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Both graphs represent the Casimir force density versus separation distance. a The solid curves represent roughness for different values of the anisotropic scaling parameter z = 1, 2, 3, with a constant temperature T = 100. The dashed curve represents the case without
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We investigate the Casimir effect of a rough membrane within the framework of the Hořava–Lifshitz theory in \(2+1\) dimensions. Quantum fluctuations are induced by an anisotropic scalar field subject to Dirichlet boundary conditions. We implement a coordinate transformation to render the membrane completely flat, treating the remaining terms associ...
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Context 1
... Casimir force density is obtained deriving the Casimir energy density with respect to the separation a. Henceforth, we proceed with numerical calculations. In the Fig. 2, we present the force density versus separation distance graphs for the case of a membrane with periodic border (see example in Eq. (3.10)), considering the effects of temperature and the anisotropic scaling factor. In the Fig. 2a, an increase in the magnitude of the force is observed, which is compared to the case zero temperature. ...
Context 2
... Casimir energy density with respect to the separation a. Henceforth, we proceed with numerical calculations. In the Fig. 2, we present the force density versus separation distance graphs for the case of a membrane with periodic border (see example in Eq. (3.10)), considering the effects of temperature and the anisotropic scaling factor. In the Fig. 2a, an increase in the magnitude of the force is observed, which is compared to the case zero temperature. Likewise, it is noted that as the anisotropic scaling factor increases, the magnitude of the force density also increases. In contrast to the case zero temperature, there is a contribution to the Casimir force density for even values ...
Context 3
... the force is observed, which is compared to the case zero temperature. Likewise, it is noted that as the anisotropic scaling factor increases, the magnitude of the force density also increases. In contrast to the case zero temperature, there is a contribution to the Casimir force density for even values of the anisotropic scaling factor. In the Fig. 2b, when the temperature increases for a constant value of the anisotropic scaling factor, there is an increase in the magnitude of the force. Therefore, the factors contributing to the intensification magnitude of the Casimir force density are the roughness, the anisotropic factor and ...
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In this paper, we explore the Casimir effect of a rough membrane within the framework of theories that break Lorentz symmetry. We consider two constant aether vectors: one time-like and other space-like, simultaneously. We employ an appropriate change of coordinates such that the membrane assumes a completely flat border and the remaining terms ass...
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Citations
... Additionally, the leading-order Casimir energy for a massless scalar field confined by DBC between two rough membranes in 2+1 dimensions has been documented in Refs. [36,37]. However, calculating the radiative correction to the Casimir energy for a self-interacting scalar field confined between two rough membranes under these boundary conditions is original and constitutes a novel contribution of this paper. ...
... To account for the roughness properties of the membranes in the eigenvalue, we use a perturbative approach [36,37]. This results in the eigenvalue being expressed as follows: ...
In this paper, we calculate the radiative correction to the Casimir energy for both massive and massless Lorentz-violating scalar fields confined between two membranes with rough surfaces in a 3+1 dimensional spacetime. The computations are performed for four types of boundary conditions: Dirichlet, Neumann, Periodic, and Mixed. A crucial element of our approach involves the use of position-dependent counterterms to incorporate the influence of boundaries within the renormalization program. To manage the divergences that emerge in the Casimir energy calculations, we apply the Box Subtraction Scheme (BSS) along with the cutoff regularization technique. We present and discuss results for various degrees of membrane roughness, emphasizing the consistency of our findings with theoretical expectations.
... For a more realistic application, we treat the roughness as a perturbation of the at case. By performing an appropriate change of coordinates, the parallel plates are now a at surface and the remaining terms associated with the roughness are incorporated into the potential term [29][30][31]. To obtain the spectrum of eigenvalues, we apply perturbation theory and use the ζ-function for the regularization process [32]. ...
... We present an explicit case where the perturbation of parallel plates exhibits periodic behavior. As in previous works, we nd that the Casimir energy and force vanish when the anisotropic scaling factor takes even values, independently of any other physical factors in the result [28,30]. Contributions from roughness up to the second order in perturbations, as well as the magnetic eld, are present as expected. ...
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... However, in principle, this phenomenon can also occur for other types of relativistic quantum fields, such as scalar and fermionic fields, when subjected to specific boundary conditions. Studies on the Casimir effect considering scalar quantum fields have been conducted in various contexts, including different boundary conditions, temperature, and Lorentz symmetry violation [10][11][12][13][14][15][16][17]. In the context of fermionic quantum fields, several studies have been developed for different scenarios [18][19][20][21][22][23][24][25][26][27][28]. ...
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... These parameters de¯ne a privileged direction in space{time, thus leading to the violation of Lorentz symmetry. The Lagrangian density of the modi¯ed Klein{Gordon theory by two orthogonal unit aether vectors is given by With these modi¯cations on the Klein{Gordon equation, we extend the study done in Ref. 28 with the presence of aether vectors. 29,30 Our aim is to investigate the Casimir e®ect resulting from the quantum°u ctuations of a scalar¯eld acting on a rough membrane embedded in a°at three-dimensional space{time manifold including two orthogonal unit time-like and space-like aether vectors, simultaneously. ...
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... The Casimir force densities are given by In both scenarios, the e®ects generated by temporal and spatial aether vectors directly impact the primary term a À3 , making their e®ects signi¯cant. In contrast, roughness only in°uences the secondary term a À5 (see Ref. 28), hence the e®ects of the perturbative coe±cients 1;2 arise independently of . The magnitude of the parallel force density is always greater than the magnitude of the perpendicular force density by a proportional factor 2 ð3 À 1 Þ (see Fig. 1). ...
In this paper, we explore the Casimir effect of a rough membrane within the framework of theories that break Lorentz symmetry. We consider two constant aether vectors: one time-like and other space-like, simultaneously. We employ an appropriate change of coordinates such that the membrane assumes a completely flat border and the remaining terms associated with the roughness are considered as part of the potential. Quantum fluctuations are induced by a scalar quantum field subject to Dirichlet boundary conditions. The spectrum is obtained through perturbation theory and regularized using the ζ-function method. We provide an explicit example of a membrane with periodic boundaries. The presence of aether vectors has a significant impact on the dominant term of the Casimir effect, while roughness only affects the secondary terms. Additionally, we examine the finite-temperature case.
... The Casimir eect has been analyzed within the framework of Ho rava gravity theory in a prior study [22]. Extensions to Klein-Gordon and fermionic eld theories have been formulated, known as Ho rava-Lifshitz type theories [23,24,25,26,27,28]. Other research on Lorentz violation has incorporated Aether-like vectors into the Lagrangian, setting a preferred direction. ...
... With these modications on the Klein-Gordon equation, we extend the study done in [28] with the presence of Aether vectors [29,30]. Our aim is to investigate the Casimir eect resulting from the quantum uctuations of a scalar eld acting on a rough membrane embedded in a at threedimensional spacetime manifold including two orthogonal unit timelike and spacelike Aether vectors, simultaneously. ...
... In both scenarios, the eects generated by temporal and spatial Aether vectors directly impact the primary term a −3 , making their eects signicant. In contrast, roughness only inuences the secondary term a −5 (see Ref. [28]), hence the eects of the perturbative coecients σ 1,2 arise independently of ϵ. The magnitude of the parallel force density is always greater than the magnitude of the perpendicular force density by a proportional factor σ 2 (3 − σ 1 ) (see Figure 1). ...
We explore the Casimir effect of a rough membrane within the framework
of theories that break Lorentz symmetry. We consider two constant Aether
vectors: one timelike and other spacelike, simultaneously. We employ an
appropriate change of coordinates such that the membrane assumes a completely flat border and the remaining terms associated with the roughness
are considered as part of the potential. Quantum fluctuations are induced
by a scalar quantum field subject to Dirichlet boundary conditions. The
spectrum is obtained through perturbation theory and regularized using the
ζ–function method. We provide an explicit example of a membrane with
periodic boundaries. The presence of Aether vectors has a significant impact
on the dominant term of the Casimir effect, while roughness only affects the
secondary terms. Additionally, we examine the finite-temperature case
In this work, I study the Casimir effect caused by a charged and massive scalar field that violates Lorentz symmetry in an aether-like and CPT even manner, by direct coupling between the field derivatives and two fixed orthogonal unit vectors. I call this a “double” Lorentz violation. The field will satisfy either Dirichlet or mixed boundary conditions on a pair of plane parallel plates. I use the generalized zeta function technique to examine the effect of a uniform magnetic field, perpendicular to the plates, on the Casimir energy and pressure. Three different pairs of mutual directions of the unit vectors are possible: timelike and spacelike perpendicular to the magnetic field, timelike and spacelike parallel to the magnetic field, spacelike perpendicular and spacelike parallel to the magnetic field. I fully examine all of them for both types of boundary conditions listed above and, in all cases and for both types of boundary conditions, I obtain simple expressions of the zeta function, Casimir energy and pressure in the three asymptotic limits of strong magnetic field, large mass, and small plate distance.
