A Vector type of Unruh-DeWitt-like detector

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by  De-Chang Dai

A Vector type of Unruh-DeWitt-like detector
De-Chang Dai
Institute of Natural Sciences, Shanghai Key Lab for Particle Physics and Cosmology,
and Center for Astrophysics and Astronomy, Department of Physics and Astronomy,
Shanghai Jiao Tong University, Shanghai 200240, China
7 WestudyatypeofanUnruh-DeWitt-likedetectorbasedonavectorratherthanscalarfield. This
1 detectorhastwoenergystatesandproducesLarmorradiationwhenthereisnoenergygapbetween
0 them. This setup indicates that Larmor radiation and Unruh radiation are two counterparts of
2 the same phenomenon. Larmor radiation is observed in the inertial frame, while Unruh radiation
is observed in an accelerated frame. The accelerated observer sees that his detector absorbed a
n
particle inside its own accelerated horizon, while the Larmor radiation is the companion particle
a
J whichleavestheaccelerated observer’shorizon. Sincethedetectionisbasedontheelectromagnetic
field,thistypeof detectorismuchcloser totherealworld thanastandard Unruh-DeWittdetector
0
which is coupled to a scalar field.
2

PACSnumbers:
]
h
p
- I. INTRODUCTION mor radiationin the non-relativisticregime. This proves
n
thatLarmorradiationisbasicallyUnruhradiation. They
e
arethesamephenomenaobservedbydifferentobservers.
g Unruh-Dewitt detector is a theoretical setup initially
. introducedbyUnruhandDeWitt[1–6]tostudyphenom- Thisresultagreeswithearlierstudies[17]. Wethenapply
s
the detector setup to a long time uniformly accelerated
c ena of quantum fields in curved spacetimes, or equiva-
si lently phenomena observed by an accelerated observer. crehlaartgeeddtdoetUenctrourh. OteumrpreersautlutrsehoTws=thaat,tbhuetspitesctforurmm iiss
y Among the other things, this setup reveals that a uni- 2π
h formly accelerated observer in Minkowski space should different from the scalar field.
p
observe a thermal spectrum[1] with temperature T =
[ u
a , where a is the acceleration. This thermal radiation
2π II. UNRUH DEWITT DETECTOR
1 is called the Unruh radiation. Equivalently, one can use
v thesamesetuptostudythermalradiationseenbyastatic
5 Unruh De-Witt detector is considered to be a point-
detectorinSchwarzschild[7]andde-Sitter[8]spacetimes.
3 like detector which moves in a relativistic spacetime on
The root of Unruh radiation is acceleration. Since an
6 a smooth path x(τ) (where τ is the detector’s proper
8 eternally accelerating observer observes a horizon (a re-
time). This detector usually consists of the groundstate
0 gionofspacetimefromwhichno signalcanemerge),Un-
(|0 > ) and an excited state |1 > , with eigen energies
. ruh radiation is considered to be equivalent to Hawking D D
1 0 and ω respectively. The detector is usually coupled to
radiation which is caused by a black hole event horizon.
0
a scalerfield, φ, or the derivativeof the scalarfield, ∂ φ.
7 Since it is very difficult (if possible at all) to create a τ
Asanexample,herewecoupleittoφforsimplicity. The
1 black hole in a laboratoryto observeHawking radiation,
interaction is described by the following Hamiltonian
: Unruhradiationcanserveasagoodalternativetostudy
v
i aspects of Hawking radiation. Recently, it was proposed H =cµ(τ)φ(x(τ)). (1)
X that a high intensity laser can test the Unruh radiation int
r throughitsstrongelectricfieldwhichcausesacceleration Here, c is a small coupling constant andµ is a monopole
a ofalargemagnitude[9–11]. However,thisproposalisnot moment operator. The probability of the transition rate
straightforward,and different studies disagree about the from |0 >D to |1 >D can be found from the first order
existence of Unruh radiationcreatedinthis way [12–16]. perturbation theory
More work is certainly needed to clarify all the issues.
P(ω)=c2| <1|µ(0)|0> |2F(ω). (2)
D D
Oneoftheproblemsintheoriginalproposalisthatthe
modelinquestionwasbasedonascalarfield. Incontrast, Here, F(ω) is the detector’s response function
when studying radiation from electrons one has to con- ∞
sider photons, which are vector particles. However, so F(ω)= dτdτ′e−iω(τ−τ′) <0|φ(x(τ))φ(x(τ′))|0>(3)
Z
far only two types of Unruh-DeWitt-like detectors were −∞
constructed. One is a detector coupled to a scalar field
|0> is φ’s ground state. The unit time response is
[1, 2], while the other is coupled to the derivative of a
∞
scalar field[12]. In this paper we extend the construct F˙(ω)| = dτ′e−iω(τ−τ′) <0|φ(x(τ))φ(x(τ′))|0>(4)
to a vector field. In our analysis, we reproduce the Lar- τ Z
−∞
2
Ifthe detectorundergoesauniformlinearacceleration IV. LARMOR RADIATION
a in z-direction as
We now consider a small magnitude acceleration in a
sinh(aτ)
t = (5) non-relativistic limit, i.e. ∆x<<∆t and ∆t≈∆τ. The
a
surviving term in this limit in equation 13 is the term
cosh(aτ)
z = (6) which includes only the z-component
a
x = y =0 (7) <0γ|Az(x(τ))Az(x(τ′))|0γ >
∞
, then the unit time response becomes ≈ 1 ke−ik(τ−τ′)dk (15)
6π2 Z
0
1 ω
F˙(ω)= . (8)
If ω =0, then equation 13 becomes
2πe2πω/a−1
From here we can read-off the temperature T = a . e2 ∞ |a˜(k)|2
2π P = dk (16)
Therefore, an accelerated observer sees a thermal bath j 6π2 Z k
0
radiation.
a˜(k)= a(t)e−iktdt (17)
Z
III. VECTOR TYPE UNRUH-DEWITT
This is nothing else but non-relativistic Larmor radia-
DETECTOR
tion [18]. This implies that Larmor radiation is just a
special case of Unruh effect. The difference is that Lar-
In this section we want to extend the construct to a morradiationis observedinMinkowskispacetime, while
vector field type of detector. This detector is coupled to Unruh effect is observed in a co-moving spacetime( fig.
avectorfield,Aµ,inparticulartheelectromagneticfield. 1). Indeed, we can arrive to the same conclusion from
Such detector cannot be of monopole type. It must in- the equivalence principle. Larmor radiation and Unruh
cludeavectorcurrent,jµ(τ),whichcouplestothevector radiation are just two complementary parts of the same
field through the following Hamiltonian phenomenon - Unruh radiation is observed by the accel-
eratedobserverinside his own horizon,while Larmorra-
H =eA ((τ))jµ((τ)) (9)
int µ diationistheradiationthatleavestheobserver’shorizon.
Larmor radiation is then detected by a non-accelerated
Here,
observer in Minkowski space.
jt = vt(τ)δ(x)δ(y)δ(z−z(τ)) (10) Larmor
jz = vz(τ)δ(x)δ(y)δ(z−z(τ)) (11)
radiation
jx = jy =0 (12) horizon
t
where vα is the four velocity. We consider the detector
whichmovesinz-directiononly,andhastwoeigenstates,
|0> and|1> ,witheigenenergies0andω respectively.
j j
Theprobabilityoftransitionratefrom|0> to|1> can
j j
again be found from first order perturbation theory
Unruh
∞ radiation
P (ω) = e2 dτdτ′v (τ)v (τ′)e−iω(τ−τ′)
j Z α β
−∞ x
× <0γ|Aα(x(τ))Aβ(x(τ′))|0γ >, (13)
where |0γ > is Aµ’s ground state.
<0γ|Aα(x(τ))Aβ(x(τ′))|0γ >
= d3~k ǫα(k)ǫβ(k)e−ik·(x(τ))−x(τ′)). (14)
Z 2k(2π)3 FIG. 1: An accelerated particle creates a horizon in its co-
movingframe. Thisparticleemitsavectorfieldwhichpasses
Here, ǫα(k) is the polarization of the vector field, and throughitshorizonandbecomesLarmorradiation. However,
it is transverse to kµ. The term ǫα(k)ǫβ(k) should not thecomovingobservertreatsthesamevectorfieldastheUn-
be simplified to ηαβ since it is the external field which ruh radiation.
includes only two degree of freedoms.
3
V. LONG TIME UNIFORM ACCELERATION This new detector can give more realistic results in the
context of radiation from a charged accelerated device.
We now consider the detector which is uniformly ac- If the detector does not have an energy gap between its
celerated for long time but not eternally. Its trajectory eigen states, the result is the same as the Larmor radi-
can be approximately described by equations 5 to 7. In ation. This confirms that Unruh radiation and Larmor
this case radiation are two parts of the same phenomenon which
are observed by different observers[17]. Unruh radiation
<0γ|Az(x(τ))Az(x(τ′))|0γ > is seen by an accelerated observer and Larmor radiation
−1 ∞ ke−ik(τ−τ′) is seen by an inertial frame observer. In other words,
≈ dk (18) Unruh radiation is observed within the accelerated ob-
6π2 Z−∞ e2πk/a−1 server’s horizon, while Larmor radiation is the part that
falls into the accelerated observer’s horizon.
ThedenominatorisinthePlanckianform,soT = a can
2π We also considered Unruh radiation for a long-time
be treated as a thermal temperature. The transmission
(but finite) uniformly accelerated observer. The effect is
rate, equation 13, becomes
shownin equation19. Radiationis still characterizedby
e2 ∞ |a˜(k+ω)|2 k Unruh temperature T = a , but also depends on ω+k.
Pj =−6π2 Z−∞ (k+ω)2 e2πk/a−1dk (19) Thisimpliesthatsomeoft2hπeenergyisextractedfromthe
acceleratedsourcetotheexciteddetector. Thisenergyis
Theresponsedependsalsoonω+k. Thisimpliesthatthe radiated outside of the horizon (Larmor radiation part).
energy, k+ω, is extracted from the accelerating source.
ω part is absorbed by the detector and k is released to
the eventhorizonanddisappearsfromdetector’sside. It
Acknowledgments
becomesLarmorradiationafteritcrossthehorizon. One
finds thatequation19reducestonon-relativisticLarmor
radiation as a → 0. However, the Unruh temperature D.C.DaiwassupportedbytheNationalScienceFoun-
is related to k instead of ω, which is what a scalar type dation of China (Grant No. 11433001 and 11447601),
detector predicts. Similar result has been found in two National Basic Research Program of China (973 Pro-
energy-level-atomcase[19]. gram 2015CB857001), No.14ZR1423200 from the Office
of Science and Technology in Shanghai Municipal Gov-
ernment, the key laboratory grant from the Office of
VI. CONCLUSION Science and Technology in Shanghai Municipal Govern-
ment (No. 11DZ2260700) and the Program of Shanghai
WereviewedthestandardUnruh-DeWittdetectorand Academic/TechnologyResearchLeaderunderGrantNo.
extendittoavectortypeofUnruh-DeWitt-likedetector. 16XD1401600.
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