# Revisiting Vacuum Energy in Compact Spacetimes

S R Haridev<EMAIL_ADDRESS>Prasant Samantray
<EMAIL_ADDRESS>Department of Physics, BITS-
Pilani Hyderabad Campus, Jawahar Nagar, Shamirpet Mandal, Secunderabad 500078,
India

###### Abstract

We revisit the calculation of vacuum energy density in compact spacetimes. For
the general case of $k$ compact spatial dimensions in $p+k$ dimensional
Minkowski spacetime, we calculate the Casimir force on a piston placed in the
presence of external electromagnetic fields. We observe that while the
presence of a strong external magnetic field reduces the intensity of the
Casimir force, the presence of an electric field enhances it instead. We
discuss various physical cases, and also derive the particle production rate
in the presence of such external fields in compact spacetimes. We observe that
the pair production rate is enhanced in the presence of a piston.

Casimir Force, Vacuum energy, Higher dimensional theory

††preprint: APS/123-QED

##  I Introduction

Almost a century ago Kaluza unified general relativity with electromagnetism
by invoking a fifth extra compact spatial dimension [1, 2]. While the idea was
profound, it did not exactly take off as a fundamental framework of our
universe. Decades later superstring theory - the most promising candidate for
unifying all the known forces at present, required the existence of compact
extra spatial dimensions to mandate the theory to be mathematically
consistent. From a phenomenological standpoint these string theories come
tantalizingly close to explaining the particle content of our observable
universe. Well almost. This naturally renewed interest in spacetimes with
extra spatial dimensions. For example, also see [2, 3].

Tangentially, in standard quantum field theory in Minkowski spacetime, the
zero point energy is infinite and is usually subtracted out by shifting the
datum of the Hamiltonian. This is done via the prescription of normal ordering
and the infinite zero point energy or the vacuum energy has no physical
meaning. However, in the presence of compact extra spatial dimensions, this
zero point/vacuum energy has non-trivial meaning and in principle can have
observational signature like the force on a plate, or on a pair of plates
placed at a distance etc. This inevitably ties the physics of compact
dimensions with the well known Casimir effect. Along with the Unruh effect and
the Schwinger effect, the Casimir effect is one of the cornerstones of
standard quantum field theory which has been experimentally verified. The
result of this effect is that if two parallel plates are kept at a distance,
they attract each other by exerting a force known famously as the Casimir
force [4]. Casimir effect/force is the result of introducing a length
scale/boundary condition on the quantum field which otherwise is generally
absent. Since compactification introduces a natural length scale, it is
expected that this will naturally lead to a non-trivial zero point/vacuum
energy even in the absence of any other additional boundary conditions. In
fact, there has been a host of literature on these issues [5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16]. Additionally placing a pair of plates in such a
compactified spacetime leads to some interesting results and the reader is
urged to refer to [17, 18, 19, 20, 21, 22, 23, 24, 25] for more details. In
particular, [5] considered the Casimir force on a plate placed in Minkowski
space with extra compactified dimension and calculated a seemingly counter-
intuitive repulsive Casimir force in the asymptotic limit. The same author a
couple of years later considered a piston but this time in a finite box in the
compact spacetime, and concluded that the repulsive force was neutralized in
the asymptotic limit [6]. In [8] the authors considered the extra dimension
with electromagnetic field and computed an analytic expression for Casimir
force. They also compared with the experimental data available and placed
upper bounds on the size of extra dimensions. In fact, Casimir effect has also
been studied in the context of stabilizing the extra compact dimensions, early
universe cosmology and also in the context of string theory. Further
commentaries and investigations on all these issues and more can be looked up
in [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37].

While most previous works on this issue have focused using the mode-sum
approach and subsequent regularization procedures to extract the vacuum energy
density and calculate Casimir force on plates. We in the present work use the
method of effective action via the heat kernel method to do the same. This
method allows for approximation schemes which allows us to calculate the
vacuum energy in various different setting with a considerable degree of
accuracy and ease as we shall see in later sections. We also use this
formalism to calculate the Casimir force on a piston placed along the compact
spatial dimension which is more direct and straightforward as compared with
the standard mode-sum approach in the earlier approaches. We later generalize
our approach to including electromagnetic fields, and also looking at particle
production.

The organization of this paper is as follows. In section II we discuss the
mathematical tools used throughout the paper. In section III we calculate the
vacuum energy of a massive complex scalar field in $d$ dimensional spacetime
with $k$ compact spatial dimensions. We use this result to calculate the
Casimir force per unit area on a piston placed in the same manifold. In
sections IV and V, we extend these calculations to include the presence of a
constant magnetic field perpendicular to the plates and an electric field
parallel to the plates, respectively. We also calculate the Schwinger pair
production in higher dimensions with and without the piston in such compact
spacetimes. Finally, we summarize our results in section VI and conclude with
a discussion.

##  II General mathematical frame work

This section gives a general outline of the mathematical tools we used
throughout the paper. Throughout this paper we works in $\hbar=c=1$ and our
metric convention is $(-,+,+,+)$. Consider a complex scalar field of mass $m$
and charge $q$ coupled to a vector potential $A_{\mu}$. The Lagrangian density
of the system is

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle-\left(D_{\mu}\phi\right)^{*}\left(D^{\mu}\phi\right)-m^{2}\phi^{2},$
(1)

where $D_{\mu}=\partial_{\mu}-iqA_{\mu}$ is the gauge derivative. We work in
Lorentz gauge ($\partial_{\mu}A^{\mu}=0$) and in the Euclidean time
($\tau=it$). The Euclidean effective potential $V_{E}$ for the theory is

$\displaystyle V_{E}=\int_{0}^{\infty}\frac{K(s;x,x)}{s}ds,$ (2)

where $K(s;x,x)$ is the heat kernel $K(s;x,x^{\prime})$ in the coinciding
limit ($x\rightarrow x^{\prime}$). Heat kernel satisfies the differential
equation

$\displaystyle\hat{D}K\left(s;x,x^{\prime}\right)=-\frac{\partial}{\partial
s}K\left(s;x,x^{\prime}\right),$ (3)

with boundary condition

$\displaystyle K(0;x,x^{\prime})$ $\displaystyle=$
$\displaystyle\delta(x-x^{\prime}),$ (4)

and

$\displaystyle\hat{D}$ $\displaystyle=$
$\displaystyle-\partial_{\mu}\partial_{\mu}+2iqA_{\mu}\partial_{\mu}+q^{2}A^{2}+m^{2}.$
(5)

We solved Eq. (3) with boundary condition (4) by expanding the heat kernel in
Fourier basis as

$\displaystyle K(s;x,x^{\prime})\propto\sumint e^{ipx}\;\tilde{k}(s,p,...),$
(6)

where the summation is over the Fourier modes along the compact dimensions.
Substituting the result of the heat kernel in Eq. 2 gives the Euclidean
effective potential from which one can deduce the vacuum energy of the field
and the corresponding Casimir pressure exerted on a parallel plate.

##  III Complex scalar field with constant gauge potential

To begin with, we consider a massive complex scalar field operating in $1+3$
spacetime dimension with one spatial dimension compactified ($x_{1}\sim
x_{1}+L$). As the simplest case, we consider the case of periodic boundary
conditions for the complex scalar field along the compact dimension. Since the
topology is non-trivial, we can not gauge transform away the gauge potentials
along the compact dimensions. So, we take the background gauge field to be
$A_{\mu}=(0,a,0,0)$. For this gauge potential, we can solve Eq. (3) by writing
the heat kernel in the Fourier modes (Eq. 6), which gives the Euclidean
effective potential as

$\displaystyle V_{E}$ $\displaystyle=$
$\displaystyle\frac{1}{8L\pi^{\frac{3}{2}}}\sum_{n=-\infty}^{\infty}\int_{0}^{\infty}\frac{ds}{s^{\frac{5}{2}}}e^{-M_{n}^{2}s},$
(7)

where $M_{n}^{2}=m^{2}+\left(\frac{2\pi n}{L}-qa\right)^{2}$. Using analytic
continuation of gamma function

$\displaystyle V_{E}$ $\displaystyle=$
$\displaystyle\frac{4\pi^{2}}{3L^{4}}\sum_{n=-\infty}^{\infty}\left(\left(n-\frac{qaL}{2\pi}\right)^{2}+\left(\frac{mL}{2\pi}\right)^{2}\right)^{\frac{3}{2}}.$
(8)

We use the result from complex analysis for the infinite sum [38] which states

$\displaystyle\sum_{n=-\infty}^{\infty}\left((n+\beta)^{2}+\alpha^{2}\right)^{-\lambda}$
$\displaystyle=$
$\displaystyle\sqrt{\pi}\frac{\Gamma\left(\lambda-\frac{1}{2}\right)}{\Gamma(\lambda)}\alpha^{(1-2\lambda)}$
(9) $\displaystyle+4\sin(\lambda\pi)f_{\lambda}(\alpha,\beta),$

where

$\displaystyle f_{\lambda}(\alpha,\beta)$ $\displaystyle=$
$\displaystyle\mathcal{R}\left\\{\int_{\alpha}^{\infty}dx\frac{(x^{2}-\alpha^{2})^{-\lambda}}{e^{2\pi
x+2\pi i\beta}-1}\right\\}.$ (10)

Here $\mathcal{R}$ stands for the real part of the complex-valued function.
The summation in Eq. (9) is convergent for $\mathcal{R}(\lambda)>1/2$, but we
have an analytic continuation of the same on the right side of the equation,
which is valid for all the values of $\lambda$. Using the result we can write
Eq. (9) as

$\displaystyle V_{E}$ $\displaystyle=$
$\displaystyle\frac{m^{4}}{16\pi^{2}}\Gamma(-2)+\frac{16\pi^{2}}{3L^{4}}f_{-\frac{3}{2}}(\alpha,\beta),$
(11)

where $\alpha=(mL/2\pi)$ and $\beta=-(qaL/2\pi)$. First term in the potential
corresponds to the contribution from $1+3$ dimensional flat topology, which
one discards by renormalizing the vacuum energy of $1+3$ dimensional Minkowski
spacetime to zero. Then the minimized Lorentzian effective potential
($V=-V_{E}$) is (appendix A)

$\displaystyle V_{min}=-\frac{16\pi^{2}}{3L^{4}}f_{-\frac{3}{2}}(\alpha,0).$
(12)

Changing the integration variable in Eq. (10) to $y=x/\alpha$ and using the
modified Bessel function of second kind $K_{\nu}(z)$ we can write

$\displaystyle f_{\lambda}(\alpha,0)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(1-\lambda)}{\pi^{1-\lambda}\alpha^{\lambda-\frac{1}{2}}}\sum_{n=1}^{\infty}n^{\lambda-\frac{1}{2}}K_{\lambda-\frac{1}{2}}(2\pi
n\alpha).$ (13)

Similarly, one can calculate the vacuum energy of a complex scalar field
operating in $d$ dimensional spacetime with $k$ compact spatial dimensions.
For $k$ compact spatial dimensions, we have $k$ summations in Eq. 8 and each
of the summation is converging (see appendix A). So we have $k!$ ways of doing
this summation. Considering this fixes the symmetry in exchanging of compact
dimensions ($L_{i}\leftrightarrow L_{j}$). ’sym’ in the below equations
corresponds to this symmetry fixing. Then the renormalized effective potential
for a complex scalar field in $d$ dimensional spacetime with $k$ compact
spatial dimension is

$\displaystyle V_{ren}$ $\displaystyle=$
$\displaystyle-\frac{4\pi\hbar}{\mathcal{V}_{k}}\sum_{s=1}^{k}\Bigg{[}\frac{(\pi)^{\frac{d-s}{2}}}{\Gamma\left(\frac{d-s}{2}+1\right)}\left(\frac{1}{L_{k+1-s}}\right)^{d-s}\left(\prod\limits_{r=1}^{k-s}L_{r}\right)$
(14)
$\displaystyle{{\sideset{}{{}^{*}}{\sum}}}_{l}f_{{}_{\frac{s-d}{2}}}\left(\alpha_{k+1-s}^{k},\beta_{k+1-s}\right)\Bigg{]}+\mbox{sym},$

where $\sideset{}{{}^{*}}{\sum}$ is a multiple summation and we have to sum
over all $l\in(n_{k},...,n_{k+2-s})|s\geq 2$ in the range $(-\infty,\infty)$,
$\mathcal{V}_{k}=L_{1}...L_{k}$ and

$\displaystyle\alpha_{k+1-s}^{k}$ $\displaystyle=$ $\displaystyle
L_{k+1-s}\Bigg{(}L_{k+2-s}^{-2}\left(n_{k+2-s}+\beta_{2}\right)^{2}...$ (15)
$\displaystyle+L_{k}^{-2}\left(n_{k}+\beta_{k}\right)^{2}+\left(\frac{m}{2\pi}\right)^{2}\Bigg{)}^{\frac{1}{2}},$

where $\beta_{k}=-qa_{k}L_{k}/2\pi$. To get Eq. (14) we used the mirror
identity $\Gamma(-x)\sin(-\pi x)=\pi/x\Gamma(x)$. The renormalized effective
potential obtains its minimum value when $a=0$ (see appendix A). Then using
Eq. (13) the vacuum energy of a massive complex scalar field is

$\displaystyle
V_{min}=-\frac{4\hbar}{L^{d}}\sum_{s=1}^{k}\sideset{}{{}^{*}}{\sum}_{l}$
$\displaystyle\sum_{n=1}^{\infty}\left(N_{k+1-s}^{k}\right)^{\frac{d-s+1}{2}}n^{\frac{s-d-1}{2}}$
(16) $\displaystyle K_{\frac{d-s+1}{2}}(2\pi nN_{k+1-s}^{k})+\mbox{sym},$

where

$\displaystyle N_{k+1-s}^{k}$ $\displaystyle=$
$\displaystyle\sqrt{n_{k+2-1}^{2}+...+n_{k+1-s}^{2}+\left(\frac{mL}{2\pi}\right)^{2}}.$
(17)

In Eq. (17) we assume $L_{1}\approx L_{2}...\approx L_{k}\approx L$. The
function $z^{\lambda}K_{\lambda}(z)$ decays rapidly as $z$ varies from $0$ to
$\infty$, this allows us to approximate the sum $\sideset{}{{}^{*}}{\sum}$ by
its leading order as

$\displaystyle V_{min}\approx$
$\displaystyle-\frac{4\hbar}{\mathcal{V}_{k}}\sum_{s=1}^{k}\left(\frac{\prod_{r=1}^{k-s}L_{r}}{L_{k+1-s}^{d-s}}\right)\left(\frac{mL_{k+1-s}}{2\pi}\right)^{\frac{d-s+1}{2}}\times$
(18)
$\displaystyle\sum_{n=1}^{\infty}n^{\frac{s-d-1}{2}}K_{\frac{d-s+1}{2}}(nmL_{k+1-s})+\mbox{sym}.$

This approximation is valid even for compact dimensions of difference size.
Also using the result $\lim_{z\rightarrow
0}z^{\nu}K_{\nu}(z)=2^{\nu-1}\Gamma(\nu)$ we can use Eq. (18) for calculating
the vacuum energy of massless scalar field in the same spacetime. For the case
of a masseless scalar field, this approximation gives a relative error of
$1.3\%$, $3.9\%$ and $9\%$ for
$\mathbb{R}^{(1,1)}\times\mathbb{S}^{1}\times\mathbb{S}^{1}$,
$\mathbb{R}^{(1,3)}\times\mathbb{S}^{1}\times\mathbb{S}^{1}$ and
$\mathbb{R}^{(1,3)}\times\mathbb{S}^{1}\times\mathbb{S}^{1}\times\mathbb{S}^{1}$,
respectively when we compare the result with Eq. (16). The relative error
increases as we increase the number of dimensions but we can consider higher
order corrections in the same way. We use this result (Eq. 18) for calculating
the Casimir force in the coming sections as we are interested in the
qualitative change of Casimir force due to background fields and extra compact
dimensions.

###  III.1 Casimir force on a piston

Armed with our expression for $V_{min}$ (Eq. 18), we now turn our attention to
calculate the Casimir force on a piston (or two parallel plates) placed in
various compact spacetime dimensions. The energy density $\rho$ for a complex
scalar field satisfying Dirichlet boundary condition (i.e., the field
vanishing at boundaries $x=0$ and $x=L$) can also inferred from our results by
replacing $L$ by $2L$ (where $L\in\\{L_{1},...,L_{k}\\}$)[36, 39] as

$\displaystyle\rho(L)$ $\displaystyle=$ $\displaystyle V_{min}(2L).$ (19)

For calculating the Casimir force on a piston we consider the finite box
boundary condition as shown in Fig. 1 and eventually take the limits
$l_{1},l_{2},l_{3}\rightarrow\infty$. The importance of taking the finite box
boundary condition is well discussed in [5, 6, 40, 35, 41].

Figure 1: A Rectangular piston

The Casimir force has contributions from both regions 1 and 2 as

$\displaystyle\frac{F}{\mathcal{V}_{k}}$ $\displaystyle=$
$\displaystyle-\frac{d}{dL}\left(L\rho_{1}\right)-\frac{d}{dL}\left((l_{1}-L)\rho_{2}\right)\Big{|}_{l_{1}\rightarrow\infty},$
(20)

where $\rho_{1}$ and $\rho_{2}$ are the energy densities in the region I and
II respectively. Considering $d=4$ and $k=1$ in Eq. (18) and using Eq. (19)
and (20)

$\displaystyle F$ $\displaystyle=$
$\displaystyle-\frac{m^{2}}{4L^{2}\pi^{2}}\sum_{n=1}^{\infty}\left(\frac{2mL}{n}K_{1}(\xi)+\frac{3}{n^{2}}K_{2}(\xi)\right),$
(21)

where $\xi=2mnL$. This gives the force per unit area on the piston due to the
vacuum fluctuations of a massive complex scalar field placed in a Minkowski
spacetime. Eq. (21) is a known result in the literature [39], and in the
limit, $m\rightarrow 0$ this gives the standard result first obtained by H B G
Casimir [4]. Similarly, we can calculate the force per unit area on the piston
in the presence of extra compact dimensions (as shown in Fig. 2), and each
extra compact dimension adds corrections to Eq. (21). The correction vanishes
in the limit $R\rightarrow 0$, where $R$ is the size of the extra compact
dimension. Also, the limit $R\rightarrow\infty$ gives the results for $d$
dimensional flat topology [42].

Figure 2: Shows the variation of Casimir force with respect to $Lm$. In this
plot we assume $R=L$.

##  IV Constant magnetic field perpendicular to the piston

Consider a complex scalar field in $1+3$ spacetime where one of the spatial
dimension is compact under the identification $x_{1}\sim x_{1}+L$. We choose
the guage potential $A_{\mu}=(0,a,0,Bx_{2})$ such that there exist a constant
magnetic field $B$ along the compact dimension along with a constant guage
potential $a$. One can solve Eq. (3) with Eq. (4) by expanding
$K(s;x,x^{\prime})$ in Fourier modes. Heat kernel in the coinciding limit
($x\rightarrow x^{\prime}$) is

$\displaystyle K(s;x,x)$
$\displaystyle=\frac{qB}{L}\left(\frac{qB}{2\pi\sinh(2qBs)}\right)^{\frac{1}{2}}\times$
$\displaystyle\sum_{n=-\infty}^{\infty}\int\frac{d\omega}{2\pi}\frac{dy}{2\pi}e^{-\omega^{2}s}\exp\left(-\tanh(qBs)qBy^{2}\right)e^{-M_{n}^{2}s},$

where
$M_{n}^{2}=m^{2}+\frac{4\pi^{2}}{L^{2}}\left(n-\frac{qaL}{2\pi}\right)^{2}$.
Using standard Gaussian integral

$\displaystyle K(s;x,x)$ $\displaystyle=$
$\displaystyle\frac{qB}{8\pi^{\frac{3}{2}}L}\frac{1}{\sqrt{s}\sinh(qBs)}\sum_{n=-\infty}^{\infty}e^{-M_{n}^{2}s}.$
(23)

Substituting Eq. (23) in Eq. (2) gives the Euclidean effective potential as

$\displaystyle V_{E}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi^{3/2}L}\sum_{n=-\infty}^{\infty}\int_{0}^{\infty}\frac{ds}{s^{5/2}}\frac{qBs}{\sinh(qBs)}e^{-M_{n}^{2}s}.$
(24)

For a weak magnetic field the renormalized (see appendix B) Euclidean
effective potential is

$\displaystyle
V_{Eren}\approx\frac{7q^{4}B^{4}}{2880\pi^{3/2}L}\sum_{n}\int_{0}^{\infty}dss^{3/2}e^{-M_{n}^{2}s},$
(25)

where we neglect the higher order terms in $B$. Using Gamma function

$\displaystyle V_{Eren}$ $\displaystyle=$
$\displaystyle\frac{7q^{4}B^{4}}{3840L\pi}\left(\frac{L}{2\pi}\right)^{5}\sum_{n=-\infty}^{\infty}\left(\frac{m^{2}L^{2}}{4\pi^{2}}+\left(n-\frac{qaL}{2\pi}\right)^{2}\right)^{-\frac{5}{2}}.$
(26)

Using Eq. (9) we can do the summation which leads to,

$\displaystyle V_{Eren}$ $\displaystyle=$
$\displaystyle\frac{7B^{4}q^{4}}{5760m^{4}\pi^{2}}+\frac{7B^{4}L^{4}q^{4}}{30720\pi^{6}}f_{\lambda}(\alpha,\beta),$
(27)

where $\alpha=mL_{1}/2\pi$ , $\beta=-qaL_{1}/2\pi$ and $\lambda=5/2$. The
first term in $V_{Eren}$ is the contribution from $1+3$ flat spacetime [43].
Similar to the above section III, we choose the constant gauge potential
($a$), such that it minimizes the Lorentzian effective potential. This gives
the vacuum energy of a complex scalar field as

$\displaystyle V_{min}$ $\displaystyle=$
$\displaystyle-\frac{7B^{4}q^{4}}{5760m^{4}\pi^{2}}\left(1+m^{2}L^{2}\sum_{n=1}^{\infty}n^{2}K_{2}(\xi/2)\right),$
(28)

where we used Eq. (13). Including the higher order magnetic field contribution
we can rewrite the vacuum energy as

$\displaystyle V_{min}$ $\displaystyle=$
$\displaystyle\sum_{k=2}^{\infty}\frac{(2^{2k}-2)}{16\pi^{2}(2k)!}m^{4-4k}B^{2k}q^{2k}B_{2k}\Gamma(2k-2)$
$\displaystyle+\sum_{k=2}^{\infty}\frac{(1-2^{1-2k})}{\pi^{2}(2k)!}\left(\frac{L}{m}\right)^{2k-2}B^{2k}q^{2k}B_{2k}\sum_{n=1}^{\infty}n^{2k-2}K_{2k-2}(\xi/2),$

where $B_{k}$ is the $(k)^{th}$ Bernoulli number. This matches with the
results in [44]. First part of the equation is the contribution from flat
space topology [45]. Now we calculate the Casimir force acting on piston as
discussed in section III.1. Substituting Eq. (28) in Eq. (19) and using Eq.
(20) , the Casimir force per unit area on the piston is

$\displaystyle F$ $\displaystyle=$
$\displaystyle\frac{7B^{4}q^{4}L^{2}}{5760m^{4}\pi^{2}}\sum_{n=1}^{\infty}\left(3\xi^{2}K_{2}(\xi)-\frac{1}{2}\xi^{3}\left[K_{1}(\xi)+K_{3}(\xi)\right]\right).$
(30)

where $\xi=2mnL$. Similarly, we calculate the Casimir force per unit area on
the piston with extra compact spatial dimensions, and the results are as shown
in the Fig. 3.

Figure 3: Shows the qualitative behaviour of Casimir force in the presence of
weak external field and extra compact dimensions. On the y-axis $A$ can be
electric field $E$ parallel to the plates or magnetic field $B$ perpendicular
to the plates. In this plot we assume that the size of extra compact dimension
is same as the seperation between the plates.

##  V Electric field parallel to the piston and the Schwinger effect

Consider the guage potential $A_{\mu}=(0,a,0,E\tau)$, a constant guage
potential along the compact dimension and a constant electric field along the
non compact dimension. Proceeding the same way as discussed in the previous
section IV, the effective potential is

$\displaystyle V$ $\displaystyle=$
$\displaystyle-\frac{1}{8\pi^{3/2}L_{1}}\sum_{n=-\infty}^{\infty}\int_{0}^{\infty}\frac{ds}{s^{5/2}}\frac{qEs}{\sin(qEs)}e^{-M_{n}^{2}s}.$
(31)

This explicitly shows the eletric-magnetic duality in the effective potential
[46, 47]. Following the calculations in appendix B and section IV, for a weak
electric field $E\rightarrow 0$, one can calculate the Casimir force per unit
area on the piston as

$\displaystyle F$ $\displaystyle=$
$\displaystyle\frac{7E^{4}q^{4}}{5760m^{4}\pi^{2}}\sum_{n=1}^{\infty}\left(3\xi^{2}K_{2}(\xi)-\frac{1}{2}\xi^{3}\left[K_{1}(\xi)+K_{3}(\xi)\right]\right).$
(32)

Similar calculations can be extended to $d$ dimensional spacetime with $k$
compact spatial dimensions. For the case of one extra compact dimension, the
corresponding Casimir force is

$\displaystyle F=-\frac{7E^{4}q^{4}}{11520\pi^{2}m^{4}}$
$\displaystyle\sum_{n=1}^{\infty}\Big{(}\xi^{2}K_{2}(\xi)+\frac{Rm}{\sqrt{2\pi}}\xi^{\frac{3}{2}}K_{\frac{3}{2}}(\xi)$
(33)
$\displaystyle-\xi^{3}K_{1}(\xi)-\frac{Rm}{\sqrt{2\pi}}\xi^{\frac{5}{2}}K_{\frac{1}{2}}(\xi)\Big{)},$

where $\xi=2mnL$ as usual and $R$ is the size of extra compact dimension. A
plot showing the influence of extra compact dimension on the Casimir force is
given in Fig. 3. In the presence of electric field $E$, the effective
potential has an imaginary contribution coming from the poles at
$s_{k}=k\pi/qE$ where $k\in N$ (Eq. 31). The imaginary part of the effective
potential is (see appendix C)

$\displaystyle Im(V)$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{2(2\pi)^{3}}e^{\frac{-k\pi
m^{2}}{qE}}\left(\frac{qE}{k}\right)^{2}\Theta\left(3,0,e^{\frac{-qEL^{2}}{4k\pi}}\right),$
(34)

where $\Theta$ is the elliptic theta function. Here we used
$M_{n}^{2}=m^{2}+\frac{4\pi^{2}}{L^{2}}\left(n-\frac{qaL}{2\pi}\right)^{2}$
and also took $a=0$ as that choice minimizes the potential. The imaginary part
of the effective potential gives the number of pairs of charged scalar
particles created by the electric field per unit time per unit volume. In the
limit $L\rightarrow\infty$, $\Theta\rightarrow 1$ and gives the standard
Schwinger effect result in $3+1$ dimension [48, 49].

Figure 4: Shows the variation of imaginary part of the effective potential
with respect to the separation between the parallel plates.

Presence of extra compact dimensions, along with the piston, add the number of
$\Theta$ functions corresponding to each extra compact dimensions. But if we
consider extra compact dimensions without the piston, i.e.,
$A_{\mu}=(0,0,0,E\tau,a)$. The corresponding effective potential is

$\displaystyle
V=-\frac{qE}{16\pi^{2}R}\sum_{n=-\infty}^{\infty}\int_{0}^{\infty}\frac{ds}{s^{2}}\frac{e^{-M_{n}^{2}}s}{\sin(qEs)},$
(35)

where $R$ is the size of extra compact dimension and $M_{n}^{2}=(\frac{2\pi
n}{R}-qa)^{2}+m^{2}$. There are no branch cuts in the integral and using the
results in [48, 43]

$\displaystyle Im(V)$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{4(2\pi)^{3}}e^{\frac{-k\pi
m^{2}}{qE}}\left(\frac{qE}{k}\right)^{\frac{5}{2}}\Theta\left(3,0,e^{\frac{-qER^{2}}{4k\pi}}\right).$
(36)

The limit $R\rightarrow\infty$ gives the Schwinger effect in $1+4$ flat
dimension. In the same way, one can calculate the pair production in a higher
compact and non-compact dimensions, with or without pistons.

##  VI Summary and conclusion

In short, Eq. 18, 30, 33, 34 and 36 are the central results of this paper,
hitherto unreported in previous literature.

Summarizing, this paper considers the influence of extra compact dimensions
and external background gauge fields on the Casimir force on a piston using
effective action method. We discuss the influence of both magnetic fields
perpendicular to the piston, as well as the electric field parallel to the
piston in the presence of extra compact dimensions. In the case of an electric
field, we also considered the effect of parallel plates and extra compact
dimensions on the Schwinger particle production process.

In section III.1, we examine the effect of extra compact dimensions on Casimir
forces. A similar study for the massless scalar field using the mode sum
approach is presented in [6]. For massive scalar fields in lower and non-
compact dimensions, our results match with the conclusions in [39, 42, 50].
The Casimir force acting between the parallel plates is always attractive
irrespective of the mass, the number of extra compact dimensions, and the size
of extra compact dimensions (Figure 2). However, the presence of extra compact
dimensions decrease the force of attraction between the parallel plates. This
may be because of the following reason. In the presence of extra dimensions,
the vacuum fluctuations can escape into these dimensions, resulting in
decreased modes in the presence of a piston. This hypothesis is supported by
the observation that the Casimir force per unit area per unit length of extra
compact dimension is maximum when the size of the extra compact dimension is
less than the separation between the plates.The influence of the extra compact
dimensions is maximum in the regime $Lm<1$ (Figure 2). Casimir force is
maximal for a massless scalar field and decreases as mass increases. Casimir
force vanishes to zero as $m\rightarrow\infty$ as there are no more quantum
fluctuations in this limit. Corrections in Casimir force per unit area on the
piston is more significant for large compact dimensions. Similar conclusions
can be obtained for massless scalar fields.

Experimentally, the Casimir effect has been observed for a parallel plate
separation of the order $10^{-7}$m [51, 52]. For the Higgs field (mass
$\approx 10^{-25}$Kg [53]), using the above values, the Casimir force per unit
area on the piston without any extra compact dimension is $-4.318\times
10^{-8}N$. Now from phenomenological models, if we consider the upper bound on
the size of one extra compact dimension as $R\leq 10$ nm [8], we see from our
Eq. (34) that the presence of one extra compact dimension almost halves the
intensity of attraction to $-2.262\times 10^{-8}N$. These qualitative numbers
can in fact provide us with a test bed to probe for existence of compact extra
dimensions.

Additionally, in sections IV and V, we explore the influence of external
background fields on the Casimir force in the presence of extra compact
dimensions. If we take the limit $B$ to infinity in Eq. (24), the effective
potential vanishes to zero. From this, we conclude that a high magnetic field
always inhibits the Casimir force as mentioned in [44]. But in the presence of
a high electric field parallel to the piston, the effective potential is
instead enhanced as seen from Eq. 31. However, in the weak field
approximation, both electric and magnetic field enhance the Casimir force of
attraction as observed from Eqs. (32 and 30). Also, in the weak field
approximation, there is a repulsive contribution to the Casimir force for
$Lm<1$ (Figure 3). For the case of a weak magnetic field, this repulsive
contribution is noted in [54]. Figure 3 illustrates that even though the extra
compact dimension does not change the qualitative behavior of Casimir force,
it does decrease its magnitude. The electric field can also produced charged
pairs of particle from the vacuum, called the Schwinger effect. The Schwinger
effect still awaits experimental confirmation in the laboratory as it requires
extreme electric fields. Our results 34 show that the presence of parallel
plates can enhance the pair production. This observation can help in the
experimental realization of the Schwinger effect. Additionally, we can deduce
from 36 that the presence of extra compact dimensions also influences the
Schwinger pair product.

The present formalism can be further extended for studying twisted fields in
similar manifolds, and also for investigating Casimir effect at finite
temperatures which will be discussed in later investigations.

##  Appendix A Behaviour of $f_{\lambda}(\alpha,\beta)$

###  A.1 The maxima and minima of the function $f_{\lambda}(\alpha,\beta)$
with respect to $\beta$

We are interested in finding the maximum of $f_{\lambda}(\alpha,\beta)$ with
respect to $\beta$ when $\lambda<0$. From Eq. (9)

$\displaystyle\frac{\partial
f}{\partial\beta}=2\pi\mathcal{I}\left\\{\int_{\alpha}^{\infty}dx\frac{(x^{2}-\alpha^{2})^{-\lambda}e^{2\pi
x+2\pi i\beta}}{\left(e^{2\pi x+2\pi i\beta}-1\right)^{2}}\right\\}\;,$ (37)

where $\mathcal{I}$ gives the imaginary part of the function inside. But,

$\displaystyle\frac{e^{2\pi x+2\pi i\beta}}{\left(e^{2\pi
x+2i\pi\beta}-1\right)^{2}}=\Big{(}$
$\displaystyle\left(2\cos(2\pi\beta)\cosh(2\pi x)-2\right)$ (38)
$\displaystyle+2i\sin(2\pi\beta)\sinh(2\pi x)\Big{)}^{-1}.$

Which gives,

$\displaystyle\frac{\partial f}{\partial\beta}$ $\displaystyle=$
$\displaystyle-4\pi\int_{\alpha}^{\infty}dx\left(x^{2}-\alpha^{2}\right)^{-\lambda}\times$
$\displaystyle\frac{\sin(2\pi\beta)\sinh(2\pi
x)}{\left(2\cos(2\pi\beta)\cosh(2\pi
x)-2\right)^{2}+\left(2\sin(2\pi\beta)\sinh(2\pi x)\right)^{2}}.$

Then,

$\begin{split}\frac{\partial f}{\partial\beta}=0\hskip 14.22636pt\forall\hskip
14.22636pt\beta=\frac{n}{2}\hskip 14.22636pt\mbox{where}\hskip
7.11317ptn\in\mathbf{Z}.\\\ \end{split}$ (39)

Checking the second derivative at these critical points,

$\displaystyle\frac{\partial^{2}f}{\partial\beta^{2}}\Big{|}_{\beta=\frac{n}{2}}$
$\displaystyle=$ $\displaystyle
4\pi\int_{\alpha}^{\infty}dx\left(x^{2}-\alpha^{2}\right)^{-\lambda}\times$
$\displaystyle\frac{2\pi(-1)^{n+1}}{\left(2\cos(2\pi\beta)\cosh(2\pi
x)-2\right)^{2}+\left(2\sin(2\pi\beta)\sinh(2\pi x)\right)^{2}}\;.$

Which gives

$\mbox{Sign}\left(\frac{\partial^{2}f}{\partial\beta^{2}}\right)=(-1)^{n+1}=(-1)^{2\beta+1}\hskip
14.22636pt\forall\hskip 7.11317ptn\in\mathbf{Z}.$

From this we can conclude that $f$ is maximum when $\beta=n$ and is minimum
when $\beta=n+\frac{1}{2}$.

###  A.2 Convergence of $f_{\lambda}(\alpha,\beta)$

From appendix A.1, the maximum of $f_{\lambda}(\alpha,\beta)$ with respect to
$\beta$ is given by

$f_{-\lambda_{max}}(\alpha)=\int_{\alpha}^{\infty}dx\frac{\left(x^{2}-\alpha^{2}\right)^{\lambda}}{e^{2\pi
x}-1}\;,$ (40)

with $\lambda,\alpha>0$. By inspection and from Fig. 5 below we can see that,

$\frac{x^{2\lambda}}{e^{2\pi
x}-1}\geq\frac{\left(x^{2}-\alpha^{2}\right)^{\lambda}}{e^{2\pi x}-1}.$ (41)

But

$\int_{0}^{\infty}dx\frac{x^{\lambda}}{e^{2\pi
x}-1}=\frac{1}{(2\pi)^{\lambda+1}}\Gamma\left(1+\lambda\right)\mbox{PolyLog}\left(1+\lambda,1\right).$

Then using comparison test for convergence, we can conclude that
$f_{\lambda_{max}}(\alpha)$ is converging, from which we can conclude that (9)
is converging.

Figure 5: Graphical representation of (41) for $\alpha=1$ and $\lambda=6$.

###  A.3 Behaviour of the function $f_{\lambda}(\alpha,\beta)$ with respect to
$\alpha$

From Eq. (9)

$\begin{split}f_{-\lambda}(\alpha,\beta)=\int_{\alpha}^{\infty}dx\frac{\left(x^{2}-\alpha^{2}\right)^{\lambda}\left(e^{2\pi
x}\cos(2\pi\beta)-1\right)}{\left(e^{2\pi
x}\cos(2\pi\beta)-1\right)^{2}+e^{4\pi x}\sin^{2}(2\pi\beta)}.\end{split}$

Which gives,

$\displaystyle\frac{\partial f_{-\lambda}(\alpha)}{\partial\alpha}$
$\displaystyle=$ $\displaystyle-2\alpha\lambda\int_{\alpha}^{\infty}dx\times$
$\displaystyle\frac{\left(x^{2}-\alpha^{2}\right)^{\lambda-1}\left(e^{2\pi
x}\cos(2\pi\beta)-1\right)}{\left(e^{2\pi
x}\cos(2\pi\beta)-1\right)^{2}+e^{4\pi x}\sin^{2}(2\pi\beta)}.$

In our case interest $\lambda\geq 1$(which is half the spacetime dimensions)
and $\alpha\geq 0$. So $f_{\lambda}(\alpha,\beta)$ is a monotonically
decreasing function if $\cos(2\pi\beta)>0$ and is monotonically increasing
function if $\cos(2\pi\beta)<0$. But in both cases the magnitude of the
integral will be maximum when $\alpha=0$. From the plots below (Fig. 6) we can
see that the $\sum_{n_{2}}f_{\lambda}\left(\alpha_{n_{1}n_{2}},\beta\right)$
is maximum when $\cos(2\pi\beta)>0$, which gives the positive value for each
term in the sum and $\beta\in\mathbb{Z}$. Also the fast decay of the function
shows that the major contribution comes when $\alpha=0$. For higher summations
also this factor remains the same, which is checked numerically.

Figure 6: Shows the variation of $f_{\lambda}(\alpha,\beta)$ with respect to
$\alpha$ for different values of $\lambda$ ($\lambda=-7,-8,-9$ and $-10$ in
plots A, B, C, and D respectively). In each plot we choose $\beta$ such that
$\cos(2\pi\beta)>0$ for one and $\cos(2\pi\beta)<0$ for other.

##  Appendix B Renormalization

The Euclidean effective potential is

$\displaystyle V_{E}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\frac{ds}{s}K(s;x,x)$ (43) $\displaystyle=$
$\displaystyle\frac{1}{8\pi^{3/2}L_{1}}\sum_{n=-\infty}^{\infty}\int_{0}^{\infty}\frac{ds}{s^{5/2}}\frac{qBs}{\sinh(qBs)}e^{-M_{n}^{2}s}.$

In the limit $B\rightarrow 0$

$\displaystyle V_{E}|_{B\rightarrow 0}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi^{3/2}L_{1}}\sum_{n=-\infty}^{\infty}\int_{0}^{\infty}\frac{ds}{s^{5/2}}e^{-M_{n}^{2}s}.$
(44)

Subtracting this will leads to the new effective potential as

$\displaystyle V1_{E}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi^{3/2}L_{1}}\sum_{n=-\infty}^{\infty}\int_{0}^{\infty}\frac{ds}{s^{5/2}}e^{-M_{n}^{2}s}\left(\frac{qBs}{\sinh(qBs)}-1\right).$

Near $s=0$ the effective potential is given by

$\displaystyle V_{1}\approx$
$\displaystyle\frac{1}{8\pi^{3/2}L_{1}}\sum_{n}\frac{B^{2}q^{2}}{6}\int_{0}^{\infty}\frac{ds}{\sqrt{s}}e^{-M_{n}^{2}s}$
(45)
$\displaystyle-\frac{7}{360}q^{4}B^{4}\int_{0}^{\infty}dss^{3/2}e^{-M_{n}^{2}s}+O(B^{6}),$

where we took $V=-V_{E}$. The first term is proportional to $B^{2}$ which
corresponds to the classical Lagrangian. The total Lagrangian for the system
is

$\displaystyle L_{tot}$ $\displaystyle=$ $\displaystyle
L_{0}+V=L_{0}+L_{c}+\left(V-L_{c}\right),$ (46)

where

$\displaystyle L_{c}$ $\displaystyle=$
$\displaystyle\frac{-1}{8\pi^{3/2}L_{1}}\sum_{n}\frac{B^{2}q^{2}}{6}\int_{0}^{\infty}\frac{ds}{\sqrt{s}}e^{-M_{n}^{2}s}\equiv
B^{2}Z.$ (47)

Now subtracting out $L_{c}$ from $V_{1}$ and redefining the field strength and
charges as

$\displaystyle B^{\prime}$ $\displaystyle=$
$\displaystyle(1+Z)^{\frac{1}{2}}B\;\;;q^{\prime}=(1+Z)^{-\frac{1}{2}}q.$ (48)

Then the renormalized effective potential is

$\displaystyle V_{ren}$ $\displaystyle=$
$\displaystyle-\frac{1}{8\pi^{3/2}L_{1}}\sum_{n}\frac{7}{360}q^{4}B^{4}\int_{0}^{\infty}dss^{3/2}e^{-M_{n}^{2}s}$
$\displaystyle+O(B^{6}),$

where we wrote $B$ and $q$ for $B^{\prime}$ and $q^{\prime}$.

##  Appendix C Contour integral

Consider

$\displaystyle
I=\int_{0}^{\infty}\frac{ds}{s^{5/2}}\frac{qEs}{\sin(qEs)}e^{-M_{n}^{2}s}$
(50)

This has branch points at $s=0$ and $s=\infty$. Also has singularities at
$s_{k}=\frac{k\pi}{qE}$ where $k\in N$. But the singularity at $s=0$ is
already considered for calculating Casimir energy or the real part of the
effective potential. Then the contribution to the imaginary part of the
effective potential is coming from other poles.

Figure 7: The contour of s integration for Eq. 50

Taking

$\displaystyle f(z)$ $\displaystyle=$
$\displaystyle\frac{1}{z^{5/2}}\frac{qEz}{\sin(qEz)}e^{-M_{n}^{2}z},$ (51)

and using Cauchy’s theorem we can say that

$\displaystyle\oint f(z)dz=2\pi
i\sum_{k=1}^{\infty}(-1)^{k}\left(\frac{qE}{k\pi}\right)^{\frac{3}{2}}e^{-\frac{M_{n}^{2}k\pi}{qE}},$
(52)

along the closed contour given in the Fig. 7. From this we can write

$\displaystyle\oint f(z)dz$ $\displaystyle=$
$\displaystyle\int_{c_{1}}f(z)dz+\int_{s_{1}+\epsilon}^{s_{2}-\epsilon}f(z)dz+...$
$\displaystyle+\int_{-\infty}^{s_{n}+\epsilon}+\int_{\tilde{c_{n}}}f(z)dz+...\int_{s_{1}-\epsilon}^{\epsilon}f(z)dz.$

In the fourth quadrant or in the lower half plane the function will have an
extra $2\pi$ phase which gives

$\displaystyle f(z)$ $\displaystyle=$
$\displaystyle\frac{e^{-i\pi}}{z^{5/2}}\frac{qEz}{\sin(qEz)}e^{-M_{n}^{2}z}=-f(z).$
(54)

Then in the limit $\epsilon\rightarrow 0$ this gives

$\displaystyle\oint f(z)dz$ $\displaystyle=$ $\displaystyle
2I+\int_{c_{1}}f(z)dz+\int_{c_{2}}f(z)dz+...$
$\displaystyle+\int_{c_{n}}f(z)dz+\int_{\tilde{c_{n}}}f(z)+...\int_{\tilde{c_{1}}}f(z)dz.$

Then for integration over $c_{1}$, take $z=s_{1}+\epsilon e^{i\theta}$

$\displaystyle\int_{c_{1}}f(z)dz$ $\displaystyle=$
$\displaystyle\int_{\pi}^{0}\frac{i\epsilon
e^{i\theta}d\theta}{s_{1}^{5/2}}\frac{qEs_{1}}{\cos(qEs_{1})\epsilon
e^{i\theta}}e^{-M_{n}^{2}s_{1}},$ (56) $\displaystyle=$
$\displaystyle-i\frac{\pi
qEe^{-M_{n}^{2}s_{1}}}{s_{1}^{\frac{3}{2}}\cos(qEs_{1})}.$

The integration over $c_{1}$ and $\tilde{c_{1}}$ add up to zero, similarly for
all other $c_{n}^{\prime}s$. From this we can write

$\displaystyle I=\pi
i\sum_{k=1}^{\infty}(-1)^{k}\left(\frac{qE}{k\pi}\right)^{\frac{3}{2}}e^{-\frac{M_{n}^{2}k\pi}{qE}},$
(57)

##  Appendix D Mode sum approach

The infinite sum in Eq. (9) leads to

$\displaystyle\sum_{n=-\infty}^{\infty}\left(n^{2}+\alpha^{2}\right)^{-\lambda}$
$\displaystyle=$
$\displaystyle\frac{\sqrt{\pi}\Gamma\left(\lambda-\frac{1}{2}\right)\alpha^{1-2\lambda}}{\Gamma(\lambda)}$
$\displaystyle+4\sin(\lambda\pi)f_{\lambda}(\alpha,0),$

and

$\displaystyle\sum_{n=1}^{\infty}\left(n^{2}+\alpha^{2}\right)^{-\lambda}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\alpha^{-2\lambda}+\frac{\sqrt{\pi}\Gamma\left(\lambda-\frac{1}{2}\right)\alpha^{1-2\lambda}}{2\Gamma(\lambda)}$
$\displaystyle+2\sin(\lambda\pi)f_{\lambda}(\alpha,0).$

The energy of the complex scalar field with one extra compact dimension and
having two plates separated by a distance of $L$ is given by

$\begin{split}E&=\int\frac{d^{2}k}{(2\pi)^{2}}\sum_{n=1}^{\infty}\sum_{n_{1}=-\infty}^{\infty}\sqrt{k_{1}^{2}+k_{2}^{2}+\frac{n^{2}\pi^{2}}{L^{2}}+\frac{4n_{1}^{2}\pi^{2}}{L_{1}^{2}}}\;,\\\
&=\frac{B\left(1,-\frac{3}{2}\right)}{4\pi}\sum_{n=1}^{\infty}\sum_{n_{1}=-\infty}^{\infty}\left(\frac{n^{2}\pi^{2}}{L^{2}}+\frac{4n_{1}^{2}\pi^{2}}{L_{1}^{2}}\right)^{\frac{3}{2}}\;,\end{split}$
(60)

where we used the integral representation of Beta function. Considering the
summation in both order leads to

$\displaystyle
E=\frac{-1}{12\pi}\Bigg{(}\sum_{n=1}^{\infty}\sum_{n_{1}=-\infty}^{\infty}$
$\displaystyle\left(\frac{n^{2}\pi^{2}}{L^{2}}+\frac{4n_{1}^{2}\pi^{2}}{L_{1}^{2}}\right)^{\frac{3}{2}}$
$\displaystyle+\sum_{n_{1}=-\infty}^{\infty}\sum_{n=1}^{\infty}\left(\frac{n^{2}\pi^{2}}{L^{2}}+\frac{4n_{1}^{2}\pi^{2}}{L_{1}^{2}}\right)^{\frac{3}{2}}\Bigg{)}.$

Using Eq. (D) and Eq. (D)

$\displaystyle\sum_{n=1}^{\infty}\sum_{n_{1}=-\infty}^{\infty}\left(\frac{n^{2}\pi^{2}}{L^{2}}+\frac{4n_{1}^{2}\pi^{2}}{L_{1}^{2}}\right)^{\frac{3}{2}}$
$\displaystyle=\frac{9L_{1}\zeta(5)}{32\pi L^{4}}$
$\displaystyle+\frac{32\pi^{3}}{L_{1}^{3}}\sum_{n=1}^{\infty}f_{-\frac{3}{2}}\left(\frac{L_{1}n}{2L},0\right),$
$\displaystyle\sum_{n_{1}=-\infty}^{\infty}\sum_{n=1}^{\infty}\left(\frac{n^{2}\pi^{2}}{L^{2}}+\frac{4n_{1}^{2}\pi^{2}}{L_{1}^{2}}\right)^{\frac{3}{2}}$
$\displaystyle=$
$\displaystyle\frac{\pi^{3}}{120}\left(\frac{1}{L^{3}}-\frac{8}{L_{1}^{3}}\right)$
$\displaystyle+\frac{9L\zeta(5)}{\pi L_{1}^{4}}$
$\displaystyle+\frac{4\pi^{3}}{L^{3}}\sum_{n=1}^{\infty}f_{-\frac{3}{2}}\left(\frac{2Ln}{L_{1}},\right).$

where we have used $\Gamma(-2)\zeta(-4)=\frac{3\zeta(5)}{4\pi^{4}}$. This
gives,

$\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}}{180}\left[\frac{1}{L_{1}^{3}}-\frac{1}{8L^{3}}\right]-\frac{3L\zeta(5)}{4\pi^{2}L_{1}^{4}}-\frac{3L_{1}\zeta(5)}{128\pi^{2}L^{4}}$
$\displaystyle-$
$\displaystyle\frac{\pi^{2}}{3L^{3}}\sum_{n=1}^{\infty}f_{-\frac{3}{2}}\left(\frac{2Ln}{L_{1}},0\right)-\frac{8\pi^{2}}{3L_{1}^{3}}\sum_{n=1}^{\infty}f_{-\frac{3}{2}}\left(\frac{L_{1}n}{2L},0\right).$

Now considering the system in a finite box as in Fig. 1, the energy in region
I is given by

$\displaystyle E(L)$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}}{180}\left[\frac{1}{L_{1}^{3}}-\frac{1}{8L^{3}}\right]-\frac{3\zeta(5)L_{1}}{128\pi^{2}L^{4}}-\frac{3\zeta(5)L}{4\pi^{2}L_{1}^{4}}$
$\displaystyle-$
$\displaystyle\frac{8\pi^{2}}{3L_{1}^{2}}\sum_{n=1}^{\infty}f_{-\frac{3}{2}}\left(\frac{nL_{1}}{2L},0\right)-\frac{\pi^{2}}{3L^{3}}\sum_{n_{1}=1}^{\infty}f_{-\frac{3}{2}}\left(\frac{2Ln_{1}}{L_{1}},0\right).$

Similarly the energy of region II is given by

$\displaystyle E(l_{1}-L)$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}}{180}\left[\frac{1}{L_{1}^{3}}-\frac{1}{8(l_{1}-L)^{3}}\right]-\frac{3\zeta(5)L_{1}}{128\pi^{2}(l_{1}-L)^{4}}$
$\displaystyle-$
$\displaystyle\frac{3\zeta(5)(l_{1}-L)}{4\pi^{2}L_{1}^{4}}-\frac{8\pi^{2}}{3L_{1}^{2}}\sum_{n=1}^{\infty}f_{-\frac{3}{2}}\left(\frac{nL_{1}}{2(l_{1}-L)},0\right)$
(65)
$\displaystyle-\frac{\pi^{2}}{3(l_{1}-L)^{3}}\sum_{n_{1}=1}^{\infty}f_{-\frac{3}{2}}\left(\frac{2(l_{1}-L)n_{1}}{L_{1}},0\right).$

Now calculating the Casimir force as in [6],

$\displaystyle F$ $\displaystyle=$
$\displaystyle-\frac{d}{dL}\left(E(L)+E(l_{1}-L)\right)\big{|}_{l_{1}\rightarrow\infty}\;,$
(66) $\displaystyle=$
$\displaystyle-\frac{\pi^{2}}{480L^{4}}-\frac{3\zeta(5)L_{1}}{32\pi^{2}L^{5}}$
$\displaystyle-\frac{\pi^{2}}{L^{4}}\sum_{n=1}^{\infty}f_{-\frac{3}{2}}\left(\frac{2Ln}{L_{1}},0\right)-\frac{4\pi^{2}}{L^{2}L_{1}^{2}}\sum_{n=1}^{\infty}n^{2}f_{-\frac{1}{2}}\left(\frac{2Ln}{L_{1}},0\right)$
$\displaystyle+\frac{2\pi^{2}}{L_{1}L^{3}}\sum_{n=1}^{\infty}n^{2}f_{-\frac{1}{2}}\left(\frac{nL_{1}}{2L},0\right).$

This matches with the result we got before upto the approximation, i.e., the
first two terms matches and we neglect other terms in our approximation.

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