Canonical Quantization of the U(1) Gauge Field
in the Rindler Coordinates

Shingo Takeuchi

Faculty of Environmental and Natural Sciences, Duy Tan University, Da Nang,
Vietnam

This paper describes the canonical quantization of the U(1) gauge field across
all four regions in the Rindler coordinates in the Lorentz covariant gauge. In
the four regions (future, past, left and right Rindler-wedges) in the Rindler
coordinates, defining the gauge-fixed Lagrangian in the Lorentz covariant
gauge, which is composed of the U(1) gauge field, the $B$-field and ghost
fields. Since the U(1) gauge and $B$-fields are decoupled from the ghost
fields by the property of the U(1) gauge theory, the U(1) gauge field and the
$B$-field are examined in this study.

Then, by solving the equations of motion obtained from the gauge-fixed
Lagrangian, the solutions of each mode of the U(1) gauge field and the
$B$-field can be obtained. Following this, with the Klein-Gordon inner-product
defined in the Rindler coordinates, the normalization constants of each of
those mode-solutions are determined.

Subsequently, formulating the canonical commutation relations of the U(1)
gauge field and its canonical conjugate momentum, the commutation relations of
the coefficient of each mode-solution in each direction of the U(1) gauge
field in each region of the Rindler coordinates are obtained. From these, it
can be seen that those coefficients have physical meaning as
creation/annihilation operators. The polarization vectors in each region of
the Rindler coordinates are also given in this study.

From these, it is shown that the Minkowski ground state can be given as the
outer-product of the quantum states multiplied by the creation operators of
the U(1) gauge field in the left and right Rindler-wedge states. Then,
obtaining the density matrix of the U(1) gauge theory in the right Rindler-
wedge from that, it is shown that the U(1) gauge field in the constant
accelerated system feels the Unruh temperature as well.

##  1 Introduction

From the analysis of the uniformly accelerated motion, the Unruh temperature
is derived as $T_{U}=\hbar a/2\pi ck_{\rm B}\approx 4\times
10^{-23}\,a/$(cm/$s^{2}$)[K] [Fulling:1972md, Davies:1974th, Unruh:1976db].
Since the Unruh temperature which can currently be produced is less than
$3$[K] CMB, its detection remains extremely challenging. While detecting the
Unruh temperature is a technologically important issue. It is also important
as experimental confirmation of Hawking radiation, and in terms of the problem
of how an event which is observed in the accelerated systems can be observed
in an inertial system.

Recent studies in the experimental field have touched upon the following
topics: BEC [Retzker], neutrino oscillation [Dvornikov:2015eqa,
Blasone:2018czm, Blasone:2020vtm], anti-Unruh effect [Chen:2021evr,
Garay:2016cpf, Liu:2016ihf, Brenna:2015fga], cold atoms [Kosior:2018vgx,
Rodriguez-Laguna:2016kri], Berry phases [Quach:2021vzo, Martin-
Martinez:2010gnz], Casimir effect [Marino:2014rfa], classical analog
[Leonhardt:2017lwm] and others [Lynch:2019hmk, Lochan:2019osm].

In addition, the effects of the thermal excitation of the Unruh effect on the
phenomena currently reproducible in the laboratory have been analyzed as
follows: analysis of critical temperatures of the phase transitions
[Ohsaku:2004rv, Ebert:2006bh, Castorina:2007eb, Takeuchi:2015nga]; analysis of
the thermal radiation from a particle in constant accelerated motion
[Schutzhold:2006gj, Schutzhold:2009scb][Iso:2010yq, Oshita:2015xaa,
Iso:2016lua]; analysis of the Schwinger effect in a constant accelerated
systems [Kim:2016dmm, Kaushal:2022las]; and analysis of quantum corrections in
the energy-momentum tensors of gases in constant accelerated systems
[Prokhorov:2019cik, Prokhorov:2019hif, Prokhorov:2019yft]. Additionally, the
Hagedorn transition in string theory due to the Unruh effect has also been
analyzed [Parentani:1989gq].

Theoretically, the coordinates of a constant accelerated motion are
generalized to the Rindler coordinates, which is equivalent to the geometry in
the neighborhood of the event-horizon of the Schwarzschild black hole, and in
which the Killing horizons exist. Therefore, the thermal radiation and the
entropy akin to the Hawking radiation and the Bekenstein-Hawking entropy,
respectively, can be considered in the Rindler coordinates. In addition, since
the Rindler coordinates is the issue of the coordinates, it can be considered
at any points in the spacetime based on the general coordinate transformation
invariance. Exploiting these properties of the Rindler coordinates, it can be
shown that Einstein’s equation can be derived as a state equation at any point
in spacetime [Jacobson:1995ab].

Furthermore, since spacetime is causally separated by the Killing horizon in
the Rindler coordinates, quantum entanglement arises between those regions in
the Rindler coordinates. Analysis of the entanglement entropy (EE) between the
separated regions in the Rindler coordinates has been performed in many
studies, and as a result, the area-proportional low is obtained, based on
which the suggestion that the Bekenstein-Hawking entropy would be a kind of EE
between the inside and outside of the black hole can be obtained
[Kabat:1994vj, Dowker:1994fi, Iellici:1996gv].

In addition, the Rindler coordinates are extended to the AdS space: the Unruh
temperature in the AdS space has been given in [Deser:1997ri], and the AdS/CFT
given with the AdS space patched by the Rindler coordinates has been studied
in [Parikh:2012kg, Fareghbal:2014oba, Sugishita:2022ldv].

In terms of studies on the properties of the Rindler coordinates, the
following studies seem particularly intriguing: the upper bound for
acceleration [Rovelli:2013osa] and another on inversion between the bosonic
and fermionic statistics occurring in the odd dimensional Rindler coordinates
[Terashima:1999xp].

A review of the literature shows that until now, the canonical quantization
(CQ) of the gauge fields in the Rindler coordinates has not been properly
performed yet, while that of scalar and spinor fields has been done
[Higuchi:2017gcd][Soffel:1980kx, Ueda:2021nln]. Studies treating the U(1)
gauge field in the Rindler coordinates can be found as cited in [StudyGauge].
However, in these studies, the normalization constants (NC) of the mode-
solutions contain some speculation and have not been obtained exclusively from
the analysis, or the discussion proceeds without the NC. Furthermore, it
appears that the solutions for the gauge field in all of the directions in
spacetime have not been explicitly obtained.

The NC have not been determined until now likely because the integrals in
Appendix.LABEL:r3g67kb in this study could not be calculated. Since those
integrals appear in the Klein-Gordon (KG) inner-products between the mode-
solutions of the U(1) gauge field in the Rindler coordinates, what those
integrals cannot be calculated leads to the following four situations; 1)
their NC cannot be determined, 2) accordingly, the canonical commutation
relations (CCR) cannot be robustly formulated (it is possible to formally
write down the CCR, if the form of the KG inner-product is known*** If the KG
inner-product (LABEL:stwgi) is known, from the demand that the commutation
relations of the creation/annihilation operators can be finally obtained like
(LABEL:roere), it can be seen that the form of the CCR is given as
(LABEL:reawv).; however, since the mode-solutions of the field are not
completely determined due to the lack of the NC, the concrete CCR, the CCR for
each mode-solution, cannot be completely known), 3) analysis performed using
the KG inner-products cannot be done without ambiguities or speculation, and
4) the creation/annihilation operators cannot be taken out. All of these would
finally lead to the first situation mentioned.

In these situations, we will calculate the integrals as demonstrated in
Appendix.LABEL:r3g67kb. Subsequently, utilizing these integrals, we determine
the normalization constants (NC) of the mode-solutions of the U(1) gauge field
across all directions of spacetime within each of the four regions: future,
past, left, and right Rindler-wedges (see Fig.1). This determination is based
on the condition that all mode-solutions are mutually orthonormal under the
Klein-Gordon (KG) inner-product. In this study, these mode-solutions of the
U(1) gauge field are obtained by solving the equations of motion derived from
the Lagrangian formulated in the Lorentz-covariant gauge within the Rindler
coordinates.

Next, by formulating the CCR of the U(1) gauge field within each of the four
regions in the Rindler coordinates, we examine the U(1) gauge field within
these CCR using the mode-expanded form based on the normalized mode-solutions
we have obtained. Subsequently, employing the Klein-Gordon (KG) inner-product,
we derive the commutation relations of the creation/annihilation operators for
the coefficients of the mode-solutions of the U(1) gauge field.

Subsequently, we demonstrate that the Minkowski ground state manifests as a
quantum entangled state of the left and right Rindler-wedges, excited by the
creation operators of the U(1) gauge field. By deriving the density matrix
from this entangled state, we illustrate that the U(1) gauge field within a
constant accelerated system will also experience the Unruh temperature.

The following section discusses the organization of this paper. In Sec.2, the
Rindler coordinates used in this study are reviewed. In Sec.3, the Lagrangian
of the U(1) gauge theory in the Lorentz covariant gauge in the Rindler
coordinates is obtained. In Sec.4, the mode-solutions of the U(1) gauge field
and the $B$-field in each region in the Rindler coordinates and their
normalization constants are obtained.

In Sec.LABEL:bywbd, the canonical quantization of the U(1) gauge field is
computed for each region in the Rindler coordinates, through which the
commutation relations which the coefficients of each mode-solution satisfy are
finally obtained. From this, it can be confirmed that the coefficients of the
each mode-solution have roles as creation/annihilation operators in each
region in the Rindler coordinates.

In Sec.LABEL:yervd, we address the creation/annihilation operators of the U(1)
gauge field, providing the polarization vector for $(S,L,\pm)$-directions.
From the expression of this vector, it becomes apparent that the region where
the polarization vector can be defined is constrained within each region, as
the norms of the 1-particle states in the $S$\- and $L$-directions can
potentially be negative. The origins of these restrictions and their
implications are discussed.

In Sec.LABEL:utempu1rc, the density matrix of the U(1) gauge field in the
right Rindler-wedge is obtained. Regarding each subsection, in
Secs.LABEL:usdvea and LABEL:ugfred, ${\bm{a}}^{{\rm(M)}\perp}_{\vec{q}}$ is
obtained in terms of ${\bm{a}}^{{\rm(R)}\perp}_{\vec{q}}$ and
$({\bm{a}}^{{\rm(R)}\perp}_{\vec{q}})^{\dagger}$ by analyzing the Bogoliubov
coefficients (${\bm{a}}^{{\rm(M)}\perp}_{\vec{q}}$ and
${\bm{a}}^{{\rm(R)}\perp}_{\vec{q}}$ represent the annihilation operators in
the Minkowski and Rindler coordinates, respectively). In Sec.LABEL:eppwte,
from the condition ${\bm{a}}^{{\rm(M)}\perp}_{\vec{q}}|0_{\rm M}\rangle=0$,
the ground state in the Minkowski coordinates is constituted by the Rindler
states, from which in Sec.LABEL:yedvsh, the density matrix is obtained and it
is shown that the U(1) gauge field in a constant accelerated system will also
feel the Unruh temperature.

In Sec.LABEL:Summary, summarizing this study, the future directions of the
study are discussed. In Appendix.LABEL:buobhs, the analysis from (64) of the
Lagrangian to the Lagrangian in the Lorentz-covariant gauge (40) is noted. In
Appendix.LABEL:r3g67kb, the integral formulas, which are essential for the
calculations of the KG inner-products to obtain the NC in Sec.LABEL:brpnevt,
are given.

##  2 The Rindler coordinates used in this study

In this section, we review the Rindler coordinates used in this study.

###  2.1 $ds^{2}$ in the each region in the Rindler coordinates

We begin with the $4$-dimensional Minkowski spacetime:

$\displaystyle ds^{2}=dt^{2}-\sum_{i=1}^{3}(dx^{i})^{2}.$ (1)

For the Minkowski coordinates in (1), we consider the following four
coordinate transformations:

$\displaystyle\cdot\quad x^{0}=$ $\displaystyle\,\,a^{-1}e^{a\tilde{\xi}}\sinh
a\tilde{\tau},\quad\hskip 11.38109ptx^{1}=-a^{-1}e^{a\tilde{\xi}}\cosh
a\tilde{\tau},$ (2a) $\displaystyle\cdot\quad x^{0}=$
$\displaystyle\,\,a^{-1}\,e^{a\xi}\sinh a\tau,\quad\hskip
9.95845ptx^{1}=+a^{-1}\,e^{a\xi}\cosh a\tau,$ (2b) $\displaystyle\cdot\quad
x^{0}=$ $\displaystyle\,+a^{-1}e^{a\eta}\cosh a\zeta,\quad
x^{1}=a^{-1}e^{a\eta}\sinh a\zeta,$ (2c) $\displaystyle\cdot\quad x^{0}=$
$\displaystyle\,-a^{-1}e^{a\tilde{\eta}}\cosh a\tilde{\zeta},\quad
x^{1}=a^{-1}e^{a\tilde{\eta}}\sinh a\tilde{\zeta}.$ (2d)

where

  * •

the regions which are transformed to by (2a) and (2b) are the left and right
Rindler-wedges (LRW and RRW), respectively, and the regions transformed to by
(2c) and (2d) are the future and past Rindler-wedges (FRW and PRW),
respectively.

On the left and right in Fig.1, the LRW/RRW and FRW/PRW are shown,
respectively.

  * •

We have used $(\tau,\xi)$, $(\tilde{\tau},\tilde{\xi})$, $(\zeta,\eta)$ and
$(\tilde{\zeta},\tilde{\eta})$ as the coordinates in the LRW, RRW, FRW and PRW
respectively, as shown in Fig.1.

  * •

In the Rindler coordinates, $a$ is considered to be fixed.

  * •

In the LRW and RRW, there are world-lines of the constant accelerated motion;
$\xi$ and $\tilde{\xi}$ label those, and $\tau$ and $\tilde{\tau}$ parametrize
them.

On the other hand, in the FRW and PRW, it is assumed that there is no object
performing constant accelerated motion. However, in the FRW and PRW, the lines
labeled by $\eta$ and $\tilde{\eta}$ and parametrized by $\zeta$ and
$\tilde{\zeta}$ exist as lines analogous to the world-lines in the LRW and
RRW, which are plotted in Fig.1.

Applying (2) to (1), $ds^{2}$ in each region can be obtained as follows:

$\displaystyle
ds^{2}=\left\\{\begin{array}[]{ll}\\!e^{2a\tilde{\xi}}(d\tilde{\tau}^{2}-d\tilde{\xi}^{2})-(dx^{\perp})^{2}&\\!\\!\textrm{in
LRW},\\\\[4.2679pt]
\\!e^{2a\xi}(d\tau^{2}-d\xi^{2})-(dx^{\perp})^{2}&\\!\\!\textrm{in
RRW},\\\\[4.2679pt]
\\!e^{2a\eta}(d\eta^{2}-d\zeta^{2})-(dx^{\perp})^{2}&\\!\\!\textrm{in
FKR},\\\\[4.2679pt]
\\!e^{2a\tilde{\eta}}(d\tilde{\eta}^{2}-d\tilde{\zeta}^{2})-(dx^{\perp})^{2}&\\!\\!\textrm{in
PKR},\end{array}\right.$ (7)

where $x^{\perp}\equiv(x^{2},x^{3})$.

Figure 1:  These figures represent the Rindler coordinates on the Minkowski
coordinates. The coordinates in each region are defined in (7) and (15). On
the left, those of LRW and RRW are represented, and in the right, those of the
FRW and PRW are represented. $(U,V)$ are the light-cone coordinates defined in
(21), and “the $\xi\to-\infty$ line” (and other three) are defined in Sec.2.4,
which are generally referred to as the Killing horizon.

Now, let us introduce $\rho$ defined as follows:

$\displaystyle\rho\equiv\left\\{\begin{array}[]{lcll}\\!a^{-1}e^{a\tilde{\xi}}&\textrm{and}&a^{-1}e^{a\xi}&\\!\textrm{in
the LRW and RRW,}\\\\[4.2679pt]
\\!a^{-1}e^{a\eta}&\textrm{and}&a^{-1}e^{a\tilde{\eta}}&\\!\textrm{in the FKR
and PKR.}\end{array}\right.$ (10)

Considering this $\rho$, $ds^{2}$ in (7) can be given as follows:

$\displaystyle
ds^{2}=\left\\{\begin{array}[]{ll}\\!a^{2}\rho^{2}d\tilde{\tau}^{2}-d\rho^{2}-(dx^{\perp})^{2}&\\!\textrm{in
the LRW,}\\\\[4.2679pt]
\\!a^{2}\rho^{2}d\tau^{2}-d\rho^{2}-(dx^{\perp})^{2}&\\!\textrm{in the
RRW,}\\\\[4.2679pt]
\\!d\rho^{2}-a^{2}\rho^{2}d\zeta^{2}-(dx^{\perp})^{2}&\\!\textrm{in the
FRW,}\\\\[4.2679pt]
\\!d\rho^{2}-a^{2}\rho^{2}\tilde{\zeta}^{2}-(dx^{\perp})^{2}&\\!\textrm{in the
PRW.}\end{array}\right.$ (15)

From the above expressions, it can be seen that the $\tau$\- and
$\tilde{\tau}$-directions are the Killing vectors in the RRW and LRW, and the
$\zeta$\- and $\tilde{\zeta}$-directions are the Killing vectors in the FRW
and PRW. Therefore, in the RRW and LRW, the Hamiltonians are defined for the
$\tilde{\tau}$\- and $\tau$-constant surfaces and in the FRW and PRW, the
Hamiltonians are defined for the $\zeta$\- and $\tilde{\zeta}$-constant
surfaces. Therefore, in the Rindler coordinates in the RRW and LRW,
$\tilde{\tau}$ and $\tau$ coordinate play the roles of the time, and in the
Rindler coordinates in the FRW and PRW, the $\tilde{\zeta}$ and $\zeta$
coordinates play the roles of time.

As mentioned above, since $\zeta$ and $\tau$ play the role of time in the FRW
and RRW, respectively, we perform the Euclideanization as $\tau\to-i\tau$ and
$\zeta\to-i\zeta$ in (15). As a result, the $ds^{2}$ of the RRW and FRW in
(15) are written as follows:

$\displaystyle ds_{\rm
E}^{2}=\left\\{\begin{array}[]{ll}\\!a^{2}\rho^{2}d\tau^{2}+d\rho^{2}+(dx^{\perp})^{2}&\\!\textrm{in
the RRW,}\\\\[4.2679pt]
\\!d\rho^{2}+a^{2}\rho^{2}d\zeta^{2}+(dx^{\perp})^{2}&\\!\textrm{in the
FRW.}\\\ \end{array}\right.$ (18)

where $ds_{\rm E}^{2}=-ds^{2}$. It can be seen that in (18), the $\tau$\- and
$\zeta$-directions are periodic by $\beta=2\pi/a$, the inverse of which agrees
with the Unruh temperature in the constant accelerated system with $a$. For
the LRW and PRW, the Euclideanization is performed in the same way as for the
RRW and FRW, and the same Unruh temperature can be obtained in the LRW and
PRW.

###  2.2 Solutions in the LRW (PRW) from the solutions in the FRW (PRW)

If the solutions in the FRW and RRW are known, solutions in the LRW and PRW,
which are parity-symmetric, can be immediately obtained. In this subsection,
this is discussed.

If a vector solution in the FRW is known, denoting it as
$f^{\mu}(\zeta,\eta)$, it can be seen from (2) and Fig.1 that the vector
solution in the PRW being in a party-symmetric relation with
$f^{\mu}(\zeta,\eta)$ for the $x^{0}=0$ line can be obtained simply by
exchanging $(\zeta,\eta)$ to $(\tilde{\zeta},\tilde{\eta})$ leaving the forms
of the functions of the solutions as

For the solution in the FRW to obtain the solution in the PRW:

$\displaystyle\bullet\quad\\!\\!$
$\displaystyle(f^{0},f^{1},f^{\perp})\to(f^{0},f^{1},f^{\perp})\quad\\!\\!\textrm{with}(\zeta,\eta)\to(\tilde{\zeta},\tilde{\eta}).$
(19)

In the same way, denoting as $g^{\mu}(\tau,\xi)$ for a given vector solution
in the RRW, the party-symmetric vector solutions in the LRW for
$g^{\mu}(\zeta,\eta)$ for the $x^{1}=0$ line can be obtained simply by
exchanging $(\tau,\xi)$ to $(\tilde{\tau},\tilde{\xi})$ as:

For the solution in the RRW to obtain the solution in the LRW:

$\displaystyle\bullet\quad\\!\\!$
$\displaystyle(f^{0},f^{1},f^{\perp})\to(f^{0},-f^{1},f^{\perp})\quad\\!\\!\textrm{with}(\tau,\xi)\to(\tilde{\tau},\tilde{\xi}).$
(20)

###  2.3 Light-cone coordinates

We define the light-cone coordinates as:

$\displaystyle U\equiv x^{0}-x^{1},\quad V\equiv x^{0}+x^{1},$ (21)

where $x^{0}$ and $x^{1}$ are the Minkowski coordinates in (1). We show these
in Fig.1, from which we can see that

$\displaystyle U\left\\{\begin{array}[]{ll}\\!>0&\\!\textrm{in LRW and
FRW,}\\\\[4.2679pt] \\!<0&\\!\textrm{in RRW and PRW,}\end{array}\right.\quad
V\left\\{\begin{array}[]{ll}\\!<0&\\!\textrm{in LRW and FRW,}\\\\[4.2679pt]
\\!>0&\\!\textrm{in RRW and PRW.}\end{array}\right.$ (26)

Based on (2a) and (2b), we can represent $U$ and $V$ in terms of the LRW and
RRW coordinates ($\tilde{\tau}$, $\tilde{\xi}$) and ($\tau$, $\xi$) as
follows:

$\displaystyle(U,V)=\left\\{\begin{array}[]{lcl}\\!a^{-1}(e^{a\tilde{u}},\,-e^{-a\tilde{v}})&\\!\textrm{in
LRW
with}\\!&(\tilde{u},\tilde{v})\equiv(\tilde{\tau}+\tilde{\xi},\tilde{\tau}-\tilde{\xi}),\\\\[4.2679pt]
\\!a^{-1}(-e^{-au},\,e^{av})&\\!\textrm{in RRW
with}\\!&(u,v)\equiv(\tau-\xi,\tau+\xi),\\\\[4.2679pt]
\\!a^{-1}(e^{-as},\,e^{at})&\\!\textrm{in FRW
with}\\!&(s,t)\equiv(-\zeta+\eta,\zeta+\eta),\\\\[4.2679pt]
\\!-a^{-1}(e^{-a\tilde{s}},\,e^{a\tilde{t}})&\\!\textrm{in PRW
with}\\!&(\tilde{s},\tilde{t})\equiv(\tilde{\zeta}+\tilde{\eta},-\tilde{\zeta}+\tilde{\eta}).\end{array}\right.$
(31)

###  2.4 The lines asymptotically approached to by taking $\xi$, $\eta$,
$\tilde{\xi}$ and $\tilde{\eta}$ closer to $-\infty$

####  2.4.1 The lines at $\xi$ closer to $-\infty$

Let us consider taking $\xi$ in (2b) closer to $-\infty$. Then, let us take
$|\tau|$ to $\infty$ and refer to the lines asymptotically approached to at
that time as “the $\xi\to-\infty$ line”. Note that the exact limit taking
$\xi$ to $-\infty$ is excluded by the definition of the Rindler coordinates.

Then, at any points on $x^{0}>0$ or $x^{0}<0$ on the $\xi\to-\infty$ line, it
can be seen that $\tau$ and $\xi$ in (2b) should be related as

$\displaystyle|\tau|\sim|\xi|,$ (32)

otherwise, on the $\xi\to-\infty$ line, neither $t$ nor $x^{1}$ can be finite
(therefore, the motion of the object on the $\xi\to-\infty$ line in the finite
$\tau$ is all packed into the neighborhood of $x^{0}=0$). This can be seen
from the forms of $x^{0}$ and $x^{1}$ in (2b)††† $x^{0}$ and $x^{1}$ in (2b)
can be roughly written as  $\displaystyle x^{0}$ $\displaystyle\sim
e^{a\xi}\sinh a\tau\sim e^{a(\xi+\tau)}-e^{-a(-\xi+\tau)},$ (33a)
$\displaystyle x^{1}$ $\displaystyle\sim e^{a\xi}\cosh a\tau\sim
e^{a(\xi+\tau)}+e^{-a(-\xi+\tau)}.$ (33b) Then, on the $\xi\to-\infty$ line,
if (32) is satisfied, $x^{0}$ can take finite values behaves as follows:
$\displaystyle\displaystyle
x^{0}\sim\left\\{\begin{array}[]{ll}e^{a(\xi+\tau)}-e^{-2a\infty}&\textrm{for
$\tau\sim-\xi$,}\\\\[4.2679pt] e^{-2a\infty}-e^{-a(-\xi+\tau)}&\textrm{for
$\tau\sim+\xi$,}\end{array}\right.$ (36) where “$\xi\sim\mp\tau$” mean that
$|\xi|$ and $|\tau|$ are in the same order. $x^{1}$ also can take finite
values on the $\xi\to-\infty$ line, if (32) is satisfied, by the same logic. .

Therefore, on the $\xi\to-\infty$ line, since $|\tau|$ approaches $\infty$,
$x^{0}/x^{1}=\tanh(a\tau)$ approaches $\pm 1$. The lines obtained by the
$x^{0}/x^{1}=\pm 1$ are the $\pm 45$-degree straight diagonal lines in the
$x^{1}\geq 0$ region in Fig.1.

These $\pm 45$-degree straight diagonal lines are generally referred to as the
past and future Killing horizons for $x^{0}<0$ and $x^{0}>0$, respectively. In
this paper, we refer to the lines obtained by taking $\xi$ closer to
$-\infty$, which asymptote to those $\pm 45$-degree straight diagonal lines,
as “the $\xi\to-\infty$ line”, as noted on the lefthand side of Fig.1.

####  2.4.2 The lines at $\eta$ closer to $-\infty$

By the same logic in Sec.2.4.1, when $\eta$ in (2c) is taken closer to
$-\infty$, it can be seen that $\eta$ and $\zeta$ should be related as
follows:

$\displaystyle|\eta|\sim|\zeta|,$ (37)

and $x^{0}/x^{1}=\tanh(a\zeta)$ asymptotes to $\pm 1$.

The lines given by these $x^{0}/x^{1}=\pm 1$ are the $\pm 45$-degree straight
diagonal lines in the $x^{0}\geq 0$ region in Fig.1, where $-1$ and $+1$
correspond to the left and right future Killing horizons respectively.

In this paper, we refer to the lines obtained by taking $\eta$ closer to
$-\infty$ as “the $\eta\to-\infty$ line”, which asymptote to those $\pm
45$-degree straight diagonal lines, as noted on the righthand side of Fig.1.

####  2.4.3 The lines at $\tilde{\tau}$ or $\tilde{\eta}$ closer to $-\infty$

By the same logic in Secs.2.4.1 and 2.4.2, we refer to the lines obtained by
taking $\tilde{\tau}$ and $\tilde{\eta}$ closer to $-\infty$ as “the
$\tilde{\tau}\to-\infty$ line” and “the $\tilde{\eta}\to-\infty$ line”,
respectively, as noted in both sides of Fig.1.

##  3 The Lorentz covariant U(1) gauge field Lagrangians in the Rindler
coordinates

In this section, the Lagrangian densities (referred to as the Lagrangian) of
the U(1) gauge field in the Lorentz-covariant gauge in the four regions in the
Rindler coordinates are obtained. Since the analysis to obtain the Lorentz
covariant U(1) gauge field Lagrangian in the LRW and PRW can be performed in
the same way as those in the FRW and RRW, this section mainly focuses on the
analysis concerning the FRW and RRW. The result is noted in (40).

First, let us begin with the actions of the U(1) gauge field in each of the
four regions in the Rindler coordinates:

$\displaystyle S=\int\\!d^{4}x\,\sqrt{-g}\,{\cal L}_{\rm U(1)},\quad{\cal
L}_{\rm U(1)}=-F_{\mu\nu}F^{\mu\nu}/4,$ (38)

where the integral region is the RRW, FRW, LRW or PRW given by the coordinates
in (15), and $\nabla_{\mu}$ are covariant derivatives given by the metrices
(15), and
$F_{\mu\nu}=\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$.

The Lagrangian in the Lorentz-covariant gauge given by
$\partial_{\mu}A^{\mu}=0$ in the Minkowski coordinates is known to be given as

$\displaystyle\tilde{{\cal L}}^{(M)}_{\rm U(1)}={\cal L}_{\rm
U(1)}+B\,\partial_{\mu}A^{\mu}+B^{2}/2+i\,\bar{c}\,\partial_{\mu}\partial^{\mu}\,c,$
(39)

where $B$ is the auxiliary field (the $B$-field) and $c$, $\bar{c}$ are the
ghost fields. From this, we are able to derive the Lagrangian in the Lorentz-
covariant gauge in each of the four regions in the Rindler coordinates by
replacing the differentials with the gravitational covariant derivatives as

$\displaystyle\tilde{{\cal L}}^{(R)}_{\rm U(1)}={\cal L}_{\rm
U(1)}+B\,\nabla_{\mu}A^{\mu}+B^{2}/2+i\,\bar{c}\,\nabla_{\mu}\nabla^{\mu}\,c,$
(40)

where the coordinates in the one above refer to those in (15).

However, since (39) is derived through the Lagrangian gauge-fixed by the
Coulomb gauge (a non-covariant gauge), we will check if (40) can be obtained
in the Rindler coordinates in the following subsection. As noted earlier, in
the following subsections, we mainly focus on the analysis concerning the FRW
and RRW.

Upon performing the analysis to obtain (40) in the following subsections, we
note the values of the Christoffel’s in each region in the Rindler
coordinates:

$\displaystyle\cdot\quad\Gamma^{0}_{01}$
$\displaystyle=\Gamma^{0}_{10}=\rho^{-1},\quad\Gamma^{1}_{00}=a^{2}\rho,\quad\\!\textrm{others
$=0$ in the RRW and LRW,}$ (41a) $\displaystyle\cdot\quad\Gamma^{0}_{11}$
$\displaystyle=a^{2}\rho,\quad\Gamma^{1}_{01}=\Gamma^{1}_{10}=\rho^{-1},\quad\\!\textrm{others
$=0$ in the FRW and PRW.}$ (41b)

Let us now evaluate the rationale behind solely considering $A_{\mu}$ and $B$
in this study. In this study, the U(1) gauge field is considered. In such a
case, the Faddeev-Popov determinant (FPd) does not include the gauge field (as
indicated in (LABEL:eartd)). As a result, the partition functions of the ghost
and the gauge fields decouple each other as $Z=Z_{A_{\mu},B}\,Z_{c,\bar{c}}$.
Therefore, we may consider only $A_{\mu}$ and $B$. With this in mind, let us
focus only on $A_{\mu}$ and $B$ in the subsequent discussion.

###  3.1 The Hamiltonian in the RRW and FRW

Beginning with ${\cal L}_{\rm U(1)}$ in (38), the dynamical variables are
$A^{\mu}$, and $x^{0}$ and $x^{1}$ play the role of the time-coordinate in the
FRW and RRW, respectively, as mentioned under (15). Therefore, the canonical
conjugate momenta in the RRW and FRW are derived as follows:

$\displaystyle\cdot\quad\\!\\!\frac{\partial{\cal L}_{\rm
U(1)}}{\partial(\partial_{0}A^{k})}=-F^{0}{}_{k}=g^{00}F_{k0}\equiv\pi_{k}\quad\textrm{in
the RRW},$ (42a) $\displaystyle\cdot\quad\\!\\!\frac{\partial{\cal L}_{\rm
U(1)}}{\partial(\partial_{1}A^{k})}=-F^{1}{}_{k}=g^{11}F_{k1}\equiv\pi_{k}\quad\textrm{in
the FRW},$ (42b)

where $k=1,\perp$ in RRW and $k=0,\perp$ in the FRW (in what follows, the
indices written in lower case Latin alphabets refer to this). When $\pi_{k}$
are defined as above, $\pi^{k}$ are given as follows:

$\displaystyle\cdot\quad\\!\\!\pi^{k}=F^{k0}=\frac{\partial{\cal L}_{\rm
U(1)}}{\partial(\partial_{0}A_{k})}\quad\textrm{in the RRW},$ (43a)
$\displaystyle\cdot\quad\\!\\!\pi^{k}=F^{k1}=\frac{\partial{\cal L}_{\rm
U(1)}}{\partial(\partial_{1}A_{k})}\quad\textrm{in the FRW},$ (43b)

$\pi_{k}$ and $\pi^{k}$ have physical meaning as the electric field:

$\displaystyle\cdot\quad\\!\\!\pi_{k}=E_{k},\quad\pi^{k}=E^{k}\quad\textrm{in
the RRW,}$ (44a)
$\displaystyle\cdot\quad\\!\\!\pi_{k}=E_{k},\quad\pi^{k}=E^{k}\quad\textrm{in
the FRW.}$ (44b)

As the electric fields have been defined, let us define the magnetic field in
each region:

$\displaystyle\cdot\quad\\!\\!B^{k}$
$\displaystyle\equiv-\varepsilon^{ijk}F_{ij}/2,\quad
B_{k}=\varepsilon_{ijk}F^{ij}/2\quad\textrm{in the RRW,}$ (45a)
$\displaystyle\cdot\quad\\!\\!B^{k}$
$\displaystyle\equiv-\varepsilon^{ijk}F_{ij}/2,\quad
B_{k}=\left\\{\begin{array}[]{ll}\displaystyle-\varepsilon_{ij0}F^{ij}/2&\textrm{for
$k=0$}\\\\[4.2679pt] \displaystyle+\varepsilon_{ij\perp}F^{ij}/2&\textrm{for
$k=\perp$}\end{array}\right.\quad\textrm{in the FRW,}$ (45d)

where $k$ in $B_{k}$ is the same as that in $B^{k}$ in (45a) and (45d),
respectively. Since $i,j,k=1,\perp$ and $=0,\perp$ in the RRW and FRW
respectively, it can be seen that $\varepsilon^{ijk}=-\varepsilon_{ijk}$ in
the RRW, and $\varepsilon^{ijk}=\varepsilon_{ijk}$ in the FRW.

Regarding (42)-(45), the Lagrangian (38) can be written as

$\displaystyle\cdot\quad\\!\\!{\cal L}_{\rm U(1)}=$
$\displaystyle-E^{k}(\nabla_{k}A_{0}-\nabla_{0}A_{k})-(B_{k}^{2}-g_{00}E_{k}^{2})/2\quad\\!\\!\textrm{for
the RRW},$ (46a) $\displaystyle\cdot\quad\\!\\!{\cal L}_{\rm U(1)}=$
$\displaystyle-E^{k}(\nabla_{k}A_{1}-\nabla_{1}A_{k})-(B_{k}^{2}-g_{11}E_{k}^{2})/2\quad\\!\\!\textrm{for
the FRW},$ (46b)

where $B_{k}^{2}=B_{k}B^{k}$ ($E_{k}^{2}$, $E_{0}^{2}$ and $E_{\perp}^{2}$ are
likewise), and $g_{00}$ and $g_{11}$ refer to those in (15).

The first term in the r.h.s. of (46a) can be rewritten as

$\displaystyle\cdot\quad\int_{\rm
RRW}\\!d^{4}x\sqrt{-g}\,E^{k}\nabla_{k}A_{0}$ $\displaystyle=-\int_{\rm
RRW}\\!d^{4}x\sqrt{-g}\,\partial_{k}E^{k}A_{0},$ (47a)
$\displaystyle\cdot\quad\int_{\rm
FRW}\\!d^{4}x\sqrt{-g}\,E^{k}\nabla_{k}A_{1}$ $\displaystyle=-\int_{\rm
FRW}\\!d^{4}x\sqrt{-g}\,\partial_{k}E^{k}A_{1},$ (47b)

where in the calculation (47a), expressing $E^{k}$ as $F^{k0}$, we have used
$\nabla_{k}A^{0}=(\sqrt{-g})^{-1}\partial_{k}(\sqrt{-g}A^{0})$ and assumed the
boundary condition that the fields vanish at the infinite far region which is
given by $\xi$ or $\eta$ to $\infty$. By that boundary condition, $\int_{\rm
RRW}d^{4}x\,\partial_{k}(F^{k}{}_{0}\sqrt{-g}A^{0})$ vanishes (note that as
can be seen from (7), the coordinates of the RRW and FRW reduce to the
Minkowski coordinates by $a\to 0$, but do not by $\xi$ or $\eta$ to $\infty$).
The calculation of (47b) has been performed likewise. We perform these
rewritings of (47) to change the coefficients of $A_{0}$ and $A_{1}$ to
numbers for the convenience in the calculation (60).

With (47a) and (47b),

$\displaystyle\cdot\quad\\!\\!\textrm{(\ref{bv64r1})}=E^{k}\nabla_{0}A_{k}+\partial_{k}E^{k}A_{0}-(B_{k}^{2}-g_{00}E_{k}^{2})/2,$
(48a)
$\displaystyle\cdot\quad\\!\\!\textrm{(\ref{bv64r2})}=E^{k}\nabla_{1}A_{k}+\partial_{k}E^{k}A_{1}-(B_{k}^{2}-g_{11}E_{k}^{2})/2.$
(48b)

With (48), the Hamiltonian densities (referred to as the Hamiltonian) can be
obtained as follows:

$\displaystyle\cdot\quad\\!\\!{\cal H}=\nabla_{0}A_{\mu}\,\pi^{\mu}-{\cal
L}_{\rm
U(1)}=\nabla_{0}A^{0}\,\pi_{0}-\partial_{k}E^{k}A_{0}+(B_{k}^{2}-g_{00}E_{k}^{2})/2\quad\\!\\!\textrm{for
the RRW,}$ (49a) $\displaystyle\cdot\quad\\!\\!{\cal
H}=\nabla_{1}A_{\mu}\,\pi^{\mu}-{\cal L}_{\rm
U(1)}=\nabla_{1}A^{1}\,\pi_{1}-\partial_{k}E^{k}A_{1}+(B_{k}^{2}-g_{11}E_{k}^{2})/2\quad\\!\\!\textrm{for
the FRW,}$ (49b)

where $\pi_{0}$ in the RRW and $\pi_{1}$ in the FRW are vanished by the
constraint conditions in Sec.3.2.

###  3.2 The constitution of path-integral in RRW and FRW

In the system with the Hamiltonians (49), there are two constraint conditions
in each region, which we denote as $\phi^{(i)}$ $(i=1,2)$:

$\displaystyle\cdot\quad\\!\\!\phi^{(1)}$
$\displaystyle\equiv\pi_{0}=0,\quad\phi^{(2)}\equiv\partial_{k}E^{k}=0\quad\textrm{for
the RRW,}$ (50a) $\displaystyle\cdot\quad\\!\\!\phi^{(1)}$
$\displaystyle\equiv\pi_{1}=0,\quad\phi^{(2)}\equiv\partial_{k}E^{k}=0\quad\textrm{for
the sFRW,}$ (50b)

where $\pi_{0}$ and $\pi_{1}$ above are defined in (42), which are always
vanished as can be seen from (42); the value of $\phi^{(2)}$, as given above,
can be seen from the condition that the time-development of $\phi^{(1)}$
generated by the Hamiltonians (49) should be vanished‡‡‡ The Hamiltonians
(49a) and (49b) can be written using $\phi^{(2)}$ as
$\displaystyle\cdot\quad\\!\\!\textrm{(\ref{bvery1})}=$
$\displaystyle\,\nabla_{0}A^{0}\,\pi_{0}-\phi^{(2)}A_{0}+(B_{k}^{2}-g_{00}E_{k}^{2})/2,$
(51a) $\displaystyle\cdot\quad\\!\\!\textrm{(\ref{bvery2})}=$
$\displaystyle\,\nabla_{1}A^{1}\,\pi_{1}-\phi^{(2)}A_{1}+(B_{k}^{2}-g_{00}E_{k}^{2})/2.$
(51b) The terms $\phi^{(2)}A_{0}$ and $\phi^{(2)}A_{1}$ contribute to the
time-developments of $\phi^{(1)}$ generated by the Hamiltonians (49a) and
(49b) in the RRW and FRW, respectively. .

For (50), we take the Coulomb gauge, and denote $\chi^{(i)}$ $(i=1,2)$, as
follows:

$\displaystyle\cdot\quad\\!\\!\chi^{(1)}$ $\displaystyle\equiv
A^{0}=0,\quad\chi^{(2)}\equiv\nabla_{k}A^{k}=0\quad\textrm{for the RRW,}$
(52a) $\displaystyle\cdot\quad\\!\\!\chi^{(1)}$ $\displaystyle\equiv
A^{1}=0,\quad\chi^{(2)}\equiv\nabla_{k}A^{k}=0\quad\textrm{for the FRW,}$
(52b)

where the gauge considered as the Coulomb gauge in the Minkowski coordinates
is $\partial_{\mu}A^{\mu}$, and $\chi^{(2)}$ in the ones above are the
extension of this to the Rindler coordinates; therefore, $\nabla_{\mu}$ in
$\chi^{(2)}$ are the gravitational ones, which do not include the gauge field.

Denoting (50) and (52) together as
$\tilde{\phi}\equiv\\{\phi^{(1)},\phi^{(2)},\chi^{(1)},\chi^{(2)}\\}\equiv\\{\tilde{\phi}^{(i)}\\}$
($i=1,2,3,4$), the Poisson bracket for $\tilde{\phi}$ can be derived for each
$x^{0}$ in the RRW or $x^{1}$ in the FRW as follows:

$\left[\begin{array}[]{c}\\{\tilde{\phi}^{(i)}(x),\tilde{\phi}^{(j)}(y)\\}_{\rm
P.B.}\end{array}\right]=\left[\begin{array}[]{cccc}0&0&-1&0\\\
0&0&0&-\partial^{k}\nabla_{k}\\\ 1&0&0&0\\\
0&\nabla_{k}\partial^{k}&0&0\end{array}\right]\delta^{3}(\vec{x}-\vec{y}),$
(53)

where the Poisson bracket and $\delta^{3}(\vec{x}-\vec{y})$ are given as
follows:

$\displaystyle\\{X(t,\vec{x}),Y(t,\vec{y})\\}_{{\rm P.B.}}$
$\displaystyle=\left(\frac{\partial X(t,\vec{x})}{\partial
A^{\mu}(t,\vec{x})}\frac{\partial
Y(t,\vec{y})}{\partial\pi_{\mu}(t,\vec{y})}-\frac{\partial
X(t,\vec{x})}{\partial\pi_{\mu}(t,\vec{y})}\frac{\partial
Y(t,\vec{x})}{\partial A^{\mu}(t,\vec{y})}\right)\delta^{3}(\vec{x}-\vec{y}),$
$\displaystyle\delta^{3}(\vec{x}-\vec{y})$
$\displaystyle=\left\\{\begin{array}[]{ll}\displaystyle\delta(x^{1}-y^{1})\,\delta^{2}(x^{\perp}-y^{\perp})&\textrm{for
RRW,}\\\\[4.2679pt]
\displaystyle\delta(x^{0}-y^{0})\,\delta^{2}(x^{\perp}-y^{\perp})&\textrm{for
FRW.}\end{array}\right.$ (54c)

$X$ and $Y$ represent fields, and $t$ means $x^{0}$ and $x^{1}$ for the RRW
and the FRW, respectively. Since ${\rm
Det}\big{[}\\{\tilde{\phi}(x),\tilde{\phi}(y)\\}_{\rm P.B.}\big{]}$ is non-
vanishing, $\tilde{\phi}$ forms a second-class constraint.

When the constraints in (50) and (52) are imposed in the phase space with
$(A^{\mu},\pi_{\mu})$ as its coordinates, the probability amplitude from the
infinite-past ground state to the infinite-future ground state in the RRW and
FRW can be written by the following path-integrals:

$\displaystyle\cdot\quad\\!\\!T_{\rm RRW}$ $\displaystyle\equiv\int\\!{\cal
D}\\!A\,{\cal
D}\pi\prod_{x\in{\textrm{RRW}}}\big{[}\delta[\phi^{(1)}]\,\delta[\phi^{(2)}]\,\delta[\chi^{(1)}]\,\delta[\chi^{(2)}]\big{]}\cdot\prod_{x^{0}}M_{c}$
$\displaystyle\times\exp\big{[}i\int_{\textrm{RRW}}\\!d^{4}x\,\sqrt{-g}\,{\cal
L}_{\rm path}\big{]}\quad\\!\\!\textrm{for the RRW,}$ (55a)
$\displaystyle\cdot\quad\\!\\!T_{\rm FRW}$ $\displaystyle\equiv\int\\!{\cal
D}\\!A\,{\cal
D}\pi\prod_{x\in{\textrm{FRW}}}\big{[}\delta[\phi^{(1)}]\,\delta[\phi^{(2)}]\,\delta[\chi^{(1)}]\,\delta[\chi^{(2)}]\big{]}\cdot\prod_{x^{1}}M_{c}$
$\displaystyle\times\exp\big{[}i\int_{\textrm{FRW}}\\!d^{4}x\,\sqrt{-g}\,{\cal
L}_{\rm path}\big{]}\quad\\!\\!\textrm{for the FRW,}$ (55b)

where

${\cal L}_{\rm
path}\equiv\left\\{\begin{array}[]{l}\nabla_{0}A^{\mu}\,\pi_{\mu}-{\cal
H}={\cal L}_{\rm U(1)}\quad\textrm{for the RRW,}\\\\[4.2679pt]
\nabla_{1}A^{\mu}\,\pi_{\mu}-{\cal H}={\cal L}_{\rm U(1)}\quad\textrm{for the
FRW,}\end{array}\right.$

${\cal H}$ is defined in (49a) and (49b), respectively, and ${\cal L}_{\rm
U(1)}$ is given in (48a) and (48b) respectively. $M_{c}$ and ${\cal
D}\\!A\,{\cal D}\pi$ are defined as

$\displaystyle M_{c}$ $\displaystyle\equiv{\rm
Det}\big{[}\\{\phi^{(i)}(x),\chi^{(j)}(y)\\}_{\rm P.B.}\big{]}={\rm
Det}\big{[}\left[\begin{array}[]{cccc}-1&0\\\
0&-\partial^{k}\nabla_{k}\end{array}\right]\delta^{3}(\vec{x}-\vec{y})\big{]},\quad$
(56c) $\displaystyle{\cal D}\\!A\,{\cal D}\pi$
$\displaystyle=\displaystyle{\cal C}\,\prod_{\mu=0}^{3}\,\prod_{x\in{\rm
RRW/FRW}}dA^{\mu}(x)\,d\pi_{\mu}(x),$

where ${\cal C}$ means some constant and $x$ refers to the coordinates in the
RRW or FRW. ${\rm Det}$ is the functional determinant of the functions on the
three-dimensional space at constant time in the RRW or FRW.

###  3.3 Analysis of path-integrals to obtain the Lagrangian in the Lorentz
covariant gauge in the RRW and FRW

Let us integrate out $\pi_{0}$ in the RRW and $\pi_{1}$ in the FRW in (55).
This can be performed readily as $\phi^{(1)}=\pi_{0}$ and $\pi_{1}$ in the RRW
and FRW, respectively, as in (50), and as a result, (55) can be written as

$\displaystyle\cdot\quad\\!\\!T_{\textrm{RRW}}=$ $\displaystyle\int\\!{\cal
D}\\!A\,{\cal
D}\pi_{k}\prod_{x\,\in\,\textrm{RRW}}\delta[\phi^{(2)}]\,\delta[\chi^{(1)}]\,\delta[\chi^{(2)}]\cdot\prod_{x^{0}}M_{c}$
$\displaystyle\times\,\exp\big{[}i\int_{\textrm{RRW}}\\!d^{4}x\,\sqrt{-g}\,{\cal
L}_{\rm U(1)}|_{\pi_{0}=0}\big{]},$ (57a)
$\displaystyle\cdot\quad\\!\\!T_{\textrm{FRW}}=$ $\displaystyle\int\\!{\cal
D}\\!A\,{\cal
D}\pi_{k}\prod_{x\,\in\,\textrm{FRW}}\delta[\phi^{(2)}]\,\delta[\chi^{(1)}]\,\delta[\chi^{(2)}]\cdot\prod_{x^{1}}M_{c}$
$\displaystyle\times\,\exp\big{[}i\int_{\textrm{FRW}}\\!d^{4}x\,\sqrt{-g}\,{\cal
L}_{\rm U(1)}|_{\pi_{1}=0}\big{]}$ (57b)

Since ${\cal L}_{\rm U(1)}$ in (57) and (57) do not include $\pi_{0}$ and
$\pi_{1}$, respectively, as can be seen in (48), we may write ${\cal L}_{\rm
U(1)}|_{\pi_{0}=0}$ and ${\cal L}_{\rm U(1)}|_{\pi_{1}=0}$ as ${\cal L}_{\rm
U(1)}$. Therefore, in what follows, we write those as ${\cal L}_{\rm U(1)}$.

Now, introducing a functional variable $\eta=\eta(x)$, we give
$\delta[\phi^{(2)}]$ by a functional integral:

$\displaystyle\delta[\phi^{(2)}]=\int{\cal
D}\eta\,\exp\big{[}i\int_{\textrm{RRW$/$FRW}}\\!d^{4}x\,\sqrt{-g}\,\eta\,\phi^{(2)}\big{]}.$
(58)

Then, (57) and (57) can be written as

$\displaystyle\cdot\quad\\!\\!\textrm{(\ref{gsrtx1})}=$
$\displaystyle\int\\!{\cal D}\\!A\,{\cal D}\pi_{k}\int{\cal
D}\eta\,\prod_{x\in\textrm{RRW}}\delta[\chi^{(1)}]\,\delta[\chi^{(2)}]\cdot\prod_{x^{0}}M_{c}$
$\displaystyle\times\,\exp\big{[}i\int_{\textrm{RRW}}\\!d^{4}x\,\sqrt{-g}\,\big{\\{}{\cal
L}_{\rm U(1)}+\eta\,\phi^{(2)}\big{\\}}\big{]},$ (59a)
$\displaystyle\cdot\quad\\!\\!\textrm{(\ref{gsrtx2})}=$
$\displaystyle\int\\!{\cal D}\\!A\,{\cal D}\pi_{k}\int{\cal
D}\eta\,\prod_{x\in\textrm{FRW}}\delta[\chi^{(1)}]\,\delta[\chi^{(2)}]\cdot\prod_{x^{1}}M_{c}$
$\displaystyle\times\,\exp\big{[}i\int_{\textrm{FRW}}\\!d^{4}x\,\sqrt{-g}\,\big{\\{}{\cal
L}_{\rm U(1)}+\eta\,\phi^{(2)}\big{\\}}\big{]},$ (59b)

where

$\displaystyle\cdot\quad\\!\\!{\cal L}_{\rm U(1)}+\eta\,\phi^{(2)}=$
$\displaystyle\cdots+(A_{0}+\eta)\phi^{(2)}+\cdots\quad\textrm{for the RRW,}$
(60a) $\displaystyle\cdot\quad\\!\\!{\cal L}_{\rm U(1)}+\eta\,\phi^{(2)}=$
$\displaystyle\cdots+(A_{1}+\eta)\phi^{(2)}+\cdots\quad\textrm{for the FRW.}$
(60b)

In the equations above, we have written only the part which is important in
the following analysis. Now we can integral out $A^{0}$ for the RRW in (59).
Then, since $\chi^{(1)}=A^{0}$ as in (52a), the term concerning $A_{0}$ in
(60a) disappears. However, we can take $\eta$ as $A^{0}$, by which the term
which has just disappeared is now reintroduced. In the same way, in the FRW in
(59), we integrate out $A^{1}$ and take $\eta$ as $A^{1}$. As a result, (59)
and (59) can be given as follows:

$\displaystyle\cdot\quad\\!\\!\textrm{(\ref{gastrl1})}=$
$\displaystyle\int\\!{\cal D}\\!A\,{\cal
D}\pi_{k}\\!\prod_{x\in\textrm{RRW}}\delta[\chi^{(2)}]\cdot\prod_{x^{0}}M_{c}\cdot\exp\big{[}i\int_{\textrm{RRW}}\\!d^{4}x\,\sqrt{-g}\,{\cal
L}_{\rm U(1)}\big{]},$ (61a)
$\displaystyle\cdot\quad\\!\\!\textrm{(\ref{gastrl2})}=$
$\displaystyle\int\\!{\cal D}\\!A\,{\cal
D}\pi_{k}\\!\prod_{x\in\textrm{FRW}}\delta[\chi^{(2)}]\cdot\prod_{x^{1}}M_{c}\cdot\exp\big{[}i\int_{\textrm{FRW}}\\!d^{4}x\,\sqrt{-g}\,{\cal
L}_{\rm U(1)}\big{]}.$ (61b)

Considering (48a) and (48b) as the expression of ${\cal L}_{\rm U(1)}$ in
(61a) and (61b), respectively, ${\cal L}_{\rm U(1)}$ includes the terms
$\partial_{k}E^{k}A_{0}$ and $\partial_{k}\pi^{k}A_{1}$, respectively. We can
restate those terms back by rewriting (47a) and (47b) in reverse. As a result,
the component of ${\cal L}_{\rm U(1)}$ in (61a) and (61b) are provided,
respectively, as

$\displaystyle\cdot\quad\\!\\!\textrm{${\cal L}_{\rm U(1)}$ in
(\ref{vtiof1})}=$
$\displaystyle\,F_{0k}\,\pi^{k}-(B_{k}^{2}-g_{00}\,\pi_{k}^{2})/2$
$\displaystyle=$
$\displaystyle\,-\frac{g_{00}}{2}\sum_{k=1,\perp}((\pi_{k}+g^{00}F_{0k})(\pi_{k}+g^{00}F_{0k})-(g_{00})^{2}F_{0k}F_{0k})-B_{k}^{2}/2$
$\displaystyle=$
$\displaystyle\,-\frac{g_{00}}{2}\sum_{k=1,\perp}(\pi_{k}+g^{00}F_{0k})(\pi_{k}+g^{00}F_{0k})+{\cal
L}_{\rm U(1)},$ (62a) $\displaystyle\cdot\quad\\!\\!\textrm{${\cal L}_{\rm
U(1)}$ in (\ref{vtiof2})}=$
$\displaystyle\,F_{1k}\,\pi^{k}-(B_{k}^{2}-g_{11}\,\pi_{k}^{2})/2$
$\displaystyle=$ $\displaystyle\,\frac{g_{11}}{2}\Big{(}\hskip
11.38109pt(\pi_{0}+g^{11}F_{1k})(\pi_{0}+g^{11}F_{1k})-(g^{11})^{2}F_{10}F_{10}$
$\displaystyle\hskip
22.76219pt-(\pi_{\perp}+g^{11}F_{1k})(\pi_{\perp}+g^{11}F_{1k})+(g^{11})^{2}F_{1\perp}F_{1\perp}\Big{)}-B_{k}^{2}/2$
$\displaystyle=$
$\displaystyle\,\frac{g_{11}}{2}((\pi_{k}+g^{11}F_{10})(\pi_{k}+g^{11}F_{10})-(\pi_{\perp}+g^{11}F_{1\perp})(\pi_{\perp}+g^{11}F_{1\perp}))+{\cal
L}_{\rm U(1)}.$ (62b)

where $B_{k}^{2}=B_{k}B^{k}$ and $\pi_{k}^{2}=\pi_{k}\pi^{k}$ as mentioned in
(46). If $A^{\mu}$ and $\pi_{\mu}$ were not independent of each other and were
in the relations (42a) and (42b) respectively, the r.h.s. of (62) and (62)
reduce to $0+{\cal L}_{\rm U(1)}$.

Then, $\pi_{k}$ in (61a) and (61b) can be integrated out (by redefining
$\pi_{k}$). As a result,

$\displaystyle\textrm{(\ref{vtiof1})}=$ $\displaystyle\int\\!{\cal
D}\\!A\prod_{x\in\textrm{RRW}}\delta[\chi^{(2)}]\cdot\prod_{x^{0}}M_{c}\cdot\exp\big{[}i\int_{\textrm{RRW}}d^{4}x\,\sqrt{-g}\,{\cal
L}_{\rm U(1)}\big{]},$ (63a) $\displaystyle\textrm{(\ref{vtiof2})}=$
$\displaystyle\int\\!{\cal
D}\\!A\prod_{x\in\textrm{FRW}}\delta[\chi^{(2)}]\cdot\prod_{x^{1}}M_{c}\cdot\exp\big{[}i\int_{\textrm{FRW}}d^{4}x\,\sqrt{-g}\,{\cal
L}_{\rm U(1)}\big{]}.$ (63b)

Once we have obtained the form $\int DA\,\Omega_{\textrm{Coulomb}}\,e^{i\int
d^{4}x{\cal L}_{\rm U(1)}}$ such as the ones above
($\Omega_{\textrm{Coulomb}}$ denotes the parts
$\prod[\delta(\chi^{(2)})]\cdot\prod M_{c}$ in the ones above), by proceeding
in the same way as the case of the Minkowski coordinates, we can replace the
part of $\Omega_{\textrm{Coulomb}}$ with the one by the Lorentz-covariant
gauge, and obtain the Lagrangian in Lorentz covariant gauge in the RRW and
FRW, which is (40). Since the analysis in the Minkowski coordinates is well-
known, we proceed with the analysis from (63) to (40) in
Appendix.LABEL:buobhs.

###  3.4 The Lorentz covariant U(1) gauge field Lagrangians in the LRW and PRW

Since the metrices of the FRW (RRW) and PRW (LRW) are mathematically the same,
as seen in (15), if the path-integrals in the FRW and RRW can be given as in
(63), the path-integrals which correspond to those in the PRW and LRW can be
given as

$\displaystyle T_{\rm LRW}=$ $\displaystyle\int\\!{\cal
D}\\!A\prod_{x\in\textrm{LRW}}\big{[}\delta(\chi^{(2)})\big{]}\cdot\prod_{x^{0}}M_{c}\cdot\exp\big{[}i\int_{\textrm{LRW}}d^{4}x\,\sqrt{-g}\,{\cal
L}_{\rm U(1)}\big{]},$ (64a) $\displaystyle T_{\rm PRW}=$
$\displaystyle\int\\!{\cal
D}\\!A\prod_{x\in\textrm{PRW}}\big{[}\delta(\chi^{(2)})\big{]}\cdot\prod_{x^{1}}M_{c}\cdot\exp\big{[}i\int_{\textrm{PRW}}d^{4}x\,\sqrt{-g}\,{\cal
L}_{\rm U(1)}\big{]},$ (64b)

where the coordinates in (64a) and (64b) refer to the LRW and PRW given by
(15); therefore, $x^{0}$ and $x^{1}$ mean $\tilde{\tau}$ and $\tilde{\zeta}$
in (15), respectively. Then, performing the analysis in the same way as (63a)
and (63b), the Lagrangians of the U(1) gauge field in the Lorentz covariant
gauge in the LRW and PRW can be obtained as (40).

##  4 The solutions of the U(1) gauge field and the $B$-field in each region
in the Rindler coordinates

In the previous section, the gauge-fixed Lagrangian was obtained as given in
(39). In this section, by solving the equations of motion obtained from that
Lagrangian, the mode-solutions of the U(1) gauge field and the $B$-field in
the RRW and FRW can be obtained. The results are noted in (LABEL:ba54nwy) and
(LABEL:b543es), and the forms of fields as the solution are noted in
(LABEL:w4aeea) and (LABEL:weh28a). The method for solving the equations of
motion is noted under (LABEL:weh28a). The forms of the U(1) gauge field and
the $B$-field as the solutions in the LRW and PRW and their mode-solutions are
given in Sec.LABEL:nnklf, while the normalization constants of the mode-
solutions in the four regions are given in Sec.LABEL:brpnevt.

###  4.1 The solution of the U(1) gauge field and the $B$-field in the RRW and
FRW

From the Lagrangian (39), the equations of motion can be obtained as follows:

$\displaystyle 0$ $\displaystyle=-\nabla_{\mu}F^{\mu\nu}+\partial^{\nu}B,$
(65a) $\displaystyle 0$ $\displaystyle=\nabla_{\mu}A^{\mu}+B.$ (65b)

From these, the equations of motion that the fields should satisfy in the RRW
are derived as follows:

The equations of motion in the RRW:
$\displaystyle(g^{\mu\nu}\partial_{\mu}\partial_{\nu}-3\rho^{-1}\partial_{1})A^{0}$
$\displaystyle=-2a^{-2}\rho^{-3}\partial_{0}A^{1},$ (66a)
$\displaystyle(g^{\mu\nu}\partial_{\mu}\partial_{\nu}-3\rho^{-1}\partial_{1}-\rho^{-2})A^{1}$
$\displaystyle=-2\rho^{-1}(B-\partial_{\perp}A^{\perp}),$ (66b)
$\displaystyle(g^{\mu\nu}\partial_{\mu}\partial_{\nu}-\rho^{-1}\partial_{1})A^{\perp}$
$\displaystyle=0,$ (66c)
$\displaystyle\partial_{0}A^{0}+\partial_{1}A^{1}+\partial_{\perp}A^{\perp}+\rho^{-1}A^{1}+B$
$\displaystyle=0,$ (66d)
$\displaystyle(g^{\mu\nu}\partial_{\mu}\partial_{\nu}-\rho^{-1}\partial_{1})B$
$\displaystyle=0,$ (66e) where, combining (65a) and (65b),
$\nabla_{\nu}\nabla^{\nu}A^{\mu}=0$ can be obtained, from which (66a)-(66c)
can be obtained (in obtaining (66b), (66d) is used). From (65b), (66d) can be
obtained. Multiplying the entire (65a) by $\nabla_{\nu}$,
$\nabla_{\mu}\nabla^{\mu}B=0$ can be obtained, from which (66e) can be
obtained. Non-zero $\Gamma^{\mu}_{\nu\lambda}$ are noted in (41). The $0$\-
and $1$-directions correspond to $\tau$\- and $\rho$-directions, and $\perp$
correspond to $2,3$-directions. In the same way, the equations of motion for
the fields in the FRW can be derived from (65) as follows: