Area scaling from entanglement in flat space quantum field theory Introduction Area scaling of quantum fluctuations Unruh radiation and Holography.

Area scaling from entanglement in flat space quantum field theory

•Introduction

•Area scaling of quantum fluctuations

•Unruh radiation and Holography

Black hole thermodynamicsJ. Beckenstein (1973)

S. Hawking (1975)

S ATH

S = ¼ A

An ‘artificial’ horizon.

VV in

out

xdrΟO d

V

V )(

00outin Tr

)( VinOTr

0

Entropy: Sin=Tr(inlnin)

inoutina aA 0

out

)()( kout

kin TrTr

Sin=Sout

Srednicki (1993)

00

,,,, ba

ba AbaA

ba

ba AbaA,,

*TAA

c

cc 00

,,,, ba

ba cAbaAc

,,b

bb AA

†AA

00outTr 00inTr

Entanglement entropy of a sphere

xdH 422 ||

jmljml

jmljmljml j

ll

jjj

a ,,

2,,2

2

1,,,,2

2,,

)1(

12

11

out

in00outin Tr

Ent

ropy

R2

Srednicki (1993)

Other Thermodynamic quantities

Heat capacity: 2:: VinV ETrC

More generally: 2VinOTr

A?

A?

A different viewpoint

inout

xdrOO d

V

V )(

00 VO

00outin Tr

)( VinOTr

0

=

No accessRestricted measurements

Area scaling of fluctuationsR. Brustein and A.Y. , (2004)

OaV1

ObV2V1

Assumptions:

ayx yxyOxO O

||

1)()(

0||

V2

V V

dd yxddyOxO )()( ba

byx yxyOxO O

||

1)()(

||

OaV1

2

Area scaling of correlation functions

OaV1

ObV2

= V1 V2 Oa(x) Ob(y) ddx ddy

= V1 V2 Fab(|x-y|) ddx ddy

= D() Fab() d

D()= V V (xy) ddx ddy

Geometric term:

Operator dependent term

= D() 2g() d

= - ∂(D()/d-1) d-1 ∂g() d

Geometric termD()=V1 V2 (xy) ddx ddy

V1V2

x

y

= (r) ddr ddR

Rr ddR A2)

(r) ddr d-1 +O(d)

D()=C2 Ad + O(d+1)

Geometric termD()= (r) ddr ddR

R

r ddR V + A2)

(r) ddr d-1 +O(d)

D()=C1Vd-1 ± C2 Ad + O(d+1)

V1=V2

Area scaling of correlation functions

OaV1

ObV2

= V1 V2 Oa(x)Ob(y) ddx ddy

= V1 V2 Fab(|x-y|) ddx ddy

= D() Fab() d

= D() 2g() d

∂ (D()/d-1)

= - ∂(D()/d-1) d-1 ∂g() dUV cuttoff at ~1/

D()=C1Vd-1 + C2 Ad + O(d+1)

A

Energy fluctuations

yxdqdpddeEE

EE

qpEE ddddyxqpi

qp

qpdVV

)()(

2

221 )2(

1

8

100

yxddyHxHEE ddVV 0)()(000 21

)(xF

))())(2(2())(1(8

2)1(

)1(4

321

)1(2

xdxddx

dd

dd

d

qpddeEE

EE

qp ddyxqpiqp

qpd

)()(

2

2)2(

1

8

1

inoutdd

d

VV AAd

dd

EE

124

2

21

22

2

23

21

00

qpddeaaaaEE

qp

aaaaEExH

ddxqpiqqpp

qp

qqppqpd

)(††

††2

:)

(:

)2(

1

4

1:)(:

Intermediate summary

0O0 V

V

VTr(inOV)

0O0 2V

Tr(inOV2)

Finding in

''00')'','(

DLdtExp ][00

(x,0)=(x)

00

x

t

’(x)’’(x)

Trout (’’’in(’in,’’in) =

in’in’’in Exp[-SE] D

(x,0+) = ’in(x)(x,0-) = ’’in(x)

(x,0+) = ’in(x)out(x)(x,0-) = ’’in(x)out(x)

Exp[-SE] DDout

’’in(x)

’in(x)

DLdtExp ][)'','(

(x,0+)=’(x)

(x,0-)=’’(x)

DLdtExp ][)'','(

(x,0+)=’(x)

(x,0-)=’’(x)

Finding rho

x

t

’in(x)

’’in(x)

in’in’’in Exp[-SE] D

(x,0+) = ’in(x)(x,0-) = ’’in(x)

’| e-K|’’

Kabbat & Strassler (1994)

Rindler space(Rindler 1966)

ds2 = -dt2+dx2+dxi2

ds2 = -a22d2+d2+dxi2

t=/a sinh(a)x=/a cosh(a)

Acceleration = a/Proper time =

x

t

= const

=const

HR = Kx

Unruh Radiation(Unruh, 1976)

x

tds2 = -a22d2+d2+dxi

2

= 0

a≈ a+i2

Avoid a conical singularity

Periodicity of Greens functions

Radiation at temperature 0 = 2/a

R= e-HR= e-K= in

Schematic picture

VEVs in V of Minkowski space

V V

Observer in Minkowski space with d.o.f restricted to V

Canonical ensemble in Rindler space(if V is half of space)

0O0 V Tr(inOV)= Tr(ROV)=

Other shapesR. Brustein and A.Y., (2003)

in’in’’in Exp[-SE] D

(x,0+) = ’in(x)(x,0-) = ’’in(x)

x

t

’’in(x)

’in(x)

=’in|e-H0|’’out

d/dt H0 = 0

SE = 0H0dt

(x,t), (x,t), +B.C.

H0=K, in={x|x>0}

Evidence for bulk-boundary correspondence

V1

OV1OV2 A1A2

OV

1 OV

2

V2

OV

1 OV

2

V1 V2 OV1OV2- OV1OV2

Pos. of V2

Pos. of V2

R. Brustein D. Oaknin, and A.Y., (2003)

A working example0

1 ])([])([

A

d

V

d xdxJExpxdxJExp

A A

dd

dyxddyx 110

1)()(

V V

dd

dyxddyx )()(

V V

ddd yxdd

yx1

1

V V

ddd yxdd

yx3

1

A A

ddd yxdd

yx11

31

V

mdd

d

nn

V

xdxdTrTr m ......... 11

A

mdd

d

nn

A

xdxdTrTr m 11

10

1......... 1

Large N limit )()...(()( 1 xxdiagx N

R. Brustein and A.Y., (2003)

Summary

V

Area scaling of Fluctuations due to entanglement

Unruh radiation andArea dependent thermodynamics

A

Boundary theory for fluctuations

Statistical ensembledue to restriction of d.o.f

V

A Minkowski observer restricted to part of space will observe:•Radiation.•Area scaling of thermodynamic quantities.•Bulk boundary correspondence*.

Speculations

Theory with horizon(AdS, dS, Schwarzschild)

A

Boundary theory for fluctuations

V

Area scaling of Fluctuations due to entanglement

Statistical ensembledue to restriction of d.o.f

V

?

??

Israel (1976)Maldacena (2001)

Fin