Unruh effect and Holography Shoichi Kawamoto (National Taiwan Normal University) with Feng-Li Lin(NTNU), Takayuki Hirayama(Kyoto Sangyo U.) and Pei-Wen

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.


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)


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)


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”


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


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


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


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.


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


1 The world line with a constant has a constant

acceleration.

Accelerated observer in Minkowski space = Static observer in Rindler space(Comoving frame)


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


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


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:


Quantum Field Theory on Rindler space

LR RR

x1

Move to Rindler coordinates:


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


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:


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


Plan

1. Introduction: Review of AdS/CFT & Unruh Effect

2. Uniformly accelerated string and comoving frame

3. Investigating various quantities

4. Conclusion


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



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


Comoving coordinates for uniformly accelerated string (Xiao)

25

222

223

22

21

2222

AdS 55

dRduu

RdxdxdxdtuRds

S

asinh22 aerat


aeru 1

acosh221

aerax

Now the open string configuration: with

r


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)


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.


III. Calculation of various quantities

Boundary stress tensor Quark-anti Quark potential Some phase transition involving mesons


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


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


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)


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.)


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.


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


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


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


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)


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.


Thank you

Documents

Unruh effect and Holography Shoichi Kawamoto (National Taiwan Normal University) with Feng-Li Lin(NTNU), Takayuki Hirayama(Kyoto Sangyo U.) and Pei-Wen