$κ-$Minkowski spacetime and a uniformly accelerating observer

0.23 MB·English

by  Hyeong-Chan Kim

κ Minkowski spacetime and a uniformly accelerating observer
−
Hyeong-Chan Kim∗ and Jae Hyung Yee†
Deptartment of Physics, Yonsei University, Seoul 120-749, Republic of Korea.
Chaiho Rim‡
Department of Physics and Research Institute of Physics and Chemistry,
Chonbuk National University, Chonju 561-756, Korea.
(Dated: February 2, 2008)
We analyze the response of a detector with a uniform acceleration α in κ−Minkowski spacetime
using the first order perturbation theory. The monopole detector is coupled to a massless complex
scalar field in such a way that it is sensitive to the Lorentz violation due to the noncommutativity
of spacetime present in the κ−deformation. The response function deviates from the thermal dis-
7
tribution of Unruh temperature at the order of 1/κ and vanishes exponentially as the proper time
0
of the detector exceeds a certain critical time, a logarithmic function of κ. This suggests that the
0
Unruhtemperature becomes not only fuzzy butalso eventually decreases to zero in this model.
2

n PACSnumbers: 11.15.Tk,05.70.Ln
a Keywords: noncommutative spacetime, κ−Minkowski spacetime, Lorentz Symmetry breaking, Unruh tem-
J perature

8

1
v
4 I. INTRODUCTION
5
0 In the last decade, there has been a great interest in attempting to explain the cosmic observational data as a
1
quantum gravitational effect. As a theoretical framework to study the quantum gravity effects phenomenologically,
0
several field theories on noncommutative spacetimes [1, 2, 3, 4, 5, 6, 7, 8] have been studied.
7
In this directionofresearch,one commonaspect is the introductionof deformedsymmetries andresults in Lorentz
0
/ symmetrybreaking. Thisisreformulatedinnoncommutativespacetimeswhereadimensionalparameterisintroduced,
h
related to the Planck mass. This dimensional parameter is expected to suppress the Lorentz symmetry breaking in
t
- the commutative limit. However, careful estimates suggest that there may exist a strong fine-tuning problem in
p
noncommutative spacetime approach at a one-loop level [9]. In addition, unitarity of the theory is in question in
e
h noncommutative spacetime field theories [10, 11]. This consideration, however, does not decrease the interest in
: noncommutative spacetime field theories. On the contrary, one needs to find a good candidate for a realistic model.
v
The κ-Minkowski spacetime introduces a dimensionful deformation parameter κ, whose natural choice is to put
i
X κ = M , the Planck mass [12, 13, 14]. The κ Minkowski spacetime respects rotational symmetry and appears
P
−
r to be a good candidate to study the quantum gravity effect. Scalar field theory has been studied by introducing
a the differential structure in κ Minkowski space [12, 15]. The κ deformation is extended to the curved space with
− −
κ Robertson-Walker metric and is applied to cosmic microwave background radiation in [16].
−
On the other hand, it is an interesting question how the quantum gravity effect changes the structures of vacua
andparticleincurvedspacetime. Unruh[17, 18]calculatedthe responseofa particledetectormovingwithauniform
acceleration, under the assumption that the state of field is initially in its vacuum state, i.e., the Minkowski vacuum
and the field interacts only with the detector. It is shown that the detector responds as if it would have remained
un-accelerated but immersed in a heat bath at temperature acceleration/2π, the Unruh temperature. This is called
the Unruh effect and has been studied further in [19]. The idea has been extended to include back-reaction problem
and generalized to realistic detectors [20, 21].
The Unruh effect is interesting since it gives an analogy of the Hawking radiation in blackhole spacetime due to
the thermal formof the transitionprobability in the presence of eventhorizon. In this point of view, the accelerating
frame (Rindler spacetime) is regarded as a simplest toy model simulating the radiation effect of a blackhole. Since
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
‡Electronicaddress: [email protected]
2
it is not easy to calculate the field theory in curved-noncommutative spacetime containing a blackhole directly, one
may instead, seek for the effect on an accelerating frame first.
In the present paper, we study the response function of a uniformly accelerating monopole particle detector which
interacts with massless complex scalar field in κ Minkowskispacetime. Since the primary effect of κ deformationis
− −
the Lorentzsymmetrybreaking,we designaparticle detectormodelsothatitis sensitiveto this symmetryviolation.
In Sec. II, we briefly summarize the complex scalar field theory in κ Minkowski spacetime using the differential
−
structure given in [12]. We derive the Feynman propagator from the action and propose Wightman functions. In
Sec. III,weconsiderauniformlyacceleratingdetectorinteractingwithmasslesscomplexscalarfieldandcalculatethe
transition amplitude of the detector. It is found that the Lorentz symmetry violation effect appears to the response
function at O(1/κ). We summarize the results in Sec. IV.
II. SCALAR FIELD THEORY IN κ-MINKOWSKI SPACETIME
In this section, we briefly introduce the κ deformation of the Minkowski spacetime and construct the scalar field
−
theory. Some details can be found in Ref. [12]. Here we use the signature of spacetime metric, (+, , , ).
− − −
A. κ−Minkowski spacetime
Thetimeandspacecoordinatesarenotcommutinginκ-Minkowskispacetimebutsatisfythecommutationrelations
i
[xˆ0,xˆi]= xˆi, [xˆi,xˆj]=0, i,j =1,2,3. (1)
κ
κ isa positiveparameterwhichrepresentsthe deformationofthe space-time. Its HopfalgebraH is describedby the
x
co-product
∆(xˆµ)=xˆµ 1+1 xˆµ.
⊗ ⊗
The κ-deformed Poincar´e algebra is constructed using commuting four-momenta [pµ,pν] = 0 and the dual Hopf
algebra H . The co-product of four momenta is given as
p
∆(p0)=p0 1+1 p0, ∆(pi)=pi e−p0/κ+1 pi. (2)
⊗ ⊗ ⊗ ⊗
Exponential operator e−ip·xˆ is the basic ingredient which transforms the theory in space-time coordinates to the
theory in momentum space. Here xˆ=(xˆ0,xˆ), p=(p0,p) and p xˆ p0xˆ0 p xˆ. Its ordering is defined as
· ≡ − ·
:e−ip·xˆ : e−ip0xˆ0eip·xˆ. (3)
≡
Multiplication of two ordered exponentials follows from Eqs. (1) and (3):
:e−ipxˆ ::e−iqxˆ :=:ei(pe−q0/κ+q)xˆ−i(p0+q0)xˆ0 :, (4)
which is indicated in the four momentum addition rule described by the coproduct (2).
To find the differential calculus one differentiates Eq. (1) and finds
i
[τ0,xˆi]+[xˆ0,τi]= τi, [τi,xˆj]+[τj,xˆi]=0,
κ
where τµ = dxˆµ is the exterior derivatives along the four dimensional space-time direction. One may choose the
commutation relations
i
[τ0,xˆi]= τi, [τi,xˆ0]=0, (5)
κ
and fix the unique differential structure through Jacobi’s identities:
1 i i 4
[τµ,xˆν]= ηµντ5 ηµντ0+ η0µτν, [τ5,xˆµ]= τµ. (6)
4 − κ κ −κ2
3
Thisdemonstratesthatthedifferentialcalculusisnotclosedin4-dimensionsbutneedsanewexteriorderivativealong
the fifth direction τ5 [22, 23]. One may identify τ5 as
3i
τ5 =[τµ,x ]+ τ0.
µ
κ
The covariance property under the action of κ deformed Poincar´e group was established in Ref. [12, 14] and a
−
five-dimensional bi-covariant differential calculus was considered in Ref. [12].
Using τA with A=0,1,2,3,5,one writes the derivative of the exponential function as
d:e−ipxˆ : = i:χ (p)e−ipxˆ :τA (7)
A
−
p p2
χ (p) = κ sinh 0 + ep0/κ
0 κ 2κ2
(cid:20) (cid:21)
χ (p) = p ep0/κ
i i
i
χ (p) = M2(p).
5 −8 κ
M2(p) is the first Casimir operator of the algebra (2),
κ
p 2
M2(p)= 2κsinh 0 p2ep0/κ, (8)
κ 2κ −
(cid:16) (cid:17)
whichisinvariantunderthedeformedPoincar´etransformations. Asκ ,thisreducestothecommutativerelation,
→∞
M2 (p)=p2 p2 =m2. For finite κ, the on-shell three-momentum is bounded from above by p2 κ2 [24] since
∞ 0− ≤
p2 M2
=1 2+ κ e−p0/κ+e−2p0/κ
κ2 − κ2
(cid:18) (cid:19)
andM2 isanon-negativeconstant. p0 goestoinfinitywhenthemomentumreachesthemaximalvalueκ. Inaddition,
κ
defining 5-dimensional momentum
=(χ ,χ~,κ+4iχ /κ),
A 0 5
P
we realize that the momentum space constitutes a 4-dimensional de Sitter spacetime:
( )2 ( )2 ( )2 = κ2.
0 i 5
P − P − P −
B. Scalar field representation
Scalar field in κ-Minkowski spacetime is represented using the κ-deformed Fourier transformation. Suppose Φ˜(p)
is a classical function in commuting four-momentum space. The scalar field on the κ-Minkowski spacetime is then
defined as
Φ(xˆ)= Φ˜(p) :e−ipxˆ : (9)
Zp
d4p
where denotes . This scalar field on the non-commuting coordinates smoothly reduces to the commut-
p (2π)4
Z
ing casRe if κ . The definition also allows us to transform the integration over non-commuting coordinates
→ ∞
unambiguously into the ones with commuting momentum space integration if one uses the delta-function relation
(2π)4δ4(p)= :e−ipxˆ : (10)
Zxˆ
where = d4xˆ. For example, the multiplication of fields integrated over xˆ is expressed as
xˆ
R R
Φ2(xˆ)= Φ˜(p)Φ˜( p0, pep0/κ). (11)
− −
Zxˆ Zp
4
One may obtain a conjugate field if one defines the conjugation of ordered exponential as
:e−ipxˆ : † =e−ip·xˆeip0xˆ0 =:e−i(ep0/κ)p·xˆ+ip0xˆ0 : . (12)
(cid:16) (cid:17)
With this, we have
† †
Φ†(xˆ) =Φ(xˆ), Φ (xˆ)Φ (xˆ) =Φ†(xˆ)Φ†(xˆ), (13)
1 2 2 1
(cid:16) (cid:17) (cid:16) (cid:17)
and the conjugate scalar field is given as
†
Φ†(xˆ) = Φ˜(p)† :e−ip·xˆ : Φ˜c(p) :e−ip·xˆ : (14)
≡
Zp (cid:16) (cid:17) Zp
Φ˜c(p) = e3p0/κΦ˜†( p , ep0/κp).
0
− −
Φ†(p)representsthe ordinarycomplexconjugateofΦ(p)forclassicalfieldandhermitianconjugateforquantumfield.
Likewise in Eq. (11), one may write the multiplication of fields as
Φ†(xˆ)Φ (xˆ) = Φ˜†(p)Φ˜ (p) (15)
1 2 1 2
Zxˆ Zp
†
Φ (xˆ)Φ (xˆ) = Φ†(xˆ)Φ†(xˆ)= Φ˜c( pep0/κ, p )Φ˜c(p).
1 2 2 1 2 − − 0 1
Zxˆ(cid:16) (cid:17) Zxˆ Zp
Ifthe scalarfieldΦ(xˆ)is real,we needΦ(xˆ)† =Φ(xˆ). This,inturngivesΦ˜c(p)=Φ˜(p). (Ourdefinitiondiffers from
the one in Ref. [12]. There, Φ˜c(p) is replaced by Φ˜†(p). )
The partial derivative of field is defined from the partial derivative of the exponential functions in Eq. (7),
∂ˆ :e−ip·xˆ := iχ (p) :e−ip·xˆ : . (16)
µ µ
−
The adjoint derivative ∂ˆ† is obtained from the relation
µ
Φ (xˆ)∂ˆ Φ (xˆ)= ∂ˆ†Φ (xˆ) Φ (xˆ)
1 µ 2 µ 1 2
Zxˆ Zxˆ(cid:16) (cid:17)
which leads to
∂ˆ† :e−ip·xˆ : = iχ†(p) :e−ip·xˆ : (17)
µ − µ
χ†(p) = χ ( p0, pep0/κ).
µ µ − −
This results in a useful relation
†
∂ˆ Φ(xˆ) = ∂ˆ†Φ†(xˆ). (18)
µ − µ
(cid:16) (cid:17)
C. Free field action of a complex field
The free field action of a complex scalar field in κ Minkowski spacetime can be written in analogy with the
−
commutative one as
S = (∂ˆ†Φ†(xˆ))∂ˆµΦ(xˆ) m2Φ†(xˆ)Φ(xˆ) . (19)
µ −
Zxˆh i
Here the fifth directionis omitted andevaluatedin 4-dimensionsonly. This actionhas notSO(4,1)invariancein [15]
but the deformed Poincar´e symmetry is respected. (See Eq. (20) below). This action can be written in momentum
space representation by using the Fourier transform (9),
S = Φ˜†(p)∆−1(p)Φ˜(p) (20)
F
Zp
M2(p)
∆−1(p) = M2(p) 1+ κ m2+iǫ ,
F κ 4κ2 −
(cid:20) (cid:18) (cid:19) (cid:21)
5
where ǫ is the usual small number prescription for the Feynman propagator. This action shows that the non-
commutative spacetime modifies the propagator for the free fields defined in the commutative coordinate space
φ(x)= e−ip·xΦ˜(p). (21)
Zp
The number ofpoles in ∆ are,however,infinitely many onthe complex momentumplane. Explicitpole positions
F
are given as ω± =ω±+inκπ with n an arbitrary integer;
n
m2 iǫ m2+p2 iǫ
ω± = κln 1+ − − . (22)
− r κ2 ∓r κ2 !
For the massless case, ω± reduces to
p iǫ
ω± = κln(1 | |− ). (23)
− ∓ κ
ω± represent the two stable on-shell spectra and due to the restriction p2 < κ2, ω+ is positive and ω− is negative.
| |
The existence of the two stable spectra suggests that one may define the Minkowski vacuum 0 which reduces to
M
| i
commutative vacuum when κ . Then the time-ordered product is given as
→∞
G (x,y)= 0 Tφ(x)φ†(y)0 = e−ip·(x−y)i∆ (p)=G (x y), (24)
F M M F F
h | | i −
Zp
while
0 Tφ(x)φ(y)0 = 0 Tφ†(x)φ†(y)0 =0.
M M M M
h | | i h | | i
The asymmetry of the pole positions of the Feynman propagator in Eq. (20), ω+ +ω− = κln(1 p2/κ2) = 0,
demonstrates that the Lorentz symmetry is broken at the order of 1/κ. It is natural to defi−ne ω+ −as the par6ticle
spectrumandω− astheanti-particleone. Inthiscase,thevacuum 0 doesnotrespecttheparticleandantiparticle
M
| i
symmetry.
In addition, it is noted that when ∆x0 > 0, G (∆x) creates not only the excitation with ω+ but also unstable
F
ones with ω+ and ω− with n negative integers. Thus we may define particle spectra as ω+, ω+ and ω− with n < 0.
n n n n
Similarly, when ∆x0 < 0, G (∆x) creates the stable excitation with ω− as well as unstable ones with ω+ and ω−
F n n
with n positive integers, which indicates that ω−, ω+ and ω− with n>0 represent anti-particle spectra.
n n
The positive Wightman function W (x,y) is defined as
+
W (x,y) = 0 φ(x)φ†(y)0 . (25)
+ M M
h | | i
W (x,y) measures the amplitude to create particles including the unstable ones and is defined to be the Feynman
+
propagator when ∆x0 >0. Thus it can be represented as
d3p e−iω(x0−y0)+ip·(x−y)
W (x,y)=W (x y)= . (26)
+ + − (2π)32p 1+M2(ω,p)/(2κ2)
ω=ωX+,ωn±<0Z | | κ
This result can be written formally as
M2
W (∆x)= e−ip·∆x2πδ M2 1+ κ m2 .
+ κ 4κ2 −
Zp+ (cid:18) (cid:18) (cid:19) (cid:19)
Here indicates that the integral over p0 includes not only the real mass-shell position ω+ but also the complex
p+
ones, ω+ and ω− with n<0. The delta function integration is evaluated using the property
Rn n
∂M2 p2
κ = κ 1 ep0/κ κe−p0/κ = 2p . (27)
∂p − κ2 − ± | |
0 (cid:12)ω± (cid:20) (cid:18) (cid:19) (cid:21)ω±
(cid:12)
Likewise, the negative Wightman(cid:12)function W (x,y) is defined as
(cid:12) −
W (x,y) = 0 φ†(y)φ(x)0 =W (x y) (28)
− M M −
h | | i −
d3p e−iω(x0−y0)+ip·(x−y)
= .
(2π)32p 1+M2(ω,p)/(2κ2)
ω=ωX−,ωn±>0Z | | κ
6
The explicit form of W (∆x) for massless case is needed in the next section. When κ∆x0 1, the complex
±
| | ≫
mass-shell contributions decay exponentially, representing the unstable excitations. As a result, among the infinitely
many contributions to W (∆x), the main contribution comes from the ω± part. After the angular integrations and
±
rescaling p by κ, we have
| |
κ
W (∆x) = (g (∆x) g (∆x)), (29)
+ 8iπ2 ∆x + − −
| |
1
g (∆x) = dz eiκ[log(1−z)∆x0±z|∆x|].
±
Z0
Using the result in Appendix A, we have, to the order of 1/κ,
1 i∆x 3(∆x )2+(∆x)2) −1
W (∆x) = ξ 0 0 +O(κ−2), (30)
+ −4π2 − κ ξ
(cid:26) (cid:18) (cid:19)(cid:27)
where ξ =(∆x )2 (∆x)2. For W (∆x), we have
0 −
−
κ
W (∆x) = (h (∆x) h (∆x)) (31)
− 8iπ2 ∆x + − −
| |
1
h (∆x) = dz eiκ(log(1+z)∆x0±z|∆x|).
±
Z0
To the order of 1/κ, we have
1 i∆x 3(∆x )2+(∆x)2) −1
W (∆x) = ξ 0 0 +O(κ−2). (32)
− −4π2 − κ ξ
(cid:26) (cid:18) (cid:19)(cid:27)
Here we assume that κ∆x 1 and ξ 1, even though W (∆x) holds for ξ 0 in the commutative limit. In
0 ±
≫ ≫ ∼
addition, it is noted that the Lorentz symmetry breakingis reflected in this resultsince W ( ∆x)=W (∆x) at the
− +
− 6
order of 1/κ.
III. PARTICLE DETECTOR INTERACTING WITH A MASSLESS SCALAR FIELD IN
κ−MINKOWSKI SPACE
Suppose a particle detector is moving along a world line Xµ(τ), where τ is the detector’s proper time. Unruh [17]
calculated the response of the particle detector moving with a uniform acceleration: If the detector interacts with a
free massless field, and the system initially lies in the Minkowski vacuum state, then the detector responds as if it
were immersed in a heat bath in an un-accelerated frame whose temperature turns out to be acceleration/2π, the
Unruh temperature. This effect is called the Unruh effect.
In the Minkowski spacetime, we will let the detector interact with a massless complex scalar field through the
detector’s monopole moments, (τ) and †(τ). The interaction is written as
M M
S = c dτ δ4(x X(τ)) †(τ)φ(x)+φ†(x) (τ) (33)
I
− M M
Z Zx (cid:16) (cid:17)
= dτ †(τ)φ(X(τ))+φ†(X(τ)) (τ)
M M
Z (cid:16) (cid:17)
where c is a small coupling constant.
Inthe κ-Minkowskispacetime,the complex scalarfield is affectedby the non-commutativenature ofthe spacetime
and its action is written as Eq. (20) with vanishing mass. For the detector part, on the other hand, we assume that
it experiences the time evolution under the ordinary quantum mechanics while moving along the commutative world
line Xµ(τ).
The detector can be coupled to the field in various ways. To find the possibility we consider a κ-deformed delta
function
δ(4)(xˆ X(τ))= :e−ip(xˆ−X(τ)) := eipX(τ) :e−ipxˆ : . (34)
−
Zp Zp
7
This delta function gives the property
Φ†(xˆ)δ(4)(xˆ X(τ))= Φ˜†(p)eip·X(τ) =Φ†(X(τ)). (35)
−
Zxˆ Zp
However, the κ-deformed delta function is not self-conjugate, since
†
δ(4)†(xˆ X(τ))= eipX(τ) :e−ip·xˆ : = e3p0/κeip0X0(τ)−iep0/κp~·X~(τ) :e−ipxˆ : = δ(xˆ X(τ)).
− 6 −
Zp (cid:16) (cid:17) Zp
Thus we need
Φ(X(τ))= δ(4)†(xˆ X(τ))Φ(xˆ). (36)
−
Zxˆ
Thanks to this property of the delta function, we may rewrite the interaction Eq. (33) for the interaction in the
κ Minkowski spacetime as
−
S = c dτ †(τ) δ†(xˆ X(τ))Φ(xˆ)+ (τ) Φ†(xˆ)δ4(xˆ X(τ)) . (37)
I
M − M −
Z (cid:20) Zxˆ Zxˆ (cid:21)
Thischoiceisthesimplestone,inthesensethatcomplicatedcoproductoftheκ Minkowskispacetimedoesnotappear
−
in the interaction term. There exist other choices of interaction, which may give more complicated non-commutative
effect but this possibility is not considered in the present paper.
Supposethedetectorliesinitiallyinitsgroundstate E ofaquantummechanicalhamiltonianH . Asthedetector
0 0
| i
moves along a trajectory, it will in general find itself undergoing a transition from the initial to an excited state E
| i
with E > E . In addition, the field might be affected by the detector since the field is coupled to the detector and
0
there will be a field contribution to the transition.
We assume for simplicity that the field initially is in the ground state 0 , the vacuum state with respect to
M
| i
the Minkowski spacetime and finally makes a transition to an excited state φ , which may be assumed to be a one
| i
particle state. Then we expect the whole transition amplitude of the system A to be evaluated by the first order
f←i
perturbation theory:
τ0
A = ic E,φ dτ †(τ) Φ˜(p)e−ipX(τ)+ (τ) Φ˜†(p)eipX(τ) 0 ,E (38)
f←i M 0
h | M M | i
Z−∞ (cid:20) Zp Zp (cid:21)
where τ is the time when the detector and the field reach the final state φ,E . Using the time evolution of the
0
| i
monopole moment operator (τ)
M
(τ)=eiH0τ (0)e−iH0τ, †(τ)=eiH0τ †(0)e−iH0τ, (39)
M M M M
A may factorize to give
f←i
τ0
A = ic E †(0)E dτei(E−E0)τ e−ipX(τ) φΦ˜(p)0 (40)
f←i 0 M
h |M | i h | | i
Z−∞ Zp
τ0
+ic E (0)E dτei(E−E0)τ eipX(τ) φΦ˜†(p)0 .
0 M
h |M | i h | | i
Z−∞ Zp
We first consider the detector moving along the world line with a constant velocity v, the inertial world line,
X0 =t=γτ, X(τ)=X(0)+vγτ,
where γ =1/ 1 v2/c2. Then the integral in (40) with τ becomes
0
− →∞
p
A = ic E †(0)E 2πδ E E γ(p0 p v) eip·x φΦ˜(p)0
f←i 0 0 M
h |M | i − − − · h | | i
Zp
(cid:0) (cid:1)
+ic E (0)E 2πδ E E +γ(p0 p v) e−ip·x φΦ˜†(p)0 .
0 0 M
h |M | i − − · h | | i
Zp
(cid:0) (cid:1)
The expectation values φΦ˜(p)0 ( φΦ˜†(p)0 ) is nonzero only for the case of p <0 (p >0) as in the commu-
M M 0 0
h | | i h | | i
tative case. In addition, since E >E and p p v, the delta functions make A vanish, which reflects the fact
0 0 f←i
| |≥ ·
that the κ-Minkowski spacetime has the time-translational invariance.
8
To explorethe casewhenthe detectorfollowsa uniformly acceleratingpathwith accelerationα, whose coordinates
are chosen as
X0(τ)=α−1sinhατ, X1(τ)=α−1coshατ, X2(τ)=0=X3(τ), (41)
we consider the transition probability,
A 2 =c2 M (E,E ) (E E )+M (E,E ) (E E ) , (42)
f←i + 0 + 0 − 0 − 0
| | F − F −
XE (cid:16) (cid:17)
where we sum over intermediate states using the completeness of the basis φ and
{| i}
1
M (E,E )= E †(0)E 2 E (0)E 2 .
± 0 0 0
2 |h |M | i| ±|h |M | i|
(cid:0) (cid:1)
Note that M (E,E ) is always larger than M (E,E ). is the part due to the response of the field and is given
+ 0 − 0 ±
| | F
by
(E) = τ0 dτ τ0 dτ′e−iE(τ−τ′)f (X(τ),X(τ′)),
± ±
F
Z−∞ Z−∞
where
f (X(τ),X(τ′)) = 0 Φ(X(τ))Φ†(X(τ′)) Φ†(X(τ))Φ(X(τ′)) 0
± M M
h | ± | i
= W (X(τ) X(τ′)) W (X(τ′) X(τ))=f (X(τ) X(τ′)).
+ (cid:2) − ±(cid:3)
− ± − −
In the Lorentz invariant systems such as in the commutative field theory, (E) vanishes since W (∆X) =
− +
F
W ( ∆X). In our case, (E) does appear at the order of 1/κ.
− −
− F
Excitation rate R(τ ,E) is the rate of the transition probability,
0
dA 2
R(τ ,E) | f←i| =c2 M (E,E )S (τ ,E)+M (E,E )S (τ ,E) , (43)
0 + 0 + 0 − 0 − 0
≡ dτ
0
XE (cid:16) (cid:17)
where S (τ ,E) is called the response function [26] representing the Unruh effect:
± 0
d (E)
±
S (τ ,E) F (44)
± 0
≡ dτ
0
τ0 τ0
= dτeiE(τ−τ0)f (X(τ ),X(τ))+ dτe−iE(τ−τ0)f (X(τ),X(τ )).
± 0 ± 0
Z−∞ Z−∞
Shifting τ by τ we have
0
0 0
S (τ ,E) = dτ eiEτ f (X(τ ),X(τ +τ ))+ dτ e−iEτ f (X(τ +τ ),X(τ )) (45)
± 0 ± 0 0 ± 0 0
Z−∞ Z−∞
∞
= dτ e−iEτ f (X(τ τ /2+τ/2),X(τ τ /2 τ/2)).
± 0 0
−| | −| | −
Z−∞
For the massless case, f (∆X) is given by
±
1
f (∆X) = +O(κ−2), (46)
+ −2π2ξ
∆X0 3(∆X0)2+(∆X)2
i
f (∆X) = +O(κ−2). (47)
− −2π2κ (cid:16) ξ3 (cid:17)
Along the uniformly accelerating path in Eq. (41), ∆X =X(τ) X(τ′) is parameterized as as
−
2 α(τ τ′) α(τ +τ′)
∆X0 = sinh − cosh , (48)
α 2 2
2 (cid:16)α(τ τ′)(cid:17) (cid:16)α(τ +τ′)(cid:17)
∆X1 = sinh − sinh
α 2 2
(cid:16) (cid:17) 4 (cid:16) α(τ(cid:17) τ′)
ξ = (∆X0)2 (∆X1)2 = sinh2 − ,
− α2 2
(cid:16) (cid:17)
9
and we have
α2 1 1
f (∆X) = +O , (49)
+ −8π2 (sinh2 α(τ2−τ′)) (cid:18)κ2(cid:19)
iα3 4cosh3 α(τ+τ′) coshα(τ+τ′) 1
f (∆X) = 2 − 2 +O .
− −8π2κ( sinh3 α(τ2−τ′) ) (cid:18)κ2(cid:19)
Thus f in Eq. (45) is given explicitly in terms of the proper-time parametrization
±
α2 1
f (X((τ τ /2+τ/2),X(τ τ /2 τ/2)) = +O(κ−2),
+ 0−| | 0−| | − −8π2sinh2(ατ)
2
iα3 cosh3ατ cosh3ατ +2coshατ coshατ
f (X((τ τ /2+τ/2),X(τ τ /2 τ/2)) = 0 2 0 2
− 0−| | 0−| | − −8π2κ( sinh3(ατ)
2
sinh3ατ (2coshατ +1)+2sinhατ
ǫ(t) 0 0 +O(κ−2).
− sinh2(ατ) )
2
0.4
0.3
s
0.2
s
0.1 BE
0.5 1 1.5 2 2.5 3
2E/α
FIG. 1: The distribution functions sBE(2E/α) and s(2E/α).
We evaluate the response function using contour integration. To do this we need to detour the contour so that we
exclude the singularity around τ =0. Using the contour integration result given in Appendix, we finally have
α 1
S (τ ,E) = s (2E/α)+O , (50)
+ 0 2π BE κ2
(cid:18) (cid:19)
α α α 2E
S (τ ,E) = (9cosh(3ατ )+2coshατ ) (cosh(3ατ )+2cosh(ατ )) s (2E/α)
− 0 0 0 0 0 BE
−2π 2κ 2E − α
(cid:26)(cid:20) (cid:21)
1 1
+ [3sinh(3ατ )+2sinh(ατ )]s(2E/α) +O , (51)
π 0 0 κ2
(cid:27) (cid:18) (cid:19)
where
ζ ∞ dk ζ k 1 ζ k+1
s (ζ) = , s(ζ)= | − | | | . (52)
BE eπζ 1 k eπζ|k−1| 1 − eπζ|k+1| 1
− Z0 (cid:16) − − (cid:17)
The distribution function s (2E/α) is the Bose-Einstein one, finite at E = 0 and decays exponentially for large
BE
E. On the other hand, s(2E/α) shows no definite statistics but fuzziness, vanishes at E =0, and decreases slowly as
s(2E/α) α/(4π2E) for large E. The behaviors of s and s are plotted in Fig. 1.
BE
∼
It is to be notedthat S only reproducesthe commutative resultandpresents no O(1/κ) correction. Lorentz sym-
+
metry breaking appears in S at O(1/κ). Here the correctionterm shows the preferred-frame effect, the dependency
−
of the detector time τ .
0
10
Since the transition probability in Eq. (43) also depends on how the detector couples to the complex scalar field,
one may tune the detector without modifying the field theory of the massless complex scalar field. Thus if one tune
the detector so that M is hermitian (M = 0), then the detector does not see the Lorentz violation at O(1/κ).
−
Since M (E,E ) M (E,E ) and M is sensitive to the violation of the Lorentz symmetry, one may consider the
+ 0 − 0 −
≥| |
detector with M (E,E )=M (E,E ), maximally sensitive to the violation of Lorentz symmetry.
+ 0 − 0
In this maximal case, the response function is given by
S (E,τ )=S (E,τ )+S (E,τ ). (53)
max 0 + 0 − 0
An explicit example of the response function with the restrictions given in Eqs. (55) and (56) below is plotted in
Fig. 2. As seen in the figure, the response function deviates slightly from the Bose-Einstein result.
0.3
0.25
0.2
2π 2E
S ( )
α max α
0.15
0.1
0.05
0
0 0.5 1 1.5 2 2.5 3
2E/α
FIG. 2: Plot of the response function Smax(τ0,E). α is set as 5·10−5κ. The shaded curve represents the commutative case,
the Bose-Einstein result. The solid curves denote the response function at time ατ0 =0, ατ0 =1, ατ0 =2, respectively, from
top tobottom. At each time scale, thecurves are shown for the valid energy range of the relevant perturbation result.
The Lorentz symmetry breaking results in τ dependence and the excitation rate depends on the proper-time. At
0
τ 0, S is given by s term only since sinh(ατ ) = 0. Since the O(1/κ) correction term must be smaller than
0 − BE 0
the∼O(κ0) term, we have the range of validity of the results (50) and (51),
α2
E κ. (54)
κ ≪ ≪
The maximal response function becomes
E 1 α 11α 6E
S (τ =0,E) 1 .
max 0 ≃ π e2πE/α 1 − 2κ 2E − α
− (cid:20) (cid:18) (cid:19)(cid:21)
At ατ O(1), the infrared partof S is dominated by s term and the ultravioletpart is dominated bys term.
0 − BE
∼
For 1 ατ < ατ = (ln(2κ/(3α)))/3, the proper time contribution becomes large and thus the range of validity is
0 c
≪
limited as
9α 2E 2κ
e3ατ0 e−3ατ0. (55)
2κ ≪ α ≪ α
(At τ = τ the left and right-hand side become equal and the range is empty). In the UV part of the spectrum,
0 c
another restriction appears by comparing the behavior of s(2E/α) and s (2E/α):
BE
α 2 4π3κ
e2πE/α e−3ατ0. (56)
2E ≪ 3α
(cid:16) (cid:17)
Whenτ τ ,wecannotusetheresultofW inEqs.(30)and(32). Thisisbecauseafterthistime,ifτ τ ,
0 c ± 0
≫ ∼ →∞
the four vector X(τ) X(τ ) is not time-like but becomes almost light-like. Therefore, the series expansion used in
0
−
Appendix is not valid anymore. In the limit, X(τ) X(τ ) 0 with X(τ ) , the coefficient (a b) in the
0 0
| − | → → ∞ −