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

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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.

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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?

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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)

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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

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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

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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:

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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

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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.

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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?

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Plan

1. Introduction: Unruh effect in field theory

2. Uniformly accelerated string and comoving frame

3. Introducing mesons

4. Conclusion and Discussion

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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

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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

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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

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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)

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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)

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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 =

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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 −=

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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

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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

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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...

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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...

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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??

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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?

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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

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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

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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.

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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

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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!

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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.

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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?

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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!