Physics Letters B 807 (2020) 135567
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Physics Letters B
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A non-perturbative approach to the scalar Casimir effect with Lorentz symmetry violation ∗ C.A. Escobar a, , A. Martín-Ruiz b, O.J. Franca b, Marcos A. G. Garcia c a Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, Ciudad de México 01000, Mexico b Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, A. Postal 70-543, 04510 Ciudad de México, Mexico c Instituto de Física Teórica (IFT), UAM-CSIC, Campus de Cantoblanco, 28049 Madrid, Spain a r t i c l e i n f o a b s t r a c t
Article history: We determine the effect of Lorentz invariance violation in the vacuum energy and stress between two Received 30 April 2020 parallel plates separated by a distance L, in the presence of a massive real scalar field. We parametrize the Received in revised form 23 May 2020 Lorentz-violation in terms of a symmetric tensor h μν that represents a constant background. Through the Accepted 16 June 2020 Green’s function method, we obtain the global Casimir energy, the Casimir force between the plates and Available online 20 June 2020 the energy density in a closed analytical form without resorting to perturbative methods. With regards Editor: A. Volovichis √ ˜ μν to the pressure, we find that Fc (L) = F0(L)/ −det h , where F0 is the Lorentz-invariant √expression, ˜ μν ˜ nn Keywords: and L is the plate separation rescaled by the component of h normal to the plates, L = L/ −h . We Casimir effect also analyze the Casimir stress including finite-temperature corrections. The local behavior of the Casimir Thermal Casimir effect energy density is also discussed. Lorentz violation © 2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.
1. Introduction ical systems [13]. Among theoretical extensions one can list its generalization to spacetimes with non-trivial topologies [14,15], The existence of a zero-point vacuum energy is one of the dynamical boundary conditions [16], and non-Euclidean space- main tenets of the quantum formulation of the laws that we be- times [17–19]. In the latter case it offers an independent derivation lieve govern our Universe. In a Quantum Field Theory (QFT), the of the Hawking temperature from particle production from black presence of fluctuating zero-point fields implies the existence of a holes [20,21]. Modifications of the Casimir effect in the presence non-vanishing macroscopic force between the boundaries that de- of weak gravitational fields have been extensively studied [22–28]. limit a spatial region [1], due to the difference in the spectrum In this context, the Casimir effect can potentially provide clues on of quantized field modes inside and outside this region. When the connection between zero-point fluctuations and the cosmolog- the boundaries of this delimited spatial domain take the form of ical constant [29–32]. two parallel plates, this manifestation of the vacuum fluctuation is The Casimir effect stands as a potential handle to distinguish known as the Casimir effect [2]. The computation of the Casimir between Lorentz-invariant and Lorentz-violating formulations of force in QFT is a standard textbook exercise [3–5], and its exis- QFT. Lorentz invariance (LI) is one of the cornerstones behind QFT tence, in the case of Quantum Electrodynamics, has been verified and general relativity, and to date there are no experimental signs to a high precision [6–9]. of a departure from it [33]. Nevertheless, the quantum nature of The Casimir effect is now behind many experimental and theo- the spacetime at distances of the order of the Planck length (P ) retical pursuits. It is used as a tool to place constraints on Yukawa- has been shown to provide mechanisms that can lead to viola- type interactions [10,11], and it has been suggested as a poten- tion of LI in certain formulations of quantum gravity [34–36]. As tial probe for the detection of feebly-interacting axion-like dark an example, spontaneous Lorentz symmetry breaking can occur matter [12]. Casimir forces cannot be neglected at the nanoscale, within some string theories [35]. Therefore, a better understand- and must be accounted for in the design of microelectromechan- ing on the consequences of the breakdown of LI at scales larger than P would provide valuable information about the microscopic structure of spacetime. In this Letter we explore the manifesta- * Corresponding author. tion of the spontaneous breakdown of LI, induced by a constant E-mail addresses: carlos_escobar@fisica.unam.mx (C.A. Escobar), background tensor, on the Casimir effect for a real massive scalar [email protected] (A. Martín-Ruiz), [email protected] (O.J. Franca), [email protected] field between two parallel conductive plates in flat spacetime. We (M.A.G.Garcia). also explore how thermal corrections are affected by the Lorentz https://doi.org/10.1016/j.physletb.2020.135567 0370-2693/© 2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 2 C.A. Escobar et al. / Physics Letters B 807 (2020) 135567
symmetry breaking. Our work provides a generalization of previ- Here z> (z<) is the greater (lesser) between z and z . The coeffi- ous studies of LI violation in the Casimir context [37,38]. cients ξ0,1 denote the following combinations of energy-momenta and the Lorentz-violating tensor, 2. The model 1 03 ¯i3 ξ0 = (h ω − h k⊥¯), (6) Arguably, the most straightforward way to implement Lorentz h33 i violation is by means of the introduction of a tensor field with a 1/2 1 03 ¯i3 2 33 2 ξ1 = (h ω − h k⊥¯) − h γ , (7) non-zero vacuum expectation value (VEV). When coupled to the |h33| i Standard Model fields the spontaneous symmetry breaking, in- duced by the non-zero VEV, is manifested as preferential directions where on the spacetime, leading to a breakdown of LI. In the case of a 2 00 2 0¯i ¯i¯j 2 real scalar field in flat spacetime, with the Minkowski metric with γ = h ω − 2h ωk⊥ + h k⊥ k⊥ − m . (8) ¯i ¯i ¯j signature (+, −, −, −), we parametrize this coupling in the follow- μν → μν ing form, The LI limit is recovered by taking h η , which implies → 2 → 2 − 2 − 2 ξ0 0 and ξ1 ω k⊥ m . It is worth noting that the case 1 μν 1 2 2 with Neumann conditions can be trivially recovered by replacing L = h ∂μφ∂νφ − m φ . (1) 2 2 sin → cos in the numerator of (5). The determination of the Casimir energy and stress requires not Here h μν is a symmetric tensor that represents a constant back- ground, independent of the spacetime position, and which does only the GF for the two plate setup, but also the GF in the absence not transform as a second order tensor under active Lorentz trans- of plates and a single plate. For the former, we find 1 formations. Naturally, causality, the positive energy condition and iξ (z−z ) stability impose restrictions on the components of h μν . i e 0 − g (z, z ) =− eiξ1(z> z<) , (9) v 33 2ξ1 h Consider now the following set-up: a pair of parallel, conduc- tive plates, orthogonal to the zˆ-direction, located at z = 0 and while for the latter, z = L, on which Dirichlet boundary conditions apply for the field φ. − = = = = iξ0(z z ) That is, φ(z 0) φ(z L) 0. We now solve for the scalar field e iξ (z −L) g|(z, z ) =− sin[ξ (z − L)]e 1 > . (10) 33 1 < between the plates, applying the Green’s function technique [40]. ξ1h Namely, we are interested in computing the time-ordered, vac- uum two-point correlation function G(x, x ) =−i0|T φ(x)φ(x )|0, In order to quantify the Casimir effect, we need an expression which as is well known (see e.g. [41]) satisfies the Green’s function for the vacuum expectation value for the stress-energy tensor of μν = μα ν − μνL (GF) equation the scalar field, T h ∂αφ∂ φ η . In terms of the GF, it can be generically computed as [40] (4) O G(x, x ) = δ (x − x ). (2) x T μν=−i lim hμα ν G x x − μνL (11) ∂α ∂ ( , ) η , Here, in the configuration space, the modified Klein-Gordon oper- x→x ator has the following explicit form, while the VEV of the Lagrangian density can be written as L = 1 μν 2 ¯ ¯¯ ¯ −i lim → h ∂ ∂ − m G(x, x ). O = 00 2 + 0i + i j + 03 + i3 x x 2 μ ν x h ∂0 2h ∂0∂¯i h ∂¯i∂ ¯j 2h ∂0∂3 2h ∂¯i∂3 Substitution of (4)leads to the following expressions for the + h33∂2 + m2 (3) z energy density and the pressure in the zˆ-direction, ¯ ¯ with i, j = 1, 2. In the chosen coordinate system, the GF is invari- 2 ant under translations in the (xˆ, yˆ )-plane. Taking advantage of this 00 dω d k⊥ 00 2 0¯i T =−i lim h ω − h ωk⊥¯ symmetry, we can express the GF in terms of the Fourier transform z→z 2π (2π)2 i in the direction parallel to the plates, 03 +ih ω∂z g(z, z ) −L, (12) 2 d k⊥ ik⊥·(x⊥−x ) dω −iω(t−t ) 2 G(x, x ) = e ⊥ e g z, z ; ,k⊥ , i dω d k⊥ 2 ω 33 2 33 (2π) 2π T =− lim γ − h ∂z∂z g(z, z ). 2 z→z 2π (2π)2 (4) (13) where k⊥ = (kx, ky) and x⊥ = (x, y). Henceforth we will drop the where explicit dependence on ω, k⊥ of g for simplicity. After substitution of (4)into (2), and straightforward integration of the resulting 1D 2 i dω d k⊥ 2 33 2 L=− lim + h ∂ ∂ boundary problem, an exact solution for the reduced GF between γ z z 2 z →z 2π (2π)2 the plates can be found, − 33 − [ − ] ih ξ0(∂z ∂z) g(z, z ). (14) −iξ (z −z) sin(ξ1z<) sin ξ1(z> L) g (z, z ) = e 0 . (5) 33 h ξ1 sin(ξ1 L) 3. Casimir effect with Lorentz symmetry violation
1 Single derivative terms, such as iφuμ∂ φ with uμ a constant 4-vector, can be μ With the VEV of the stress-energy tensor at hand, we now pro- reduced to surface terms, which in absence of topological effects do not have phys- ical contributions [39]. ceed to compute the global Casimir energy and the Casimir stress 2 An analogous step-by-step procedure can be found in [40]. upon the plates in the presence of Lorentz-invariance violation. C.A. Escobar et al. / Physics Letters B 807 (2020) 135567 3
3.1. Global Casimir energy where K2(x) is the second-order Bessel function of the second kind. Note that the Lorentz-violating result reduces trivially to the The renormalized vacuum energy stored between the parallel LI one as h33, h →−1, which would be the case for hμν → ημν . plates can be computed formally as the difference between the Although the sum which appears in (21)does not have an ana- zero-point energy in the presence of the boundary, T 00 , and that lytical closed form, it can be reduced to simple expressions in the 00 of the free vacuum, T v . Namely, large and small mass limits, ⎧ L ⎪ π 2 m2 ⎨⎪ − , mL˜ 1 , 00 00 33 ˜ ˜ EC (L) = T −T v dz. (15) h 1440L3 96L E (L) − × (22) C ⎪ 2 0 h ⎪ m π − ˜ ⎩⎪ e 2mL , mL˜ 1 . 2 ˜ ˜ We begin by evaluating T 00 . As a first step, it can be noted after 16π L mL a cursory computation that the contribution from the VEV of L The massless case is trivially recovered taking the m → 0 limit in in (12)is L-independent and will therefore not contribute to the the previous equation. Casimir pressure.3 After simplification, the remaining terms in (12) can be rearranged to lead to the following expression, 3.2. Stress on the plates 2 00 dω d k⊥ 00 2 0¯i We now proceed to determine the Casimir stress upon the plate T =−i h ω − h ω k⊥¯ 2 (2 )2 i at z = L by direct evaluation of the normal-normal component π π 33 03 of the stress-energy tensor (13). Denoting by T the vacuum − h ξ g (z, z). (16) 33 ω 0 stress due to the confined scalar field, and by T | the stress due to the field above the plate, we can write The term inside the brackets in the previous equation is a quadratic form in (ω, kx, ky), with coefficients given by the compo- 33 33 FC (L) =T −T |. (23) nents of h μν . This quadratic form is different from that appearing | 33| 2 In a similar fashion to the previous computation of the Casimir in the argument of the GF, h ξ1 , and this makes the evaluation of (16)a non-trivial task. However, a closed-form solution may energy, all it takes to calculate these stresses is to substitute the be obtained by diagonalization of the latter quadratic form, map- corresponding reduced GFs into (13), and to repeat the quadratic | 33| 2 = 2 − 2 − 2 − 2 form diagonalization procedure. A key difference in this analysis is ping it into a mimic of the LI case, h ξ1 ω kx ky m , 4 where primed quantities correspond to the rotated frequency and the fact that the Lagrangian density does contribute to the stress. momenta. Further performing a Wick rotation ω → iζ , it can be Nevertheless, despite this relative complication, a straightforward shown that (16)is equivalent to the following expression, calculation using (5) and (10)yields 2 ˜ 2 1 dζ d k ⊥ sinh(γ z˜) sinh[γ (z˜ − L)] 33 1 dζ d k⊥ γ ˜ T 00 = √ ζ 2 , T =−√ coth(γ L), − 2π (2π)2 ˜ −h 2π (2π)2 2 h γ sinh(γ L) (24) 2 (17) 1 dζ d k⊥ γ T 33| =−√ . 2 2 2 2 2 μν −h 2π (2π) 2 where now γ = ζ + k⊥ + m , h ≡ det h , and Each stress contains an L-independent divergent term that is can- z L z˜ = √ , L˜ = √ . (18) celed by the regularization provided by (23). Substitution of these −h33 −h33 expressions into (23), and following the same steps that lead to An entirely analogous procedure can be followed to evaluate Eq. (20), produces 00 the vacuum energy density T v , making use in this case of the ∞ √ 2 2 + 2 corresponding GF (9). For it we obtain 1 ˜ 1 1 τ τ m FC (L) = √ F0(L) = √ dτ . (25) −h −h 4π 2 2L˜ τ 2+m2 − 2 2 e 1 1 dζ d k⊥ ζ 0 T 00 =−√ . (19) v 2 −h 2π (2π) 2γ In this expression F0 denotes the LI result. Expectedly, the stress in the LI violating result is proportional to the stress in the ab- Finally, substituting into (15), integrating with respect to z and sence of Lorentz violation, but evaluated at the rescaled length L˜ . dropping an L-independent constant term leads to the following For a vanishing√ scalar field mass, Eq. (25)reduces to FC (L)|m=0 = expression for the vacuum energy between the plates, −π 2/(480L˜ 4 −h). As a consistency check, one can verify that the Casimir energy 33 2 2 h dζ d k⊥ ζ ˜ ˜ (21) and the stress (25)are connected by the elementary relation EC (L) =− L [coth (γ L) − 1]. (20) h 2π (2π)2 2γ ∂E (L) F =− C E C (L) . (26) This resulting integral can be recognized as the LI result, 0, ∂ L rescaled by the factor h33/h, with a rescaled separation between the plates (18)[40]. Integration gives 3.3. Local effects h33 m2 h33 ∞ 1 In Section 3.1 we derived an expression for the global Casimir ˜ ˜ 00 00 EC (L) = E0(L) =− K2(2mnL), (21) energy by computing the integral of T −T v in the region h 8π 2 L˜ h n=1 n2
4 L also plays a fundamental role regarding the behavior of the field near the L dω d2k⊥ 3 More precisely, L dz = (1/2i) . boundaries, see Section 3.3. 0 2π (2π)2 4 C.A. Escobar et al. / Physics Letters B 807 (2020) 135567 between the plates by means of the GF method. Although alter- the presence of a non-trivial h μν they are in general non-zero, al- native methods exist to evaluate EC [42], the power of the GF though they can also be related to the 00 and 33 components by procedure arises clearly when studying the local energy density, symmetry arguments. A cursory computation provides the follow- which in turn reveals the divergent structure of the theory. The ing general expression for the VEV of the stress-energy tensor, computation of T μν is the goal of this section. α3 We begin with the energy density per unit volume between 2h T μν=− (η μα + nμnα)nν T 00−T 33− f (z) the plates. Without dropping in this case the contribution of h33 L (which was discarded in the global analysis due to its L- +(η μν + nμnν) T 00+nμnν T 33 . (32) independence after integration), the same analysis that led to (17) μ = in this case gives Here n (0, 0, 0, 1) is the unit vector perpendicular to the plates, 00 33 and T and T are given by Eqs. (28) and (24), respectively. 2 1 dζ d k⊥ Clearly, in the LI limit the first term vanishes and we recover the T 00=−√ − 2 (2 )2 usual structure of the vacuum stress [43]. h π π 2 2 2 ˜ ˜ ζ k⊥ + m cosh[γ (2z − L)] 3.4. Finite temperature effects × coth(γ L˜) + . (27) 2γ 2γ sinh(γ L˜) The Casimir effect, as described in the previous sections, is a The introduction of the polar coordinates k⊥ = ρ cos θ, ζ = ρ sin θ, manifestation of the fluctuations of the φ field in the vacuum. where ρ ∈[0, ∞) and θ ∈[−π/2, π/2], leads to the following re- However, any realistic parallel plate setup will necessarily be im- sult mersed in a bath with a temperature above absolute zero. It is ∞ therefore crucial to determine the effect that thermal fluctuations 1 1 4 2 00=− √ ρ would have in the Casimir stress. Luckily, in our relatively simple T ∗ ∗ ˜ 12π 2 −h γ e2γ L − 1 scenario, the stress at T > 0case can be determined in a straight- 0 forward manner. ∗ ∗ 2 2γ z˜ 2γ (L˜−z˜) In the Matsubara formalism of finite temperature QFT, the ρ ∗ e + e + (2 2 + m2) d . (28) ∗ γ ∗ ˜ ρ Casimir stress at nonzero temperature can be obtained from γ e2γ L − 1 − ∞ Eq. (24)upon the replacement dζ/2π → β 1 , together n=−∞ ∗ Here γ = ρ2 + m2, and we have discarded an L-independent with mapping the imaginary frequency ζ to the discrete Matsubara ≡ = term. Denoting by U the z-independent term in the previous ex- frequency ζn 2πn/β [44]. Here β 1/kB T , with kB the Boltz- mann constant. These substitutions yield pression, one can easily show that U = EC /L. Similarly, a straight- +∞ forward change of variables allows us to write the z-dependent 2 1 d k⊥ γn term of (28), which we denote by f (z), as follows, F (L; T ) =− √ , (33) C 2 ˜ β h (2π) 2γn L − ∞ n=−∞ e 1 1 1 =− √ 2 − ˜ 2 f (z) y (2mL) = 2 + 2 + 2 192π 2 L˜ 4 −h where γn ζn k⊥ m . Although this expression lacks a 2mL˜ closed form in terms of elementary functions, we can gain some − e yz/L + e y(1 z/L) insight of its behavior in the massless case for small temperature × 2y2 + (2mL˜)2 dy. (29) e y − 1 and large temperature (classical) limits. For low temperature, the above expression for the pressure takes the form In the massless limit, this function can be expressed in terms of = ∞ + −s 2 the Hurwirtz zeta function, ζ(s, a) n=0(n a) , π 1 4 60 −4π 2/s FC (L; T 1) ≈− √ 1 + s − se , ˜ 4 48π 4 π 2 1 (h33)2 480L h f (z) =− √ [ζ(4, z/L) + ζ(4, 1 − z/L)] . (30) (34) 16π 2 L4 −h where s = 4πk T L˜ 1. Clearly this result is consistent with the Therefore we have found that T 00 = U + f (z). U encodes B Nernst heat theorem, since the associated entropy vanishes as s the part of the vacuum energy resulting in an observable force, goes to zero. whereas f (z) corresponds to a local, divergent effect that does not In the opposite regime, at high temperatures, all terms in the contribute to the pressure, as the L-independence of the following sum of Eq. (33)except the n = 0term are exponentially sup- integral confirms pressed, resulting in L ∞ 1 h33 dx ζ(3)k T k T s2 2 2 2 2 B B −s f (z)dz =− x − 4m (x + 2m ) . (31) FC (L; T 1) ≈− √ − √ 1 + s + e , (35) 48π 2 h x 8π L˜ 3 h 4π L˜ 3 h 2 0 2m where here s 1. The leading term can also be obtained from the In the massless case this divergence is quartic as z approaches the Helmholtz free energy for Lorentz-violating massless bosons. plates, as can be appreciated from Eq. (30). For a generic mass the The results of equations (34) and (35) exhibit an interesting complex form of (29)prevents us from analytically determining behavior as a function of the Lorentz√ violating parameter h33 the degree of divergence. through the rescaled length L˜ = L/ −h33. When Lorentz invari- We turn now to the evaluation of the VEV for the remaining ance is mildly broken, h33 ≈−1, and hence the conditions s 1 μν components of T . Owing to the symmetry of the setup, these and s 1 correspond to low and high temperatures, respectively. components can be easily determined. For example, rotational in- However, when Lorentz symmetry breaking is not negligible, such variance around the z-axis immediately implies that T 11 =T 22. conditions are relaxed and possibly flipped. For example, when − Moreover, after an explicit calculation we find that T 11 =−T 00. h33 ≈ 0 , the condition s 1can be fulfilled even for low tem- The off-diagonal components of T μν vanish in the LI limit, but in peratures. C.A. Escobar et al. / Physics Letters B 807 (2020) 135567 5
4. Summary and discussion which generically leads to anisotropies, will play an analogous role to that of a background in empty space. In the present work we have obtained explicit expressions for the Casimir energy and force between two parallel conductive Declaration of competing interest plates, arising from the vacuum fluctuations of a massive real scalar field, in the presence of a generic background defined by The authors declare that they have no known competing finan- the tensor h μν in Eq. (1). This background is motivated by theo- cial interests or personal relationships that could have appeared to ries in which the breakdown of Lorentz invariance manifests itself influence the work reported in this paper. as the non-vanishing vacuum expectation value of a fundamental field. Acknowledgements Since no deviation from Lorentz invariance has been experi- mentally observed yet, the perturbative expansion h μν = η μν + A.M.-R. acknowledges support from DGAPA-UNAM Project No. kμν is justified. Here η μν is the Minkowski metric and kμν is a IA101320. C.A.E. is supported by a UNAM-DGAPA postdoctoral fel- constant tensor whose components are much smaller than one lowship and Project PAPIIT No. IN111518. M.A.G.G. is supported |k μν| 1. Working to first order in k μν , it is possible to prove by the Spanish Agencia Estatal de Investigación through the grants that the Lorentz-violating theory described by Eq. (1)can be trans- FPA2015-65929-P (MINECO/FEDER, UE), PGC2018095161-B-I00, IFT formed into the standard Lorentz-invariant theory by an appro- Centro de Excelencia Severo Ochoa SEV-2016-0597, and Red Con- μ = μ − 1 μ ν solider MultiDark FPA2017-90566-REDC. O. J. F. acknowledges sup- priate change of spacetime coordinates x x 2 k ν x [45]. In this new coordinate system it is relatively straightforward to port from DGAPA-UNAM Project No. IN103319. evaluate the Casimir energy. It is given by the Lorentz-invariant re- sult, albeit with a redefinition of the separation between the plates References and a global multiplicative factor arising from the Jacobian of the transformation. Let us discuss our result in Eq. (25)in this ap- [1] V. Mostepanenko, N. Trunov, The Casimir effect and its applications, Sov. Phys. Usp. 31 (1988) 965–987, https://doi .org /10 .1070 /PU1988v031n11ABEH005641. proximation. One can verify that the global multiplicative factor, √ [2] H. Casimir, On the attraction between two perfectly conducting plates, K. Ned. − 1/ h in Eq. (25), corresponds to the square root of the Jacobian, Akad. Wet. Proc. 51 (1948) 793. 1 ˜ ≈ + 33 [3] A. Zee, Quantum Field Theory in a Nutshell, Princeton University Press, Prince- whereas L L(1 2 k ) is precisely the transformed distance be- tween plates. 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