Josephson junction formed in the wormhole space time

from the analysis for the critical temperature of BEC

Shingo Takeuchi

†Phenikaa Institute for Advanced Study and Faculty of Basic Science,
Phenikaa University, Hanoi 100000, Vietnam

In this study, considering some gas in the Morris-Thorne traversable wormhole
space time, we analyze the critical temperature of the Bose-Einstein
condensate in the vicinity of its throat. As a result, we obtain the result
that it is zero. Then, from this result, we point out that an analogous state
to the Josephson junction is always formed at any temperatures in the vicinity
of its throat. This would be interesting as a gravitational phenomenology.

##  1 Introduction

The issues concerning the wormhole space times as the solution and the
phenomena of these are one of the problems that have been investigated so much
until now. Since this paper will treat a phenomenological issue in the
wormhole space times, we refer some phenomenological works on the wormhole
space times; gravitational lensing [1, 2, 3, 4, 5, 6, 7, 8], shadows [9, 10,
11], observation [12], Casimir effect [13], teleportation [14], collision of
two particles [15], and creation of a traversable wormhole [16].

However, in these years, studies regarding wormhole space times are performed
so energetically. Its reason would be that recently it has turned out that the
Einstein-Rosen bridge (ER bridge) [17] gives some kinds of the EPR pair [18,
19, 20, 21], which plays very important role in the literature of the
information paradox [22]. It would be also its reason that there are common
features and behaviors between AdS2 wormholes and the SYK models, from which
currently we can perform various interesting studies [23, 24, 25, 26].

Also, there are studies to construct the graphene wormhole in the material
physics from some brane configurations in the superstring theory [27, 28].
From these studies, Chern-Simon current in the graphene wormhole is studied in
[29].

In this study, considering the Morris-Thorne traversable wormhole (traversable
wormhole) [30], we consider some situation that some gas fills the whole space
time of that, where it is assumed Bose-Einstein condensate (BEC) can be formed
at some temperature in this gas. Then, we point out that an analogous state to
the Josephson junction is always formed at any temperatures except for zero in
the vicinity of its throat.

For this, we first analyze the critical Unruh temperature of the gas regarding
BEC in the Rindler, then check that it can agree to the critical temperature
obtained in the flat Euclid space.

The motivation of this is to check the rightness of our analytical method to
obtain the critical temperature of BEC in curved space times. Actually, our
analytical method in the curved space time is different from the usual one in
the flat Euclid space [31] to treat curved space times. However, since the
space time for the accelerated system can be regarded as the flat space with
the temperature same with the Unruh temperature if Euclideanized (see the text
given under (4)), the critical Unruh temperature obtained in the Rindler space
should agree to that in the flat Euclid space.

Then, considering the same gas in the traversable wormhole, we analyze its
critical temperature concerning BEC in the vicinity of its throat.

Here, it is known that if the number of its spatial dimensions is 2 or less in
the flat Euclid space, the effective potential always gets diverged for the
contribution of the infrared region and BEC is not formed111 In the $1+d$
dimensional flat Euclid space, the particle density at the critical point of
BEC is given as $\displaystyle
d=\int\frac{d^{d}k}{(2\pi)^{d}}(\frac{1}{e^{\beta(\sqrt{k^{2}-m^{2}}-\mu)}-1}-\frac{1}{e^{\beta(\sqrt{k^{2}-m^{2}}+\mu)}-1})$
in [31], where $\mu=m$. Then we can see this is diverged for the contribution
of small $k$, if $d=1$ or $2$. At this time, the critical temperature becomes
$0$ [32], which means the state is normal at any temperature. . Then, since
the spatial part of the vicinity of the throat in the traversable wormhole
space time becomes effectively 1 dimensional (as for this point, see (92),
then just take the limit: $r\to v$). we can expect that the critical
temperature of BEC in the vicinity of the throat is always zero. (Of course,
what the dimension effectively becomes 1 depends on the coordinate system we
take. Therefore, surely we could expect the zero critical temperature in the
vicinity of the throat if the dimension becomes 1, however it would be
considered it is not its essential reason. We will give comment on our result
in the end of Sec.4.6 from this point of view.)

Then,

  * •

If the critical temperature is always zero in the vicinity of the throat,
since the throat exists in the form to separate the wormhole space time from
the another side of the wormhole space time, the normal state also appears in
the form to separate the wormhole space time from the another side of the
wormhole space time like Fig.1.

Figure 1:  Penrose diagram for the ($t$, $r$) part of the traversable wormhole
space time. The dotted line represents the throat of the wormhole. The yellow
and blue parts represent the normal and superconductor states we expect to
appear, respectively. As the normal state is appearing in the form to separate
the space time, we can expect that an analogous state to the Josephson
junction is formed in the vicinity of the throat.

  * •

The far region of the wormhole space time is asymptotically flat space time.
Therefore, the critical temperature of BEC in the far region is given by the
one calculated in the flat space time.

  * •

Then, considering by extrapolating between the results in the throat region
and the far region, we can obtain an expectation that an analogous state to
the Josephson junction is formed at any temperatures in the vicinity of the
throat.

Since this could be considered as a gravitational phenomenology, this would be
interesting. This would be also interesting because we can consider a
possibility that the Josephson current flows from the one side of the wormhole
to another side. Of course, the wave function of the current may be damped
when it tunnels, and the Josephson current does not exist in practice.
Currently we could not say any explicit things on this point from the analysis
in this study, which is a future work. We discuss this in Sec.5.

The wormholes we can consider would be the ER bridge and the traversable
wormhole space time. Then, since the ER bridge is, briefly saying, given by
patching two Schwarzschild black holes together, the Josephson current will
flow out from the event horizon. Hence, it would be lightlike, and could not
be observed at any points in the outside of the event horizon (in the regions
I and III in the Penrose diagram) except for the future timelike infinity.
Therefore, it might not be suitable to consider the ER bridge if we consider
about the Josephson current.

The traversable wormhole is free from this problem as we can depict some
timelike flow of the current going across the throat part in Fig.1. However it
needs the exotic matter (some matter to violate the energy condition) [33]
such that the traversable wormhole space time can satisfy the Einstein
equation, which we explicitly show in Sec.4.2. This is a critical problem as
it means that the traversable wormhole space time is not realizable
realistically. For some recent studies on the exotic matter, see [34, 35, 36,
37, 38, 39, 40, 41]. In this study, we consider the traversable wormhole upon
knowing this problem. If we care this problem in this study, we could mention
that it appears for some reason and can exist for some time, then this study
is the one for that time.

We mention the organization of this paper. We analyze the critical Unruh
temperature of the gas regarding BEC in the Rindler space (Sec.2) in the
Euclid space (Sec.3) in the traversable wormhole space time (Sec.4). Then in
Sec.5, we point out that an analogous state to the Josephson junction is
formed in the vicinity of its throat. In Sec.5, we obtain the phase structure
for the BEC/normal states transition in the ER bridge. In Appendix.B, we note
some points in the mechanism for the formation of BEC in this study.

##  2 Critical temperature of BEC in accelerating system

###  2.1 Rindler coordinates and Unruh temperature

As a theoretical model, we consider some gas that entirely fills the Minkowski
spacetime and performs an uniformly accelerated motion with an acceleration
$a$ for one direction. Based on (1) 222 Solving
$\frac{d}{dt}(\frac{mv}{\sqrt{1-\beta^{2}}})=ma$ with the condition $v=0$ at
$t=0$ and the general relation between the Minkowski time and proper time:
$\tau=\int_{0}^{t}dt\sqrt{1-\beta^{2}}$, the trajectory of an accelerated
motion can be obtained as  $\displaystyle
t=\frac{c}{a}\sinh\frac{a\tau}{c},\quad
x=\frac{c^{2}}{a}\,(\cosh\frac{a\tau}{c}-1),$ (1) where we can check with the
$t$ and $x$ above, we can obtain $(\frac{mv}{\sqrt{1-\beta^{2}}})=mat$. Toward
this, there are several ways for how to take the coordinate system along an
accelerated motion [42]. , we denote the coordinates of the gas as

$\displaystyle t=\frac{\xi}{a}\sinh a\zeta,\quad x=\frac{\xi}{a}\cosh a\zeta,$
(2)

which are a kind of the Rindler coordinates, and at $\xi=1$, $\zeta$ becomes
the proper time of the object performing the uniformly accelerated motion $a$.
Then, the squared line element can be given as follows:

$\displaystyle ds^{2}=$
$\displaystyle\,\xi^{2}d\zeta^{2}-a^{-2}d\xi^{2}-dx_{\perp}^{2},$
$\displaystyle=$ $\displaystyle\,\rho^{2}d\tau^{2}-d\rho^{2}-dx_{\perp}^{2},$
(3)

where $\rho\equiv a^{-1}\xi$ and $\tau\equiv a\zeta$.

Here, $(y,z)$ are common with those in the Minkowski coordinates and we use
the notation $x_{\perp}\equiv(y,z)$ in what follows. Note that we take not a
sphere coordinates but a plane coordinates for the $x_{\perp}$-direction.

Euclideanize the coordinate system with $\tau\to i\tau$, and regard it as
periodic. Then, from the no conical singularity condition:
$\frac{\textrm{circumferential
length}}{\textrm{radius}}\big{|}_{\rho=0}=2\pi$, the period in the
$\zeta$-direction can be determined as $2\pi/a$. Therefore, the gas in the
Rindler coordinate system with the proper time $\zeta$ can be would be at the
following temperature:

$\displaystyle T_{U}=a/2\pi=1/2\pi\rho,$ (4)

which agrees to the Unruh temperature.

Here, in (2.1), when we put $\rho$ to $1$, $d\rho^{2}$ vanishes. However, let
us define $d\rho^{2}$ as the squared line element perpendicular to the
original $\rho$-direction. Then, since the period of $\tau$-direction is
$2\pi$, the period in terms of $\zeta$ becomes $2\pi/a$. Hence, when we put
$\rho$ to $1$, (2.1) can be regarded as the $D=1+3$ flat Euclid space with the
Euclid finite temperature $T_{E}=2\pi/a$ for the one with the proper time
$\zeta$. Therefore, a space time for the accelerated system can be considered
to be equivalent to the flat space with the temperature same with the Unruh
temperature if Euclideanized, so that the critical Unruh temperature obtained
in the Rindler space would be considered to agree to that in the flat Euclid
space.

We can see from (4) that locating at different $\rho$ means having different
Unruh temperature. Therefore, we should be careful when we perform the
$\rho$-integral. For this point, we will consider that the Rindler coordinates
are individually applied for each particle comprising the gas, and all the
particles will have the same value with regard to $\rho$. As for the treatment
of $\rho$ in the analysis in this study, we will consider the effective
potential at some $\rho$ as in (57)333 In [42], there is some comment for
another problem in order for an object with some volume performing an
accelerating motion to keep its shape..

###  2.2 Hamiltonian in finite density, probability amplitude and
Euclideanization

We start with the following Lagrangian density for the complex scalar field
for the particles comprising the gas that we mention in Sec.2.1 that fills the
whole space:

$\displaystyle{\cal
L}=\,g^{\mu\nu}\partial_{\mu}\phi^{*}\partial_{\nu}\phi-m^{2}\phi^{*}\phi,$
(5)

where $\phi=\frac{1}{\sqrt{2}}(\phi_{1}+i\phi_{2})$ ($\phi_{1,2}$ are real
scalar fields), Indices $\mu$, $\nu$ and $g^{\mu\nu}$ refer to the Rindler
coordinates ($\eta$, $\rho$, $x_{\perp}$) and the metrices in (2.1).

We define $(\pi,\pi^{*})$ as the canonical momenta for ($\phi^{*}$, $\phi$),
and ($\bar{\pi}$, $\bar{\pi}^{*}$) as those with lower indices:

$\displaystyle(\pi,\pi^{*})$ $\displaystyle\equiv(\frac{\partial{\cal
L}}{\partial(\partial_{\eta}\phi^{*})},\frac{\partial{\cal
L}}{\partial(\partial_{\eta}\phi)})=(g^{\eta\eta}\partial_{\eta}\phi,g^{\eta\eta}\partial_{\eta}\phi^{*}),$
(6) $\displaystyle(\bar{\pi},\bar{\pi}^{*})$
$\displaystyle\equiv(g_{\eta\eta}\pi,g_{\eta\eta}\pi^{*})=(\partial_{\eta}\phi,\partial_{\eta}\phi^{*}).$

Corresponding to $\phi=\frac{1}{\sqrt{2}}(\phi_{1}+i\phi_{2})$, $\pi_{1,2}$
and $\bar{\pi}_{1,2}$ are defined as

$\displaystyle\displaystyle\pi_{1,2}\equiv
g^{\eta\eta}\partial_{\eta}\phi_{1,2},\quad\bar{\pi}_{1,2}\equiv
g_{\eta\eta}\pi_{1,2}=\partial_{\eta}\phi_{1,2}.$ (7)

With these, the Hamiltonian density associated with the Lagrangian density (5)
is given as

$\displaystyle{\cal H}$
$\displaystyle=\pi\,\partial_{\eta}\phi+\pi^{*}\,\partial_{\eta}\phi^{*}-{\cal
L}$
$\displaystyle=\pi^{*}\bar{\pi}-g^{ij}\partial_{i}\phi^{*}\partial_{j}\phi+m^{2}\phi^{*}\phi,$
(8)

where $i,j=\rho,x_{\perp}$. Then, we can write the Hamiltonian in the grand
canonical ensemble, ${\cal H}-\mu_{R}\,q$, as

$\displaystyle{\cal H}-\mu_{R}\,q=$
$\displaystyle\,\frac{1}{2}(g^{\eta\eta}(\bar{\pi}_{1}\bar{\pi}_{1}+\bar{\pi}_{2}\bar{\pi}_{2})-g^{ij}(\partial_{i}\phi_{1}\partial_{j}\phi_{1}+\partial_{i}\phi_{2}\partial_{j}\phi_{2})+m^{2}(\phi_{1}^{2}+\phi_{2}^{2}))$
$\displaystyle-\mu_{R}(\pi_{2}\phi_{1}-\pi_{1}\phi_{2}),$ (9)

where $\mu_{R}$ is the chemical potential and
$q\equiv-i\,g^{\eta\eta}(\phi\,\partial_{\eta}\phi^{*}-\phi^{*}\,\partial_{\eta}\phi)$
is the particle density.

With (2.2), we can write the probability amplitude as

$\displaystyle Z=$ $\displaystyle\int\\!{\cal D}\bar{\pi}{\cal
D}\bar{\pi}^{*}{\cal D}\phi\exp\big{[}i\,\int_{\rm
I}\\!d^{4}x\,\gamma_{R}\,(g^{\eta\eta}\bar{\pi}\partial_{\eta}\phi+g^{\eta\eta}\bar{\pi}^{*}\partial_{\eta}\phi^{*}-({\cal
H}-\mu_{R}\,q))\big{]}$ $\displaystyle=$ $\displaystyle\,\,{\cal
C}_{R}\int\\!{{\cal D}\phi_{1}}{{\cal
D}\phi_{2}}\exp\big{[}\frac{i}{2}\int_{\rm
I}\\!d^{4}x\,\gamma_{R}\,(g^{\eta\eta}(\partial_{\eta}\phi_{1}+\mu_{R}\,\phi_{2})^{2}+(\partial_{i}\phi_{1})^{2}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\,\,+g^{\eta\eta}(\partial_{\eta}\phi_{2}+\mu_{R}\,\phi_{1})^{2}+(\partial_{i}\phi_{2})^{2}-m^{2}({\phi_{1}}^{2}+{\phi_{2}}^{2}))\big{]},$
(10)

where ${\rm I}$ means the whole Rindler wedge I, $\gamma_{R}\equiv\sqrt{-\det
g_{\mu\nu}}$ and we have performed the following redefinitions for the
canonical momenta:

$\displaystyle\bar{\pi}_{1}-(\partial_{\eta}\phi_{1}-\mu_{R}\,\phi_{2})\rightarrow\bar{\pi}_{1},$
(11a)
$\displaystyle\bar{\pi}_{2}-(\partial_{\eta}\phi_{2}+\mu_{R}\,\phi_{1})\rightarrow\bar{\pi}_{2}.$
(11b)

${\cal C}_{R}$ is given as

$\displaystyle{\cal C}_{R}\equiv\int\\!{\cal D}\pi_{1}{\cal
D}\pi_{2}\exp\big{[}i\int_{\Omega_{\rm
I}}\\!d^{4}x\,\gamma_{R}\,g_{\eta\eta}((\pi_{1})^{2}+(\pi_{2})^{2})\big{]}.$
(12)

Since ${\cal C}_{R}$ is some number irrelevant of $\mu_{R}$ and we finally
take the derivative with regard to $\mu_{R}$ to get the e.v. of the particle
density, we can ignore ${\cal C}_{R}$ in our analysis in what follows.

We perform Euclideanization:

$\displaystyle\eta\to-i\tau.$ (13)

At this time,

$\displaystyle(\pi_{\alpha},\bar{\pi}_{\alpha})\to
i(\pi_{\alpha},\bar{\pi}_{\alpha}),$ (14a) $\displaystyle q\to iq,$ (14b)
$\displaystyle ds^{2}\to-\rho^{2}d\tau^{2}-d\rho^{2},\quad 0\leq\tau<2\pi,$
(14c) $\displaystyle g_{\eta\eta}\to g_{\tau\tau},\quad
g_{\tau\tau}=g_{\eta\eta},$ (14d)

where there is no changes in the contents between $g_{\tau\tau}$ and
$g_{\eta\eta}$ except for the notations. Here, the Euclidean temperature is
identical with the Unruh temperature $T_{U}$ in (4). There is a problem that
the temperature depends on the space. As for this, it would be considered that
$\rho$ is always fixed to some value corresponding to the fact particles are
performing an uniformly accelerated motion.

Under the Euclideanization (13) with (14), $Z$ in (2.2) can be rewritten as

$\displaystyle Z=\int\\!{\cal D}\phi_{1}{\cal
D}\phi_{2}\exp\Big{[}\\!-\frac{1}{2}\int\\!d^{3}x\\!\int_{0}^{\beta_{R}}\\!d\tau\rho\,\gamma_{R}\,\left(\\!\begin{array}[]{cc}\phi_{1}&\phi_{2}\end{array}\\!\right)\\!\left(\\!\begin{array}[]{cc}\hat{G}_{R}+M_{R}^{2}&-2ig^{\tau\tau}\mu_{R}\,\partial_{\eta}\\\
2ig^{\tau\tau}\mu_{R}\,\partial_{\eta}&\hat{G}_{R}+M_{R}^{2}\end{array}\\!\right)\\!\left(\\!\begin{array}[]{c}\phi_{1}\\\
\phi_{2}\end{array}\\!\right)\Big{]},$ (20)

where $\beta_{R}\equiv 2\pi$,
$\hat{G}_{R}\equiv\,-g^{\tau\tau}\partial_{\tau}\partial_{\tau}+\gamma_{R}^{-1}g^{ij}\partial_{i}(\gamma_{R}\partial_{j})$
and $M_{R}^{2}\equiv m^{2}-g^{\tau\tau}\mu_{R}^{2}$.

###  2.3 Description for BEC and upper limit of chemical potential

In order to express the fields in the superconductor state, we separately
rewrite $\phi_{1,2}$ as the e.v. part and the original $\phi_{1,2}$ like the
following [31]:

$\displaystyle\phi_{1}\,\equiv$
$\displaystyle\,\sqrt{2}\,\alpha\,\cos\Theta+\tilde{\phi}_{1},$ (21a)
$\displaystyle\phi_{2}\,\equiv$
$\displaystyle\,\sqrt{2}\,\alpha\,\sin\Theta+\tilde{\phi}_{2}.$ (21b)

where $\alpha$ and $\Theta$ mean the absolute value and the phase of e.v., and

$\displaystyle\alpha\left\\{\begin{array}[]{l}=0\quad\textrm{corresponds to
the normal state,}\\\\[2.84526pt] \not=0\quad\textrm{corresponds to the BEC
state.}\\\ \end{array}\right.$ (24)

Rewriting $Z$ in (20) employing the expressions in (21), we can obtain the
following $Z$:

$\displaystyle Z=$ $\displaystyle\exp[-\alpha^{2}\beta_{R}V_{\perp}\int
d\rho\,\rho\,\gamma_{R}\,M_{R}^{2}]\int\\!{\cal D}\tilde{\phi}_{1}{\cal
D}\tilde{\phi}_{2}\exp\bigg{[}$
$\displaystyle-\frac{1}{2}\int_{0}^{\infty}\\!d\rho\,\rho\int_{-\infty}^{\infty}\\!d^{2}x_{\perp}\\!\int_{0}^{\beta_{R}}\\!d\tau\,\gamma_{R}\left(\\!\begin{array}[]{cc}\tilde{\phi}_{1}&\tilde{\phi}_{2}\end{array}\\!\right)\\!\left(\\!\begin{array}[]{cc}\hat{G}_{R}+M_{R}^{2}&2ig^{\tau\tau}\mu_{R}\,\partial_{\eta}\\\
-2ig^{\tau\tau}\mu_{R}\,\partial_{\eta}&\hat{G}_{R}+M_{R}^{2}\end{array}\\!\right)\\!\left(\\!\begin{array}[]{c}\tilde{\phi}_{1}\\\
\tilde{\phi}_{2}\end{array}\\!\right)\bigg{]},$ (30)

where $V_{\perp}\equiv\int dx_{\perp}^{2}=\int dk_{\perp}^{2}$.

Now, let us look at the contribution from the zero mode in the pass-integral
part in (2.3), which we can write as

$\displaystyle\int\\!{\cal D}\tilde{\phi}_{1}{\cal
D}\tilde{\phi}_{2}\exp[-\frac{M_{R}^{2}}{2}(\tilde{\phi}_{1}^{2}+\tilde{\phi}_{2}^{2})]^{V_{I}},$
(31)

where the configurations generated by $\int{\cal
D}\tilde{\phi}^{\prime}_{1}{\cal D}\tilde{\phi}^{\prime}_{2}$ are only the
constant configurations irrelevant of the coordinates, and $V_{I}$ means
$\int_{0}^{\infty}\\!d\rho\int_{-\infty}^{\infty}\\!d^{2}x_{\perp}\\!\int_{0}^{\beta_{R}}\\!d\tau\,\gamma_{R}$.
Then we can see that (34) can converge when $M_{R}^{2}$ is positive, whereas
diverge when $M_{R}^{2}$ is negative or zero. Therefore, we can write as

$\displaystyle\textrm{$Z$ in
(\ref{bjskf})}\left\\{\begin{array}[]{l}\textrm{converges for
$M_{R}^{2}>0$},\\\\[2.84526pt] \textrm{diverges for $M_{R}^{2}\leq
0$}.\end{array}\right.$ (34)

From (34), we can see that there is the upper bound for the value that the
chemical potential can take as

$\displaystyle\mu_{R}^{c}=m/\sqrt{g^{\tau\tau}}=m/a.$ (35)

Here, the one above is evaluated at $\xi=1$, therefore, we can see from (2.1)
that the critical Unruh temperature obtained from the following analysis with
(35) is associated with the one having $\zeta$ as its proper time and
performing the uniformly accelerated motion $a$.

Here, we explain how the BEC is formed in our system. When decreasing the
Unruh temperature from some high temperatures (therefore, $\alpha$ is $0$)
keeping the e.v. of the particle density constant, it turns out that the
chemical potential should rise (see (83)). However, as in (35), there is the
upper limit for the value the chemical potential can take. Therefore, finally
$\alpha$ should start to have some finite value to keep the e.v. of the
particle density constant. Like this, at some lower Unruh temperature,
$\alpha$ becomes finite and BEC is formed. (For more description on this
issue, see Appendix.B.)

###  2.4 Effective potential (1)

We can diagonalize the shoulder in $Z$ in (2.3) as

$\displaystyle Z=$
$\displaystyle\exp[-\alpha^{2}\beta_{R}V_{\perp}\int_{0}^{\infty}d\rho\,\rho\,\gamma_{R}M_{R}^{2}]\int\\!{\cal
D}\tilde{\phi}^{\prime}_{1}{\cal D}\tilde{\phi}^{\prime}_{2}\exp\bigg{[}$
$\displaystyle-\frac{1}{4}\int_{0}^{\infty}\\!d\rho\,\rho\int_{-\infty}^{\infty}\\!d^{2}x_{\perp}\\!\int_{0}^{\beta_{R}}\\!d\tau\,\gamma_{R}\left(\\!\begin{array}[]{cc}\tilde{\varphi}^{\prime}{}^{*}&\tilde{\varphi}^{\prime}\end{array}\\!\right)\\!\left(\\!\begin{array}[]{cc}\hat{G}_{R+}+M_{R}^{2}&0\\\
0&\hat{G}_{R-}+M_{R}^{2}\end{array}\\!\right)\\!\left(\\!\begin{array}[]{c}\tilde{\varphi}^{\prime}\\\
\tilde{\varphi}^{\prime}{}^{*}\end{array}\\!\right)\bigg{]},$ (41)

where $\hat{G}_{R\pm}\equiv\hat{G}_{R}\pm 2g^{\eta\eta}\mu_{R}$ and
$\frac{1}{\sqrt{2}}\left(\\!\begin{array}[]{c}\tilde{\varphi}^{\prime}\\\
\tilde{\varphi}^{\prime}{}^{*}\end{array}\\!\right)\equiv
U^{-1}\left(\\!\begin{array}[]{c}\tilde{\phi}_{1}\\\
\tilde{\phi}_{2}\end{array}\\!\right)$. $U$ is given as
$\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}i&-i\\\ 1&1\end{array}\right)$.
The difference arisen in the path-integral measure by the transformation $U$
is just some constant, which we can ignore.

We express $\phi_{\alpha}$ $(\alpha=1,2)$ by the plain wave expansion for the
$x_{\perp}$-directions remaining the $\rho$-direction as

$\displaystyle\tilde{\phi}_{\alpha}(\rho,\eta,x^{\perp})=\frac{1}{(2\pi)^{2}\beta_{R}}\sum_{n=-\infty}^{\infty}\int
dk_{\perp}^{2}\tilde{\varphi}_{\alpha,n}(\rho,k_{\perp})e^{-i(\omega_{n}\eta\,+\,k_{\perp}x^{\perp})}.$
(42)

Then $\hat{G}_{0}\equiv\frac{1}{2}(\hat{G}_{R+}+\hat{G}_{R-}+2M_{R}^{2})$ in
$Z$ in (2.4) is given as444 $\hat{G}_{0}$ appears in (2.4) as follows:
$\displaystyle\left(\\!\begin{array}[]{cc}\tilde{\varphi}^{\prime}{}^{*}&\tilde{\varphi}^{\prime}\end{array}\\!\right)\\!\left(\\!\begin{array}[]{cc}\hat{G}_{R+}+M_{R}^{2}&0\\\
0&\hat{G}_{R-}+M_{R}^{2}\end{array}\\!\right)\\!\left(\\!\begin{array}[]{c}\tilde{\varphi}^{\prime}\\\
\tilde{\varphi}^{\prime}{}^{*}\end{array}\\!\right)=(\hat{G}_{R+}+\hat{G}_{R-}+2M_{R}^{2})|\tilde{\varphi}^{\prime}|^{2}.$
(48)

$\displaystyle\hat{G}_{0}\to
G_{0}=g^{\tau\tau}\omega_{n}^{2}+\gamma_{R}^{-1}g^{\rho\rho}\partial_{\rho}(\gamma_{R}\partial_{\rho})+g^{\perp\perp}k_{\perp}^{2}+M_{R}^{2}\equiv
G_{R}+M_{R}^{2}$ (49)

with $(\partial_{\eta},\partial_{\perp})\to-i(\omega_{n},k_{\perp})$;
$\gamma_{R}^{-1}g^{\rho\rho}\partial_{\rho}(\gamma_{R}\partial_{\rho})$ is
still operator. As a result, $Z$ in (2.4) can be given as

$\displaystyle Z=$
$\displaystyle\exp[-\alpha^{2}\beta_{R}V_{\perp}\int_{0}^{\infty}d\rho\,\rho\,\gamma_{R}M_{R}^{2}]\int\\!{\cal
D}\tilde{\phi}^{\prime}_{1,n}{\cal D}\tilde{\phi}^{\prime}_{2,n}\exp\bigg{[}$
$\displaystyle-\frac{1}{2\beta_{R}}\sum_{n=-\infty}^{\infty}\int_{0}^{\infty}\\!d\rho\,\rho\int_{-\infty}^{\infty}\\!d^{2}k_{\perp}\,\gamma_{R}(G_{R}+M_{R}^{2})((\tilde{\phi}^{\prime}_{1,n}(\rho,k_{\perp}))^{2}+(\tilde{\phi}^{\prime}_{2,n}(\rho,k_{\perp}))^{2})\bigg{]},$
(50)

where the general formula: $\beta^{-1}\int_{0}^{\beta}d\tau
e^{i\tau(\omega_{m}-\omega_{n})}=\delta_{mn}$ has been used. Then, performing
the functional integral for $\tilde{\phi}^{\prime}_{\alpha,n}$, we can get the
following $Z$:

$\displaystyle Z=\exp[-\alpha^{2}\beta_{R}V_{\perp}\int
d\rho\,\rho\,\gamma_{R}M_{R}^{2}]\prod_{n=-\infty}^{\infty}\prod_{\rho}{\rm
Det}(\frac{\pi\,\rho\Delta\rho\,\gamma_{R}}{2\beta_{R}}(G_{R}+M_{R}^{2}))^{-1},$
(51)

where from (2.4) to (51), we have rewritten the integral $\int d\rho$ as
$\sum_{\rho}\Delta\rho$ ($\rho$ takes all the real numbers from $0$ to
$\infty$), and ${\rm Det}$ is the one with regard to the $k_{\perp}$-space for
each $\rho$, where we have written the reason for why we separate by each
$\rho$ in Sec.2.1.

Defining the free energy $F_{R}$ as $Z=\exp(-\beta_{R}F_{R})$,

$\displaystyle F_{R}=$
$\displaystyle\,\alpha^{2}V_{\perp}\int_{0}^{\infty}d\rho\rho\,\gamma_{R}M_{R}^{2}+\frac{1}{\beta_{R}}\sum_{n=-\infty}^{\infty}\sum_{\rho}\rho\,{\rm
Tr}\,{\rm Log}(G_{R}+M_{R}^{2}),$ (52)

where we have ignored ${\cal N}\,{\rm
Log}(\frac{\pi\Delta\rho\,\gamma_{R}}{2\beta_{R}})$ (where ${\cal N}={\rm
Dim}(n)\times{\rm Dim}(\rho)\times{\rm Dim}(k_{\perp})$) as it is irrelevant
of $\mu_{R}$. We can express $F_{R}$ in (52) as

$\displaystyle F_{R}=$
$\displaystyle\,\alpha^{2}V_{\perp}\int_{0}^{\infty}d\rho\,\rho\,\gamma_{R}M_{R}^{2}+\frac{V_{\perp}}{\beta_{R}}\sum_{n=-\infty}^{\infty}\sum_{\rho}\rho\,\gamma_{R}\int\frac{dk_{\perp}^{2}}{(2\pi)^{2}}\int_{0}^{M_{R}^{2}}d\Delta^{2}\widetilde{D}_{\Delta^{2}}(\rho,n,k_{\perp}),$
(53)

where

$\displaystyle\widetilde{D}_{\Delta^{2}}(\rho,k_{\eta},k_{\perp})$
$\displaystyle\equiv\left(G_{R}+\Delta^{2}\right)^{-1}=(g^{\tau\tau}\omega_{n}^{2}+\gamma_{R}^{-1}g^{\rho\rho}\partial_{\rho}(\gamma_{R}\partial_{\rho})+g^{\perp\perp}k_{\perp}^{2}+\Delta^{2})^{-1},$
(54)

where

$\displaystyle\omega_{n}=n.$ (55)

This is because $\tau$ is periodic with the period $2\pi$ as can be seen
(14c), therefore $\omega_{n}$ is given as $\omega_{n}=2\pi n/\beta_{R}$
$(\beta_{R}=2\pi)$. (Here, the Euclidean temperature is identical with the
Unruh temperature (4))

In (53), we have ignored ${\rm Log}\,G_{R}$ as it is irrelevant of
$\mu_{R}$555 ${\rm Log}\,G_{R}$ appears in (53) as follows:
$\displaystyle{\rm
Log}(G_{R}+M_{R}^{2})=\int_{0}^{M_{R}^{2}}d\Delta^{2}\left(G_{R}+\Delta^{2}\right)^{-1}+{\rm
Log}G_{R}.$ (56) . Further,
$\widetilde{D}_{\Delta^{2}}(\rho,k_{\eta},k_{\perp})$ in (53) includes some
operator, but for now we suppose that it is some numbers.

Considering that the (53) divided by $\rho$ is the effective potential for the
particles performing a uniformly accelerated motion determined by $\rho$, we
consider to express $F_{R}/V_{\perp}$ for each $\rho$ as
$\Gamma_{R}^{(\rho)}$, then,

$\displaystyle\Gamma_{R}^{(\rho)}$
$\displaystyle=\alpha^{2}\gamma_{R}M_{R}^{2}+\frac{1}{\beta_{R}}\sum_{n=-\infty}^{\infty}\gamma_{R}\\!\int_{-\infty}^{\infty}\\!\frac{dk^{2}}{(2\pi)^{2}}\int^{M_{R}^{2}}_{0}\\!\\!\\!d\Delta^{2}\,\widetilde{D}_{\Delta^{2}}(\omega_{n},\rho,k).$
(57)

###  2.5 $\widetilde{D}_{\Delta^{2}}(\omega_{n},\rho,k)$

$\widetilde{D}_{\pm,\Delta^{2}}(\rho,\omega_{n},k_{\perp})$ in (54) is given
as some operator. In this subsection, we obtain its expression as numbers. For
this purpose, we define
$\widetilde{D}_{\pm,\Delta^{2}}(\rho,\omega_{n},k_{\perp})$ as

$\displaystyle
D_{\Delta^{2}}(x-x^{\prime})=\int\\!\frac{d^{3}k}{(2\pi)^{3}}\,\widetilde{D}_{\Delta^{2}}(\rho-\rho^{\prime},k_{\eta},k_{\perp})\,e^{i\,(\omega_{n}\,(\eta-\eta^{\prime})-k\\!_{\perp}(x\\!_{\perp}-x\\!_{\perp}^{\prime}))}.$
(58)

$D_{\Delta^{2}}(x-x^{\prime})$ should satisfy the following identity:

$\displaystyle(g^{\tau\tau}\omega_{n}^{2}+\gamma_{R}^{-1}g^{\rho\rho}\partial_{\rho}(\gamma_{R}\partial_{\rho})+g^{\perp\perp}k_{\perp}^{2}+\Delta^{2})D_{\Delta^{2}}(x-x^{\prime})=\gamma_{R}^{-1}\delta^{4}(x-x^{\prime}),$
(59)

where the operator in l.h.s. in (59) is taken from (53). From (59), we can
obtain the relation that $\widetilde{D}_{\pm,\Delta^{2}}$ should satisfy as

$\displaystyle\big{(}\,\hat{{\cal
F}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-}\omega_{n}^{2}\,\big{)}\,\widetilde{D}_{\pm,\Delta^{2}}(\rho-\rho^{\prime},k_{\eta},k_{\perp})=-\rho\,\delta(\rho-\rho^{\prime}),$
(60) $\displaystyle\hat{{\cal
F}}\equiv\rho^{2}\partial_{\rho}^{2}+\rho\,\partial_{\rho}-\rho^{2}\kappa^{2},\quad\kappa^{2}\equiv(k_{\perp}^{2}+\Delta^{2}).$

Based on (60), we obtain
$\widetilde{D}_{\Delta^{2}}(\rho-\rho^{\prime},k_{\eta},k_{\perp})$ in what
follows.

To obtain $\widetilde{D}_{\Delta^{2}}(\rho-\rho^{\prime},k_{\eta},k_{\perp})$,
we focus on the fact that $\hat{{\cal F}}$ is the operator of the following
eigenvalue equation with eigenvalue $(i\lambda)^{2}$:

$\displaystyle\hat{{\cal
F}}\,\Theta_{\lambda}(\rho\kappa)=(i\lambda)^{2}\,\Theta_{\lambda}(\rho\kappa),$
(61) $\displaystyle\Theta_{\lambda}(\rho\kappa)\equiv
C_{\lambda}K_{i\lambda}(\rho\kappa),\quad
C_{\lambda}\equiv\frac{1}{\pi}\sqrt{2\lambda\sinh(\pi\lambda)},$

where $K_{\alpha}(x)$ is the second kind modified Bessel function, and
$\Theta_{\lambda}(\rho,k)$ satisfies the following normalized orthogonal
relation666 It is written in [43, 44] as
$\int_{0}^{\infty}\frac{dx}{x}\,K_{\alpha}(x)\,K_{\beta}(x)=\frac{\pi^{2}\delta(\alpha-\beta)}{2\beta\sinh\pi\beta}$,
which we derive in Appendix.A. Writing $x$ with $ay$, l.h.s. can be changed to
$\int_{0}^{\infty}\frac{dy}{y}\,K_{\alpha}(ay)\,K_{\beta}(ay)$ without
changing the r.h.s. :

$\displaystyle\int_{0}^{\infty}\frac{d\rho}{\rho}\,\Theta_{\lambda^{\prime}}(\rho\kappa)\,\Theta_{\lambda}(\rho\kappa)=\delta(\lambda^{\prime}-\lambda).$
(62)

$\lambda$ is real number, therefore $\Theta_{\lambda}(\rho\kappa)$ can form a
set of infinite dimensional orthogonal system. Then, by taking
$\Theta_{\lambda}(\rho\kappa)$ as a set of the orthogonal bases, let us
formally write
$\widetilde{D}_{\pm,\Delta^{2}}(\rho-\rho^{\prime},k_{\eta},k_{\perp})$ in the
expanding form as

$\displaystyle\widetilde{D}_{\Delta^{2}}(\rho-\rho^{\prime},k_{\eta},k_{\perp})=\int_{0}^{\infty}\\!d\lambda\,f_{\lambda}\,\Theta_{\lambda}(\rho\kappa),$
(63)

where $f_{\lambda,\pm}$ are the coefficients of each independent direction,
$\Theta_{\lambda}(\rho\kappa)$, specified by $\lambda$, which are to be
obtained in what follows.

To obtain $f_{\lambda}(\omega,\rho^{\prime})$, we consider two quantities:
$\langle\widetilde{D}_{\Delta^{2}}|\hat{{\cal
F}}|\Theta_{\lambda}(\rho\kappa)\rangle$ and
$\langle\Theta_{\lambda}(\rho\kappa)|\hat{{\cal
F}}|\widetilde{D}_{\Delta^{2}}\rangle$, which should be equivalent each other,
however each of these expressions can be given as

$\displaystyle\int_{0}^{\infty}\frac{d\rho}{\rho}\,\widetilde{D}_{\Delta^{2}}\,\hat{{\cal
F}}\,\Theta_{\lambda}(\rho\kappa)=-\lambda^{2}\int_{0}^{\infty}\frac{d\rho}{\rho}\,\widetilde{D}_{\Delta^{2}}\,\Theta_{\lambda}(\rho\kappa),$
(64a)
$\displaystyle\int_{0}^{\infty}\frac{d\rho}{\rho}\,\Theta_{\lambda}(\rho\kappa)\,\hat{{\cal
F}}\widetilde{D}_{\Delta^{2}}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\omega_{n}^{2}}\int_{0}^{\infty}\frac{d\rho}{\rho}\,\Theta_{\lambda}(\rho\kappa)\widetilde{D}_{\Delta^{2}}-\Theta_{\lambda}(\rho^{\prime}\kappa),$
(64b)

where we have used (61) and (60). From the condition
$(\ref{DFTheta})-(\ref{ThetaFD})=0$ where777 (66) is calculated as follows:
$\displaystyle(\ref{DFTheta})-(\ref{ThetaFD})$
$\displaystyle=\big{(}\\!-\lambda^{2}+\omega_{n}^{2}\big{)}\int\frac{d\rho}{\rho}\widetilde{D}_{\Delta^{2}}\,\Theta_{\lambda}(\rho\kappa)+\Theta_{\lambda}(\rho^{\prime}\kappa)$
$\displaystyle=\big{(}\\!-\lambda^{2}+\omega_{n}^{2}\big{)}\int\frac{d\rho}{\rho}\cdot\int\\!d\lambda^{\prime}\,f_{\lambda^{\prime}}\Theta_{\lambda^{\prime}}(\rho\kappa)\cdot\Theta_{\lambda}(\rho\kappa)+\Theta_{\lambda}(\rho^{\prime}\kappa)$
$\displaystyle=\big{(}\\!-\lambda^{2}+\omega_{n}^{2}\big{)}\int\\!d\lambda^{\prime}\cdot\int\frac{d\rho}{\rho}\,\Theta_{\lambda^{\prime}}(\rho\kappa)\Theta_{\lambda}(\rho\kappa)\cdot
f_{\lambda^{\prime}}+\Theta_{\lambda}(\rho^{\prime},k)$
$\displaystyle=\big{(}\\!-\lambda^{2}+\omega_{n}^{2}\big{)}\int\\!d\lambda^{\prime}\cdot\delta(\lambda^{\prime}-\lambda)\cdot
f_{\lambda^{\prime}}+\Theta_{\lambda}(\rho^{\prime}\kappa)$
$\displaystyle=\big{(}\\!-\lambda^{2}+\omega_{n}^{2}\big{)}\,f_{\lambda}+\Theta_{\lambda}(\rho^{\prime}\kappa).$
(65)

$\displaystyle(\ref{DFTheta})-(\ref{ThetaFD})=\big{(}\\!-\lambda^{2}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-}\omega_{n}^{2}\big{)}\,f_{\lambda}+\Theta_{\lambda}(\rho^{\prime}\kappa),$
(66)

$f_{\lambda,\pm}(\rho-\rho^{\prime},k_{\eta},k_{\perp})$ can be determined as

$\displaystyle
f_{\lambda}=\frac{\Theta_{\lambda}(\rho^{\prime}\kappa)}{\lambda^{2}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>(67)

Using (67) in (63), we can write $\widetilde{D}_{\Delta^{2}}$ as

$\displaystyle\widetilde{D}_{\Delta^{2}}(\rho-\rho^{\prime},k_{\eta},k_{\perp})=\int_{0}^{\infty}\\!d\lambda\,\frac{\Theta_{\lambda}(\rho^{\prime}\kappa)\Theta_{\lambda}(\rho\kappa)}{\lambda^{2}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+}\omega_{n}^{2}}=\int_{0}^{\infty}\\!d\lambda\,\frac{C_{\lambda}^{2}K_{i\lambda}(\rho^{\prime}\kappa)K_{i\lambda}(\rho\kappa)}{\lambda^{2}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>(68)

This expression does not include operators, which is some numbers in
principle.

###  2.6 Effective potential (2)

In the previous subsection we have obtained some concrete expression of
$\widetilde{D}_{\Delta^{2}}(\rho-\rho^{\prime},k_{\eta},k_{\perp})$ as in
(68). Using it, we can write $\Gamma^{(\rho)}_{R}$ in (57) as888 We have used
$\displaystyle\sum_{n=-\infty}^{\infty}\frac{1}{\lambda^{2}+n^{2}}=\frac{\pi}{\lambda}\coth(\pi\lambda)$
and
$\displaystyle\frac{1}{\beta_{R}}\left(\frac{\sqrt{2\lambda}}{\pi}\right)^{2}\int_{-\infty}^{\infty}\frac{dk^{2}}{(2\pi)^{2}}=\frac{\lambda}{2\pi^{4}}\int_{0}^{\infty}dkk$..

$\displaystyle\Gamma^{(\rho)}_{R}=\alpha^{2}\gamma_{R}M_{R}^{2}+\frac{\gamma_{R}}{2\pi^{3}}\int_{0}^{\infty}\\!d\lambda\,\cosh(\pi\lambda)\int_{0}^{\infty}\\!dkk\,\Psi,$
(69)

where

$\displaystyle\Psi$
$\displaystyle\equiv\int^{M_{R}^{2}}_{0}\\!\\!d\Delta^{2}\,K_{i\lambda}^{2}(\kappa\rho),\quad(\kappa^{2}\equiv
k_{\perp}^{2}+\Delta^{2},\quad M_{R}^{2}\equiv
m^{2}-g^{\eta\eta}\mu_{R}^{2}).$ (70a)

Considering to perform the derivative with regard to $\mu_{R}$, we pick up the
$\mu_{R}$-dependent parts in $\Psi$ by expanding it around $\mu_{R}=0$999 In
this section we have performed the expansion around $\mu_{R}=0$ as in (2.6).
However, if we expanded around $\mu_{R}=\mu_{R}^{c}$, we can obtain the
following effective potential:
$\displaystyle\Gamma^{(\rho)}_{R}=\alpha^{2}a_{c}^{-1}M^{2}+\frac{a_{c}}{\pi^{3}}\int_{0}^{\infty}d\lambda\cosh\pi\lambda\int_{0}^{\infty}dkkK_{i\lambda}^{2}(\frac{m}{a}\mu_{R}-\frac{m^{2}}{a^{2}}).$
(71) Therefore, there is no difference in the e.v. of the particle density we
can obtain finally, .  as

$\displaystyle\Psi=$
$\displaystyle\,\Psi\Big{|}_{\mu_{R}=0}+\frac{\partial\Psi}{\partial\mu_{R}}\Big{|}_{\mu_{R}=0}\,\mu_{R}+\frac{1}{2}\frac{\partial^{2}\Psi}{\partial\mu_{R}^{2}}\Big{|}_{\mu_{R}=0}\,\mu_{R}^{2}+{\cal
O}(\mu_{R}^{4})$ $\displaystyle=$
$\displaystyle\Psi_{0}-a^{2}K_{i\lambda}^{2}(\kappa\rho)\Big{|}_{\Delta^{2}=M^{2}}\,\mu_{R}^{2}+{\cal
O}(\mu_{R}^{4})$ (72)

where101010 We have calculated (74b) and (74c) using the following formula:
$\displaystyle\frac{\partial}{\partial\beta}\int_{\alpha}^{\beta}dxf(x)=\lim_{\Delta\beta\to
0}\frac{1}{\Delta\beta}(\int_{\alpha}^{\beta+\Delta\beta}dxf(x)-\int_{\alpha}^{\beta}dxf(x))=\lim_{\Delta\beta\to
0}\frac{1}{\Delta\beta}(f(\beta+\Delta\beta)\Delta\beta+{\cal
O}(\Delta\beta^{2})).$ (73) ,

$\displaystyle\Psi_{0}$
$\displaystyle\equiv\int_{0}^{m^{2}}\\!\\!d\Delta^{2}\,K_{i\lambda}^{2}(\kappa\rho),$
(74a) $\displaystyle\frac{\partial\Psi}{\partial\mu_{R}}\bigg{|}_{\mu_{R}=0}$
$\displaystyle=\frac{\partial
M_{R}^{2}}{\partial\mu_{R}}\frac{\partial\Psi}{\partial
M_{R}^{2}}\bigg{|}_{\mu_{R}=0}=0,$ (74b)
$\displaystyle\frac{1}{2}\frac{\partial^{2}\Psi}{\partial\mu_{R}^{2}}\bigg{|}_{\mu_{R}=0}$
$\displaystyle=\frac{1}{2}\frac{\partial}{\partial\mu_{R}}\bigg{(}\frac{\partial
M_{R}^{2}}{\partial\mu_{R}}\frac{\partial\Psi}{\partial
M_{R}^{2}}\,\bigg{)}\bigg{|}_{\mu_{R}=0}=-a^{2}K_{i\lambda}^{2}(\kappa\rho)\Big{|}_{\Delta^{2}=m^{2}}.$
(74c)

With (2.6), we can write $\Gamma^{(\rho)}_{R}$ in (69) as

$\displaystyle\Gamma^{(\rho)}_{R}=\alpha^{2}\gamma_{R}M_{R}^{2}+\frac{\gamma_{R}}{2\pi^{3}}\int_{0}^{\infty}d\lambda\,\cosh(\pi\lambda)\int_{0}^{\infty}\\!dkk\,\Big{(}\Psi_{0}-a^{2}\mu_{R}^{2}\,K_{i\lambda}^{2}(\kappa\rho)\Big{|}_{\Delta^{2}=m^{2}}\Big{)}.$
(75)

We are going to finally assign the critical value $\mu_{R}^{c}=m/a_{c}$ in
(34) to $\mu_{R}$ in (2.6), then take the leading order in the expansion of
$\mu_{R}^{c}$. For this, now we use two symbols, $\mu_{R}$ and
$\mu_{R}^{\prime}$:

  * •

$\mu_{R}$: chemical potential on which the derivative regarding $\mu_{R}$ can
act; finally the value $m/a_{c}$ is assigned,

  * •

$\mu_{R}^{\prime}$: just a symbol for the value $m/a_{R}^{c}$, on which the
derivative regarding $\mu_{R}$ does not act on.

Then, $\Psi_{0}$ and
$a^{2}\mu_{R}^{2}\,K_{i\lambda}^{2}(\kappa\rho)\Big{|}_{\Delta^{2}=m^{2}}$ can
be expanded with regard to $\mu_{R}^{\prime}$ as

$\displaystyle\Psi_{0}=$
$\displaystyle\,\,a_{c}^{2}\,K_{i\lambda}^{2}(k/a_{c})\,{\mu_{R}^{\prime}}^{2}+{\cal
O}(\mu_{R}^{\prime}{}^{4}),$ (76a) $\displaystyle
a_{c}^{2}\mu_{R}^{2}\,K_{i\lambda}^{2}(\kappa\rho)\Big{|}_{\Delta^{2}=m^{2}}=$
$\displaystyle a_{c}^{2}\mu_{R}^{2}\,(K_{i\lambda}^{2}(k/a_{c})+{\cal
O}(\mu_{R}^{\prime}{}^{2})).$ (76b)

Using (76), we can obtain $\Gamma^{(\rho)}_{R}$ at the critical moment given
by (2.6) as

$\displaystyle\Gamma^{(\rho)}_{R}=\alpha^{2}a_{c}^{-1}M_{R}^{2}+\frac{a_{c}^{3}}{4\pi^{2}}(\mu_{R}^{\prime}{}^{2}-\mu_{R}^{2})\int^{\infty}_{0}d\lambda\lambda\coth(\pi\lambda),$
(77)

where we have used
$\displaystyle\int^{\infty}_{0}dkkK_{i\lambda}^{2}(k/a_{c})$ is given as
$\frac{a_{c}^{2}\pi\lambda}{2\sinh\pi\lambda}$.

Let us evaluate the $\lambda$-integral in (77).

$\displaystyle\int^{\infty}_{0}d\lambda\lambda\coth(\pi\lambda).$ (78)

First, we can see that (78) is diverged if it is evaluated as it is.
Therefore, we consider to do some regularize toward (78)111111 The divergences
also appear in other works for the critical acceleration for the spontaneous
symmetry breaking [45, 46, 47] and the $D=1+3$ Euclid space at finite
temperature [31]. In these, the divergences are ignored supposing that some
regularization, e.g. a mass renormalization, the UV-cutoff and so on, could
work, though it is not shown explicitly. . For this purpose, we consider to
pull out some constant in $\coth(\pi\lambda)$. Therefore, expressing
$\coth(\pi\lambda)$ as

$\displaystyle\coth(\pi\lambda)=1+\frac{2}{e^{2\pi\lambda}-1},$ (79)

we exclude “$1$”. Then, once putting the upper limit of the integral as
$\Lambda$ we perform the integral, then we get as

$\displaystyle\int^{\Lambda}_{0}d\lambda\lambda(\coth(\pi\lambda)-1)=-\frac{1}{12}-\Lambda^{2}+\frac{\Lambda\log(1-e^{2\pi\Lambda})}{\pi}+\frac{\text{Li}_{2}(e^{2\pi\Lambda})}{2\pi^{2}},$
(80)

where $\Lambda\to\infty$ finally. Then excluding “$-\frac{1}{12}$” in (80), we
take the $\Lambda$ to $\infty$. As a result we get as

$\displaystyle\lim_{\Lambda\to\infty}(-\Lambda^{2}+\frac{\Lambda\log(1-e^{2\pi\Lambda})}{\pi}+\frac{\text{Li}_{2}(e^{2\pi\Lambda})}{2\pi^{2}})=\frac{1}{6}.$
(81)

Then, using this result, we can give the effective potential
$\Gamma_{R}^{(\rho)}$ as

$\displaystyle\Gamma_{R}^{(\rho)}=\alpha^{2}a_{c}^{-1}M_{R}^{2}+\frac{a_{c}^{3}}{24\pi^{2}}(\mu_{R}^{\prime}{}^{2}-\mu_{R}^{2}).$
(82)

###  2.7 Critical Unruh temperature for BEC

We can obtain the e.v. of the particle density according to
$d_{R}=-\partial\Gamma_{R}^{(\rho)}/\partial\mu_{R}$ as

$\displaystyle
d_{R}=2\alpha^{3}a_{c}\mu_{R}+\frac{a_{c}^{3}}{12\pi^{2}}\mu_{R},$ (83)

From (83), we can see that if we decrease the Unruh temperature from some high
temperatures fixing the e.v. of the particle density to some constant, the
value of the chemical potential should rise.

Let us consider to reach the critical Unruh temperature by decreasing the
acceleration gradually from some high accelerations where BEC is not formed.
Then, from the explanation under (35), we can obtain the critical acceleration
from (83) as121212 (84) can be written in the MKS units as
$a^{c}=\,2\,\pi\,\sqrt{c^{3}\hbar\,3d_{R}/m}\,\approx
16.763\,\sqrt{2\pi}\,\sqrt{d_{R}/m}\,[{\rm cm}/{\rm s}^{2}],$ where $m$ is the
mass of the particle comprising the gas and $d_{R}$ is its density.

$\displaystyle a^{c}=2\,\pi\,\sqrt{3d_{R}/m},$ (84)

where $\alpha=0$ and $\mu_{c}=m/a^{c}$ have been assigned in (83). Here, note
the comment given under (35).

Using the relation between Unruh temperature and acceleration, $a=2\pi T_{U}$,
we can obtain the critical Unruh temperature as $T_{U}=\sqrt{3d_{R}/m}$. This
result is consistent with the critical temperature for BEC obtained from the
different analytical method in the $D=1+3$ flat Euclid space at finite
temperature [31].

##  3 Critical Unruh temperature of BEC in flat Euclid space at finite
temperature

In this section, we obtain the critical temperature in the $D=1+3$ flat
Euclidean space at finite temperature from our analysis in the previous
section just by exchanging the space time for the $D=1+3$ flat Euclidean space
at finite temperature.

Therefore, as the calculation way in this section is basically the same with
that in the previous section, we in this section describe only the points in
the case of the $D=1+3$ flat Euclidean space at finite temperature.

###  3.1 Exchange the back ground space for Euclidean space

First, we exchange the Rindler space (2.1) for the flat $D=1+3$ Euclidean
space at finite temperature, which can be done by 1) putting $\rho$ to $1$,
then 2) Euclideanizing $\tau$-direction like (13) then periodizing it by the
arbitrary period $\beta_{E}$. The (2.1) in which these 2 manipulations are
performed can be written as

$\displaystyle ds_{E}^{2}=-d\tau^{2}-d\rho^{2}-dx_{\perp}^{2},$ (85)

where the $\tau$-direction is periodic with the arbitrary period $\beta_{E}$.

###  3.2 Effective potential

Employing (85), we proceed with the analysis in previous section. Then, the
following $\Gamma_{E}$ can be obtained instead of (57):

$\displaystyle\Gamma_{E}=$
$\displaystyle\,\,\alpha^{2}M_{E}^{2}+\frac{2}{\pi^{2}\beta_{E}}\int_{-\infty}^{\infty}\frac{dk^{2}\\!_{\perp}}{(2\pi)^{2}}\int_{0}^{\infty}\\!d\lambda\,\lambda\,\sinh(\pi\lambda)\sum_{n=-\infty}^{\infty}\\!\frac{1}{\lambda^{2}-\omega_{n}^{2}}\Psi_{E},$
(86)

where

$\displaystyle\omega_{n}\equiv\frac{2\pi}{\beta_{E}}n,\quad\Psi_{E}\equiv\int^{M_{E}^{2}}_{0}\\!\\!\\!d\Delta^{2}\,K_{i\lambda}^{2}(\kappa),\quad(\kappa^{2}\equiv
k_{\perp}^{2}+\Delta^{2},\quad M_{E}^{2}\equiv m^{2}-\mu_{E}^{2}),$ (87)

and $K_{\alpha}(x)$ mean the second-kind Bessel function, and as for
$\omega_{n}$, we use the same notation with the one in the previous
subsection.

From $M_{E}^{2}$ above, we can see that the upper bound of the value that the
chemical potential in the case of the flat $D=1+3$ Euclidean space at finite
temperature is given as

$\displaystyle\mu_{E}^{c}=m.$ (88)

Expanding (86) to the second-order of the value of the critical chemical
potential in the same way we have done in Sec.2.6, we can obtain the following
$\Gamma_{E}$:

$\displaystyle\Gamma_{E}=\,\alpha^{2}M_{E}^{2}+\frac{1}{4\pi^{2}}\int_{0}^{\infty}\\!d\lambda\,\lambda\,\coth(\frac{\beta_{E}\lambda}{2})(\mu_{E}^{\prime
2}-\mu_{E}^{2}),$ (89)

where $\mu_{E}^{\prime}$ is the same meaning with that in Sec.2.6, but in the
current case, $\mu_{E}^{\prime}$ is given by $m$ corresponding to (88).

Now, we evaluate
$\displaystyle\int_{0}^{\infty}\\!d\lambda\,\lambda\,\coth(\beta_{E}\lambda/2)$
in (89). If we performed the integration as it is, it would be diverged.
Hence, we do some regularization. As $\coth x$ can be rewritten as
$\displaystyle 1+{2}/(e^{2x}-1)$, we subtract the constant “1” as some
regularization, then evaluate it as
$\displaystyle\int_{0}^{\infty}\\!d\lambda\,\lambda\,(\coth(\beta_{E}\lambda/2)-1)=2\pi^{2}/3\beta_{E}^{2}$.
Using this, we can obtain the following $\Gamma_{E}$:

$\displaystyle\Gamma_{E}=\,\alpha^{2}M_{E}^{2}+\frac{1}{6\beta_{E}^{2}}(\mu_{E}^{\prime
2}-\mu_{E}^{2}).$ (90)

###  3.3 Critical temperature

From (90), calculating the density according to
$d_{E}=-\frac{\partial\Gamma_{E}}{\partial\mu_{E}}$, then putting $\alpha=0$
and $\mu_{E}^{c}=m$ corresponding to the critical moment, we can obtain the
relation between the temperature and density at the critical moment as

$\displaystyle T_{E}^{c}=\sqrt{3d_{E}/m}.$ (91)

This can agree to the critical temperature for BEC in the $D=1+3$ flat Euclid
space at finite temperature in [31], however our way to obtain this result is
different from [31].

##  4 BEC in the traversable wormhole space time

###  4.1 Traversable wormhole space time

In this section, we consider the traversable wormhole given from considering
two of the following space time:

$\displaystyle
ds^{2}=dt^{2}-\frac{dr^{2}}{1-v^{2}/r^{2}}-r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (92)

then attaching the parts of $r=v$ of these [33]. Therefore, the position of
the throat is located at $r=v$ and the range of $r$ in one side of the
wormhole space time is

$\displaystyle v\leq r\leq\infty,\quad(\textrm{the throat is located at
$r=v$}).$ (93)

In order that the wormhole (92) can satisfy the Einstein equation, the exotic
matter, some matter to violate the energy condition, is needed [33] as shown
in the next section, however we do not consider it in this study.

###  4.2 On the fact that matter to violate the energy condition is needed

(92) leads to the following Einstein tensor:

$\displaystyle
R_{\mu\nu}-\frac{g_{\mu\nu}}{2}R=\left(\begin{array}[]{cccc}-\frac{v^{2}}{r^{4}}&0&0&0\\\
0&\frac{v^{2}}{r^{2}\left(v^{2}-r^{2}\right)}&0&0\\\
0&0&\frac{v^{2}}{r^{2}}&0\\\ 0&0&0&\frac{v^{2}\sin^{2}\theta}{r^{2}}\\\
\end{array}\right).$ (98)

From the result above, we can see that the energy density of our scalar field
should be negative, which means that our scalar field is tachyonic or to break
the energy condition. However, since such tachyonic matter cannot exist, (92)
cannot be a solution realistically. Therefore, (92) will not be realized. This
study treats (92) upon knowing this. If we care this problem in this study, we
could mention that the wormhole space time appears for some reason and exists
for some time, then this study is the one for that time.

Indeed, this problem is general in the context of the traversable wormhole
space time. For the references for the recent studies on this, see Sec.1.

###  4.3 Probability amplitude and its Euclideanization

In the case that the wormhole (92) is taken as our space time, the probability
amplitude corresponding to (2.2) is given as

$\displaystyle Z=$ $\displaystyle\,\,{\cal C}_{w}\\!{{\cal D}\phi_{1}}{{\cal
D}\phi_{2}}\exp\big{[}\frac{i}{2}\int\\!d^{4}x\,\gamma_{w}\,(j^{tt}(\partial_{t}\phi_{1}+\mu_{w}\,\phi_{2})^{2}+(\partial_{i}\phi_{1})^{2}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\,\,+j^{tt}(\partial_{t}\phi_{2}+\mu_{w}\,\phi_{1})^{2}+(\partial_{i}\phi_{2})^{2}-m^{2}({\phi_{1}}^{2}+{\phi_{2}}^{2}))\big{]},$
(99)

where $i=r,\theta,\phi$ and $j_{\mu\nu}$ refer to the metrics in (92) and
$\gamma_{w}\equiv\sqrt{-\det j_{\mu\nu}}$. ${\cal C}_{w}$ is ignorable as well
as Sec.2.2.

We perform the Euclideanization,

$\displaystyle t\to-i\tau.$ (100)

At this time,

$\displaystyle(\pi_{\alpha},\bar{\pi}_{\alpha})\to
i(\pi_{\alpha},\bar{\pi}_{\alpha}),$ (101a) $\displaystyle
dt^{2}\to-d\tau^{2},$ (101b) $\displaystyle j_{tt}\to
j_{\tau\tau}\quad(j_{tt}=j_{\tau\tau}),$ (101c)

then $S^{1}$ compactify the $\tau$-direction with the period $\beta_{w}$. As a
result, the momenta in the $\tau$-direction can be written as

$\displaystyle 2\pi n/\beta_{w}\equiv\omega_{n},$ (102)

and it can be considered that we can consider the gas at the temperature given
as $\beta_{w}^{-1}$ in the space given by the space part of (92). We write
$\phi_{\alpha}$ ($\alpha=1,2$) as

$\displaystyle\phi_{\alpha}(\tau,r,\theta,\phi)=\frac{1}{\beta_{w}}\sum_{n=-\infty}^{\infty}\varphi_{\alpha,n}(r,\theta,\phi)e^{-i\omega_{n}\tau},\quad(\alpha=1,2)$
(103)

As a result, $Z$ in (4.3) can be written as

$\displaystyle Z=\int\\!{\cal D}\phi_{1}{\cal
D}\phi_{2}\exp\Big{[}\\!-\frac{1}{2}\int\\!d^{3}x\\!\int_{0}^{\beta}\\!d\tau\,\gamma_{w}\,\left(\\!\begin{array}[]{cc}\phi_{1}&\phi_{2}\end{array}\\!\right)\\!\left(\\!\begin{array}[]{cc}\hat{G}_{w}+M_{w}^{2}&-2ig^{\tau\tau}\mu_{w}\,\partial_{\tau}\\\
2ij^{\tau\tau}\mu_{w}\,\partial_{\tau}&\hat{G}_{w}+M_{w}^{2}\end{array}\\!\right)\\!\left(\\!\begin{array}[]{c}\phi_{1}\\\
\phi_{2}\end{array}\\!\right)\Big{]},$ (109)

where
$\hat{G}_{w}\equiv\,-j^{\tau\tau}\partial_{\tau}\partial_{\tau}+\gamma_{w}^{-1}j^{ij}\partial_{i}(\gamma_{w}\partial_{j})$
and $M_{w}^{2}\equiv m^{2}-g^{\tau\tau}\mu_{0}^{2}$. From the $M_{w}^{2}$
above, we can see that the upper bound of the value of the chemical potential
in the case of the traversable wormhole at finite temperature is

$\displaystyle\mu_{w}^{c}=m.$ (110)

###  4.4 Effective potential in the vicinity of the throat

In the same way with (21), putting $\phi_{\alpha}$ ($\alpha=1,2$) as

$\displaystyle\phi_{\alpha}$
$\displaystyle\equiv\sqrt{2}\,\alpha\,\cos\Theta+\tilde{\phi}_{\alpha},$
(111a) $\displaystyle\phi_{\alpha}$
$\displaystyle\equiv\sqrt{2}\,\alpha\,\sin\Theta+\tilde{\phi}_{\alpha},$
(111b)

then proceeding with the calculation from (109) in the same way with Sec.2.4,
we can obtain the following $Z$:

$\displaystyle Z=$
$\displaystyle\exp[-\alpha^{2}\beta_{w}V_{3}M_{w}^{2}]\int\\!{\cal
D}\tilde{\phi}^{\prime}_{1,n}{\cal
D}\tilde{\phi}^{\prime}_{2,n}\exp\bigg{[}-\frac{1}{2\beta_{w}}\sum_{n=-\infty}^{\infty}\int_{0}^{\infty}\\!dr\int_{-\infty}^{\infty}\\!d^{2}k_{\perp}\,\gamma_{w}\hat{G}_{w}(\tilde{\phi}_{1,n}^{2}+\tilde{\phi}_{2,n}^{2})\bigg{]},$
(112)

where $V_{3}\equiv\int dx^{3}\gamma_{w}$.

We switch our viewpoint from the entire region from $r=0$ to $\infty$ to only
the vicinity of the throat. Therefore, replacing the $r$ in
$\gamma_{w}\hat{G}_{w}$ in (112) with $v+r$, we take $r$ up to $r^{0}$-order
in $\gamma_{w}\hat{G}_{w}$ in (112) as

$\displaystyle\gamma_{w}\hat{G}_{w}=\frac{v^{5/2}\sin\theta}{\sqrt{2r}}(-\partial_{\tau}^{2}+M_{w}^{2})-\frac{\sqrt{v}\sin\theta}{\sqrt{2r}}(v\partial_{r}-\hat{L}^{2})+{\cal
O}(r^{0}),$ (113)

where $\hat{L}^{2}$ is the squared angular-momentum operator:

$\displaystyle\hat{L}^{2}\equiv-(\frac{\partial_{\theta}(\sin\theta\partial_{\theta})}{\sin\theta}+\frac{\partial_{\phi}^{2}}{\sin\theta^{2}}).$
(114)

Also, $V_{3}$ is given as

$\displaystyle V_{3}=\frac{v^{5/2}\sin\theta}{\sqrt{2r}}+{\cal O}(r^{0}).$
(115)

In what follows, we proceed with our analysis by focusing on the vicinity of
the throat, where the $r$ in what follows refers from $0$ to some
infinitesimal number.

We perform the expansion for the ($\theta,\phi$)-directions of $\phi_{\alpha}$
($\alpha=1,2$) with the spherical harmonics as

$\displaystyle\tilde{\phi}_{\alpha,n}(r,\theta,\phi)=\frac{1}{\beta_{w}}\sum_{n=-\infty}^{\infty}\sum_{l,m}e^{-i\omega_{n}\tau}\tilde{\varphi}_{\alpha,n,l,m}(r)Y^{l}_{m}(\theta,\phi).$
(116)

Then, proceeding with the calculation from (112) by the same way with Sec.2.4,
we can obtain the free energy, $Z=\exp(-\beta_{w}F_{w})$, as

$\displaystyle F_{w}=$ $\displaystyle\,\alpha^{2}M_{w}^{2}\frac{4\pi
v^{5/2}}{\sqrt{2}}\int\frac{dr}{\sqrt{r}}+\frac{1}{\beta_{w}}\sum_{n=-\infty}^{\infty}\sum_{r}\sum_{l,m}\int_{0}^{M_{w}^{2}}d\Delta^{2}H_{\Delta^{2}}(r,n,l),$
(117)

where $r$ in the summation takes all the real numbers from $0$ to $\infty$,
and

$\displaystyle
H_{\Delta^{2}}(r,n,l)\equiv\frac{1}{\omega_{n}^{2}-v\partial_{r}+l(l+1)+\Delta^{2}}.$
(118)

We finally obtain the critical temperature at some points where $r$ is near 0,
so we take the contribution at each $r$ in (117) as

$\displaystyle\Gamma_{w}=$ $\displaystyle\,\alpha^{2}M_{w}^{2}\frac{4\pi
v^{5/2}}{\sqrt{2r}}+\frac{1}{\beta_{w}}\sum_{n=-\infty}^{\infty}\sum_{l,m}\int_{0}^{M_{w}^{2}}d\Delta^{2}H_{\Delta^{2}}(r,n,l),$
(119)

where $r$ in (119) takes some values near $0$.

###  4.5 $H_{\Delta^{2}}(r,n,l)$

$H_{\Delta^{2}}(r,n,l)$ in (118) can be defined as the one to satisfy the
following equation:

$\displaystyle(\omega_{n}^{2}-v\partial_{r}+l(l+1)+v^{2}\Delta^{2})H_{\Delta^{2}}(r,n,l)=\gamma_{w}^{-1}\delta(r)$
(120)

where $\gamma_{w}^{-1}=\frac{\sqrt{2r}}{v^{5/2}\sin\theta}+{\cal O}(r)$.
Performing the Fourier expansion for the $r$-direction in
$\tilde{D}_{\Delta^{2}}(r,n,l)$ and $\delta(r)$ as

$\displaystyle
H_{\Delta^{2}}(r,n,l)=\frac{1}{2\pi}\int_{-\infty}^{\infty}dk\tilde{J}_{\Delta^{2}}(k)e^{-ikr},\quad\delta(r)=\frac{1}{2\pi}\int_{-\infty}^{\infty}dke^{-ikr},$
(121)

it can be seen that $\tilde{J}(k)$ is given as

$\displaystyle\tilde{J}_{\Delta^{2}}(k)$
$\displaystyle=\frac{\gamma_{w}^{-1}|_{r=r_{0}}}{iv(k-ik_{0})}\quad(\textrm{$r_{0}$
is taken to 0 from the positive}),$ $\displaystyle k_{0}$
$\displaystyle\equiv\frac{1}{v}(l(l+1)+v^{2}\Delta^{2}+v^{2}\omega_{n}^{2}),$
(122)

where $r_{0}$ is taken to some values near $0$ and $ik_{0}$ is located in the
upper half-plane in the complex plane (see Fig.2).

Now we have obtained $\tilde{J}_{\Delta^{2}}(k)$ as in (4.5), we evaluate
$\tilde{H}_{\Delta^{2}}(r,n,l)$ in (121). We use the residue theorem for this,
then we can see

$\displaystyle\bullet\quad\textrm{when $r>0$, the contribution from the path
of}\left\\{\begin{array}[]{l}\textrm{the upper semi circle, $C_{+}$, is
$\infty$},\\\\[2.84526pt] \textrm{the lower semi circle, $C_{-}$, is
$0$},\end{array}\right.$ (125) $\displaystyle\bullet\quad\textrm{when $r<0$,
the contribution from the path of}\left\\{\begin{array}[]{l}\textrm{the upper
semi circle, $C_{+}$, is $0$},\\\\[2.84526pt] \textrm{the lower semi circle,
$C_{-}$, is $\infty$},\end{array}\right.$ (128)
$\displaystyle\bullet\quad\textrm{when $r=0$, the contribution from the paths
of $C_{\pm}$ are $\infty$.}$

where $C_{\pm}$ and the position of $k_{0}$ are sketched in Fig.2.

Figure 2: This figure represents the paths of $C^{\pm}$ and the position of
the pole $ik_{0}$ to calculate $H_{\Delta^{2}}(r,n,l)$ in (121) using the
residue theorem.

Finally, we can obtain as

$\displaystyle\tilde{H}_{\Delta^{2}}(r,n,l)=e^{\frac{r}{v}(\Delta^{2}+l(l+1)+\omega_{n}^{2})}-\frac{\sqrt{2r_{0}}}{v^{7/2}\sin\theta}\Theta(-r)=e^{\frac{r}{v}(\Delta^{2}+l(l+1)+\omega_{n}^{2})},$
(129)

where $\Theta(-r)$ is the step function (it is 1 or 0 for negative or positive
$r$), and our $r_{0}$ is some positive values near $0$.

Now we have obtained $\tilde{D}_{\Delta^{2}}(r,n,l)$ as in (129). Then, in
(119), performing the integration with regard to $\Delta^{2}$ , we can obtain
the following $\Gamma_{w}$:

$\displaystyle\Gamma_{w}=\alpha^{2}M_{w}^{2}\frac{4\pi
v^{5/2}}{\sqrt{2r}}+\frac{v}{\beta_{w}r}\sum_{n=-\infty}^{\infty}\sum_{l,m}(-e^{\frac{r}{v}(l(l+1)+\omega_{n}^{2})}+e^{\frac{r}{v}(l(l+1)+M^{2}+\omega_{n}^{2})}).$
(130)

###  4.6 Particle density and critical temperature in the vicinity of the
throat

From (130), according to $d_{w}=-d\Gamma_{w}/d\mu_{w}$, we can obtain the e.v.
of the particle density as

$\displaystyle
d_{w}=\frac{2\mu_{w}}{\beta_{w}}\sum_{n=-\infty}^{\infty}\sum_{l,m}e^{\frac{r}{v}(l(l+1)+m^{2}-\mu_{w}^{2}+\omega_{n}^{2})},$
(131)

where considering we close from some high temperatures to the critical
temperature, we have put $\alpha$ to 0 as well as Sec.2.7 and 3.3.

Now, we get the critical temperature in the vicinity of the throat. For this
purpose, we first expand $d_{w}$ in (131) around $r=0$ assuming $v$ is some
finite number. Then, we apply $\mu_{w}^{c}=m$. At this time, the temperature
is at the critical temperature, so we can write $\beta_{w}$ as
$\beta_{w}^{c}$. Then, we can write (131) as

$\displaystyle
d_{w}=\frac{2m}{\beta^{c}_{w}}\sum_{n=-\infty}^{\infty}\sum_{l,m}(1+\frac{2m}{v}(l(l+1)+\omega_{n}^{2})r)+{\cal
O}(r^{2}/v^{2}).$ (132)

Then, once treating the summations
$\displaystyle\sum_{n=-\infty}^{\infty}\sum_{l,m}$ in (132) as
$\displaystyle\sum_{n=-N_{0}}^{N_{0}}\sum_{l=0}^{L_{0}}(2l+1)$, ($N_{0}$ and
$L_{0}$ are taken to infinity finally), we evaluate these summations. Then, we
can obtain the critical temperature as

$\displaystyle T_{w}^{c}$
$\displaystyle=\frac{d_{w}}{2(L_{0}+1)^{2}(2mN_{0}+m)}$
$\displaystyle-\frac{(2\pi^{2}d^{2}N_{0}(N_{0}+1)+3L_{0}(L_{0}+2)(L_{0}+1)^{4}(2mN_{0}+m)^{2}}{12v(L_{0}+1)^{6}(2mN_{0}+m)^{3}}d_{w}\,r+{\cal
O}(r^{2}/v^{2}),$ (133)

We can see that this $T_{w}^{c}$ goes to $0$ when $N_{0}$ and $L_{0}$ are sent
to $\infty$, where

  * •

Temperature $T_{w}$ is Euclid temperature as in around (100), $v$ is assumed
as some finite number, and $r$ is measured from $v$ (see around (113)),

  * •

we have taken the part for small $r$ in the effective potential as in Sec.4.4,
which we consider to be effective for the phenomena in the vicinity of the
throat. Then from the analysis of it, we have obtained the result as in (4.6).

Lastly, let us mention on the physical reason for the result (4.6), which may
be considered that the dimension of the space becomes effectively 1 in the
vicinity of the throat (see Eq.(92)). However, it depends on the coordinate
system we take. Actually, the space can be 3 dimensional even in the vicinity
of the throat, if we perform a coordinate transformation to $\zeta$ given as
$r=v+\zeta^{2}/2v$. After all, the physical reason of the result (4.6) would
be that contribution from the infrared region gets diverged for the curvature
of the space in the vicinity of the throat, independently of the coordinate
system we take.

Next, we comment on the coordinate dependence of the value of the critical
temperature (4.6). Generally, the value of the Euclidean temperature would be
changed up to the definition of the time coordinate. However, the result
$T_{w}^{c}=0$ ($\beta_{w}^{c}=\infty$) at $r=0$ would not be changed as long
as we do not consider some special transformation using some inverse of $r$
toward $t$. Therefore it would be considered the critical temperature is
always 0 as long as some inertial systems are taken (see footnote in Sec.1).

##  5  Phase structure of the normal/BEC states in the traversable wormhole
space time and Josephson junction formed in vicinity of its throat

###  5.1 Phase structure

In the previous section, we have obtained the result that the critical
temperature of BEC in the gas in the vicinity of the throat is 0 as in (4.6).
On the other hand, the far region of our wormhole space time (92) is
asymptotically flat, and we have obtained the critical temperature of BEC in
$D=3+1$ Euclid flat space time at finite temperature as in (91). Summing up
these results:

  * •

$T^{c}_{w}=0$ at $r\sim v$ (vicinity of throat),

  * •

$T^{c}_{w}=\textrm{(\ref{dccile})}$ at $r=\infty$ (far flat region).

Interpolating between these two results, we can sketch a phase structure like
Fig.3.

Figure 3:  Sketch of the phase structure for the critical temperature
regarding BEC in the gas in the one side of the wormhole space time. Left and
right red solid lines respectively represent the critical temperatures in the
vicinity of the throat and the far region that we have analytically obtained.
On the other hand, red dotted line is the interpolation between these. From
this sketch, we can see that an analogous state to the Josephson junction can
be formed at any temperatures in the vicinity of the throat.

From Fig.3, we can see that an analogous state to the Josephson junction is
formed at any temperatures in the vicinity of the throat.

Then, the question, whether the Josephson current is flowing or not in the
vicinity of the throat, would arise, which we discuss in the next subsection.

###  5.2 On Josephson current to flow in the vicinity of the throat

The typical scales for the largeness of the Josephson junction and current in
the laboratories would be roughly,

1[nm]-[$\mu$m] and 1[$\mu$A]-[mA]. (134)

Then, since Josephson current is a kind of the tunneling, it is considered
that Josephson current would be more damped by the exponential as the width of
the vicinity of the throat gets greater131313 The ratio of the transmitted
wave to the incident wave in the one-dimensional space with the potential
barrier written in every textbook for the quantum mechanics is given as
$\displaystyle\left(\frac{\textrm{Transmitted wave}}{\textrm{Incident
wave}}\right)^{2}=\frac{1}{1+\frac{V_{0}^{2}\sinh^{2}\alpha
L}{4\varepsilon(V_{0}-\varepsilon)}}\sim e^{-2\alpha L}\quad\textrm{for large
$L$},$ (135) where $\varepsilon\leq V_{0}$ and
$\alpha=\sqrt{{2m}(V_{0}-\varepsilon)/{\hbar^{2}}}$, and $V_{0}$ and $L$ mean
the height and width of potential barrier. $\varepsilon$ and $m$ mean the
energy and the mass of the particles given as the wave function. . Therefore,
if the width of the normal state in the vicinity of the throat were larger
than the scale in (134), the Josephson current could not occur in practice.
Contrary, if it were in the scale of (134), the Josephson current in the
magnitude in (134) might appear in the vicinity of the throat (though it is up
to the property of the space time being in the normal state through which the
Josephson current flows).

To answer this problem, we have to analyze the width of the normal state in
the vicinity of the throat and how much the wave function is damped when it
tunnels. We cannot say any explicit things about this from the analysis in
this paper.

Normally considering, the wormhole is the astronomical object, therefore the
width of the normal state is very larger than (134). Hence, the wave function
of the Josephson current would be damped and vanish when it goes through the
normal state in the vicinity of the throat.

However, we could not exclude the possibility that it is not much damped for
some effect of the curved space. Actually, there is a thought that the Hawking
radiation is a kind of the tunneling [48]. Hence, if the Hawking radiation
exists, the Josephson current might also exist as the same tunneling
phenomenology.

One approach to this issue is to analyze the thermal de Broglie wavelength in
the curved space time, which would be a future works.

##  6 Summary

In this study, we have investigated the phase structure for the BEC/normal
states transition in the traversable wormhole space time filled by some gas to
form BEC up to temperature.

The first idea to make this work begin is something like the consideration we
mentioned in Sec.1 , and we mention the points in the mechanism for the
formation of BEC in this study in Appendix.B.

As a result, we have obtained the result that the critical temperature of the
gas for BEC is zero in the vicinity of the throat. Then, based on that result,
we have pointed out that an analogous state to the Josephson junction is
always formed in the vicinity of the throat.

There is the problem of the exotic matter in the traversable wormhole space
time as shown in Sec.4.2. This is a critical problem from the standpoint of
the realizability of the Josephson junction in this study. For some references
for this problem, see Sec.1, and we in this study could care this problem by
the consideration that it appears for some reason and can exist for some time,
then this study is the one for that time. As for the analysis for how long
time it can exist, it might come to some analysis for the unstable modes in
the classical perturbations on the wormhole space time like the analysis for
the Gregory-Laflamme instability.

If we could get these, we could reach the stage to discuss the realizability
of the Josephson junction in this study, at which we should care the following
2 practical problems: 1) Effect of the strong tidal force to the existence of
our Josephson junction formed at the vicinity of the throat, and 2) how to
actually create the situation where the space is filled by the gas. If we
could finally clear these problems, we might consider our Josephson junction,
realistically.

The result in this study means that the state of the gas is changed from the
normal state to the superconductor state at some point in the space.
Investigating where it is and how it is are future works. In addition, the
result in this study should be independent of the coordinate system we take as
mentioned in the end of Sec.4.6. Checking it is also the future work.

As one of the interesting examples in which our Josephson junction would play
a very intriguing role, author considers some scalar-gravity system, where the
scalar field is supposed to form BEC up to the temperature, c.f. [49], on the
following Euclidean five-dimensional traversable wormhole space time
($ds_{4}^{2}$ part is represented in Fig.4):

$\displaystyle ds_{5}^{2}=dt_{E}^{2}+ds_{4}^{2},\quad\textrm{where}\quad
ds_{4}^{2}\equiv\frac{dr^{2}}{1-v^{2}/r^{2}}+r^{2}d\Omega_{3}^{2},$ (136)

as an effective model for the expanding early universe including the previous
universe collapsing to the beginning of the current universe; the space time
given by (136) corresponds to the shape of the space time for the two
universes joined by the throat part corresponding to the beginning of the
current universe. Postponing the explanation for (136) to the next paragraph,
we first say that it seems we could get the five-dimensional traversable
wormhole space time as a solution regardless of the values of the cosmological
constant if it comes to the five dimension (c.f. [51]). This is because the
uniqueness theorem is the theorem in the four dimension.

In (136), $S^{3}$-direction given by $r^{2}d\Omega_{3}^{2}$ corresponds to the
three-dimensional spatial part we exist, and $r$-direction parameterizes the
time development of that $S^{3}$ space. $t_{E}$-direction is the originally
the time-direction, which is now being $S^{1}$ compactified into the imaginary
direction and prescribes the temperature of the four-dimensional part of
$ds_{4}^{2}$ (c.f. [52]).

Hence, the space given by $ds_{4}^{2}$ part corresponds to the four-
dimensional space time we exist, which is applied to the curved surface in
Fig.4, and is at some temperature determined by the period of the
$t_{E}$-direction. The scalar field to form BEC up to the temperature also
exists on the curved surface in Fig.4.

The three-dimensional space at the beginning of the current universe
corresponds to the throat part at $r=v$ in (136), which is not singular,
therefore it is considered that the space time at the beginning of the
cosmology is regularized in this model. This point is one of the points in
this model as the effective model for cosmology.

Our Josephson junction is supposed to be formed in the vicinity of the
beginning of the cosmology. It would be interesting if the boundary between
the superconductor and normal phases in our study can relate to Big Bang,
which is another point in this model (at this time, the superconductor region
would correspond to the inflation era).

In conclusion, supposing we could get the five-dimensional traversable
wormhole space time as a solution, it would be interesting to examine whether
or not the effective model above can reproduce the picture of the early
cosmology described by the standard cosmology and give some solutions for the
unsettled problems in our current cosmology (c.f. [53]).

Figure 4:  This figure represents the image of the $ds_{4}^{2}$ part of the
traversable wormhole space time given by (136) to be used as the effective
early cosmological model in the idea written in the body text (For how to
obtain this figure, see [50]). The curved surface in this figure corresponds
to the four-dimensional space time we exist; blue circle represents its
$S^{3}$ spatial space ($\Omega_{3}$-direction), and red direction is
$r$-direction and parameterizes the time development of that $S^{3}$ spatial
space. Here, throat part is the $S^{3}$ space at $r=v$, which corresponds to
the three-dimensional space at the beginning of the current cosmology. We can
see it is not singular, therefore the space at the beginning of the cosmology
is regularized. In this figure, $t_{E}$-direction is not included; it is
originally the time-direction, however which is now being $S^{1}$ compactified
into the imaginary direction and prescribes the temperature of the four-
dimensional space depicted in this figure. Hence, the four-dimensional space
depicted in this figure is supposed to be at some temperature. It would be
interesting if the boundary between the superconductor and normal phases in
our study can relate to Big Bang, where at this time, the superconductor
region corresponds to the inflation era.

##  Appendix A Derivation of (62)

The second kind modified Bessel function can be written using the first kind
modified Bessel function $J_{\alpha}(x)$ as [54]

$\displaystyle
K_{\alpha}(x)=\frac{\pi}{2}\frac{i^{\alpha}J_{-\alpha}(ix)-i^{-\alpha}J_{\alpha}(ix)}{\sin(\alpha\pi)}.$
(137)

Using this, it can be written as

$\displaystyle\int^{\infty}_{0}\frac{1}{x}K_{\alpha}(x)K_{\beta}(x)dx=\frac{i\left({\pi}/{2}\right)^{2}}{\sin(\alpha\pi)\sin(\beta\pi)}\int^{\infty}_{0}\frac{dy}{y}({\cal
A}_{1}+{\cal A}_{2}),$ (138) $\displaystyle{\cal
A}_{1}=i^{\alpha+\beta}J_{-\alpha}(y)J_{-\beta}(y)+i^{-\alpha-\beta}J_{\alpha}(y)J_{\beta}(y),$
$\displaystyle{\cal
A}_{2}=-i^{\alpha-\beta}J_{-\alpha}(y)J_{\beta}(y)-i^{-\alpha+\beta}J_{\alpha}(y)J_{-\beta}(y).$

We can check ${\cal A}_{1}+{\cal A}_{2}=0$ for $\alpha\not=\beta$, therefore
(138) is 0 for $\alpha\not=\beta$.

Next, for $\alpha=\beta$, performing the Wick rotation as $\alpha\to i\alpha$,
the part of ${\cal A}_{1}$ can be calculated as

$\displaystyle-\frac{i\left({\pi}/{2}\right)^{2}}{\sin^{2}(i\alpha\pi)}\int^{\infty}_{0}\frac{dy}{y}{\cal
A}_{1}=-\frac{\left({\pi}/{2}\right)^{2}}{\alpha\sinh(\alpha\pi)}.$ (139)

Putting $\beta$ as $\beta\to\alpha+\Delta\alpha$ where $\Delta\alpha$ is taken
to $0$ finally, the part of ${\cal A}_{2}$ can be calculated as

$\displaystyle-\frac{2i(\pi/2)^{2}}{\sin(\alpha\pi)^{2}}\int^{\infty}_{0}\frac{dy}{y}{\cal
A}_{2}=\frac{i\pi^{2}}{2\alpha+\Delta\alpha}\frac{1}{\sin(\alpha\pi)}\frac{1}{\Delta\alpha},$
(140)

where we have used the formula
$\int_{0}^{\infty}\frac{dz}{z}J_{\alpha}(z)J_{\beta}(z)=\frac{2}{\pi}\frac{\sin(\frac{\pi}{2}(\alpha-\beta))}{\alpha^{2}-\beta^{2}}$
[54]. Putting $\alpha\to i\alpha$ and $\Delta\alpha\to i\Delta\alpha$,

$\displaystyle(\ref{ksdc})=\frac{\pi^{2}}{2\alpha\sinh(\alpha\pi)}\frac{1}{\pi\Delta\alpha}.$
(141)

Summarizing the results above,

$\displaystyle(\ref{sdaps})=\left\\{\begin{array}[]{ll}0&\textrm{for
$\alpha\not=\beta$},\\\\[2.84526pt]
{\displaystyle\frac{\pi^{2}}{2\alpha\sinh(\alpha\pi)}\frac{1}{\pi\Delta\alpha}-\frac{\left({\pi}/{2}\right)^{2}}{\alpha\sinh(\alpha\pi)}}&\textrm{for
$\alpha=\beta$}.\end{array}\right.$ (144)

When $\Delta\alpha\to 0$, the results of each case in (144) can be written at
once as

$\displaystyle\int^{\infty}_{0}\frac{1}{x}K_{\alpha}(x)K_{\beta}(x)dx=\frac{\pi^{2}\delta(\alpha-\beta)}{2\alpha\sinh(\alpha\pi)},$
(145)

where we have regarded $\pi\Delta\alpha$ in (144) as $dx$, then regarded it as
$\delta(\alpha-\beta)$ (generally, the delta function $\delta(x)|_{x=0}$ is
equivalent to $dx^{-1}$, where $dx$ is the one in $\int
dx\delta(x)=dx\delta(x)|_{x=0}$).

##  Appendix B Mechanism for the formation of BEC in this study

In this appendix, the technical points in the mechanism of the formation of
BEC in this study is mentioned.

First of all, the fundamental thought in our model could be considered as the
one in the usual fundamental model of BEC like the one given in [31], and what
we have done is to apply it to curved space times. Therefore, the fundamental
form of our Hamiltonian is given by the one in the grand canonical ensemble,
$H-\mu N$.

Then, if naively considering, it appears that we may be able to move only the
chemical potential, only the number of particles or only temperature without
the rests of these not moved. However, this is wrong. The structure in the
microscopic states in the ground canonical ensemble is very complex, and these
are closely related each other as can be seen from the Bose distribution
function (meaning of variables are mentioned in the following),

$\displaystyle\langle n_{r}\rangle=\frac{1}{e^{\beta(e_{r}-\mu)}-1}.$ (146)

Therefore, this appendix begins from the derivation of (146) as confirmation.
Then based on it, the technical points in the mechanism of the formation of
BEC in this study is mentioned.

###  B.1 Bose distribution function: Relation that number of particles
increased when chemical potential increased

First, in order to obtain the form of the Hamiltonian in the ground canonical
ensemble, let us suppose that there are $M$ copied systems, then suppose
$M_{N,i}$ as the number of the systems having the energy value specified by
$N$ and $i$, $E_{N,i}$, and containing $N$ particles. These $M_{N,i}$,
$E_{N,i}$ and $N$ get the following constraints:

$\displaystyle\sum_{N,i}M_{N,i}=M,\quad\sum_{N,i}E_{N,i}M_{N,i}=E_{0},\quad\sum_{N,i}NM_{N,i}=N_{0},$
(147)

where $\sum_{N,i}$ means $\sum_{N=0}^{N_{0}}\sum_{i}$, and $M$, $E_{0}$ and
$N_{0}$ are constants.

Now, consider the $\Gamma$ space (the space that has canonical variables of
each of $N_{0}$ particles in the whole of $M$ copied systems as its
coordinates). Then, each infinitesimal region in the $\Gamma$ space
corresponds to a set of $M_{N,i}$, and if we check each infinitesimal region
of some region, we would find that there are a number of the same sets
$M_{N,i}$ (for the meaning of “the same”, read it out from the $W$ in (148)).
Here, let us suppose the Ergodic hypothesis (probability that each $M$ copied
system takes some one of $M_{N,i}$ states is always the same in the region
determined by $N_{0}$ and $E_{0}$) is held in the $\Gamma$ space. As a result,
the most appearing “the same” set of $M_{N,i}$ is considered to be the set for
the thermal equilibrium state. Therefore, let us obtain the most appearing set
of $M_{N,i}$ in the region determined by $N_{0}$ and $E_{0}$ in the $\Gamma$
space. This problem is to obtain the set of $M_{N,i}$ which maximizes the
following $W$:

$\displaystyle W=\frac{M!}{\Pi_{N,i}M_{N,i}!}.$ (148)

Then, using the method of Lagrange multiplier and Stirling’s approximation, we
can finally express all such the $M_{N,i}$ at once as

$\displaystyle\frac{M_{N,i}}{M}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{1}{\Xi}\exp[-\beta(E_{N,i}-N\mu)],$ (149)
$\displaystyle\Xi\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\sum_{N,i}\exp[-\beta(E_{N,i}-N\mu)],$ (150)

where $M$, $E_{0}$ and $N_{0}$ are supposed to be very large positive integers
to use Stirling’s approximation, and $\beta$ and $\mu$ mean the inverse
temperature and chemical potential. (149) means the probability that the
system with $N$ particles and the energy $E_{N,i}$ appears.

From here, let us suppose that the particles in the system follow the Bose
statistics. As a result, the energy values are discretized and the number of
particles to take each energy value is no limited. Then we can rewrite and
replace as

$\displaystyle N=\sum_{r=\textrm{lowest}}^{\textrm{highest}}n_{r}\quad{\rm
and}\quad\sum_{N,i}\to\sum_{N=0}^{N_{0}}\sum_{\\{n_{r}\\}},\quad
E_{N,r}\to\sum_{r=\textrm{lowest}}^{\textrm{highest}}e_{r}n_{r}.$ (151)

where “lowest” and “highest” mean those of the discretized energy levels,
“$\sum_{\\{n_{r}\\}}$” means to produce all the sets of $n_{r}$ satisfying
$N=\sum_{r}n_{r}$, and $e_{r}$ mean the values of the energy labeled by $r$
that each particles takes. Therefore, finally $\Xi$ in (150) can be given as

$\displaystyle\Xi=\sum_{N=0}^{N_{0}}\sum_{\\{n_{r}\\}}\exp[-\beta\sum_{r}\varepsilon_{r}n_{r}]=\sum_{N=0}^{N_{0}}\sum_{\\{n_{r}\\}}\prod_{r}\exp[-\beta\varepsilon_{r}n_{r}],$
(152)

where $\varepsilon_{r}\equiv e_{r}-\mu$ and $\sum_{r}$ means
$\sum_{r=\textrm{lowest}}^{\textrm{highest}}$. Then, it is known that
$\sum_{N=0}^{N_{0}}\sum_{\\{n_{r}\\}}$ can be treated as
$\prod_{r}\sum_{n_{r}}$, namely we can independently perform the summation for
each $n_{r}$ in the range $\sum_{r}n_{r}\leq N_{0}$. At this time, if $N_{0}$
is infinity, $\sum_{N=0}^{N_{0}}\sum_{\\{n_{r}\\}}$ can be treated as
$\prod_{r}\sum_{n_{r}=0}^{\infty}$. Therefore, supposing that $N_{0}$ is
infinity,

$\displaystyle\Xi\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\prod_{r}\sum_{n_{r}=0}^{\infty}\exp[-\beta\varepsilon_{r}n_{r}].$
(153)

At this time, if $\exp[-\beta\varepsilon_{r}]<1$, namely if
$\varepsilon_{r}>0$, for all $r$,

$\displaystyle\Xi=\prod_{r}(1-\exp[-\beta\varepsilon_{r}])^{-1}.$ (154)

On the other hand, if any one of $\varepsilon_{r}\leq 0$, $\Xi$ gets diverged.

Now, we obtain the e.v. of $n_{r}$, which can be written as

$\displaystyle\langle n_{r}\rangle=\frac{\sum
n_{r}\exp[-\beta\sum_{r}\varepsilon_{r}n_{r}]}{\sum\exp[-\beta\sum_{r}\varepsilon_{r}n_{r}]},$
(155)

where $\sum$ above mean $\sum_{N=0}^{N_{0}}\sum_{\\{n_{r}\\}}\prod_{r}$. Since
$\langle
n_{r}\rangle=-\frac{1}{\beta}\frac{\partial\ln\Xi}{\partial\varepsilon_{r}}$,
therefore using (154), we can obtain $\langle n_{r}\rangle$ given in (146),
and we can get a key fact in the formation of BEC in this study, the number of
the particles is increased when the chemical potential is increased.

###  B.2  Keep density of gas constant in decreasing temperature, and for this
purpose, increase chemical potential, then finally formed BEC described by the
constants

As the fundamental thought in our model, we want to keep the number of the
particles of the gas as it is when we decrease the temperature. Then, since
there is a relation (146), the chemical potential should be raised.

However, as can be seen from the text under (154), there is the upper limit
for the value the chemical potential can take. Actually, (35), (88) and (110)
would be that.

Therefore, the system needs to take some way other than the growing up of the
chemical potential when the chemical potential reaches the upper limit, which
is to have the $\alpha$ in (21) have some finite value. This is the definition
of the formation of BEC.

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