Fulling-Davies-Unruh effect and spontaneous excitation of an accelerated atom interacting with a quantum scalar field

0.15 MB·English

by  Zhiying Zhu

Fulling-Davies-Unruh effect and spontaneous excitation of an
accelerated atom interacting with a quantum scalar field
Zhiying Zhu2 and Hongwei Yu1,2,∗
1CCAST(World Lab.), P. O. Box 8730, Beijing, 100080, P. R. China
2Department of Physics and Institute of Physics,
Hunan Normal University, Changsha, Hunan 410081, China †
7 Abstract
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We investigate, from the point of view of a coaccelerated frame, the spontaneous excitation
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n of a uniformly accelerated two-level atom interacting with a scalar field in a thermal state at
a
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a finite temperature T and show that the same spontaneous excitation rate for the uniformly

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accelerated atom in the Minkowski vacuum obtained in the inertial frame can only be recovered
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v
in the coaccelerated frame assuming a thermal bath at the Fulling-Davies-Unruh temperature
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T = a/2πforwhatappearstobetheMinkowskivacuumtotheinertialobserver. Ourdiscussion
0 FDU
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0 providesanotherexampleofaphysicalprocessdifferentfromthoseexaminedbeforeintheliterature
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0 to better understand the Fulling-Davies-Unruh effect.
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PACS numbers: 04.62.+v, 03.70.+k, 42.50.Lc,
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∗ Corresponding author
† Mailing address
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I. INTRODUCTION
It is well known that the particle content of a quantum field theory is observer dependent.
According to the works of Fulling, Davies and Unruh (FDU), for a uniformly accelerated
observer, the Minkowski vacuum is seen to be equivalent to a thermal bath of Rindler
particles at a temperature T = a/2π (FDU effect [1]). The existence of the FDU effect
FDU
has been shown to be mandatory for the consistency of quantum field theory in various
cases, where it has been demonstrated that the way to obtain agreement between physical
observables calculated fromtheinertial andcoaccelerated framepoint ofview isby assuming
the FDU effect [2, 3, 4, 5, 6]. For the weak decay of a uniformly accelerated proton, the
same lifetime can be obtained in the inertial frame and the coaccelerated frame considering
the FDU effect [3]. For the bremsstrahlung effect associated with a uniformly accelerated
point charge, Higuchi, Matsas and Sudarsky [4] have shown that the same response rate of
the uniformly accelerated point charge to the Larmor radiation can be obtained in both the
inertial and coaccelerated frame assuming the FDU thermal bath with an infinite number
of zero-energy Rindler photons in the Rindler wedge, and the same has been shown to
be true in the presence of boundaries [5]. Meanwhile the generalization to the quantum
bremsstrahlung effect for a uniformly accelerated point scalar source has also been made[6].
As another example, besides the weak decay of the proton and the bremsstrahlung effect,
demonstrating the necessity of the FDU effect for the consistency of quantum field theory,
we will compute, from the point of view of a coaccelerated frame, the spontaneous excitation
rate of a uniformly accelerated two-level atom interacting with a massless scalar field, and
show that equality can only be established between the excitation rates obtained in the
inertial and the coaccelerated frames by assuming the FDU effect.
Let us note that the spontaneous excitation of a uniformly accelerated atom interacting
with a massless scalar field [7, 8] and with electromagnetic fields [9] in a Minkowski vacuum
has already been studied in the inertial frame [7] using the formalism proposed by Dalibard,
Dupont-RocandCohen-Tannoudji[10,11],which demandsasymmetricoperatororderingof
atom and field variables and allows one to separately calculate the contributions of vacuum
fluctuations and radiation reaction to the spontaneous excitation rate of the accelerated
atom. It is found that for a uniformly accelerated atom transition from ground state to
excited states becomes possible even in vacuum. This phenomenon provides a physically
appealing interpretation of the FDU effect, since it gives a transparent illustration for why
anaccelerateddetectorclicks(SeeRef.[12]foradiscussion inadifferent contextandRef.[13]
for a non-perturbative approach to study the interaction of a uniformly accelerated detector,
modelled by a harmonic oscillator which may be regarded as a simple version of an atom,
with a quantum field in (3+1) dimensional spacetime).
The purpose of the present paper is to show that the rate of change of the atomic energy
for a uniformly accelerated atom interacting with a massless scalar field in vacuum obtained
in the inertial frame for both cases with and without boundaries [7, 8] can be recovered in
the coaccelerated frame by using Fulling’s quantization in conjunction with the fact that the
2
Minkowski vacuum is a thermal state of Rindler particles at a temperature T = a/2π.
FDU
So, we will consider a uniformly accelerated two-level atominteracting with a massless scalar
field in the coaccelerated frame both in space with and without the presence of boundary,
and evaluate the rate of change of the mean atomic energy of the atom assuming a thermal
bath of Rindler particles at a finite temperature T. Since the rate of change of the atomic
energy is a scalar, the same value of this observable must be obtained in the inertial and
coaccelerated frame. We will show that only when the temperature equals the FDU value,
i.e., T = a/2π, can agreement in both frames be obtained.
II. THE GENERAL FORMALISM FOR THERMAL FLUCTUATIONS AND RA-
DIATION REACTION
We consider a two-level atom in interaction with a massless scalar field, and assume the
atomhasaconstantproperaccelerationaalongthexdirectionintheMinkowskicoordinates.
In the Rindler wedge, the line element [14] can be described by
ds2 = e2aξ(dτ2 dξ2) dy2 dz2 . (1)
− − −
The Rindler coordinates are related to the usual Minkowski coordinates by
eaξ eaξ
t = sinhaτ , x = coshaτ . (2)
a a
In the Rindler coordinates our accelerated atom is stationary on the trajectory x =
(τ,ξ,y(τ),z(τ)). The stationary trajectory guarantees that the undisturbed atom has sta-
tionary states, and + , with energies 1ω and +1ω and a level spacing ω . The
|−i | i −2 0 2 0 0
Hamiltonian that governs the time evolution of the atom with respect to the proper time τ
is written in Dick’s [15] notation
H (τ) = ω R (τ) , (3)
A 0 3
where R = 1 + + 1 . The free Hamiltonian of the quantum scalar field with
3 2| ih | − 2|−ih−|
respect to τ is
H (τ) = dω dk dk ω b† (τ) b (τ) , (4)
F Z Z yZ z ω,ky,kz ω,ky,kz
whereb† andb denotethecreationandannihilationoperatorsforaRindlerparticle
ω,ky,kz ω,ky,kz
with transverse momentum k and k and frequency ω. The quantization of the field in the
y z
Rindler wedge can be carried out by the expansion in terms of annihilation and creation
operators as
φ(x) = dω dk dk [b (τ)υ (x)+H.c.] , (5)
Z Z yZ z ω,ky,kz ω,ky,kz
3
where υ are the Rindler modes. Let us note that the positive acceleration a of the
ω,ky,kz
atom makes only the Rindler wedge R+ (x > t ) completely accessible to a coaccelerated
| |
observer [16]. Thus in the following sections, we will only consider the Rindler modes in R+.
Following Ref. [7], the Hamiltonian that describes the interaction between the atom and the
quantum field can be written
H (τ) = µR (τ)φ(x) . (6)
I 2
Here µ is a coupling constant which we assume to be small, and R = 1i(R R ), where
2 2 − − +
R = + and R = + . The coupling is effective only on the trajectory of the atom.
+ −
| ih−| |−ih |
The Heisenberg equations of motion for the dynamical variables of the atom and the field
can be derived from the Hamiltonian H = H +H +H :
A F I
d
R (τ) = iω R (τ)+iµφ(x)[R (τ),R (τ)] , (7)
± 0 ± 2 ±
dτ ±
d
R (τ) = iµφ(x)[R (τ),R (τ)] , (8)
3 2 3
dτ
d
b (τ) = iωb (τ)+iµR (τ)[φ(x),b (τ)] . (9)
dτ ω,ky,kz − ω,ky,kz 2 ω,ky,kz
We can split the solutions of these equations of motion into the “free” and “source” parts.
Let us now assume that the system is in a thermal bath of arbitrary temperature T as
seen by the coaccelerated observer (Rindler observer), so that the scalar field is in a thermal
state (as opposed to a vacuum state in the inertia frame) and the density matrix is given by
ρ = e−βHF = e−HF/T, and the atom is in the state a . Our aim is to identify and separate
| i
the contributions of quantum thermal fluctuations (as opposed to vacuum fluctuations in
the inertial frame) and radiation reaction to the rate of change of the mean atomic energy.
For this purpose, we choose a symmetric ordering between the atom and field variables, and
separate the two contributions of thermal fluctuations and radiation reaction to the rate of
change of H ( cf. Ref. [7, 10, 11]),
A
dH (τ) dH (τ) dH (τ)
A A A
= + , (10)
dτ (cid:18) dτ (cid:19) (cid:18) dτ (cid:19)
TF RR
where
dH (τ) 1
A = iω µ(φf(x)[R (τ),R (τ)]+[R (τ),R (τ)]φf(x)) , (11)
0 2 3 2 3
(cid:18) dτ (cid:19) 2
TF
representing the contribution of thermal fluctuations and
dH (τ) 1
A = iω µ(φs(x)[R (τ),R (τ)]+[R (τ),R (τ)]φs(x)) , (12)
0 2 3 2 3
(cid:18) dτ (cid:19) 2
RR
representing that of radiation reaction.
4
We can separate R and R into their free part and source part, and take the expectation
2 3
value in the field’s and atom’s states. We obtain, up to order µ2,
dH (τ) R τ d
A = 2iµ2 dτ′CF(x,x′) χA(τ,τ′) , (13)
(cid:28) dτ (cid:29) Z β dτ
TF τ0
dH (τ) R τ d
A = 2iµ2 dτ′χF(x,x′) CA(τ,τ′) , (14)
(cid:28) dτ (cid:29) Z β dτ
RR τ0
where the superscript R denotes the calculation performed in the Rindler coordinates and
= a,β representing the atom in the state a and the field in the thermal state β . The
|i | i | i | i
statistical functions CF(x,x′) and χF(x,x′) of the field at a nonzero temperature T = 1/β
β β
are defined as
1 1
CF(x,x′) = φf(x),φf(x′) = tr(ρ φf(x),φf(x′) )/tr(ρ) , (15)
β 2h{ }iβ 2 { }
1 1
χF(x,x′) = [φf(x),φf(x′)] = tr(ρ[φf(x),φf(x′)])/tr(ρ) , (16)
β 2h iβ 2
and those of the atom as
1
CA(τ,τ′) = a Rf(τ),Rf(τ′) a , (17)
2h |{ 2 2 }| i
1
χA(τ,τ′) = a [Rf(τ),Rf(τ′)] a . (18)
2h | 2 2 | i
CF (CA)iscalledthesymmetriccorrelationfunctionofthescalarfieldatafinitetemperature
β
T = 1/β (atom), χF (χA) its linear susceptibility. The explicit forms of the statistical
β
functions of the atom are given by
1
CA(τ,τ′) = a Rf(0) b 2 eiωab(τ−τ′) +e−iωab(τ−τ′) , (19)
2 |h | 2 | i| (cid:18) (cid:19)
X
b
1
χA(τ,τ′) = a Rf(0) b 2 eiωab(τ−τ′) e−iωab(τ−τ′) , (20)
2 |h | 2 | i| (cid:18) − (cid:19)
X
b
where ω = ω ω and the sum extends over a complete set of atomic states. In order to
ab a b
−
get the statistical functions for the field, we consider firstly the two point function for the
field at a nonzero temperature T = 1/β [14]
φf(x)φf(x′) = tr[ρφf(x)φf(x′)]/tr(ρ) . (21)
β
h i
From Eq. (5), one finds
∞ ∞ ∞
φf(x)φf(x′) = dω dk dk (n+1)e−nω/Tυ (x)υ∗ (x′)
h iβ Z Z yZ z(cid:20) ω,ky,kz ω,ky,kz
0 −∞ −∞ Xn=0
+ n e−nω/Tυ∗ (x)υ (x′) e−nω/T . (22)
ω,ky,kz ω,ky,kz (cid:21)(cid:30)
Xn=1 Xn=0
5
Here we assume that there exist an infinite number of Rindler particles in the thermal bath.
We can obtain
∞ ∞ ∞ eω/T
φf(x)φf(x′) = dω dk dk υ (x)υ∗ (x′)
h iβ Z Z yZ z(cid:20)eω/T 1 ω,ky,kz ω,ky,kz
0 −∞ −∞ −
1
+ υ∗ (x)υ (x′) . (23)
eω/T 1 ω,ky,kz ω,ky,kz (cid:21)
−
The statistical functions of the field can be calculated using (23).
III. EXCITATION RATE IN THE COACCELERATED FRAME IN THE UN-
BOUNDED SPACE
In this section, we will apply the previously developed formalism to calculate, in the
coacceleratedframe,therateofchangeofthemeanatomicenergyforauniformlyaccelerated
atom interacting with a massless free scalar field at finite temperature T = 1/β. In the
Rindler coordinates, the atom is static and its trajectory can be described by
τ , ξ = y(τ) = z(τ) = 0 . (24)
The wave equation for the scalar field in the Rindler coordinates is given by
∂2 ∂2 ∂2 ∂2
e−2aξ φ(x) = 0 . (25)
(cid:20) (cid:18)∂τ2 − ∂ξ2(cid:19)− ∂y2 − ∂z2(cid:21)
It can be shown that the orthonormal mode solution with a positive frequency is
1 πω 1
υ (x) = sinh1/2 K k eaξ eikyy+ikzz−iωτ , (26)
ω,ky,kz 2π2√a (cid:18) a (cid:19) iω/a(cid:18)a ⊥ (cid:19)
where k = k2 +k2 and K (x) is the Bessel function of imaginary argument. Now we can
⊥ y z ν
evaluate theptwo point function (23) for the trajectory (24), and get
1 ∞ ∞ ∞ πω k
φf(x)φf(x′) = dω dk dk sinh K2 ⊥
h iβ 4π4a Z Z yZ z (cid:18) a (cid:19) iω/a(cid:18) a (cid:19)
0 −∞ −∞
eω/T 1
e−iω(τ−τ′) + eiω(τ−τ′) . (27)
×(cid:18)eω/T 1 eω/T 1 (cid:19)
− −
With the help of the following integral,
∞ ∞ k aπ2ω
dk dk K2 ⊥ = , (28)
Z yZ z iω/a(cid:18) a (cid:19) sinh(πω)
−∞ −∞ a
we obtain
1 ∞ eω/T 1
φf(x)φf(x′) = dω ω e−iω(τ−τ′) + eiω(τ−τ′) . (29)
h iβ 4π2 Z (cid:18)eω/T 1 eω/T 1 (cid:19)
0 − −
6
The statistical functions of the field, (15) and (16), can now be written
1 ∞ 2
CF(x,x′) = dω ω 1+ e−iω(τ−τ′) +eiω(τ−τ′) , (30)
β 8π2 Z (cid:18) eω/T 1(cid:19)(cid:18) (cid:19)
0 −
1 ∞
χF(x,x′) = dω ω e−iω(τ−τ′) eiω(τ−τ′) . (31)
β 8π2 Z (cid:18) − (cid:19)
0
Note that χF(x,x′) has no temperature dependence and agrees with the linear susceptibility
β
in vacuum. This is a result of the fact that only the field commutator appears in (16) and
the linear susceptibility thus does not depend on the state of the field. With a substitution
u = τ τ′, the contributions Eq. (13) and Eq. (14) to the rate of change of the atomic
−
energy can be evaluated to get
dH (τ) R µ2 ∞ 2
A = ω a Rf(0) b 2 dω ω 1+
(cid:28) dτ (cid:29) −8π2 ab|h | 2 | i| Z (cid:18) eω/T 1(cid:19)
TF Xb 0 −
∞
du(e−iωu +eiωu)(e−iωabu +eiωabu) (32)
×Z
0
dH (τ) R µ2 ∞
A = ω a Rf(0) b 2 dω ω
(cid:28) dτ (cid:29) −8π2 ab|h | 2 | i| Z
RR Xb 0
∞
du(e−iωu eiωu)(e−iωabu eiωabu) . (33)
×Z − −
0
Here we have extended the range of integration to infinity for sufficiently long times τ τ .
0
−
After the evaluation of the integrals we obtain
dH (τ) R µ2 1 1
A = ω2 a Rf(0) b 2 +
(cid:28) dτ (cid:29) −2π(cid:20) ab|h | 2 | i| (cid:18)2 eωab/T 1(cid:19)
TF ωXa>ωb −
1 1
ω2 a Rf(0) b 2 + (34)
− ab|h | 2 | i| (cid:18)2 e|ωab|/T 1(cid:19)(cid:21)
ωXa<ωb −
for the contribution of the thermal fluctuations to the rate of change of atomic excitation
energy, and
dH (τ) R µ2 1 1
A = ω2 a Rf(0) b 2+ ω2 a Rf(0) b 2 (35)
(cid:28) dτ (cid:29) −2π(cid:18) 2 ab|h | 2 | i| 2 ab|h | 2 | i| (cid:19)
RR ωXa>ωb ωXa<ωb
for that of radiation reaction. A comparison with the spontaneous excitation rate of an
accelerated atom calculated in the inertial frame [7] shows that the contribution of radiation
reaction to the rate of change of the atomic energy is the same as that calculated in the
inertial frame, and thus is observer independent and temperature independent. While the
contributions of vacuum fluctuations are equal only when the temperature equals the FDU
value, i.e., T = a/2π. Hence the same total rate of change of the atomic energy can be
obtained both in the inertial and coaccelerated frames, only when we assume a thermal
state of Rindler particles at a temperature T = a/2π to a coaccelerated observer for what
appears to be a vacuum state to an inertial observer. This is the FDU effect.
7
IV. A SPACE-TIME WITH A REFLECTING PLANE BOUNDARY
It is well known that the presence of boundaries modifies the quantum fluctuations of
fields both in a vacuum state and a thermal state and can lead to a lot of novel effects, such
as the Casimir effect [17], the light-cone fluctuations [18], the Brownian (random) motion of
test particles caused by quantum fluctuations in vacuum and quantum fluctuations at finite
temperature [19], and so on. Therefore, the investigation of spontaneous excitation rate
of accelerated atoms interacting with the scalar field in the presence of a reflecting plane
boundary is an interesting issue and has been carried out in the inertial frame [8]. The
results obtained there show that, with the presence of a reflecting boundary, a uniformly
accelerated atom in a vacuum does not have to behave as if it were static in a thermal bath
at a temperature T = a/2π, in the sense that the spontaneous excitation rate is changed,
by the presence of the boundary, in such a way that it is different from what one would
expect for an inertial atom in a thermal bath in the same space with the boundary. We
will, however, show that the same excitation rate can be obtained in the coaccelerated frame
assuming the FDU effect.
We assume that a perfectly reflecting boundary for the scalar field is located at z = 0 in
space, the atom is being uniformly accelerated in the x direction with a proper acceleration
a at a distance z from the boundary, and the whole system is in a thermal bath at a
temperature T = 1/β. The atom’s trajectory is now described in the Rindler coordinates
by
τ , ξ = y(τ) = 0 , z(τ) = z . (36)
To satisfy the Dirichlet boundary condition, φ(x) = 0, the normalized Rindler mode
z=0
|
function with a positive frequency for the scalar field becomes
1 πω 1
υ (x) = sinh1/2 K k eaξ sin(k z)eikyy−iωτ . (37)
ω,ky,kz π2√2a (cid:18) a (cid:19) iω/a(cid:18)a ⊥ (cid:19) z
From Eq. (23) we can calculate the two-point function of the field at a finite temperature
T = 1/β for the trajectory (36) to get
1 ∞ ∞ ∞ πω k
φf(x)φf(x′) = dω dk dk sinh K2 ⊥ [1 cos(2k z)]
h iβ 4π4a Z Z yZ z (cid:18) a (cid:19) iω/a(cid:18) a (cid:19) − z
0 −∞ −∞
eω/T 1
e−iω(τ−τ′) + eiω(τ−τ′) . (38)
×(cid:18)eω/T 1 eω/T 1 (cid:19)
− −
With the help of the following integral
∞ ∞ k aπ2 sin(2ωsinh−1(az))
dk dk K2 ⊥ [1 cos(2k z)] = ω a , (39)
Z yZ z iω/a(cid:18) a (cid:19) − z sinh(πω)(cid:20) − 2z√1+a2z2 (cid:21)
−∞ −∞ a
8
we find
1 ∞ sin(2ωsinh−1(az))
φf(x)φf(x′) = dω ω a
β
h i 4π2 Z (cid:20) − 2z√1+a2z2 (cid:21)
0
eω/T 1
e−iω(τ−τ′) + eiω(τ−τ′) .
×(cid:18)eω/T 1 eω/T 1 (cid:19)
− −
(40)
From the general form (15) and (16), we can obtain the corresponding statistical functions
of the field
1 ∞ sin(2ωsinh−1(az)) 2
CF(x,x′) = dω ω a 1+ e−iω(τ−τ′) +eiω(τ−τ′) ,
β 8π2 Z (cid:20) − 2z√1+a2z2 (cid:21)(cid:18) eω/T 1(cid:19)(cid:18) (cid:19)
0 −
(41)
1 ∞ sin(2ωsinh−1(az))
χF(x,x′) = dω ω a e−iω(τ−τ′) eiω(τ−τ′) . (42)
β 8π2 Z (cid:20) − 2z√1+a2z2 (cid:21)(cid:18) − (cid:19)
0
Again χF(x,x′) has no temperature dependence as expected, it however does rely on the
β
acceleration. With a substitution u = τ τ′ and extending the range of integration to
−
infinity for sufficiently long times τ τ , we find the contributions of thermal fluctuations
0
−
(13) and radiation reaction (14) to the rate of change of the atomic energy,
dH (τ) R µ2 1 1 1
A = ω2 a Rf(0) b 2f(ω ,a,z) +
(cid:28) dτ (cid:29) −2π(cid:20) 2 ab|h | 2 | i| ab (cid:18)2 eωab/T 1(cid:19)
TF ωXa>ωb −
1 1 1
ω2 a Rf(0) b 2f(ω ,a,z) + , (43)
− 2 ab|h | 2 | i| ab (cid:18)2 e|ωab|/T 1(cid:19)(cid:21)
ωXa<ωb −
and
dH (τ) R µ2 1
A = ω2 a Rf(0) b 2f(ω ,a,z)
(cid:28) dτ (cid:29) −2π(cid:20) 2 ab|h | 2 | i| ab
RR ωXa>ωb
1
+ ω2 a Rf(0) b 2f(ω ,a,z) , (44)
2 ab|h | 2 | i| ab (cid:21)
ωXa<ωb
where
1 2ω sinh−1(az)
ab
f(ω ,a,z) = 1 sin . (45)
ab
− 2ω z√1+a2z2 (cid:18) a (cid:19)
ab
Comparing the above results with Eqs. (34) and (35), one can see that the function
f(ω ,a,z) gives the modification induced by the presence of the boundary. Letting T 0,
ab
→
we obtain the contribution of vacuum fluctuations
dH (τ) R µ2 1
A = ω2 a Rf(0) b 2f(ω ,a,z)
(cid:28) dτ (cid:29) (cid:12) −2π(cid:20) 2 ab|h | 2 | i| ab
TF(cid:12)(cid:12)T→0 ωXa>ωb
(cid:12) 1
ω2 a Rf(0) b 2f(ω ,a,z) , (46)
− 2 ab|h | 2 | i| ab (cid:21)
ωXa<ωb
9
Adding up the contributions of vacuum fluctuations (46) and radiation reaction (44) to the
Rindler atom yields the total spontaneous excitation rate,
dH (τ) R µ2
A = ω2 a Rf(0) b 2f(ω ,a,z) . (47)
(cid:28) dτ (cid:29) (cid:12) −2π ab|h | 2 | i| ab
tot(cid:12)(cid:12)T→0 ωXa>ωb
(cid:12)
This indicates that if no FDU thermal bath were assumed to exist, the Rindler atom would
be stable in its ground state with no excitation. Yet this is different from the behavior of
an inertial atom (cf. Eq. (23) in Ref. [8]), because of the appearance of the acceleration a in
function f(ω ,a,z), revealing that without assuming the FDU effect what an coaccelerated
ab
observer would see will be different from an inertial observer. This is in contrast to the case
without the presence of the boundary.
Now, let us recall the spontaneous excitation rate of a uniformly accelerated atom in
the inertial frame, with the presence of a plane boundary [8]. The contributions of vacuum
fluctuations and radiation reaction are
dH (τ) Inertial µ2 1 1
A = ω2 a Rf(0) b 2f(ω ,a,z) +
(cid:28) dτ (cid:29) −2π(cid:20) ab|h | 2 | i| ab (cid:18)2 e2πωab/a 1(cid:19)
VF ωXa>ωb −
1 1
ω2 a Rf(0) b 2f(ω ,a,z) + , (48)
− ab|h | 2 | i| ab (cid:18)2 e2π|ωab|/a 1(cid:19)(cid:21)
ωXa<ωb −
and
dH (τ) Inertial µ2 1
A = ω2 a Rf(0) b 2f(ω ,a,z)
(cid:28) dτ (cid:29) −2π(cid:20) 2 ab|h | 2 | i| ab
RR ωXa>ωb
1
+ ω2 a Rf(0) b 2f(ω ,a,z) . (49)
2 ab|h | 2 | i| ab (cid:21)
ωXa<ωb
A comparison with Eq. (43) and Eq. (44) shows that, the same as in the unbounded case,
the contribution of radiation reaction is again observer independent, however, only when we
assumeathermalbathofRindlerparticlesatthetemperatureT = a/2πforthecoaccelerated
observer, is Eq. (43) equal to Eq. (48) and the two observers agree on the total rate of the
changeoftheatomicenergyfortheacceleratedatom. Letusnotethat,duetotheappearance
of f(ω ,a,z), the total spontaneous excitation rate of a uniformly accelerated atom in the
ab
presence of a reflecting boundary deviates from the pure thermal rate in the unbounded
case. However, this discrepancy does not imply that the exact final thermal equilibrium is
not achieved [20]. As a matter of fact, since function f(ω ,z,a) is an even function of ω ,
bd bd
one can show, by the same argument as that in Ref [20], that exact thermal equilibrium will
be established at the FDU temperature T = a/2π, even though the accelerated atom
FDU
radiates and absorbs differently from an inertial atom immersed in the thermal bath.
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