Entanglement &

area thermodynamics of Rindler space Entanglement & area Entanglement & dimensional reduction

(holography)

Entanglement, thermodynamics & area

אוניברסיטת בן-גוריון

Ram Brustein

sorry, not today!

Series of papers with

Amos Yarom, BGU

(also David Oaknin, UBC)

hep-th/0302186 + to appear

Thermodynamics, Area, Holography

• Black Holes

• Entropy Bounds

– BEB

– Holographic

– Causal

• Holographic principle: Boundary theory with a limited #DOF/planck area

NG

AS

4

ERS c2

N

LS

G

AS

4

NcG

EVS

Bekenstein, Hawking

Bekenstein

Fichler & Susskind, Bousso

Brustein & Veneziano

‘thooft, Susskind

Rindler space

zt

z

0

22222 )()( dxdRdaRds

Lines of constant -constant acceleration

horizon

Addition of velocities in SR

uv

vuw

1rstrv tanh

AvArAd

dvtanh

aa

a

aeaeddt

ddx

dt

dxv

ded

)tanh(/

/

222

proper acceleration

Minkowski vacuum is a Rindler thermal state

(Unruh effect) in = z > 0out = z < 0

inoutoutin Tr //

Compare two expressions for in (by writing them as a PI)

1.

2.

22222 )()( dxdRdaRdsE

outin

TFD

1.

)(0

)(0)0,(

xz

xzx

out

in

)(

)()0,(

xoutx

xinxx

out

in

In general:

)(0

)(0)0,(

xz

xzx

out

in

)(0

)(0)0,(

xz

xzx

out

in

)(

)(

)(

0

0,0

0,0

)0,(

x

x

x

z

tz

tz

x

out

in

in

)(0,0

)(0,0)0,(

xtz

xtzx

in

in

)(0,0

)(0,0)0,(

xtz

xtzx

in

in

Result

inout

Heff – generator of time translations

Time slicing the interval [0,0]:

2.

Guess:

00

1

g

result

1. The boundary conditions are the same2. The actions are equal3. The measures are equal

effHein

0

)(0,0

)(0,0)0,(

xtz

xtzx

in

in

Results

If

Then

inoutinout

For half space Heff=HRindler ,

HRindler= boost

RH

a

20

Rindler area thermodynamicsSusskind UglumCallan WilczekKabat StrasslerDe Alwis OhtaEmparan…

Volume of optical space

Go to “optical” space

Compute using heat kernel method

In 4D:

High temperature approximation

Optical metric

In 4D

Euclidean Rindler

22

2/)2(

)()2/(

DDD

D

VT

D

DDF

22

DDV

22

|

D

DV

V

T

UC

HTT

22222 )( dxdRdRds

VVD 2

minR

Compute:

ininin OTrO

12

21

Tr

Tr

IOO

OIO

1

2

הפוך

S

MS

S S

S,T unitary

M M

M M

M

S S

M

1

o

O

Entanglement &

area thermodynamics of Rindler space Entanglement & area Entanglement & dimensional reduction

(holography)

Entanglement, thermodynamics & area

אוניברסיטת בן-גוריון

Ram Brustein

sorry, not today!

Series of papers with

Amos Yarom, BGU

(also David Oaknin, UBC)

hep-th/0302186 + to appear

inoutinout

For half space Heff=HRindler ,

HRindler= boosta

20

effHein

0

ininin OTrO

12

21

Tr

Tr

IOO

OIO

1

2

הפוך

(EV)2

•System in an energy eigenstate energy does not fluctuate •Energy of a sub-system fluctuates “Entanglement energy” fluctuations

Connect to Rindler thermodynamics

EV=

For free fields

X

For a massless field

F(x)

VD

VD

Geometry

Operator

Vanishes for the whole space!

F(x) =

F(x)

UV cutoff!!In this exampleExp(-p/)

)1(2

)1(2

~)(1

~)(1

d

d

xFx

xxFx

VD

yxr

),( rzr

For half space

00

2

E

D

002

E

00

2

E

00

2

E

Rindler specific heat

VRH

MRM CTHeTrH R 222 )(0)(0

@

E+ = … contributions from the near horizon region

MRM Ha 0:)(:0 22

)2(2)1(

)1(

0)(0)1(

0:)(:0

222

212

21

22

22

22

dd

dd

MMMRM

d

d

daV

Ed

aH

Other shapes

Heff complicated, time dependent, no simple thermodynamics, area dependence o.k.For area thermodynamics need – Thermofield double

z

t y

Entanglement and area

|0> is not necessarily an eigenstate of|0> is an entnangled state w.r.t. V

Non-extensive!, depends on boundary (similar to entanglement entropy)

Show:

Proof:

is linear in boundary area

R is the radius of the smallest sphere containing V

Show that

0)2( RDV

Need to evaluate

2k

Ik

General cutoffNumerical factors depend

on regularization

F(x)

(EV)2 for a d-dimensional sphere

V̂V

DV(x)=

K27 =

Kd

Fluctuations live on the boundary

Covariance

V1

V2 V3

-10 -5 5 10

-10

-5

5

10

V1

V1

V2

sin)]sin([,cos)]sin([ JdRRJdRR

E

lengthBoundary

The “flower”-10 -5 5 10

-10

-5

5

10

Circles 5 < R < 75R=40, dR=4, JR=20, dR=2, JR=10, dR=1, J

Increasing m

sin)]sin([,cos)]sin([ JdRRJdRR

Boundary theory ?

Express as a double derivative and convert to a boundary expression

This is possible iff

which is generally true for operators of interest

0)(

~

022

2

qq

qfij

i+j = 2 logarithmici+j = d -function

Boundary* correlation functions

(massless free field, V half space, large # of fields N)Show

First, n-point functions of single fields

Only contribution in leading order in N comes from

Then, show that in the large N limit equality holds for all correlation functions

Summary

Entanglement &

area thermodynamics of Rindler space Entanglement & area Entanglement & dimensional reduction