fully investigated.03.65.Bz,11.10.Wx,11.30.Pb,12.20.-m,12.60.JvTypeset using REVTEX1I. INTRODUCTIONThe Casimir effect is an interesting phenomenon in the sense that it provides us with oneof the primitive means of extracting the energy out of the vacuum. Since the originalwork of Casimir [1,2] a number of works have appeared in extending the result to the caseof more general topological and dynamical configurations of the boundary condition andto the circumstances at finite temperature and gravity [3]. In the studies of the Casimireffects it is common to assume the free electromagnetic field in the bounded region. Itmay be interesting to extend our arguments for fields other than the electromagnetic field.The Casimir effect due to the free fermionic fields has been investigated by several authorsand has been found to result in an attractive force under the suitable physical boundaryconditions [4,5].The supersymmetry is considered to be promising in searching for the unified theory ofelementary particles. In this connection it is natural to extend the quantum electrodynamicsto incorporate the supersymmetry. It may then be interesting to study the Casimir effectin the supersymmetric quantum electrodynamics in the hope that some evidence of super-symmetry could be observed through the Casimir effect. In the present communication wedeal with the Casimir effect by applying the path-integral formulation [6] working in thesupersymmetric quantum electrodynamics [7,8]. In the supersymmetric quantum electrody-namics we have extra particles, photinos (fermions), as superpartners of photons (bosons)and we have to discuss the contributions of photinos as well as photons in the arguments ofCasimir effects.II. PATH-INTEGRAL FORMULATIONWe start with the Lagrangian for the supersymmetric quantum electrodynamics,L = LB + LF , (2.1)withLB = −14FµνFµν − 12α(∂µAµ)2 − iη¯∂µ∂µη, (2.2)andLF = −iλσµ∂µλ¯, (2.3)where Fµν = ∂µAν − ∂νAµ with Aµ the photon field, the gauge fixing term as well as theFadeev-Popov term with Fadeev-Popov ghost η are included in the Lagrangian LB and λin LF denotes the two-component photino field. We adopt the covariant gauge with gaugeparameter α. The Fadeev-Popov ghost is usually neglected in quantum electrodynamicssince it is irrelevant as far as we consider processes without photon loops and in the normalphysical QED processes no photon loop appears. In the Casimir effect, however, we calculatethe contribution of the photon loop and so we need to take into account the Fadeev-Popovghost in order to count correctly the physical degrees of freedom of photons.2We first deal with the Casimir effect at vanishing temperature. Consider the generatingfunctional Z given byZ =∫[dAµ][dη][dη¯][dλ][dλ¯]ei∫d4xL (2.4)The free energy density ǫ is related to the generating functional Z through the relationZ = e−iǫΩ, (2.5)where Ω is the space-time volume.A. PhotonThe contribution of photons to the generating functional Z is given byZB =∫[dAµ][dη][dη¯]ei∫d4xLB . (2.6)Performing integration over Aµ, η and η¯ we obtainZB =[Det(2i∂2)] [Det(−gµν∂2 + (1− 1/α)∂µ∂ν)]−1/2, (2.7)where the overall factor of the numerical constant is neglected. We note that in momentumspace,det[gµνk2 +(1α− 1)kµkν]=1α(k20 − k21 − k22 − k23)4, (2.8)and hencelnZBΩ= −2∫ d4k(2π)4ln(−k2) + 12∫ d4k(2π)4(lnα− iπ2) +∫ d4k(2π)4ln(−k2). (2.9)Here in Eq. (2.9) the first, second and third term come from the Fµν term, the gauge termand the Fadeev-Popov term in the Lagrangian (2.2) respectively. Since the second term is aconstant which is physically irrelevant, we neglect it in the following arguments. We makethe Wick rotation in Eq. (2.9) and findlnZBΩ= −i∫d4k(2π)4ln k2, (2.10)for Euclidean momentum k. According to the relation (2.5) the free energy density ǫ isthen easily obtained by performing the integration over the fourth (time) component of theEuclidean four-momentum kµ,ǫB =∫d3k(2π)3√k2, (2.11)where we have neglected the additive constant stemming from the k4 integration. Of coursethe free energy density ǫB is a divergent quantity if one considers the infinite space region.3In the present argument of the Casimir energy we consider the region bounded by twoconducting plates. We place two infinite conducting plates perpendicular to the z directionat z = 0 and at z = L. For photons we introduce the physical boundary condition [1,2]A(z = 0) = A(z = L) = 0. (2.12)According to the condition (2.12) the z-component kz of momentum k is discretized suchthatkz =πnL, (2.13)with n an integer. Replacing the kz integration by summation on n in Eq. (2.11) we obtainǫB =12L∞∑n=−∞∫d2k(2π)2√kT2 + (πn/L)2, (2.14)where kT refers to the transversal component of k, i. e., (kx, ky). The free energy density(2.14) is a divergent quantity and we regularize it by introducing the regularization factore−τω in the integration (2.14) with ω =√kT2 + (πn/L)2,ǫB =1L∞∑n=0∫d2k(2π)2ωe−τω. (2.15)The integration and summation in Eq. (2.15) can be easily performed resulting inǫB =3π2τ 4− π2720L4. (2.16)We distinguish the free energy density for the infinite volume (Eq. (2.11)) from the onefor the bounded region (Eq. (2.15)) by denoting the former by ǫB∞. If we use the sameregularization as in Eq. (2.15), we obtain ǫB∞ = 3/(π2τ 4). The Casimir energy ǫCB then isgiven byǫCB = ǫB − ǫB∞ = − π2720L4. (2.17)B. PhotinoThe contribution of photinos to the generating functional Z is given byZF =∫[dλ][dλ¯]ei∫d4xLF . (2.18)After integrating over the photino fields we have ZF = const. × Det(i∂/). Going over to themomentum representation and performing the Wick rotation we findlnZFΩ= i∫d4k(2π)4ln k2. (2.19)4Then the photino contribution to the free energy density is given byǫF = −∫ d3k(2π)3√k2. (2.20)For the infinite volume Eq. (2.20) gives the divergent expressionǫF∞ = − 3π2τ 4, (2.21)with τ the same regularization parameter as in ǫB∞.Consider the bounded region where two parallel plates are placed perpendicular to thez axis at z = 0 and z = L respectively. The boundary condition for photino fields should bechosen by physical requirements. The physically acceptable requirement may be that thereis no fermionic outgoing flux perpendicular to the boundaries. This requirement has beenadopted to the MIT bag model long time ago [4,9]. It is satisfied by the following conditionat the boundary plates [5],in/λ = λ, (2.22)where nµ = (0,n) with n the inward normal to the boundary plates. According to thecondition (2.22) the z-component kz of momentum k is discretized such thatkz =π(2n+ 1)2L, (2.23)with n an integer.It should be noted here that the supersymmetry under consideration is explicitly brokenaccording to the fact that the boundary condition for photinos is different from that forphotons. If the supersymmetry is to be respected even in the bounded region, then of courseone has to choose a specific (but rather unphysical) boundary condition for photinos inorder to maintain the supersymmetry. In our investigation we rather choose the physicallyacceptable boundary condition for photinos and consider that the supersymmetry is anexact symmetry only for the unbounded space-time. The situation is in some similaritywith the finite temperature case where the supersymmetry is broken explicitly according tothe different boundary conditions for bosons and fermions respectively.Replacing the kz integration by summation on n in Eq. (2.20) we obtainǫF = − 12L∞∑n=−∞∫d2k(2π)2√kT2 + {π(2n+ 1)/2L}2. (2.24)The free energy density (2.24) is also divergent and we regularize it by using the sameregularization factor as in Eq. (2.15). Performing the summation and integration we obtainǫF = − 3π2τ 4− 78π2720L4. (2.25)The Casimir energy due to the fermionic effect is given by5ǫCF = ǫF − ǫF∞ = −78π2720L4. (2.26)The total Casimir energy for the supersymmetric quantum electrodynamics is given bysumming up the photon and photino contributions and readsǫC = ǫCB + ǫCF = −158π2720L4. (2.27)It is interesting to note here that the divergences appearing in ǫB and ǫF cancel out whenthey are added up and thusǫC = ǫB + ǫF . (2.28)III. TEMPERATURE DEPENDENCEWe now introduce the temperature. The Matsubara formalism [10] will be employed tointroduce temperature effects in the following calculation. We replace the integral in theenergy variable k0 by the summation where we apply the periodic boundary condition forboson fields and the anti-periodic boundary condition for fermion fields respectively:∫dk02πf(k0)→ 1β∞∑n=−∞f(ωn), (3.1)where β = 1/T with T the temperature and the energy variable k0 in the summation isreplaced by ωn which is given byωn =ωBn ≡ 2nβ π (boson),ωFn ≡ 2n+1β π (fermion).Repeating the similar calculations as in the previous zero-temperature case we obtain thetemperature dependent Casimir energy for photons and photinos respectively:ǫB(T ) =12L∞∑n=−∞[∫d2k(2π)2√k2 +2β∫d2k(2π)2ln(1− e−βωB)], (3.2)ǫF (T ) =12L∞∑n=−∞[−∫d2k(2π)2√k2 +2β∫d2k(2π)2ln(1 + e−βωF )], (3.3)where ωB and ωF are defined byωB =√kT2 + (πn/L)2, ωF =√kT2 + (π(2n+ 1)/2L)2. (3.4)Performing the integrations in Eqs. (3.2) and (3.3) with the same regularization factors asbefore we obtain for photons,6ǫB(T ) =3π2τ 4− π2720L4(3.5)−∞∑n=1(T 2n2π)2nπ2ξ coth(nπ2ξ)+(nπ2ξ)2cosech2(nπ2ξ) ,and for photinos,ǫF (T ) = − 3π2τ 4− 78π2720L4(3.6)+14πL4∞∑l=1(−1)l ξ2l2 sinh(lπ/2ξ){2ξl+ π coth(lπ2ξ)}.where ξ = L/β = LT and τ is the cut-off parameter. The total contribution of photons andphotinos in the supersymmetric quantum electrodynamics to the Casimir energy is given byǫC(T ) = ǫB(T )+ǫF (T ). We have shown the behaviors of ǫC(T ) as a function of T for L fixedin Fig. 1. Also shown in Fig. 1 are the individual contribution of photons and photinos tothe Casimir energy ǫCB(T ) and ǫCF (T ) where the divergent parts in Eqs. (3.6) and (3.7) areeliminated as in the case of the vanishing temperature. Note here that for low temperatureξ ≪ 1ǫB(T ) =3π2τ 4− π2720L4− ξ22L4{1 +2ξπ}e−π/2ξ, (3.7)and for high temperature ξ ≫ 1ǫB(T ) =3π2τ 4− π2720L4− 78π245T 4 (3.8)IV. PHOTINO MASS DEPENDENCEIn our calculations we have kept the photino massless in order to respect the supersymmetryat vanishing temperature. Experimentally, however, the massless photino has not beenobserved. One of the possibilities of explaining the situation may be that the photino couldbe extremely massive and is not be observed in the low energy experiments. The photinomass would be generated by some supersymmetry breaking mechanism. We consider thispossibility and recalculate the Casimir energy with massive photinos. The calculation isstraightforward and the result reads as follows:ǫF (T,m) = − 14πL∞∑n=−∞{αn2τ+2αnτ 2+2τ 3}e−ταn (4.1)+12πL∞∑n=−∞∞∑l=1(−1)l(T 3l3+T 2l2αn)e−lαn/T ,where αn =√m2 +(2n+12Lπ)2with m the photino mass. The behavior of the Casimir energyǫC at vanishing temperature with massive photinos is given in Fig. 2 as a function of7the photino mass m together with the individual contributions of photons and photinosrespectively.Finally we calculate the Casimir pressure by differentiating the energy ǫC(T,m) =ǫB(T ) + ǫF (T,m) in terms of distance L,p(T,m) = − ∂∂L[ǫC(T,m)L]∣∣∣T. (4.2)We have shown the behaviors of p(T,m)L4 as a function of T and m in FIG. 3 and 4respectively. It may be more transparent to show the Casimir energy and the Casimirpressure in the 2 dimensional plot with regards to T and m and those are given in FIG. 5and 6 respectively.Judging from the rather strong dependence of the Casimir energy as well as the Casimirpressure on the photino mass as shown in FIG. 5 and 6, it may be rather difficult to detectthe evidence of the supersymmetry in the presently accessible experimental situations. Wehope that future improvements of the experimental situation will resolve the difficulty.ACKNOWLEDGMENTSThe authors would like to thank Masahito Ueda for useful discussions. One of the authors(T. M.) is indebted to Grant-in-Aid for Scientific Research (C) provided by the Ministry ofEducation, Science, Sports and Culture for the financial support under the contract number08640377.8REFERENCES[1] H. B. G. Casimir, Proc. Kon. Ned. Akad. Wetensch.51 (1948) 793.[2] H. B. G. Casimir and D. Polder, Phys. Rev. 73 (1948) 360.[3] For a review see, for example, V. M. Mostepanenko and N. N. Trunov, The CasimirEffect and its Applications Clarendon Press, Oxford, 1997[4] K. Johnson, Acta Phys. Pol. B6 (1975) 865.[5] S. A. Gundersen and F. Ravndal, Ann. Phys. 182 (1988) 90.[6] A. K. Ganguly and P. B. Pal, Preprint hep-th/9803009 (1998).[7] Y. Igarashi, Phys. Rev. D30 (1984) 1812.[8] M. Nakahara, Phys. Lett. 142B (1984) 395.[9] A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn and V. F. Weisskopf, Phys. Rev. D9(1974) 3471.[10] T. Matsubara, Prog. Theor. Phys. 14(1955) 351.9FIGURES0 0.2 0.4 0.6 0.8 1LT−0.5−0.4−0.3−0.2−0.10.0εL4Lm=0photonphotinototalFIG. 1. The behavior of Casimir energy ǫCL4 as a function of temperature LT . Also shownare ǫCBL4 and ǫCFL4 respectively0 0.2 0.4 0.6 0.8 1Lm−0.030−0.025−0.020−0.015−0.010−0.0050.000εL4LT=0photonphotinototalFIG. 2. The behavior of Casimir energy ǫC(T )L4 as a function of photino mass Lm. Individualcontributions ǫCB(T )L4 and ǫCF (T )L4 respectively are also shown.100 0.2 0.4 0.6 0.8 1LT−0.10.00.10.20.30.4pL4Lm=0photonphotinototalFIG. 3. The behaviors of Casimir pressure pL4 as a function of temperature LT .0 0.2 0.4 0.6 0.8 1Lm−0.08−0.06−0.04−0.02pL4LT=0photonphotinototalFIG. 4. The behaviors of Casimir pressure pL4 as a function of photino mass Lm.110.20.40.60.81LT012345Lm-0.5-0.4-0.3-0.2-0.10eL4FIG. 5. The T and m dependence of Casimir energy ǫC(T )L4.0.20.40.60.81LT012345Lm00.10.20.3pL4FIG. 6. The T and m dependence of Casimir pressure pL4.12

Path-Integral formulation of Casimir effects in supersymmetric quantum electrodynamics

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Path-Integral Formulation of Casimir Effects in Supersymmetric Quantum
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Path-Integral Formulation of Casimir Effects in Supersymmetric Quantum Electrodynamics

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