Introduction/purpose: The quantization of the electromagnetic field gives

rise to quantum fluctuations which in turn produce a force on macroscopic boundaries. This phenomenon is called the Casimir effect.

Method: The second quantization of the electromagnetic field is employed. The Zeta function regularization technique has been applied.

Results: Because of the electromagnetic field quantization, a force on

macroscopic boundaries is observed.

Conclusions: Vacuum fluctuations due to quantum effects give macroscopic results

Nicola Fabiano

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ПРЕГЛЕДНИ РАДОВИОБЗОРНЫЕ СТАТЬИREVIEW PAPERSThe Casimir effectNicola FabianoUniversity of Belgrade, “Vinča” Institute of Nuclear Sciences - NationalInstitute of the Republic of Serbia, Belgrade, Republic of Serbia,e-mail: nicola.fabiano@gmail.com,ORCID iD:https://orcid.org/0000-0003-1645-2071DOI: 10.5937/vojtehg71-41282;https://doi.org/10.5937/vojtehg71-41282FIELD: mathematicsARTICLE TYPE: review paperAbstract:Introduction/purpose: The quantization of the electromagnetic field givesrise to quantum fluctuations which in turn produce a force on macro-scopic boundaries. This phenomenon is called the Casimir effect.Method: The second quantization of the electromagnetic field is em-ployed. The Zeta function regularization technique has been applied.Results: Because of the electromagnetic field quantization, a force onmacroscopic boundaries is observed.Conclusions: Vacuum fluctuations due to quantum effects give macro-scopic results.Key words: quantum electrodynamics, quantization, vacuum energy,Casimir effect.The Casimir effectWe are going to consider an effect arising from the electromagnetic fieldquantization. Casimir first observed that, because of quantum fluctuationsof the electromagnetic field, between two neutral parallel conducting platesseparated by a distance d and located in a vacuum, there is a force whichis attractive in this particular geometry. This is known as the Casimir ef-fect (Casimir, 1948; Casimir & Polder, 1948; Casimir, 1953). Only trans-verse modes will contribute to the energy, and, assuming the plates areperpendicular to the direction x of propagation, that component will have740VOJNOTEHNIČKIGLASNIK/MILITARYTECHNICALCOURIER,2023,Vol.71,Issue3nodes on the planes and will take discrete valuesk = (kx = nπ/d, ky, kz), n = 1, 2, . . . ;n =(dπ)k . (1)The zero point energy of this configuration is given byE =+∞∑n=112ℏωn =∑n12ℏc|kn| (2)For the sake of simplicity, we will limit ourselves to a 1 + 1 dimen-sional space, i.e. 1D, in order not to deal with ky and kz, the componentsof the electromagnetic wave vector. Therefore, the modes are given bysin(nπx/d), where n = 1, 2, . . . and the corresponding energy is ωn = πn/d.Plugging all this back into eq. (2), for the vacuum energy in one spatial di-mension, we obtain the expressionE(d) =π2ℏcd+∞∑n=1n . (3)Of course the term∑+∞n=1 n is a source of trouble being divergent. Inorder to solve this problem, let us introduce the generating function+∞∑n=1e−an (4)for a > 0, this series is, of course, convergent to11− e−a(5)and has the property that− dda+∞∑n=1e−an =+∞∑n=1ne−an (6)so that in the limit a → 0 we recover the sum of eq. (3). Besides its mathe-matical properties in the series, the parameter a plays a role as a cutofffor frequencies with a wavenumber n larger than 1/a. We will isolate thesource of divergence for a → 0, obtaining a finite value for the zero pointenergy.741Fabiano,N.,TheCasimireffect,pp.740–747Taking the derivative of eq. (5) with respect to a, we obtain+∞∑n=1ne−an = − dda(11− e−a)=1ea + e−a − 2. (7)Expanding (7) with Taylor for small a (i.e. taking into account higherfrequencies), we have1a211 + 112a2 +O(a4)=1a2(1− 112a2 +O(a4))=1a2− 112+O(a2) (8)The expression for the zero point energy (3) becomesE(d) =π2ℏcd(1a2− 112+O(a2))(9)and we observe that the divergent part goes like 1/a2, while all other re-maining terms are regular in the limit a → 0. As it stands, E does not have adefinite value. Remembering that the energy is defined up to a constant, weshould regularize it by subtracting a suitable “counterterm” EC(d) that willeventually furnish us with a finite value for E. For a discussion of the coun-terterms in Quantum Electrodynamics, consult (Fabiano, 2021), and for thevarious regularization techniques in Quantum Field Theory, see (Fabiano,2022). Defining the counterterm in the following mannerEC(d) =π2ℏcd1a2(10)that is, the sole divergent part of (9) and subtracting it to E(d), we end upwith a perfectly regularized energy value for a = 0:ER(d) = E(d)− EC(d) =π2ℏcd(− 112+O(a2))(11)What experimentalists do measure in the Casimir effect is the forceamong plates, that is F = −∂E/∂d and, of course, this value does notblow up. The complete expression for the (attractive) force between two742VOJNOTEHNIČKIGLASNIK/MILITARYTECHNICALCOURIER,2023,Vol.71,Issue3plates isF = −∂E(d)∂d= −π2ℏcd2(1a2− 112+O(a2))(12)whose regular part is just given by FR = −∂ER(d)/∂d:FR =π24ℏcd2(13)as all other terms vanish in the limit a → 0. This is the finite result for theCasimir force in 1D.To summarize, while computing the zero point energy, we stumble uponthe divergent term∑n n. In order to regularize its behavior, we introducea parameter a that goes to zero thus discovering the pole 1/a2 and otherregular terms.Then we subtract a suitably “infinite” term that cancels thedivergent part and retain the finite value for a = 0.For the sake of completeness, a more exhaustive expansion of (7) inpowers of a is given by1a2− 112+a2240− a46048+a6172800− a85322240+691 a10118879488000+O(a12), (14)where the coefficients are related to Bernoulli numbers.An alternative approach to the generating function, of course completelyequivalent, is the well–known regularization via the Riemann zeta functiondefined for ℜ(s) > 1ζ(s) =+∞∑n=1n−s (15)and extended on the whole complex plane to a meromorphic function, i.e.that is holomorphic everywhere except for a simple pole at s = 1 withresidue 1, see for instance (Fabiano, 2020). It is possible to show thatζ(−1) = − 112. (16)Three dimensional caseIn spatial 3D, that is 3 + 1 dimensions, we restore the other two com-ponents ky and kz of the wave vector which of course are not subject to743Fabiano,N.,TheCasimireffect,pp.740–747boundary conditions due to the presence of the plates. The frequency iswritten asωn = c√n2π2d2+ k2y + k2z (17)giving the expression for the energyE(d) =ℏ2· 2A∫dkydkz(2π)2+∞∑n=1ωn (18)where A is the area of the conducting plates, the factor 2 accounts for thetwo polarizations.Turning to polar coordinates in two dimensions by setting r2 = k2y + k2zand performing the angular integration, we arrive at the density of energyper surfaceE(d)A=ℏc2π+∞∑n=1∫ +∞0dr r√n2π2d2+ r2. (19)As it stands, this integral is strongly divergent. As a regularization mea-sure, we multiply everything by ω−an and eventually let a → 0. The expres-sion we obtain is given byE(d)A=ℏc1−a2π+∞∑n=1∫ +∞0dr r(n2π2d2+ r2) 1−a2, (20)and this integral is well behaved for ℜ(a) > 3. Performing the integration,we arrive atE(d)A= − ℏ2πc1−a3− a(πd)3−a +∞∑n=1n3−a, (21)the sum on the rhs is recognized to be the zeta function, ζ(a − 3), whichis not singular for a = 0 and assumes the value ζ(−3) = 1/120. Thecomplete zero point energy density in three dimensions is therefore givenby the expression:E(d)A= − π2720ℏcd3. (22)It is worth noticing that in 3D a different geometry of the plates in theCasimir effect could change the sign of the force making it, say, attractiveinstead of repulsive.744VOJNOTEHNIČKIGLASNIK/MILITARYTECHNICALCOURIER,2023,Vol.71,Issue3As a byproduct, we have just “proved” that1 + 2 + 3 + 4 + 5 + . . . = − 112. (23)The Casimir effect has been explicitly shown here for two parallel platesin 1D and 3D respectively, and the force is attractive.There are many more possible situations in which the effect couldbe observed. Lifshits (Lifshitz, 1956) studied the case of two paral-lel dielectric bodies and the effects of finite temperature; the case ofa liquid layer of separation was studied by Dzyaloshinskii, Lifshitz andPitaevskii (Dzyaloshinskii et al, 1961). They also showed that, undercertain circumstances, the Casimir force could be repulsive rather thanattractive. Schwinger (Schwinger, 1951, 1975) studied the problem;in (Schwinger, 1992a) he used a different approach from the effective ac-tion, and in (Schwinger, 1992b,c), he started the calculations for a spheri-cally shaped object.ReferencesCasimir, H.B.G. 1948. On the attraction between two perfectly conductingplates. Proc. Kon. Ned. Akad. Wet., 51, p.793 [online]. Available at:https://dwc.knaw.nl/DL/publications/PU00018547.pdf [Accessed: 18 November2022].Casimir, H.B.G. 1953. Introductory remarks on quantum electrodynamics.Physica, 19(1-12), pp.846-849. Available at:https://doi.org/10.1016/S0031-8914(53)80095-9.Casimir, H.B.G. & Polder, D. 1948. The Influence of Retardation on the London-van der Waals Forces. Physical Review, 73(4), pp.360-372. Available at:https://doi.org/10.1103/PhysRev.73.360.Dzyaloshinskii, I.E., Lifshitz, E.M. & Pitaevskii, L.P. 1961. General Theory ofvan der Waals’ Forces. Soviet Physics Uspekhi, 4(2), pp.153-176. Available at:https://doi.org/10.1070/PU1961v004n02ABEH003330.Fabiano, N. 2020. Zeta function and some of its properties. Vojnotehnički glas-nik/Military Technical Courier, 68(4), pp.895-906. Available at:https://doi.org/10.5937/vojtehg68-28535.Fabiano N. 2021. Corrections to propagators of quantum electrodynamics. Voj-notehnički glasnik/Military Technical Courier, 69(4), pp.930-940. Available at:https://doi.org/10.5937/vojtehg69-30604.Fabiano N. 2022. Regularization in quantum field theories. Vojnotehnički glas-nik/Military Technical Courier, 70(3), pp.720-733. Available at:https://doi.org/10.5937/vojtehg70-34284.745Fabiano,N.,TheCasimireffect,pp.740–747Lifshitz, E.M. 1956. The Theory of Molecular Attractive Forces between Solids.Journal of Experimental and Theoretical Physics (JETP)/Soviet Physics, 2(1),pp.73-83 [online]. Available at: http://jetp.ras.ru/cgi-bin/e/index/e/2/1/p73?a=list.[Accessed: 18 November 2022].Schwinger, J. 1951. On Gauge Invariance and Vacuum Polarization. PhysicalReview, 82(5), pp.664-679. Available at: https://doi.org/10.1103/PhysRev.82.664.Schwinger, J. 1975. Casimir effect in source theory. Letters in MathematicalPhysics, 1(1), pp.43-47. Available at: https://doi.org/10.1007/BF00405585.Schwinger, J. 1992a. Casimir energy for dielectrics. PNAS, 89(9), pp.4091-4093. Available at: https://doi.org/10.1073/pnas.89.9.4091.Schwinger, J. 1992b. Casimir effect in source theory II. Letters in MathematicalPhysics, 24(1), pp.59-61. Available at: https://doi.org/10.1007/BF00430003.Schwinger, J. 1992c. Casimir effect in source theory III. Letters in MathematicalPhysics, 24(3), pp.227-230. Available at: https://doi.org/10.1007/BF00402898.Эффект КазимираНикола ФабианоБелградский университет, Институт ядерных исследований«Винча» − Институт государственного значения для РеспубликиСербия, г. Белград, Республика СербияРУБРИКА ГРНТИ: 29.05.03 Математические методытеоретической физики,29.05.23 Релятивистская квантовая теория.Квантовая теория поля29.05.33 Электромагнитное взаимодействиеВИД СТАТЬИ: обзорная статьяРезюме:Введение/цель: Квантование электромагнитного поля по-рождает квантовые флуктуации, которые, в свою оче-редь, создают силу на макроскопических границах. Этотфеномен называется эффектом Казимира.Методы: В данной статье используется вторичное кван-тование электромагнитного поля. Также применяетсяметод регуляризации с помощью дзета-функции.Результаты: В результате квантования электромагнит-ного поля на макроскопических границах наблюдается си-ла.Выводы: Флуктуации вакуума из-за квантовых эффектовдают макроскопические результаты.Ключевые слова: квантовая электродинамика, квантова-ние, энергия вакуума, эффект Казимира.746VOJNOTEHNIČKIGLASNIK/MILITARYTECHNICALCOURIER,2023,Vol.71,Issue3Казимиров ефекатНикола ФабианоУниверзитет у Београду, Институт за нуклеарне науке „Винча” -Институт од националног значаја за Републику Србију,Београд, Република СрбијаОБЛАСТ: математикаКАТЕГОРИЈА ЧЛАНКА: прегледни радСажетак:Увод/циљ: Квантизација електромагнетног поља доводидо квантних флуктуација које заузврат производе силу намакроскопским границама. Овај феномен назива се Кази-миров ефекат.Методе: Користи се друга квантизација електромагнет-ног поља. Примењена је техника регуларизације помоћузета-функције.Резултати: Због квантизације електромагнетног пољапримећује се сила на макроскопским границама.Закључак: Флуктуације вакуума услед квантних ефекатадају макроскопске ефекте.Кључне речи: квантна електродинамика, квантизација,енергија вакуума, Казимиров ефекат.Paper received on / Дата получения работы / Датум пријема чланка: 20.11.2022.Manuscript corrections submitted on / Дата получения исправленной версии работы /Датум достављања исправки рукописа: 12.06.2023.Paper accepted for publishing on / Дата окончательного согласования работы / Датумконачног прихватања чланка за објављивање: 14.06.2023.© 2023 The Authors. Published by Vojnotehnički glasnik / Military Technical Courier(http://vtg.mod.gov.rs, http://втг.мо.упр.срб). This article is an open access article distributed underthe terms and conditions of the Creative Commons Attribution license(http://creativecommons.org/licenses/by/3.0/rs/).© 2023 Авторы. Опубликовано в ”Военно-технический вестник / Vojnotehnički glasnik / MilitaryTechnical Courier” (http://vtg.mod.gov.rs, http://втг.мо.упр.срб). Данная статья в открытом доступеи распространяется в соответствии с лицензией ”Creative Commons”(http://creativecommons.org/licenses/by/3.0/rs/).© 2023 Аутори. Објавио Војнотехнички гласник / Vojnotehnički glasnik / Military Technical Courier(http://vtg.mod.gov.rs, http://втг.мо.упр.срб). Ово је чланак отвореног приступа и дистрибуира сеу складу са Creative Commons лиценцом (http://creativecommons.org/licenses/by/3.0/rs/).747Fabiano,N.,TheCasimireffect,pp.740–747

The Casimir effect

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The Casimir effect

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