Unruh effect and
HolographyShoichi Kawamoto
(National Taiwan Normal University) with
Feng-Li Lin(NTNU), Takayuki Hirayama(NCTS)and Pei-Wen Kao (Keio, Dept. of Math.)
2nd Mini Workshop on String Theory @ KEK
2009 November 10 @ KEK
2009 Nov. 10 2nd Mini Workshop on String Theory @ KEK 2
Uniformly accelerated observer and Unruh Effect
The world-line of the observer with a constant acceleration a is given by solving
The observer feels the temperature
maFx
xmdtd
==− 21
and the solution is given by hyperbolas
Unruh—Davies—De Witt—Fulling effect
Let us discuss this phenomenon first.
2009 Nov. 10 2nd Mini Workshop on String Theory @ KEK 3
Uniformly accelerated observer and the Rindler coordinates
23
22
21
22 dxdxdxdtds +++−=
( ) 23
22
2222 dxdxddeds a +++−= ξτξ
τξ aeat a sinh1−= τξ aeax a cosh11
−=
Starting from the Minkowski metric
Coordinate transform
Rindler coordinates:
It is convenient to work with the following coordinates choice
Why it is nice?
2009 Nov. 10 2nd Mini Workshop on String Theory @ KEK 4
The Rindler coordinates as a comoving frame
τ∂∂−∂ ~11 tx xt
( ) 23
22
2222 dxdxddeds a +++−= ξτξ
Rindler coordinates:
LR RR
CDK
EDKt
x1 It covers the region (Right Rindler wedge)
The “time” translation is generated by the Killing vector
τξ aeat a sinh1−=τξ aeax a cosh1
1−= The world line with a constant ξ has a constant
acceleration.
Accelerated observer in Minkowski space = Rest observer in Rindler space
(Comoving frame)
2009 Nov. 10 2nd Mini Workshop on String Theory @ KEK 5
Vacuum, Particles and Observers
Let us briefly discuss how the accelerated observer feels a finite temperature.
Vacuum is observer dependent.
two complete sets of solutions: { })()1( xfi { })()2( xfI
Klein-Gordon equation:
complete sets [ ]∑ −=I
IIiIIii fff )*2()2(*)1( βα
( ) ( ) ijjiji ffff δ=−= **,, ( ) 0, * =ji ff
space-like hypersurface
2009 Nov. 10 2nd Mini Workshop on String Theory @ KEK 6
Vacuum, Particles and Observers II
Vacuum:10 20 00ˆ 1
)1( =ia 00ˆ 2)2( =Iadefined by
Quantum field can be expanded as
( ) ( )11)1( ˆˆ iii aaN †= 000 1)1(
1 =iN
∑=I
IiiN 22
)1(2 00 β
Bogolubov transformation: ( )∑ +=I
IIiIIii aaa †)2(*)2()1( ˆˆˆ βα
VEV of the number operator is
But,
20 is an excited states with respect to the particles of (1).
Bogolubov coefficients
positive frequency modes
2009 Nov. 10 2nd Mini Workshop on String Theory @ KEK 7
Quantum Field Theory on Minkowski space
2D massless scalar field theory: An example
KG equaton: ( ) 0)(22 =∂−∂ xxt φtiikxM
k ef ω
πω−=
41 ∞<=< kω0
( )∫∞
+=0
*ˆˆ)( Mk
Mk
Mk
Mk fafadkx †φ
Minkowski vacuum:M0 00ˆ =M
Mka
right mover:
2009 Nov. 10 2nd Mini Workshop on String Theory @ KEK 8
Quantum Field Theory on Rindler space
LR RRx1
Move to Rindler coordinates:
τξ aeat a sinh1−=τξ aeax a cosh1−=
( )2222 ξτξ ddeds a +−=
( ) 0),(22 =∂−∂ ξτφξτ
τωξ
πωRR
R
iik
R
RRk ef −=
41
KG eq.
( )∫ += *ˆˆ)( RRk
RRk
RRk
RRkR RRRR
fafadkx †φ
τωξ
πωRR
R
iik
R
LRk ef −=
41
( )∫ += *ˆˆ)( LRk
LRk
LRk
LRkR RRRR
fafadkx †φ
τξ aeat a sinh1−=
τξ aeax a cosh1−−=
R0 00ˆ0ˆ == RRRkR
LRk RR
aaThe Rindler vacuum is defined by
2009 Nov. 10 2nd Mini Workshop on String Theory @ KEK 9
Minkowski vacuum as a thermal state
Bogolubov transformation: ( )∫ += †Mk
LRkk
Mk
LRkk
LRk aadka
RRRˆˆˆ *,,, βα
⎟⎠⎞
⎜⎝⎛ −Γ⎟
⎠⎞
⎜⎝⎛=
−
aik
ka
kkie R
aik
R
akR
kk
RR
R1
2
/2/
πα
π
⎟⎠⎞
⎜⎝⎛ +Γ⎟
⎠⎞
⎜⎝⎛−
=a
ikka
kkie R
aik
R
akL
kk
RR
R1
2
/2/
πα
π
*/ Rkk
akLkk R
R
Re αβ π−=
*/ Lkk
akRkk R
R
Re αβ π−−=
So the expectation value of the number operators
(assume now the energy levels are discrete )Rii k=ω
110000 /2 −
== aMLiMM
RiM ie
NN πω
It represents the heat bath with the temperatureπ2aT =
The set { }LRk
RRk RR
ff , can be related to Minkovski ones.
Each of them cannot be written as Minkowski operators.
2009 Nov. 10 2nd Mini Workshop on String Theory @ KEK 10
Unruh effect and QCD
In QCD, there is a critical temperature Tc at which the chiral symmetry is restored.
Chiral symmetry can be restored by acceleration?
An interesting work by Ohsaku (PLB599, 2004 ).
Consider chiral restoration in 4D Nambu-Jona-Laisnio model
For Λ ∼ 1GeV, ac ~ Λ * 10-1 ac ~ 3 ∗ 1034
cm/sec2
Too big to test experimentally, but theoretical implication is intriguing.
For QCD, we may study the effect of acceleration through holographic correspondence.
What are the same (similar)? What’s the difference?
2009 Nov. 10 2nd Mini Workshop on String Theory @ KEK 11
Plan
1. Introduction: Unruh effect in field theory
2. Uniformly accelerated string and comoving frame
3. Introducing mesons
4. Conclusion and Discussion
2009 Nov. 10 2nd Mini Workshop on String Theory @ KEK 12
Uniformly accelerated string in AdS space (1/3)
Let us consider a uniformly accelerated particle (quark) on the boundary field theory.
The particle is the end point of an open string.
We are going to make a coordinate transformation which gives the comoving frame on the boundary.
Infinitely many choices!!!
a
2009 Nov. 10 2nd Mini Workshop on String Theory @ KEK 13
Uniformly accelerated string in AdS space (2/3)
We wand to take a “comoving frame” for the open string.
First determine the configuration. Consider AdS part of the metric
with boundary condition:
Exact solution to NG action has been found (Xiao)
and solve the e.o.m.
aboundary
2009 Nov. 10 2nd Mini Workshop on String Theory @ KEK 14
Comoving coordinates for uniformly accelerated string (Xiao)
( ) 25
222
223
22
21
2222AdS 5
5Ω+++++−=
×dRdu
uRdxdxdxdtuRds S
( )τα asinh22 aerat −= −
Uniformly accelerated string in AdS space (3/3)
αaeru −−= 1
( )τα acosh221
aerax −= −
Now the open string configuration: withα
r
2009 Nov. 10 2nd Mini Workshop on String Theory @ KEK 15
Generalized Rindler space (Xiao’s metric)
constant r surface
Illustrate how the new coordinates covers a part of the original AdS5
right Rindler wedge with 0 < r < a-1
(horizon = Rindler horizon + AdS horizon)
2009 Nov. 10 2nd Mini Workshop on String Theory @ KEK 16
Temperature in the comoving frame
On the boundary, the observer feels the Unruh temperature
Xiao’s metric has the horizon. And the Hawking temperature is
They coincides Boundary acceleration = Bulk Blackhole
The effect of the acceleration is completely equivalent to the gravitational force from BH?
Let us calculate some physical quantities and see whether we can see difference.
(Later we also discuss another accelerated frame)
2009 Nov. 10 2nd Mini Workshop on String Theory @ KEK 17
Boundary stress tensor
We first look at the boundary stress tensor. (Balasubramanian-Kraus, Myers)
νμ
μν δγ
δγ
tot2lim STr −
=∞→
where [ ] [ ])(8
18
1216
121
45tot γγ
πγ
ππRcc
Gxd
GRxd
GS
r+−+Θ−−Λ−−= ∫∫∫ ∞=∂MM
counter termAfter eliminating the divergences, we get (HKKL)
trace of the extrinsic curvature of the boundary
Xiao’s metric (generalized Rindler): 423/),1,1,1,3( TNpT ∝=−−−∝ εμν
Conformal thermal gas with the temperatureπ2aT =
2009 Nov. 10 2nd Mini Workshop on String Theory @ KEK 18
Quark – anti Quark potential
1x
1a2aa
21 aaa <<
α
L
r0
ε
We may calculate quark – anti quark potential in the accelerated space.
Energy given by the Wilson loop
1/a
)0,0),(,,( rrX ατμ =Wilson loop profile
Solution is given by )()(1
)(02
0
20
22
2
rhrR
rhrh
rR
=′+
′
αα 221)( rarh −=
2009 Nov. 10 2nd Mini Workshop on String Theory @ KEK 19
Profile of the Wilson loop (1/3)
)()(1
)(02
0
20
22
2
rhrR
rhrh
rR
=′+
′
ααWe first look at the profile
r
α
r0 large
The left is the profile function α(r) for variousr0 (R=a=1).
First they keep similar shape, but for large r0 ,the profile becomes more steep.
“right half” of the string profile
2009 Nov. 10 2nd Mini Workshop on String Theory @ KEK 20
Profile of the Wilson loop (2/3)
r0 against L plot: the maximal length does not occur for r0 =1/a.
The energy against quark-anti quarkdistance. First grows linearly, but finallyshows a strange behavior.
( ) ∫ ′=−= 0
00 )(2)()0(2r
drrrL ααα
2009 Nov. 10 2nd Mini Workshop on String Theory @ KEK 21
Profile of the Wilson loop (3/3)α
L
r0
Compare the energy to the straightline configuration (green ones).
So at some critical distance (=critical acceleration difference), the force between quark-anti quark is screened?
Finite temperature case, it does happen.However, our horizon is not real one!!
Energy cannot reach the other end? Loose causal relation?? Still unclear...
2009 Nov. 10 2nd Mini Workshop on String Theory @ KEK 22
Another choice of the metric?
Why do we need to stick to Xiao’s choice?
( )( ) 25
222
223
22
222222Rindler Ω+++++−= dRdu
uRdxdxddeuRds a ξτξ
τξ aeat a sinh1−= τξ aeax a cosh11
−=
A naive extension of Rindler space.
0=μνT zero temp. vacuum?? But the boundary theory
should be the finite temperature system with T=a/2π
We may choose a simpler coordinates.the same transformation as on the boundary
We also check the boundary stress tensor.
No clear answer yet...
2009 Nov. 10 2nd Mini Workshop on String Theory @ KEK 23
Comments on other’s work
Pareres-Peeters-Zamaklar(2009) computed the similar system, but using naive Rinlder space, and found also a bound for the acceleration difference.
Fig 1. JHEP 0904:015,2009
( )+−−= 222222
22 dzdd
zRds ξηκξ
ξ
They also found the maximum of the ratio of the acceleration ( ) 70.2~/ maxLR aa
Also argue that dissociation happens when aR = aL . (String reaches the Rindler horizon)
Again, what happens in the rest frame??
2009 Nov. 10 2nd Mini Workshop on String Theory @ KEK 24
Introducing D7-brane
Now we come to investigate the meson physics.Introducing meson in AdS5 is achieved by putting a probe D7-brane.
0,1,2,3
4,5,6,7
8,9
D7
D3
fundamental matters“meson” excitations
We would like to argue the chiral condensates.
First we argue, what is the appropriate setup for accelerated mesons?
2009 Nov. 10 2nd Mini Workshop on String Theory @ KEK 25
Who is moving?
There will be two ways to embed D7-brane in the Generalized Rindler space.
(I) First, embed D7-brane in the original AdS5 coord., and make the coordinate transform.
In this embedding, the quarks are “rest” in the Minkowski metric.
On the boundary field theory, Minkowski vacuum
Fluctuations are the operators on Minkowski space
M0
MO
MMM 00 OWill be calculating
After coordinate transformation, the operator is accelerated. While the vacuum seems to be thermal (to the observer).
MMM 00 Ostill ???
Embedding I
2009 Nov. 10 2nd Mini Workshop on String Theory @ KEK 26
Comoving probe brane
Embedding II
We want to have a static operator in the accelerated frame.
The other way around!
1. Moved to Xiao’s metric (generalized Rindler coord.)
2. Then embedding D7-brane to be static on this coordinate system.
This will define RO (Note: D7-brane is time dependent in the original frame. But “quarks” are static.)
Holographic calculation will be thermal one.
MRM 00 O
2009 Nov. 10 2nd Mini Workshop on String Theory @ KEK 27
D7-brane embedding (1/2)
( )( )[ ] ( )26
25
23
222
223
22
222222
22 dwdwdd
wRdxdxedhdh
Rwds a ++Ω+++++−= −
+− ρρατ α
We work with the following coordinates. ⎟⎟⎠
⎞⎜⎜⎝
⎛++=±=
+= ±
26
25
222
42
22 ,4
1,4
4 wwwwRah
wawr ρ
Ansatz: 0),( 65 == wzw ρ
∫ ′+−= −+− 2
53328
77 )(1 ρρα whhexdTS aD
D7 brane extends these 8 directions.
Then we solve the equation of motion with boundary conditions.
2009 Nov. 10 2nd Mini Workshop on String Theory @ KEK 28
D7-brane embedding (2/2)
Minkowski embedding
BH embedding
horizon
( ) ( ))(1))ln((1~)( 22
2 −− +++ ρρνρρρ OOmz
Asymptotic solution near the boundary is
Regular solution: ( ))(, mm ν
D7 reaches to the center: Minkowski embeddingD7 terminates at horizon: Blackhole embedding
In general, starting with arbitrary m and ν,the solution will diverge.
D7-brane profile z(ρ)
m
2009 Nov. 10 2nd Mini Workshop on String Theory @ KEK 29
One point function and Chiral condensate
BH embedding
Minkowski embedding
Mateos et al. JHEP 0705:067, Fig. 4
(Hirayama-Kao-SK-Lin)
AdS-BH result
2~)(ρνρ +mzSolution near the boundary:
Parameters may be identified with
mmqqmTM q ln21~,~/ +ν
It shows the phase transition behavior similar to AdS-BH case.
T/M
mm ln21
+ν
There will be a chiral restoration!
2009 Nov. 10 2nd Mini Workshop on String Theory @ KEK 30
Conclusion
(Review) To describe string with accelerated end point, the generalized Rindler coordinates is useful.Checked that it has the boundary stress tensor corresponding to thermal conformal matter.Wilson loop shows strange behavior.We have calculated various quantities of holographic QCD-like model in the generalized Rindler space.The results quite resemble AdS-BH results.
2009 Nov. 10 2nd Mini Workshop on String Theory @ KEK 31
Open problems....(a lot)
•What is the difference of the coordinates choice?•Xiao’s metric gives similar results to BH case. Other choices?They do not have to be the same. Equivalence principle?Gravitational force and acceleration are not the same in thissetup...•How to interpret the Wilson loop and q-q bar potential?•We are still checking...
1. Behavior of the fluctuations on D7-brane2. Drag force?
2009 Nov. 10 2nd Mini Workshop on String Theory @ KEK 32
Future directions
Hagedorn temperature:
Acceleration in the finite temperature field theory
There is a limiting temperature in string theory. Is there any limiting acceleration??
Generalized Rindler space from AdS-BH metric?Temperature vs. Acceleration? Acceleration horizon covers or is covered by BH horizon? What happens??
Should be lots more.... Interesting to study!