Unruh effect and Holography Shoichi Kawamoto (National Taiwan Normal University) with Feng-Li...
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Transcript of Unruh effect and Holography Shoichi Kawamoto (National Taiwan Normal University) with Feng-Li...
Unruh effectand
Holography
Shoichi Kawamoto
(National Taiwan Normal University) with
Feng-Li Lin(NTNU), Takayuki Hirayama(Kyoto Sangyo U.)
and Pei-Wen Kao (Keio, Dept. of Math.)
HEP-QIS Joint Seminar at CYCU
2010 October 19 @ CYCU
(Based on arXiv:1001.1289)
2010 Oct. 19Shoichi Kawamoto (NTNU) 2
String Theory and AdS/CFT correspondence
String theory is a candidate of quantum gravity with strings being fundamental d.o.f.
Recently, strong coupling regime of a class of (conformal) field theory can be probed by using string theory (supergravity) on a curved b.g.
AdS/CFT correspondence This is a best understood example of holography.
D-dimensional QFT vs. D+1 dimensional gravity
We start with a brief review of them.
2010 Oct. 19Shoichi Kawamoto (NTNU) 3
String Theory
A tiny string is propagating in D-dimensional space-time (target space)
Consistent quantization D=10 (with supersymmetry)
Strings are very small (almost Planck length): Looking like “particles”
But it has more internal degrees of freedom: diversity of particles
Closed strings: Gravity multiplets (graviton, dilaton, ….)
Open strings: gauge multiplets (gauge fields, gaugino,…)
D-dim.
supergravity (SUGRA)
2010 Oct. 19Shoichi Kawamoto (NTNU) 4
Dp-branes as a boundary of open strings
Dirichlet p-brane (Dp-brane) is (1+p) dimensional object on which open strings can end.
Gauge theory(open string)
p dim
9-p dim
9 dim
Open strings: In (1+p) dim. with gauge fields
(1+p) dim. U(1) (SUSY) gauge theoryclosed strings (SUGRA)
There are also freely moving closed strings (SUGRA on flat 10D)
If N number of Dp-brane are on top of each other,
N Dp )()1( NUU N Gauge symmetry is enhanced:
We have U(N) Super Yang-Mills (SYM) in (1+p) dim. on N Dp-branes.
Note: it is a source of RR (p+2)-form field strength 1 ]1[p p dxAC
(Witten)
2010 Oct. 19Shoichi Kawamoto (NTNU) 5
Dp-brane as a classical solution of gravity
Curved space(Black p-brane)
RR-flux
The same charged object can be constructed as a classicalsolution of SUGRA.
Black p-brane solution
It... •has the same RR-flux•preserves the same SUSY in extremal case.•does not have gauge symmetry (no open strings)and has only gravity d.o.f. (closed strings)•is extended version of black “hole”.
Open-Closed duality:
It is believed that these two descriptions are the different viewpoints forthe same object.
It leads to the following “duality”
2010 Oct. 19Shoichi Kawamoto (NTNU) 6
AdS/CFT correspondence (p=3 case)
N D3-branes
Gauge theory(open string)
10D SUGRA Curved space(Black 3-brane)
decouple
Flat space
Maldacena limit (N !1 , ’ ! 0)
N=4 Supersymmetric Yang-Mills String (SUGRA) in AdS5 * S5
z=1 z=0 : AdS boundary
(Strong coupling regime: gYM2N=1) (weak curvature: L4=’2=1)
Best known case: D3-branes and N=4 super Yang-Mills theory
Correspondence: Symmetry, States, correlation functions,....
decouple
2010 Oct. 19Shoichi Kawamoto (NTNU) 7
AdS/CFT correspondence 2
Symmetry: )4()2,4( SUSO Conformal Symmetry and R-symmetry
Isometry of AdS5 and S5.
(Actually, full superconformal symmetry matches.)
)(),( 00bulkCFT
)()(04
xzxZez
xxxd
OCorrelation functions:
(GKPW)
source of an operator boundary value of gravity fields
A special example:A fundamental charge An open string
boundary
bulk
2010 Oct. 19Shoichi Kawamoto (NTNU) 8
Finite temperature
We can put a blackhole in the AdS background (AdS-BH solution).
The blackhole is at a finite temperature (Hawking temperature).
The corresponding gauge theory becomes finite temperature as well (with the same temperature). Finite temperature quantum field theory.
In quantum field theory, a temperature is measured for an accelerated observer.
Unruh Effect!!
Q: How is it looking like in the gravity side?
Before then, we will recall the Unruh effect...
Hrr
2010 Oct. 19Shoichi Kawamoto (NTNU) 9
Unruh(-Davies-De Witt-Fulling) effect
The world-line of the observer with a constant acceleration a is given by solving
The observer feels the temperature
maFx
xm
dt
d
21
the solution is given by hyperbolas
23
22
21
22 dxdxdxdtds
23
22
2222 dxdxddeds a
aeat a sinh1 aeax a cosh1
1
Coordinate transform
Rindler coordinates:
There is a convenient choice of coordinates.
2010 Oct. 19Shoichi Kawamoto (NTNU) 10
The Rindler coordinates as a comoving frame
~11 tx xt
23
22
2222 dxdxddeds a
Rindler coordinates:
LR RR
CDK
EDKt
x1 It covers the region (Right Rindler wedge)
The “time” translation is generated by the Killing vector
aeat a sinh1 aeax a cosh1
1 The world line with a constant has a constant
acceleration.
Accelerated observer in Minkowski space = Static observer in Rindler space(Comoving frame)
2010 Oct. 19Shoichi Kawamoto (NTNU) 11
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( xf I
Klein-Gordon equation:
complete sets I
IIiIIii fff )*2()2(*)1(
ijjiji ffff **,, 0, * ji ff
space-like hypersurface
Assume
2010 Oct. 19Shoichi Kawamoto (NTNU) 12
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 y 000 1)1(
1 iN
I
IiiN2
2)1(
2 00
Bogolubov transformation: I
IIiIIii aaa y)2(*)2()1( ˆˆˆ
VEV of the number operator is
But,
20 is an excited states with respect to the particles of (1).
Bogolubov coefficients
positive frequency modes
2010 Oct. 19Shoichi Kawamoto (NTNU) 13
Quantum Field Theory on Minkowski space
2D massless scalar field theory: An example
KG equaton: 0)(22 xxt tiikxM
k ef
4
1 k0
0
*ˆˆ)( Mk
Mk
Mk
Mk fafadkx y
Minkowski vacuum:M0 00ˆ M
Mka
right mover:
2010 Oct. 19Shoichi Kawamoto (NTNU) 14
Quantum Field Theory on Rindler space
LR RR
x1
Move to Rindler coordinates:
aeat a sinh1 aeax a cosh1
2222 ddeds a
0),(22
RR
R
iik
R
RRk ef
4
1
KG eq.
*ˆˆ)( RRk
RRk
RRk
RRkR RRRR
fafadkx y
RR
R
iik
R
LRk ef
4
1
*ˆˆ)( LRk
LRk
LRk
LRkR RRRR
fafadkx y
aeat a sinh1
aeax a cosh1
R0 00ˆ0ˆ RRRkR
LRk RR
aaThe Rindler vacuum is defined by
2010 Oct. 19Shoichi Kawamoto (NTNU) 15
Minkowski vacuum as a thermal state
Bogolubov transformation: yMk
LRkk
Mk
LRkk
LRk aadka
RRRˆˆˆ *,,,
a
ik
k
a
kk
ie R
aik
R
akR
kk
RR
R1
2
/2/
a
ik
k
a
kk
ie R
aik
R
akL
kk
RR
R1
2
/2/
*/ Rkk
akLkk R
R
Re
*/ Lkk
akRkk R
R
Re
So the expectation value of the number operators
(assume now the energy levels are discrete )Rii k
1
10000 /2
aMLiMM
RiM ie
NN
It represents the heat bath with the temperature 2
aT
The set LRk
RRk RR
ff , can be related to Minkovski ones.
Each of them cannot be written as Minkowski operators.
details
RRRRRRMRM OO 0ˆ000 For operator with Right Rindler modes:
2010 Oct. 19Shoichi Kawamoto (NTNU) 16
Notion of “vacuum” and “particles” are observer dependent.
Observer in an accelerated frame (Rindler observer) sees the vacuum of the inertia observer (Minkowski vacuum) as a thermal b.g.
This is due to having a “horizon” (Rindler horizon) and loosing the access to the other part of spacetime.
How is this effect looking like in the holographic dual theory?
Summary of the introduction
2010 Oct. 19Shoichi Kawamoto (NTNU) 17
Plan
1. Introduction: Review of AdS/CFT & Unruh Effect
2. Uniformly accelerated string and comoving frame
3. Investigating various quantities
4. Conclusion
2010 Oct. 19Shoichi Kawamoto (NTNU) 18
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
2010 Oct. 19Shoichi Kawamoto (NTNU) 19
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
GXXg ba abdet
2010 Oct. 19Shoichi Kawamoto (NTNU) 20
Comoving coordinates for uniformly accelerated string (Xiao)
25
222
223
22
21
2222
AdS 55
dRduu
RdxdxdxdtuRds
S
asinh22 aerat
Uniformly accelerated string in AdS space (3/3)
aeru 1
acosh221
aerax
Now the open string configuration: with
r
2010 Oct. 19Shoichi Kawamoto (NTNU) 21
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)
2010 Oct. 19Shoichi Kawamoto (NTNU) 22
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 temperature = Bulk Blackhole temperature
Note: The horizon appears in the radial direction. Different from the effect of the heavy object on the accelerated direction.
We will examine the thermal properties, and see what are similar to the case with BH.
2010 Oct. 19Shoichi Kawamoto (NTNU) 23
III. Calculation of various quantities
Boundary stress tensor Quark-anti Quark potential Some phase transition involving mesons
2010 Oct. 19Shoichi Kawamoto (NTNU) 24
Boundary stress tensor
We first look at the boundary stress tensor. (Balasubramanian-Kraus, Myers)
tot2lim
ST
r
where )(8
1
8
12
16
121
45tot
Rcc
Gxd
GRxd
GS
r MM
counter term
After eliminating the divergences, we get (HKKL)
trace of the extrinsic curvature of the boundary
Xiao’s metric (generalized Rindler): 423/),1,1,1,3( aNpT
Conformal thermal gas with the temperature 2
aT
T
BT
r
2010 Oct. 19Shoichi Kawamoto (NTNU) 25
Quark – anti Quark potential
1x
1a2aa
21 aaa
L
r0
We may calculate quark – anti quark potential in the accelerated frame.
Energy is given by that of the stretched string
1/a
)0,0),(,,( rrX String profile
Solution is given by )()(1
)(02
0
20
22
2
rhr
R
rh
rh
r
R
221)( rarh
comoving frame
configuration satisfying the boundary conditions
2010 Oct. 19Shoichi Kawamoto (NTNU) 26
Limiting acceleration and screening
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 may be screened (and no limiting acceleration difference).
Maybe, energy cannot reach the other end due to causality.
Energy(=total length)
2010 Oct. 19Shoichi Kawamoto (NTNU) 27
A look at mesons: 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
First we argue, what is the appropriate setup for accelerated mesons?
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
Holographic calculation will be thermal one. MRM 00 O
: An operator corresponding to an accelerated meson.
(static for accelerated observers)
(will be replaced with curved b.g.)
2010 Oct. 19Shoichi Kawamoto (NTNU) 28
D7-brane embedding (1/2)
26
25
23
222
223
22
222222
22 dwdwdd
w
Rdxdxedhdh
R
wds a
We work with the following coordinates.
26
25
222
42
22,
41,
4
4www
w
Rah
wa
wr
Ansatz: 0),( 65 wzw
2
53328
77 )(1 whhexdTS aD
D7 brane extends these 8 directions.
Then we solve the equation of motion with boundary conditions.
2010 Oct. 19Shoichi Kawamoto (NTNU) 29
D7-brane embedding (2/2)
Minkowski embedding
BH embedding
horizon
)(1))ln((1~)( 22
2 OOmz
Asymptotic solution near the boundary is
Regular solution: )(, mm
D7 reaches to the center: Minkowski embeddingD7 terminates at horizon: Blackhole embedding
In general, starting with arbitrary m and ,the solution will diverge.
D7-brane profile z()
m
2010 Oct. 19Shoichi Kawamoto (NTNU) 30
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~)(
mzSolution near the boundary:
Parameters may be identified with
mmqqmTM q ln2
1~,~/
It shows the phase transition behavior corresponding to “meson” melting.
T/M
mm ln2
1
2010 Oct. 19Shoichi Kawamoto (NTNU) 31
Fluctuations on D7-brane
Now we consider transverse fluctuations of D7.
w5()
h
),,,()()(5 yzw
We can calculate the retarded Green’s function for this fluctuation modeand then derive the spectral function:
)( Im)( RG
1/T
2010 Oct. 19Shoichi Kawamoto (NTNU) 32
Only results: Spectral functions for mesons
8/1h
12/1h16/1h
10/1h
temperature low
high
In low temperature,there are sharp peaks(stable mesons)
For higher temperature,spectral function becomesfeatureless (meson dessociation)
2010 Oct. 19Shoichi Kawamoto (NTNU) 33
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 screening behavior. We have calculated various quantities of holographic QCD-like model in the generalized
Rindler space. The results quite resemble AdS-BH results.
2010 Oct. 19Shoichi Kawamoto (NTNU) 34
Thank you