Extra Dimensions Corrections for Fermionic Casimir
Effect in Three Dimensional Box
Heru Sukamto, Agus Purwanto
Theoretical Physics and Natural Philosophy Laboratorium, Sepuluh Nopember Institute of Technology, Surabaya, Indonesia Email: [email protected], [email protected] Received February 27, 2013; revised March 28, 2013; accepted April 25, 2013
Copyright © 2013 Heru Sukamto, Agus Purwanto. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
We want to show extra-dimensions corrections for Fermionic Casimir Effect. Firstly, we determined quantization fermion field in Three dimensional Box. Then we calculated the Casimir energy for massless fermionic field confined inside a three-dimensional rectangular box with one compact extra-dimension. We use the MIT bag model boundary condition for the confinement and M4 × S1 as the background spacetime. We use the direct mode summation method
along with the Abel-Plana formula to compute the Casimir energy. We show analytically the extra-dimension cor- rections to the Fermionic Casimir effect to forward a new method of exploring the existence of the extra dimensions of the universe.
Keywords: Casimir; Fermionic; Extra Dimensions
1. Introduction
Casimir effects, first discovered in 1948 [1], are mani- festation 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 con- ducting plates. Since spacetimes with extra dimensions are fundamental in most of theories of high energy phys- ics, 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 Randall- Sundrum spacetime [2-3]. For fermionic field, there has been calculated Casimir energy in three-dimensional box [4]. There also has been calculated Casimir energy be- tween parallel plates with compact dimensions [5]. Then, in this paper we investigate the extra-dimension correc- tion to fermionic Casimir energy in three-dimensional box to explore the existence of extra-dimensions of our uni- verse.
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 sur-
faces. Then we compute the Casimir energy by perform- ing 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 dimen- sion on the nature of Casimir energy between the con-figuration boundary that confine the field in the space- times with extra dimensions.
2. The Dirac Field 5D Confined in
Three-Dimensional Cube
We consider a quantum fermionic field on (3 + 1 + 1)-dimensional spacetimes with
4 1
M S
2 d d 2 2 2 2
d A B d d d
AB x x
manifold.
g c t y
x
s (1)
The field is assumed to satisfy the general com- pactification
,
einy R nn
x y x
0 i A
A m
(2)
where R is the size of extra dimension. The field Ψ sat- isfy 5D Dirac equation
(3)
0, , , ,1 2 3 4
A
4 5
(4) with i [6]. The positive energy solution of the 5D Dirac equation can be written respectively as
eiEteiny R
x
n
i in
,
n
n R n
m x y
E
x (5)Spinor x
2 2 3 3 1
1 1 3 3 2 2
3 3 1 1 2 2
i
i i
i
i i
e e e
k x k x
k x x
k x k x
x k xk x
1 1 2 2 3 3 3 3
3 3 3
i i i
i
i
i
ek k x k x k x
k x
k x x
are given by
1 1 12 2 3 3
i i
1 1
i i
2 2
i i
3 3
e e
e e
e e
k x k
x
n n n
k x
n n
k k
n n
x
(6)
Then we have
1 1 2 2 1 1 2 2 1 1 2 2 3
1
i i
1
1 1 2 2 3 3
i i
1 1 2 2 3 3
i i
1 1 2 2 3
2 3
2
-3
3 i
e e e
n
n n n
k k x n
k k x n
k k x n
x
x
x
k k k
k k k
k k k
k k k
x
(7)
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:
ˆ
Boundary 0i .
1 n x
(8) This boundary condition for our special case becomes
1 2 3
2
1 i , , 0, 1, 32,
k k k
x a
x x x k
, , a a a
(9) where 1 2 3 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
1 1 2
x a surface:
1 1
1 1
1 1 1 i
1 2 2 3 3
1 1 1 1
i 1
1 e
i 1
e i
i
k n
k x n
x
n R m E
k k
k n R
m E m E m E
m E
σ k
(10)
for x1a1 2,we get
1 12
1 1 1 2 2 3 3 1 1 i
1 1
1
e i
i
a k
n n
k k
m m E k k k n R
m E m k
(11) and for x1 a1 2
, we get
1 12
1 1 1 2 2 3 3 1 1 i
1 1
1
e i
i
k a
n n
k k
m m E k k k n R
m E m k
(12)
Comparing (11) with (12), we find that in order to have nontrivial solutions for
1
n
1cot 1 1
k k a m
0
m
1 ,
n
, one requires k1
to satisfy a transcendental equation
(13) by setting for massless Dirac field, thequantiza- tion condition Equation (13) yields
1 1
1
1 π 2
n a
k
(14)
3. Casimir Fermionic Energy in
Three-Dimensional Cube at
M
4×
S
1From this point, we concentrate on the massless case
with a1a2a3a
, for simplicity. By using the se- cond quantized form of Dirac field, the vacuum expecta- tion value of the free Hamiltonian can be expressed in the form
3 2
3 2 2 2
1 2 3
3 2
d 2π
FV
s m
p m
E a p p p
R
(15)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). There- fore the integrals turn into summations:
1 2 3
1 2
, , 0 2 2
s m n n
BV
n
E n n
2 2 3 2 221 1 1
2 m R n
(16)
where EBV 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
tion over half-integer numbers
i
i0 0 0 2π
d
d i
1 1
2 e t
m
t
F n tF t
F t F t
(17)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 branch-
cut 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
3
bessel
2 4
0
3 2
2 2
3
1 0
0 0 bessel 3
π 1 π 1
π 1 π 1
π
1
2, 2 2
2 2 2
2 1 2
, 2 1 2
2 3 2 π 1
2 π 1
j
m
m j
j
n j
j m
j j
m
K j
K
n
n j
2 2 2 2
2 0 0 0
2 d d d
1 π
π 1
BV
j
j m
m
m
E u t k k t u
a
2 3
2 2
2
2 2
2
2 3
bessel 2 3
, 0 0
2 1 2 1 2
1 1, 2
π
2 1 π 1
m n n
j
j
m
m
j n
j
n n
K n
2 2
1 2 1 2 m
(18)
2 π2 2
L R
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 FV and is supposed to be subtracted from
E
BV in order to obtain the Casimir energy. Second, fourth, and fifth term related to extra dimension corrections.
E
Cube
Casimir: BV FV
E E E
4 1
(19) As a check on our procedure we have computed the Casimir energy for a fermionic field between two parallel plates in M S , separated by a distance a, and ex- tradimensional size R, we obtain
3
mir 2 Bessel 4
1 0 0
3 2
3
Bessel 0
2
2 3
0 3
1
π π
π 1 π
2 2 2π π 1
π
3 2 π 1
4 π 1
2 1
2 2, 2 1
2
1 2
1 2
1 , 2
2 2
1 1
2
ll
j
m j j
j
n j
K j m
j j
j n
a
n
K
n n
j
2
1 j
m
Casi
E
1 j 2 2
2 3
Bessel 2 3
1 0
2 2
2 2
2 3
Bessel 0
3 ,
2 0
π 1
π 1
1
π 1
2
1, 2
2 2
1 1
2
2 2 1 1
1 1, 2
2π 1 2 2
m n n j
j
j
n n
n n
n
2 2
2 2
2
1 1
2
m
K j m
j
K j
j n
2,3 0
n n
Figure 1. Fermionic Casimir energy with and without a compact dimension as a function of the plates separation.
Figure 1 depicts the dependence of the FermionicCa- simir 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
2 3
2 2
2
2 3
, 0 Bessel 2 3
1 0
1 1
2
2 2 1 1
1, 2
1 π 1
2 1
2π 2
j
j n m n
n n
n n
m
K j
j
2
Casimir 2 Bessel
1 0
2 2
2 2
2 π π 1 2, 2π 1
2 2 2π 1
j ED
m j
m K E
a j j m
m
4 1
If there are no extra dimension, then the term αm2
vanishes. Then casimir energy will become as be shown by [4].
4. Conclusion
In this paper, we have investigated the extra dimensional corrections for Casimir energy in a three-dimensional box in M S due to the vacuum fluctuations of mass- less fermionic field with MIT bag boundary conditions. The Casimir energy is computed using generalized Abel- Plana summation formula. The most important result we obtain in this letter is that Fermionic Casimir energy de- pends on the size of extra dimensions.
REFERENCES
[1] H. B. G. Casimir, Koninklijke Nederlandse Akademie van Wetenschappen, Vol. 51, 1948, p. 793
[2] K. A. Milton, “The Casimir Effect: Physical Manifestations of Zero-Point Energy,” World Scientific, River Edge, 2001. [3] L. P. Theo, “Casimir Effect in Spacetime with Extra Di-
mensions-From Kaluza-Klein to Randall Sundrum Mo- dels.” ArXiv:hep-th/0907.2989v3
[4] A. Seyedzahedi, R. Saghian and S. S. Gousheh, Physical Review A, Vol. 82, 2010, Article ID: 032517.
http://dx.doi.org/10.1103/PhysRevA.82.032517
[5] F. S. Khoo and L. P. Theo, Physics Letters B, Vol. 703, 2011, pp. 199-207.
http://dx.doi.org/10.1016/j.physletb.2011.07.072
[6] P. Alejandro Sanchez, M. Anabitarte and M. Bellini, “Dirac Equation in a de Sitter Expansion for Massive Neutrinos from Modern Kaluza-Klein Theory,” 2012. ArXiv:hep-th/1112.5183v4
[7] K. Poppenhaeger, S. Hossenfelder, S. Hosmann and M. Bleicher, “The Casimir Effect in the Presence of Com- pactified Universal Extra Dimensions,” 2003.
Appendix: Abel-Plana Formula in
Calculating the Casimir Energy
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
2 3
1
2
0 , 1 2
2π 1
2
m n n
a F n
21 2, ,3
n n n m
(A1)
with
2 2
2 2
2 3 2 3
1 1
2 2
, ,
n n m n n m
The factor 2 is associated with the spin multiplicity. The branch-cut term can be calculated using the follow- ing:
2 3
2 2
2 3
2 3
2 2
2 ,
3
3 0
2
, , , ,
, , , , i
i ,
2π
,
n
m n
F
n n m n n m
n n m t
t t
n n m
a t
t
(A2)By using Equations (17) and (A2), Equation (16) turns into
2 3
2 3 2 3
2 2
2 2
2 3
, 0 0
,
2 2
2 2
2 3
0 , , 2π
2π 1 1
2 2
1 2 1
d
d 1
2 e
BV
t
m n n
m n n
n n m
m a
t m
E t t n n
t
n n
2π 2
a
(A3)
The first term is infinite and we have to use the Abel-Plana formula again for the first term. We obtain
2 3
2 3,
2 3
2π
, ,
2 2π 1
2 1
e
n n n m t
n
t n
a n
m 0
3
2
3 2 2
3
0 0 0
0
1 2
2
2 2 2
3
2
2 2 2
3 2π
0
2 2
2 2
2π 1
2
2
d d
2
1
d 1
2
π d 1
d
2 e
m n
n BV
k
m n
t m
m a
k
t k m
E t k k t n
t n
t m
a
(A4)
Again the first term is infinite and we must apply the Abel-Plana formula to obtain
2 2 2
2
3 2 2
3
2 2 2
2π
0 0
2 2
3 0
1 2
2π
0
2π
2π d
d d
e 1
2π 2
2
1
2
d
e
2π d
e 1
u m
k m
k m n
t t
m
k n
k t
t n u
t k u
a
k
t k
a
k a
2 2 2 2
2
2 2
0 0 0
2π
d 1
d d d
2 BV
m
E u t k k t u m
a
m
m
2 3 2 3
3 ,
, 0 , 2
1 2 n
m n n n m
n n
(A5)
2 2
2 1 2
2
t m
Note that all of branch cut terms is finite. On other hand the free vacuum energy is
Making appropriate changes of variables, we obtain
2
2 2
1 2 3
3 2
m R
p p
3 d d d1 2 3 2
2
2π
m FV
p E
p p p a
(A6)2 2 2
0 0 0
2 2π
d d d
FV
m
E u t k k t m
a u
(A7)Therefore when we compute the Casimir energy these two terms precisely cancel each other. That is,
2 2 2
2
3 2 2
3
2 2 2 2
2π
0 0
2
2 2 2
3 2π
0
2 2
2 2
0
1 2
2π d
d d
e 1
2π d d 1
2 e
:
2
2
1
d 1
2 Cas BV FV
u m
k m
k m n
t t
n m
E E
k t
t n E
u
t k u m
a
k
t k m
a
m
2 3
2π
2π 1
2
e t 1 2
n
n t
t n
a
2,n3 0 2,3,
m n n m
(A8)
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:
2π 12π
0
1 e
e 1 1
j t j t
j
(A9)The last term turn into
2 3
2,3 0
2 2
2π 1 2 2
2 3
, , 0
4π 1 1
1
2 d
2 e
j t j
n n m
m n n j
t t n n m
a
(A10) by using the identity
1 2 1
2 2
Bess l 0
e
2 1
, 2 π
1
du u a e u a K a
(A11)then we have
2 3
3
3 2
2 2
2
2 2
Bessel 2
1 2 1 2 1 2 1, 2π 1 2 1 2
2 1
2 π 1
m
n n
n
K j n m
j
2
0 , 0
1 n
j
m n j
(A12)In order to compute the second term of Equation (A8) we first interchange the order of integrations to obtain
2 2 3
3
0 1
2 2
2π
0 2
4π d
e 1 d
Y
k m n
m n
t t Y a
k
(A13)where 1 2
n3
m 22 2 2
Y k . Now using Equations (A9) and (A11) we obtain
2
3 2
3
2 4
2 2
3 2π
2 2 0
1 2
π d
e
1 2 1
k
m n
n m
k k n m
a
m
3
3 2 2
2 3
Bessel 3
0 0 π π
1
1 2
2 1
3 2, 2π 1
2π 1 2
j
m n j
m n
K j
a j n
(A14)
2
Bessel
0
1 0 2
π π 1 π 1 π
4 2, 2 4
2 2 2π 1 π 1
j
j
j
m j
K
a a
m
j m
41 π
2 1
j j (A15)
Finally, we arrive at
3
2
Casimir Bes
0
sel
2 4
1 0 0
3 2
3
Bessel 3
0
2
2 3
2 1
2 π π 1 2, 2π 1 π
2 2 2π 1 π 1
π
3 2 π 1
4 π 1
1
1 2
1 2
1 2
1 , 2
2 2
1 1
2
2 2
j
j
m j j
j
n j
j
m
K j m
j E
j
j n
j a
n
K
n n
2 3
2 2
Bessel 2 3
1 0
2 2
2 2
2 3
Bessel 0
3 ,
2 0
π 1
π 1
1 π 1
2
1, 2
2 2
1 1
2
2 2 1 1
1 1, 2
2π 1 2 2
m n n j
j
j
n n
n n
n
m 2 2
2
1 1
2
K j m
j
K j
j n
2,3 0
n n
