Casimir effects, first discovered in 1948 [1], are manifestation of the zero-point energies of the quantum fields and have played an important role on a variety of fields of physics. It discovered by Hendrick Casimir. He showed that zero-point fluctuations in electromagnetic fields gave rise to an attractive force between parallel, perfectly conducting plates. Since spacetimes with extra dimensions are fundamental in most of theories of high energy physics, there have been intensive activities in investigating the Casimir effect in spacetime with extra dimensions. The case of a scalar field and electromagnetic field with various boundary conditions has been studied at either zero or finite temperature, for different extra-dimensional spacetimes such as Kaluza-Klein spacetime and RandallSundrum spacetime [2-3]. For fermionic field, there has been calculated Casimir energy in three-dimensional box [4]. There also has been calculated Casimir energy between parallel plates with compact dimensions [5]. Then, in this paper we investigate the extra-dimension correction to fermionic Casimir energy in three-dimensional box to explore the existence of extra-dimensions of our universe.

This article is organized as follows: In Section 2 we present the solution to the Dirac equation in 5D subject to the MIT bag model boundary condition in all the surfaces. Then we compute the Casimir energy by performing a direct sum over all modes of the field using the Abel-Plana summation formula. As we shall show, there will be no need for any analytic continuation techniques in this case. There will be influenced from extra dimension on the nature of Casimir energy between the configuration boundary that confine the field in the spacetimes with extra dimensions.

We consider a quantum fermionic field on (3 + 1 + 1)-dimensional spacetimes with manifold.

The field is assumed to satisfy the general compactification

where R is the size of extra dimension. The field Ψ satisfy 5D Dirac equation

using the chiral representation of Dirac matrices

with [6]. The positive energy solution of the 5D Dirac equation can be written respectively as

Spinor are given by

Then we have

The MIT bag model boundary condition is usually said to imply that there is no flux of fermions through the boundary. The prevalent form of the MIT bag model boundary condition is as follows:

This boundary condition for our special case becomes

where denote the lengths of the sides of the box. Subtituting Equations (6) and (7) into Equation (9) we obtain, for example, the following two equation for surface:

for, we get

and for, we get

Comparing (11) with (12), we find that in order to have nontrivial solutions for, one requires k₁ to satisfy a transcendental equation

by setting for massless Dirac field, the quantization condition Equation (13) yields

From this point, we concentrate on the massless case with, for simplicity. By using the second quantized form of Dirac field, the vacuum expectation value of the free Hamiltonian can be expressed in the form

where summation index runs over the spin states and subscripts FV stands for free vacuum. In the presence of the boundaries, all of components of the momentum are subjected to quantization condition Equation (14). Therefore the integrals turn into summations:

where E_BV denotes the vacuum energy in the presence of the boundaries. Obviously, in both situations the vacuum energy is divergent. However, the Casimir energy, which is the difference between these two quantities, is usually expected to be finite. One usually needs to utilize a regulation prescription to give a physical meaning to such a difference. In this paper we choose a modified form of the Abel-Plana formula, which is useful for the summation over half-integer numbers

where F(z) is assumed to be an analytic function in the right half-plane. The first term is the main term of turning a sum into an integral. The second term is called branchcut term. Since we have a four sum over for Equation (16), we need to apply the Abel-Plana formula four times. The details are given in the Appendix. The final result is

with. It is extremely important to note that the only divergent quantity in Equation (18) is the first term, which is precisely the free vacuum energy and is supposed to be subtracted from in order to obtain the Casimir energy. Second, fourth, and fifth term related to extra dimension corrections.

As a check on our procedure we have computed the Casimir energy for a fermionic field between two parallel plates in, separated by a distance a, and extradimensional size R, we obtain

Figure 1 depicts the dependence of the Fermionic Casimir energy in a Three-dimensional box on a radius of extra dimension and the size of the box. It is showed that corrections’ factors increase proportional to the size of extra dimensions. For extra dimension correction, we deduce from Equation (20) that

If there are no extra dimension, then the term αm² vanishes. Then casimir energy will become as be shown by [4].

In this paper, we have investigated the extra dimensional corrections for Casimir energy in a three-dimensional box in due to the vacuum fluctuations of massless fermionic field with MIT bag boundary conditions. The Casimir energy is computed using generalized AbelPlana summation formula. The most important result we obtain in this letter is that Fermionic Casimir energy depends on the size of extra dimensions.

In this appendix we present the details of the calculations leading to our main expression for the casimir energy of a massless fermionic field confined inside cube with one extra dimension via MIT bag model boundary condition. In order to apply the Abel-Plana formula to four sum in Equation (16), we first define

with

The factor 2 is associated with the spin multiplicity. The branch-cut term can be calculated using the following:

By using Equations (17) and (A2), Equation (16) turns into

The first term is infinite and we have to use the Abel-Plana formula again for the first term. We obtain

Again the first term is infinite and we must apply the Abel-Plana formula to obtain

Note that all of branch cut terms is finite. On other hand the free vacuum energy is

Making appropriate changes of variables, we obtain

Therefore when we compute the Casimir energy these two terms precisely cancel each other. That is,

here we explain the details of the calculation of the last term and then outline the calculation for remaining terms. We expand the denominator as follows:

The last term turn into

by using the identity

then we have

In order to compute the second term of Equation (A8) we first interchange the order of integrations to obtain

where. Now using Equations (A9) and (A11) we obtain

Going through this same procedure, we can compute the first branch-cut term in Equation (A8) as follows:

Finally, we arrive at