Fermionic Casimir effect in a field theory model with violation of Lorentz-symmetry

In this paper, we evaluate the Casimir energy and pressure for a massive fermionic field confined in the region between two parallel plates. In order to implement this confinement we impose on the fermionic fields the standard MIT bag boundary on the plates. In this present analysis we consider a quantum field theory model with a CPT even and aether-like Lorentz-symmetry violation. It turns out that the fermionic Casimir energy and pressure depend on the direction of the constant vector that implements the Lorentz symmetry breaking.


I. INTRODUCTION
The shift in the zero-point fluctuations of quantum fields submitted to boundary conditions, gives rise to macroscopically observable forces between material bodies, the so-called Casimir forces. The Casimir effect was proposed by H. B. Casimir in 1948 [1], and confirmed ten years later in the experiments performed by Sparnaay [2] who considered two parallel plates configuration; moreover, Blokland and Overbeek [3] have studied a plane surface opposing to a spherical one. More recently, these measurements have been performed using torsion balances [4], atomic force microscopes [5], and high precision capacitance bridges [6] (for a general review of the Casimir effect, see also [7,8]; some recent discussions of different issues related to boundary conditions within the Casimir effect can be also found in [9][10][11]), and the scattering approach to the Casimir effect is presented in [12].
The simplest theoretical way allowing to study the Casimir effect is the case of the interaction between two parallel plates placed in the vacuum. The quantum notion of vacuum corresponds to a collection of infinite set of waves which contemplate all wavelength possibilities; however, when two parallel plates are placed in this vacuum, only specific wavelengths are possible between them.
In the usual Quantum Field Theory, the Lorentz symmetry is preserved. However, the seminal paper [13] called interest to the possibility of its breaking. The Lorentz-breaking theories have attracted a great deal of attention during the recent years, it is worth to mention the Lorentzbreaking extension of the Standard Model [14]. One of the most important directions in their study is the investigation of possible impacts of Lorentz-violating additive terms for field theory models both at classical and quantum levels. We note that this approach is based on introducing the explicit Lorentz symmetry breaking in the action. This approach should be distinguished from introducing the Lorentz-breaking boundary conditions ad hoc for the Lorentz-invariant Lagrangian, since in the case of the Lorentz-breaking terms of the Lagrangian, the existence of privileged space-time directions is an intrinsic feature of the theory, while the Lorentz-breaking boundary conditions affect only the given physical situation. A large list of studies of Lorentzbreaking boundary conditions within the Casimir effect for the spinor field is presented in [15], and for the scalar field in [16].
The study of the Casimir effect for scalar quantum fields confined between two large parallel plates in the presence of Lorentz symmetry breaking has been developed in [17]. Moreover, the thermal corrections for the corresponding Casimir energies have been investigated in [18].
In both papers, it was supposed that the scalar field obeys different boundary conditions on the plates and that the Lorentz symmetry breaking is introduced through contracting of the derivative of this field with a constant vector, u µ . Also, the Horava-Lifshitz approach [19] breaks the Lorentz symmetry, but unlike the methodology proposed in [14], in this case the Lorentz symmetry breaking is strong, being introduced through an explicit asymmetry between space and time coordinates (space-time anisotropy). Within this approach, the Casimir energy and pressure have been calculated in [20,21].
In this paper we would like to carry out the analysis of Casimir energy and pressure, considering at this time a massive quantum fermionic fields confined between two parallel plates in a Lorentz-violating model. In order to prevent crossing of plates by the fermionic current, we adopt the MIT bag model [22]. The analysis of Casimir energy, in a Lorentz-symmetric theory, for the case of the massive fermionic quantum field confined between two parallel plates in a high-dimensional spacetime have been developed in [23], and in [24] such a study has been done for a non-trivial topology for the space section. This paper is organized as follows. In section II we present the model of QFT with Lorentz symmetry violation for fermionic fields, admitting a direct coupling between an arbitrary constant vector with the derivative of the field. In section III, we calculate the Casimir energy and pressure considering the field obeying the MIT bag model boundary condition on two parallel plates, considering different directions for the arbitrary constant vector. Our most relevant results are summarized in section IV. In appendix A we present our proposal of the modified Lagrangian for the MIT bag model. Here, we assume = c = 1, and the metric signature will be taken as (+, −, −, −).

II. THE LORENTZ VIOLATING MODEL
In this section, we introduce the theoretical model which we want to investigate. In fact, this model has been proposed in [25], and is described by a massive fermionic quantum field characterized by the Lagrangian density below: (II.1) In the above equation, the dimensionless parameter λ is supposed to be much smaller than unity.
It measures intensity of the Lorentz symmetry violation of the system caused by the presence of a coupling between the derivative of the field with an arbitrary constant vector u µ defined so that u µ u µ can be equal to 1, −1 or 0. This vector introduces the privileged space-time direction.
In a certain sense, the theory is characterized by four constant parameters: λ characterizing intensity of Lorentz symmetry breaking and three independent components of u µ characterizing its direction.
Here, we will assume that the Dirac matrices γ µ are given in the following representation: where I represents the 2 × 2 identity matrix, and σ j the Pauli matrices.
The modified Dirac equation reads, The corresponding energy-momentum tensor is given by We can verify that where this asymmetry is a typical characteristic for theories with a Lorentz symmetry breaking.
Moreover, we can verify that

FECT
In this section, we will analyze the influence of the space-time anisotropy, generated by the Lorentz-breaking term, on the Casimir energy associated with a massive fermionic quantum field confined between two large parallel plates. In this way, we assume two parallel plates placed at z = 0 and z = a, as is shown in Fig. 1.
Within this study, we will consider the cases where the constant 4-vector, u µ , is either timelike or spacelike. For each case, we will obtain the solution for the Dirac equation Eq. (II.3) imposing that the field obeys the MIT bag boundary condition below, where n µ is unit vector normal outgoing to the boundaries, i.e., in our case normal to the plates.
For our configuration of plates we have consider n µ = −δ z µ for the plate at z = 0 and n µ = δ z µ at z = a.

A. Timelike vector case
In this subsection, we consider the case in that 4-vector u µ is timelike, i.e.
Assuming the time dependence for the fermionic wave function in the form e ∓iωt , for positiveand negative-energy solutions of equation (II.3) can be presented as with the dispersion relation and the two-component spinors ϕ and χ correspond to particles and antiparticles respectively and have the following explicit form: These fermionic modes are characterized by the set of quantum numbers {k x , k y , k z }, with k x , k y being continuous quantum numbers, and k z must be discrete, to satisfy the boundary conditions.
To be more precise, we will introduce a fourth quantum number.
From the boundary condition (III.1) on the plate at z = 0 we find the following relations between the spinors in (III.5): (III. 6) do not depend on λ.
Substituting into (III.1) at z = a, the fermionic functions, Eq.s (III.3), (III.5), followed by (III.6), and after some intermediate steps, the transcendental equation below is obtained, The vacuum energy is obtained by taking the vacuum expectation value ofĤ: where we have defined α n = k n a, being k n = k z , the order n root of (III.7). By the above expression, we clearly observe that the vacuum energy does not depend on the parameter λ parametrizing the intensity of the Lorentz symmetry violation. So, we stop this analysis at this point and go directly to the next cases.
for positive-negative-energy wave-functions, where now the dispersion relation is The two-component spinor reads, From the boundary condition (III.1) on the plate at z = 0, we get and (III.14) Moreover, from the boundary condition at z = a follows the transcendental equation (III.7).
For this case the vacuum energy is given by, By using the coordinate transformationk x = (1 − λ)k x , we get So the vacuum energy, in this case, is the same that of the timelike case (III.8) multiplied by the factor (1 − λ) −1 . Since in this case the Lorentz-breaking modification is trivial, we will not proceed further because this situation is a particular case of the next calculation.
Finally, let us consider the case where the 4-vector is orthogonal to plates: The positive-and negative-energy solutions to this case can read where the dispersion relation is now given by and From the boundary condition (III.1) applied on the plate at z = 0, we get As to the plate on z = a, we find The vacuum energy for this case reads Again using the definition α n = k n a, being k n = k z the n-root of (III.22) we get (III.24) By the above expression, we clearly observe that in this case, the vacuum energy depends on the parameter describing the intensity of the Lorentz violation in a non-trivial way. So, we will develop below the detailed analysis for this case.
As we can see, the vacuum energy is divergent. In order to obtain a finite and well defined result for the Casimir energy, we have to renormalize it. This can be done by using the Abel-Plana summation formula below [26]: .

(III.25)
To develop the summation over n in (III.24) we must rewrite the integrand in an appropriate manner. The steps adopted from now use the approach developed in [24]. So let us start with the denominator appearing in the left hand side of the summation (III.25). By using (III.22), it is possible to obtain the identity below: (III.26) So, we arrive at the regularized vacuum energy given by where the function f (z) has been defined as (III. 28) with the b parameter given by b = a  (1 − λ) . (III.29) The first term on the right-hand side of the equation (III.27) refers to vacuum energy in the presence of only one plate, and the second one is related with vacuum energy without plates.
So, after renormalization, the Casimir energy is given by where we have made the change of variable t = bu. At this point we can separate the integral over the u variable in two intervals: the first one is 0, k 2 x + k 2 y + m 2 and the second is k 2 x + k 2 y + m 2 , ∞ . One finds that the integral in the first interval vanishes. So, we get the following Casimir energy (III.31) Using the integral relation below [24] we find the following expression to Casimir energy dy y + bm y y + 2bm ln 1 + y y + 2bm e −2(y+bm) , (III.34) where we have made the integration by parts and the changed the variable: bx = y+bm. Another procedure adopted to develop the summation over the n in (III.24), is based on using Riemann zeta function regularization [27]. Within this approach we replace k 2 x + k 2 y + (1 − λ) 2 k 2 n , in the integrand, by [k 2 x + k 2 y + (1 − λ) 2 k 2 n ] −s . Afterwards, we develop the integration over k x and k y , and the remaining sum over n converges for all complex value of s for sufficiently large value of Re(s). Finally the Casimir energy can be obtained by evaluating the corresponding zeta function at the pole s = −1/2.
Unfortunately for massive fields there is no closed expression for the integral in (III.34).
However we can obtain approximate results for two limiting cases: am 1 and am 1, corresponding to case of small distance between plates and/or small mass, and of large distance between plates and/or large mass, respectively: (i) For the case am 1, we can expand the integrand in powers of bm. Up to the first order we get: Substituting the above expression in (III.34), we obtain III.36) and consequently, the Casimir pressure is given by (III.37) (ii) For the case am 1, the most relevant contribution in the expansion of the integrand of (III.34), is: In this approximation the Casimir energy reads, We can see, that in this case, the Casimir energy per area unit decays exponentially with am.
In figures 2 and 3, we exhibit the behavior of the Casimir energy, Eq. (III.34), and pressure, respectively, as a function of am considering different values of the parameter characterizing the intensity of the Lorentz symmetry violation, λ. By these plots we can see that both quantities depend on this parameter, in the sense that they increase when we increase λ. of area L 2 and separated by a distance a. Their result is reproduced below, where W µ,ν (z) represents the Whittaker function [28]. By using the above expression the authors of [23] found approximate results for the Casimir energy for the limits am 1 and am 1.
Their result for the limit ma 1 disagrees with ours.
On the base of the result (III.40), we can adapt this formula for our case. The corresponding expression is below: (III.41) By using [28] we write below the asymptotic expression for the Whittaker function for small argument: By using this expression we obtain (III.36).

IV. CONCLUDING REMARKS
In this paper we have investigated the fermionic Casimir energy for the aether-like CPT-even Lorentz-violating (LV) model, by admitting a direct coupling between the derivative of the field and an arbitrary constant vector. We assume that the field is confined between two parallel plates by imposing the MIT bag model boundary condition on each plate.
For each arbitrary direction of the constant vector, u µ , the Casimir energy has been analyzed.
It was observed that the dispersion relations are modified by the presence of the parameter, λ, Our final remarks about this system concern the Hermiticity properties of the Hamiltonian associated with the Lagrangian density (II.1). At the end of the Appendix we will argue that in fact the Hamiltonian modified due to the additive Lorentz-violating term is a self-adjoint operator.