Physics Letters B 807 (2020) 135567 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb 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 3 (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP . 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). μν The determination of the Casimir energy and stress requires not Here h is a symmetric tensor that represents a constant back- only the GF for the two plate setup, but also the GF in the absence ground, independent of the spacetime position, and which does of plates and a single plate. For the former, we find not transform as a second order tensor under active Lorentz trans- 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- In order to quantify the Casimir effect, we need an expression uum two-point correlation function G(x, x ) =−i0|T φ(x)φ(x )|0, 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 μν the scalar field, T = h ∂ φ∂ φ − η L. In terms of the GF, it (GF) equation α 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 ).