a holographic dual of bjorken flow -...
TRANSCRIPT
![Page 1: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/1.jpg)
Shin Nakamura
(Center for Quantum Spacetime (CQUeST) , Sogang Univ.)
Based on
S. Kinoshita, S. Mukohyama, S.N. and K. Oda,
arXiv:0807.3797
A Holographic Dual of
Bjorken Flow
![Page 2: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/2.jpg)
Motivations and background
![Page 3: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/3.jpg)
Motivation
RHIC: Relativistic Heavy Ion Collider
(@ Brookhaven National Laboratory)
http://www.bnl.gov/RHIC/inside_1.htm
Heavy ion:
e.g. 197Au
~200GeV.NNS
Quark-gluon
Plasma (QGP)
is observed.
![Page 4: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/4.jpg)
Also at LHC
Similar exp. at
• FAIR@ GSI
• NICA@ JINR
http://aliceinfo.cern.ch/Public/en/Chapter4/Chapter4Gallery-en.html
ALICE
ATLAS
CMS
ALICE
![Page 5: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/5.jpg)
Quark-gluon plasma (QGP) is created
as a time-dependent system
http://www.bnl.gov/RHIC/heavy_ion.htm
![Page 6: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/6.jpg)
QGP is a strongly interacting system
RHIC Scientists Serve Up "Perfect" LiquidNew state of matter more remarkable than predicted
-- raising many new questions
April 18, 2005
BNL press release:
Small viscosity: strong coupling
Perturbative QCD says:14
3
ln
gg
T
Perfect fluidity: zero (very small) viscosity
)1.0(Os
Teaney(2003)
Hirano-Gyulassy(2005)
Shear viscosity
The smallest value which has ever been observed.
1.04
1
s
from AdS/CFT
Kovtun-Son-Starinets (2004)
![Page 7: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/7.jpg)
Shear viscosity
L
v
A
F
L
v
v=0
fluid
Viscosity: 粘性度
Shear: 剪断
If the interaction is strong:
• the cross section is large,
• the mean free path is short,
• the momentum transfer is suppressed.
![Page 8: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/8.jpg)
Frameworks for strongly interacting
YM-theory plasma
• Lattice QCD: a first-principle computation
However, it is technically difficult to analyze
time-dependent systems.
• (Relativistic) Hydrodynamics
• It works well. However, this is an effective theory
for macroscopic physics. (entropy, temperature, pressure, energy density,….)
• Information on microscopic physics is lost.(correlation functions of operators, equation of state,
transport coefficients such as viscosity,…)
Inputs of hydrodynamics has to be given by other theories.
![Page 9: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/9.jpg)
Alternative framework: AdS/CFT
AdS/CFT
Both macroscopic and microscopic physics
of (some) strongly interacting gauge-theory
plasma can be described within a single
framework of AdS/CFT.
AdS/CFT may be useful in the research of
YM-theory plasma.
![Page 10: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/10.jpg)
However,
Time-dependent AdS/CFT has not yet been
established completely.
We construct a time-dependent AdS/CFT that describes
the Bjorken flow of plasma of large-Nc, strongly coupled
N=4 super Yang-Mills theory.
Our work
A standard model for the expanding QGP
We deal with N=4 SYM
instead of QCD.
![Page 11: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/11.jpg)
Bjorken flow:
a simple standard model of the
expansion of QGP
![Page 12: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/12.jpg)
Bjorken flow (Bjorken 1983)
• (Almost) one-dimensional expansion.
• We have boost symmetry in the CRR.
Relativistically accelerated heavy nuclei
After collision
Velocity of light
Central Rapidity Region (CRR)
Time dependence of the physical quantities are written by the proper time.
Velocity of light
“A standard model”
of QGP expansion
![Page 13: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/13.jpg)
Quark-gluon plasma (QGP) as a
one-dimensional expansion
http://www.bnl.gov/RHIC/heavy_ion.htm
![Page 14: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/14.jpg)
Local rest frame(LRF)
τ=const.
22222
dxdydds yxyt sinh,cosh 1
Minkowski spacetime
x1
t
Rapidity
Proper-time
The fluid looks static
on this frame
Boost invariance:y-independence
![Page 15: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/15.jpg)
Relativistic Hydrodynamics
• We take the local rest frame.
Our case:
Then, the stress tensor becomes diagonal:
),,,( 32 xxy
RapidityProper-time
32
32
342
000
000
00)(0
000
p
p
pT
energy density
shear viscositypressure
The bulk viscosity is zero.
22222
dxdydds
yxyt sinh,cosh 1
![Page 16: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/16.jpg)
3 independent components: xxyy TTT ,,
2 independent constraints:
0
T “Conformal invariance”
(equation of state)
Energy-momentum conservation
(hydrodynamic equation)
Only 1 independent component:
3/p
0 T
TTT
TTT
xx
yy
2
1
2
)()( T
![Page 17: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/17.jpg)
To obtain a diff. equation
)()(
3
4)(2
d
d
32
2
1 pTxx
)(3)( p
34 , TT Assumption:
only one quantity (temperature)
to determine the scale 4/3
00 0
We obtain a differential eq. for ε(τ)
![Page 18: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/18.jpg)
TTT
TTT
T
xx
yy
2
1
21
2
2
0
3/40
Solutionimportant
T~τ-1/3
Once the parameters (transport coefficients) are
given, Tμν(τ) is completely determined.
expansion w.r.t
τ-2/3
But, hydro cannot determine them.
![Page 19: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/19.jpg)
Towards time-dependent AdS/CFT
![Page 20: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/20.jpg)
AdS/CFT(Weak version)
Classical Supergravity on
4dim. Large-Nc SU(Nc)
N=4 Super Yang-Mills
at the large „t Hooft coupling
5
5 SAdS
conjecture
=
Maldacena (1997)
Strongly interacting quantum YM !!
(Anti de Sitter) (Conformal
Field Theory)
![Page 21: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/21.jpg)
AdS/CFT at finite temperature
Classical Supergravity on
AdS black hole×S5
4dim. Large-Nc strongly coupled
SU(Nc) N=4 SYM at finite temperature
(in the deconfinement phase).
conjecture
=
Witten (1998)
![Page 22: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/22.jpg)
AdS/CFT dictionary
On-shell action of
the 5d gravity theory
Effective action of
the 4d gauge theory
Expectation value of an
operator by differentiating
w.r.t. the source.A function of the
boundary conditions
5d AdS
A function of sources
For example, the boundary metricg~
The stress tensor of the plasma is given by
the geometry.
T
4d stress tensor
![Page 23: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/23.jpg)
Hydrodynamics
Hydrodynamics describes dynamics of
conserved currents such as stress tensor.
We know that the 4d stress tensor can be
read from the dual geometry.
Hydrodynamics may be given by
the dynamics of the dual geometry.
![Page 24: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/24.jpg)
Towards time-dependent AdS/CFT
![Page 25: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/25.jpg)
How to obtain the geometry?
The bulk geometry is obtained by solving
the equations of motion of super-gravity
with appropriate boundary data.
5d Einstein gravity
with Λ<0
• The boundary metric is that of the
comoving frame: 22222
dxdydds
• The 4d stress tensor is diagonal on this frame.
Bjorken‟s case:
We set (the 4d part of) the bulk metric diagonal.
(ansatz)
This tells our fluid undergoes the Bjorken flow.
![Page 26: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/26.jpg)
Time-dependent AdS/CFT
Earlier works
![Page 27: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/27.jpg)
A time-dependent AdS/CFT
A time-dependent geometry that describes Bjorken flow
of N=4 SYM fluid was first obtained within a late-time
approximation by Janik-Peschanski.
Janik-Peschanski, hep-th/0512162
They have used Fefferman-Graham coordinates:
.............)(~)(~),(~
),(~
4)4()0(
2
2
2
zggzg
z
dzdxdxzgds
stress tensor of YM4d geometry (LRF)
boundary condition to 5d Einstein‟s equation with Λ<0
geometry as a solution
![Page 28: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/28.jpg)
Unfortunately, we cannot solve exactly
They emploed the late-time approximation:
fixed with ,3/1
vz
x
x
y
y ggg ~,~,~
........)()( 3/2)2()1( vfvf
have the structure of
We discard the higher-order terms.
Janik-Peschanski
hep-th/05121624/p
z
4/)4( p
![Page 29: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/29.jpg)
Janik-Peschanski‟s result
2
2222
3
2
3
2
3
2
2 )1(1
)1(1 4
4
4
z
dzxddyd
zds
z
z
z
...)( 3/4
0 from relativistic hydrodynamics
The statement
,3/4 with ,)( 0 pp
If we start with unphysical assumption like
the obtained geometry is singular:
RR at the point gττ=0. (horizon?)
Regularity of the spacetime determines
the correct parameters.
5th coordinate
![Page 30: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/30.jpg)
Many success
• 1st order: Introduction of the shear viscosity:
• 2nd order: Determination of from the
regularity:
• 3rd order: Determination of the relaxation
time from the absence of the power
singularity:
4
1
S
S.N. and S-J.Sin, hep-th/0607123
Janik, hep-th/0610144
Heller and Janik, hep-th/0703243
For example:
same as KSS
![Page 31: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/31.jpg)
But, a serious problems came out.
• An un-removable logarithmic singularity
appears at the third order. (Benincasa-Buchel-heller-Janik, arXiv:0712.2025)
This suggests that the late-time expansion they are
using is not consistent.
![Page 32: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/32.jpg)
We have many questions, too.
• Is the time-dependent geometry really
a black hole?
No a priori reason to forbid the physics
because of the singularity.
We will try to answer these questions and
resolve the problem of the logarithmic divergence.
From the viewpoint of the original string theory,
the singularity merely means a breakdown of
the super-gravity approximation.
• Why do we need the regularity?String correction:
e.g. RRls
4
![Page 33: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/33.jpg)
Our work
![Page 34: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/34.jpg)
What is black hole?
Definition: presence of an event horizon.
It is not easy to examine the presence of event horizon
especially in time-dependent systems, in general.
(Future) Event horizon:
No signals from inside the (future) event horizon
can reach the observer at the future infinity.
We need to examine the global structure of the
spacetime to determine the location of the horizon.
![Page 35: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/35.jpg)
Another horizon
Apparent horizon:
The boundary surface of a region in which the
light ray directed outward is moving inward.
• This is defined locally.
• But, this is a coordinate-dependent concept.
• This can be different from the event horizon in
time-dependent systems.
trapped region
![Page 36: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/36.jpg)
Some comments
• Usually, if we have an apparent horizon, an
event horizon is located outside, or on top of it.
(Ref. Hawking-Ellis)
Let us examine the apparent horizon of the dual
geometry.
The presence of an apparent horizon is a
sufficient condition for the presence of
an event horizon.
![Page 37: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/37.jpg)
Location of the apparent horizonThe location of the apparent horizon is given by
,0Fe
,log ggggg yyz
zz
“expansion”
derivative along the null direction
volume element of the 3d surface
0
0 : un-trapped region
: trapped region
Normalization:
we may omit this factor
in this talk.
![Page 38: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/38.jpg)
However, for Janik-Peschanski‟s
Geometry
0/3
/3
2
92
3/44
3/44
z
zeF
Always in the un-trapped region!
3/4at p
What is going on?
Janik-Peschanski (and all other related works) are based
on Fefferman-Graham coordinates.
2
22222
z
dzCdxBdyAdds
No trapped region !
![Page 39: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/39.jpg)
Static AdS-BH
2
22222
2
2
2 )1(1
)1(140
4
40
4
40
4
z
dzdxdyd
zds
z
z
z
z
z
z
Schwarzschild coordinates
2
1
4
4
0222222
4
4
022 11 drr
rrdxdyrd
r
rrds
0
2
0
222
0 //
z
zzzzr
Fefferman-Graham coordinates
0
0
2r
z
Only outside the
horizon!
![Page 40: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/40.jpg)
AdS-BH
boundary
singularity
dynamical
apparent horizon
trapped region
un-trapped region
![Page 41: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/41.jpg)
What we have found so far:
• The Fefferman-Graham coordinates are not a good coordinate system for analysis of horizons (no trapped region included).
• The geometry inside the horizon cannot be obtained from the Janik‟s results, anyway.
• We need a better coordinate system.
![Page 42: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/42.jpg)
Better coordinates?
boundary
singularity
dynamical
apparent horizon
Eddington-Finkelstein
coordinates
trapped region
un-trapped region
Cf.
Bhattacharyya-Hubeny-Minwalla-Rangamani (0712.2456)
Bhattacharyya et. al. (0803.2526, 0806.0006)
![Page 43: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/43.jpg)
Eddington-Finkelstein coordinates
Static AdS-BH:
222
4
4
022 21 xdrdtdrdtr
rrds
• There is no coordinate singularity.
• The trapped region and the un-trapped region
are on the same coordinate patch.
(We can safely analyze the location of the horizon.)
At least for the static case,
![Page 44: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/44.jpg)
Our proposal
Parametrization of the dual geometry:2222222 ~~2~ xdgrdygrdrddgrds xxyy
We assume they depend only on τ, because of the symmetry.
We solve them under the following boundary condition:
,1~,~,1~ 2 xxyy ggg at the boudary (r= ∞).
boundary metric: 22222
dxdydds
The 5d Einstein‟s eq. gives differential equations of .~g
![Page 45: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/45.jpg)
Asymptotic solution around the
boundary
...1~
...22~
....1~
4
43
212
4
1
4
43222
4112
424
1
2
2
rffrrhg
rffrhrhg
rfrhrhg
hxx
yy
h
where h and f are functions of τ.
Only two functions f(τ) and h(τ) so far….
![Page 46: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/46.jpg)
The stress tensor
The 4d stress tensor is read from the geometry in terms of
the boundary extrinsic curvature.
(Ref. Balasubramanian-Kraus, hep-th/9902121.)
For our case:
How about h?
h(τ) turns out to be a gauge degree of freedom.
)( rr does not modify the structure of
2222222 ~~2~ xdgrdygrdrddgrds xxyy
r
NGKKrT c
21
4
2 32
2
)(2
1)(
)()(
)(
0
2
0
0
ffT
ffT
fT
xx
yy
consistent with hydro eq.
![Page 47: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/47.jpg)
The hydrodynamic equation and the
equation of state in the gravity dual
The 5d Einstein‟s equation around the boundary
yields the hydrodynamic equation
(and the equation of state).
See also, Bhattacharyya-Hubeny-Minwalla-Rangamani (0712.2456)
How much does the gravity know the fluid dynamics
of the YM plasma?
Einstein‟s eq. 0 T
![Page 48: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/48.jpg)
Then, how to obtain the transport
coefficients?
• We need a global (analytic in r) solution.
222212222222 )(12
xderdyrerdrddards ccb
• A better parametrization:
,0,0,1 cbaThe boundary condition: at r= ∞.
Can we determine f(τ) as a function of τ?
(Yes.)
Let us solve the Einstein‟s equation.
![Page 49: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/49.jpg)
Late-time approximation
It is very difficult to obtain the solution analytic in τ.
We introduce a late-time approximation by making an analogy
with what Janik-Peschanski did on the FG coordinates.
Janik-Peschanski:
τ-2/3 expansion with zτ-1/3 = v fixed.
Now, r ~ z-1.
Let us employ τ-2/3 expansion with rτ1/3 = u fixed.
Our late-time approximation
![Page 50: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/50.jpg)
More explicitly,
222212222222 )(12
xderdyrerdrddards ccb
We solve the differential equations for a(τ,u), b(τ,u), c(τ,u)
order by order:
......)()()(),( 3/4
2
3/2
10 uauauaua
(similar for b and c)zeroth order first order second order
(u=rτ1/3)
![Page 51: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/51.jpg)
The zeroth-order solution
)(2)(
122222
43/1
422
xddyrdrdd
r
wrds
This is u
• This reproduces the correct zeroth-order stress tensor
of the Bjorken flow.
3/40
1
T
• We have an apparent horizon.
4412
9wueF
trapped region if u<w.
The location of the apparent horizon: u=w+O(τ-2/3 )
2
2
0
4
8
3
cN
w
![Page 52: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/52.jpg)
The (event) horizon is necessary
)(9
58 3/2
8
82
O
u
wR
We have a physical singularity at the origin.
However, this is hidden by the apparent horizon
at u=w hence the event horizon (outside it).
OK, from the viewpoint of the cosmic censorship
hypothesis.
Not a naked singularity.
![Page 53: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/53.jpg)
The first-order solution
222212222222 )(12
xderdyrerdrddards ccb
uuw
wuwuuww
c
ub
u
uwwua
3244
2
21
21
21
1
11
5
4
0
4
1
4
11
10 )1log(
)log()log()arctan(3
1
1
3)1(
3
2
gauge degree of
freedom
c1 is regular at u=w, only when .3
10
w
![Page 54: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/54.jpg)
Regularity of c1 is necessary.
We can show
)(regular)(6
31
)(12
31
)(regular2
1
3/2
2
0
2
0
3/2
11
Owuw
w
wuw
w
Ocu
cNNR y
y
2
2,0,0,0,1
2
1 2arN : a regular space-like
unit vector
Riemann tensor projected onto a regular orthonormal basis
This has to be regular to make the geometry regular.
![Page 55: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/55.jpg)
What is this value?
422
8
3TNc 322
2
1TNs c
Gubser-Klebanov-Peet, hep-th/9602135
First law of thermodynamics
4/3
00 0
2
2
0
4
8
3
cN
w
Our definition and result:
ws
034
1
Combine all of them:
w3
10
4
1
sThe famous ratio by Kovtun-Son-Starinets (2004)
![Page 56: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/56.jpg)
Second-order results:
• We have obtained the solution explicitly, but it is
too much complicated to exhibit here.
• From the regularity of the geometry,
“relaxation time” is uniquely determined.
consistent with Heller-Janik, Baier et. al., and Bhattacharyya et. al.
2nd-order transport coefficient
)(regular21
)(3
)19(4 23/422
02
O
wuwu
wR•
![Page 57: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/57.jpg)
![Page 58: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/58.jpg)
Second-order results:
• We have obtained the solution explicitly, but it is
too much complicated to exhibit here.
• From the regularity of the geometry,
“relaxation time” is uniquely determined.
consistent with Heller-Janik, Baier et. al., and Bhattacharyya et. al.
2nd-order transport coefficient
)(regular21
)(3
)19(4 23/422
02
O
wuwu
wR•
![Page 59: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/59.jpg)
Second-order results:
• We have obtained the solution explicitly, but it is
too much complicated to exhibit here.
• From the regularity of the geometry,
“relaxation time” is uniquely determined.
consistent with Heller-Janik, Baier et. al., and Bhattacharyya et. al.
2nd-order transport coefficient
)(regular21
)(3
)19(4 23/422
02
O
wuwu
wR•
![Page 60: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/60.jpg)
All-order results:
• We can always choose (the combination of) the n-th order transport coefficients in such a way that the dual geometry is regular at u=w to the n-th order.
• The geometry becomes singular at u=w if we take other value of (the combination of) the n-th order transport coefficients.
• This means that there is no logarithmic singularity which meant the inconsistency of the analysis on the FG coordinates.
We have shown by using induction that:
Our model is totally consistent and healthy!
![Page 61: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/61.jpg)
How the induction works
n-th order Einstein‟s equation:
Diff eq. for n-th order metric
= source which contains only
k(<n)-th order metric
We can always choose an integration constant
(that corresponds to a transport coeff.)
to make cn regular.
![Page 62: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/62.jpg)
All-order results:
• We can always choose (the combination of) the n-th order transport coefficients in such a way that the dual geometry is regular except at the origin.
• The geometry becomes singular at u=w if we take other value of (the combination of) the n-th order transport coefficients.
• This means that there is no logarithmic singularity which meant the inconsistency of the analysis on the FG coordinates.
We have shown by using induction that:
Our model is totally consistent and healthy!
![Page 63: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/63.jpg)
Coordinate transformation cannot
change physics
• Regular coordinate transformation cannot
change physics.
• However, the coordinate transformation
from Janik‟s coordinates (FG coordinates)
to ours (EF coordinates) is singular at the
horizon.
![Page 64: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/64.jpg)
Area of the apparent horizon
....2log62
24
1
2
11 3/4
2
3/23 ww
wAap
• This is consistent with the time evolution of the
entropy density to the first order.
• There is some discrepancy at the second order.
However, it does not mean inconsistency immediately.
)(2
31 3/43/20
OSS
w3
10
From Hydro.
![Page 65: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/65.jpg)
Non-staticity of the loal geometry
Projected Weyl tensor
....3
3
4
....3
4
3/2
4
4
0
5
4
4
4
3/2
5
4
4
4
1
1
21
21
u
w
u
w
u
wC
u
w
u
wC
yx
yx
xx
xx
An-isotropy evolves in time.
The dual geometry is not locally static,
if we include dissipation.
![Page 66: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/66.jpg)
What we have done:
• We constructed a consistent gravity dual
of the Bjorken flow for the first time.(cf. Heller-Loganayagam-Spalinski-Surowka-Vazquez,
arXiv:0805.3774)
• Our model is a concrete well-defined
example of time-dependent AdS/CFT
based on a well-controled approximation.
![Page 67: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/67.jpg)
Hydrodynamics
Time evolution of the stress tensor
• hydrodynamic equation
(energy-momentum
conservation)
• equation of state
(conformal invariance)
• transport coefficients
Our model
5d Einstein‟s eq. at
the vicinity of the
boundary
Reguarity around
the horizon
Related to local thermal equilibrium
![Page 68: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/68.jpg)
Discussion
• The definition of the late-time approximation
is a bit artificial.
(τ-2/3 expansion with rτ1/3 = u fixed.)
We are trying to “derive” it purely within
the gravity theory.
Attractor of the differential equation?
![Page 69: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/69.jpg)
Discussion
• At this stage, the connection among our method
and other methods are not clear.
• Kubo formula:
)0,0(),,(2
1lim 4
0xyxy
ti TxtTexd
• Quasi normal modes
In this case, we impose the “ingoing boundary
condition” at the horizon.
regularity?
![Page 70: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/70.jpg)
Einstein + Penrose > Kubo + Landau
久保亮五
My hope
I am glad if string (gravity) theory can say
something nontrivial to
• hydrodynamics
• QGP
• non-equilibrium physics
• plasma instability
• turbulence
• …….
![Page 71: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/71.jpg)
Future directions
• Inclusion of R-charge.
• Can we observe plasma instability?
• What happens if we take τsmall enough?
• Breakdown of the late-time approximation.
• Breakdown of local thermal equilibrium?
• Appearance of turbulence?
• ……..
• A similar analysis based on the Sakai-Sugimoto
model.
![Page 72: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/72.jpg)
A message
• We have a very good, challenging problem
which is connected the experiments at RHIC
and LHC.
• The theoretical framework is deeply
overlapped with nuclear science, string theory,
general relativity, fluid dynamics, and perhaps
with non-equilibrium statistical physics.
Now it is time to collaborate
beyond the research fields.
![Page 73: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/73.jpg)
Supplement
![Page 74: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/74.jpg)
Some thought on the regularity
Cosmic censorship hypothesis:
Naked singularity is not created by any
physical process in the gravity theory.
The “plasma” which corresponds to the
geometry with a naked singularity is not
created by any physical process of the
YM theory.
If you find a naked singularity, such a plasma
(with your parameter) cannot be realized by
any physical process.
(Penrose, 1969)
Singularity which is not
covered by the event
horizon.
![Page 75: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/75.jpg)
Various quantities from the geometry
Stefan-Boltzmann: )(8
3 422 Hc TN
)(
2
3122
324
3/43/2
0
04/3
0
4/34/12
O
N
G
AS c
Entropy creation:
From hydrodynamics:
)(2 3/43/2
0
0
OT
SS
Not only consistent with hydro but also
more information in the holographic dual.
Integration constant is given.
Numerical coefficient is given.
![Page 76: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/76.jpg)
Entropy creation
)(23
4)( 3/43/2
0
0
O
TS
TdS
The entropy (per unit volume on the LRF) at the
infinitely far future is not determined in this framework.
(Integration constant)
AdS/CFT gives more information.
The dissipation creates the entropy.
![Page 77: A Holographic Dual of Bjorken Flow - research.ipmu.jpresearch.ipmu.jp/seminars/pdf/20080919SNakamura.pdf · YM-theory plasma •Lattice QCD: a first-principle computation However,](https://reader035.vdocuments.mx/reader035/viewer/2022081406/5f1064a87e708231d448e3b5/html5/thumbnails/77.jpg)
We need microscopic theory.
Kubo formula:
)0,0(),,(2
1lim 4
0xyxy
ti TxtTexd
We need to compute the two-point function
of the current operator.
Obtainable only from the microscopic theory.