1. Unruh effect and Holography
HEP-QIS Joint Seminar at CYCU
Unruh effect and 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.)
(Based on arXiv: )
2010 October CYCU

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. 19
Shoichi Kawamoto (NTNU)

3. Shoichi Kawamoto (NTNU)
String Theory
D-dim.
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, ….)
supergravity (SUGRA)
Open strings: gauge multiplets (gauge fields, gaugino,…)
2010 Oct. 19
Shoichi 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.
9 dim
p dim
Open strings: In (1+p) dim. with gauge fields
closed strings (SUGRA)
9-p dim
(1+p) dim. U(1) (SUSY) gauge theory
Gauge theory
(open string)
There are also freely moving closed strings
(SUGRA on flat 10D)
Note: it is a source of RR (p+2)-form field strength
If N number of Dp-brane are on top of each other,
(Witten)
N Dp
Gauge symmetry is enhanced:
We have U(N) Super Yang-Mills (SYM) in (1+p) dim. on N Dp-branes.
2010 Oct. 19
Shoichi Kawamoto (NTNU)

5. Dp-brane as a classical solution of gravity
Curved space
(Black p-brane)
The same charged object can be constructed as a classical
solution 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”.
RR-flux
Open-Closed duality:
It is believed that these two descriptions are the different viewpoints for
the same object.
It leads to the following “duality”
2010 Oct. 19
Shoichi Kawamoto (NTNU)

6. Shoichi Kawamoto (NTNU)
AdS/CFT correspondence (p=3 case)
Best known case: D3-branes and N=4 super Yang-Mills theory
N D3-branes
10D SUGRA
Curved space
(Black 3-brane)
Flat space
decouple
Maldacena limit
(N ! , a’ ! 0)
decouple
Gauge theory
(open string)
z=1
z=0 : AdS boundary
String (SUGRA) in AdS5 * S5
N=4 Supersymmetric Yang-Mills
(weak curvature: L4=la’2=1)
(Strong coupling regime: l=gYM2N=1)
Correspondence: Symmetry, States, correlation functions,....
2010 Oct. 19
Shoichi Kawamoto (NTNU)

7. AdS/CFT correspondence 2
Conformal Symmetry and R-symmetry
Symmetry:
Isometry of AdS5 and S5.
(Actually, full superconformal symmetry matches.)
(GKPW)
Correlation functions:
source of an operator
boundary value of gravity fields
A special example:
A fundamental charge
An open string
boundary
bulk
2010 Oct. 19
Shoichi Kawamoto (NTNU)

8. Shoichi Kawamoto (NTNU)
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...
2010 Oct. 19
Shoichi 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 solution is given by hyperbolas
The observer feels
the temperature
There is a convenient choice of coordinates.
Coordinate transform
Rindler coordinates:
2010 Oct. 19
Shoichi Kawamoto (NTNU)

10. The Rindler coordinates as a comoving frame
EDK
Rindler coordinates: LR RR x1
It covers the region
(Right Rindler wedge)
The “time” translation is generated by the Killing vector
CDK
The world line with a constant x has a constant acceleration.
Accelerated observer in Minkowski space = Static observer in Rindler space
(Comoving frame)
2010 Oct. 19
Shoichi Kawamoto (NTNU)

11. Vacuum, Particles and Observers
Let us briefly discuss how the accelerated observer feels a finite temperature.
Vacuum is observer dependent.
Klein-Gordon equation:
Assume
two complete sets of solutions:
complete sets
space-like hypersurface
2010 Oct. 19
Shoichi Kawamoto (NTNU)

12. Vacuum, Particles and Observers II
Quantum field can be expanded as
positive frequency modes
Bogolubov coefficients
Bogolubov transformation:
Vacuum:
defined by
VEV of the number operator is
But,
is an excited states with respect to the particles of (1).
2010 Oct. 19
Shoichi Kawamoto (NTNU)

13. Quantum Field Theory on Minkowski space
2D massless scalar field theory: An example
KG equaton:
right mover:
Minkowski vacuum:
2010 Oct. 19
Shoichi Kawamoto (NTNU)

14. Quantum Field Theory on Rindler space
Move to Rindler coordinates:
KG eq. LR RR x1
The Rindler vacuum
is defined by
2010 Oct. 19
Shoichi Kawamoto (NTNU)

15. Minkowski vacuum as a thermal state
Each of them cannot be written as Minkowski operators.
The set
can be related to Minkovski ones.
Bogolubov transformation:
So the expectation value of the number operators
(assume now the energy levels are discrete )
It represents the heat bath with the temperature
For operator with Right Rindler modes:
details
2010 Oct. 19
Shoichi Kawamoto (NTNU)

16. Summary of the introduction
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?
2010 Oct. 19
Shoichi Kawamoto (NTNU)

17. Shoichi Kawamoto (NTNU)
Plan
1. Introduction: Review of AdS/CFT & Unruh Effect
2. Uniformly accelerated string and comoving frame
3. Investigating various quantities
4. Conclusion
2010 Oct. 19
Shoichi Kawamoto (NTNU)

18. Uniformly accelerated string in AdS space (1/3)
Let us consider a uniformly accelerated particle (quark) on the boundary field theory. a
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!!!
2010 Oct. 19
Shoichi 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 a
boundary
with boundary condition:
and solve the e.o.m.
Exact solution to NG action has been found (Xiao)
2010 Oct. 19
Shoichi Kawamoto (NTNU)

20. Comoving coordinates for uniformly accelerated string (Xiao)
Uniformly accelerated string in AdS space (3/3)
Comoving coordinates for uniformly accelerated string (Xiao)
Now the open string configuration:
with a r
2010 Oct. 19
Shoichi Kawamoto (NTNU)

21. Generalized Rindler space (Xiao’s metric)
Illustrate how the new coordinates covers a part of the original AdS5
(horizon = Rindler horizon + AdS horizon)
right Rindler wedge with 0 < r < a-1
constant r surface
2010 Oct. 19
Shoichi 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. 19
Shoichi Kawamoto (NTNU)

23. III. Calculation of various quantities
Boundary stress tensor
Quark-anti Quark potential
Some phase transition involving mesons
2010 Oct. 19
Shoichi Kawamoto (NTNU)

24. Boundary stress tensor
We first look at the boundary stress tensor.
(Balasubramanian-Kraus, Myers) r
where
trace of the extrinsic curvature of the boundary
counter term
After eliminating the divergences, we get (HKKL)
Xiao’s metric (generalized Rindler):
Conformal thermal gas with the temperature
2010 Oct. 19
Shoichi Kawamoto (NTNU)

25. Quark – anti Quark potential
comoving frame e r0
1/a
We may calculate quark – anti quark potential in the accelerated frame.
Energy is given by that of the stretched string
String profile
Solution is given by
configuration satisfying the boundary conditions
2010 Oct. 19
Shoichi Kawamoto (NTNU)

26. Limiting acceleration and screening
Compare the energy to the straight
line configuration (green ones). L
Energy (=total length) r0
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.
2010 Oct. 19
Shoichi 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 D7
“meson” excitations
8,9
First we argue, what is the appropriate setup for accelerated mesons?
fundamental matters
4,5,6,7
1. Moved to Xiao’s metric (generalized Rindler coord.) D3
2. Then embedding D7-brane to be static on this coordinate system.
(will be replaced with curved b.g.)
(static for accelerated observers)
This will define
: An operator corresponding to an accelerated meson.
Holographic calculation will be thermal one.
2010 Oct. 19
Shoichi Kawamoto (NTNU)

28. Shoichi Kawamoto (NTNU)
D7-brane embedding (1/2)
We work with the following coordinates.
D7 brane extends these 8 directions.
Ansatz:
Then we solve the equation of motion with boundary conditions.
2010 Oct. 19
Shoichi Kawamoto (NTNU)

29. Shoichi Kawamoto (NTNU)
D7-brane embedding (2/2)
D7-brane profile z(r)
Asymptotic solution near the boundary is
Minkowski embedding
In general, starting with arbitrary m and n,
the solution will diverge.
BH embedding
horizon m
Regular solution:
D7 reaches to the center: Minkowski embedding
D7 terminates at horizon: Blackhole embedding
2010 Oct. 19
Shoichi Kawamoto (NTNU)

30. One point function and Chiral condensate
(Hirayama-Kao-SK-Lin)
T/M
Solution near the boundary:
Parameters may be identified with
It shows the phase transition behavior corresponding to “meson” melting.
Minkowski embedding
AdS-BH result
Mateos et al. JHEP 0705:067, Fig. 4
BH embedding
2010 Oct. 19
Shoichi Kawamoto (NTNU)

31. Fluctuations on D7-brane
w5(r)
Now we consider transverse fluctuations of D7.
1/T rh r
We can calculate the retarded Green’s function for this fluctuation mode and then derive the spectral function:
2010 Oct. 19
Shoichi Kawamoto (NTNU)

32. Only results: Spectral functions for mesons
temperature low
In low temperature, there are sharp peaks
(stable mesons)
high
For higher temperature,
spectral function becomes
featureless (meson dessociation)
2010 Oct. 19
Shoichi Kawamoto (NTNU)

33. Shoichi Kawamoto (NTNU)
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. 19
Shoichi Kawamoto (NTNU)

34. Shoichi Kawamoto (NTNU)
Thank you
2010 Oct. 19
Shoichi Kawamoto (NTNU)