Area scaling in Minkowski space

•Rindler space thermodynamics.

•Area scaling in Minkowski space.

•Bulk - boundary correspondence.

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

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

Thermodynamics in curved space

ggL dtxdS d L

’,’’= ’| e-H|’’ e- SE +… D

(x,0)=’(x)

(x,)=’’(x)

LH

L

S=Tr( ln ) A

-F=ln(Tr()) A

A different method to obtain Kabat and Strassler, 1994)

''00')'','(

DLdtExp ][00

(x,0)=(x)

DLdtExp ][)'','(

(x,0+)=’(x)

(x,0-)=’’(x)

x

t

’(x)’’(x)

00

A different method to obtain

’’’ Exp[-SE] D

(x,0+)=’(x)

(x,0-)=’’(x)

Trout (’’’ Exp[-SE] DDout

(x,0+) = ’in(x)out(x)(x,0-) = ’’in(x)out(x)in’in’’in Exp[-SE] D

(x,0+) = ’in(x)(x,0-) = ’’in(x)

in(’in,’’in) =

in out

A different method to obtain

x

t

’in(x)

’’in(x)

in’in’’in Exp[-SE] D

(x,0+) = ’in(x)(x,0-) = ’’in(x)

’| e-HR|’’

Intermediate summary I

0’,’’= ’| e-HR|’’

x

t

0’,’’= ’| e-HR|’’

More relations in Minkowski space

0O0 V

OV= V O ddxV

VTr(inOV)

outina aA 0

Schematic picture

Q.M. in V of Minkowski space

V V

Statistical Mechanicsin Minkowski space with d.o.f restricted to V

Statistical mechanics In Rindler space(if V is half of space)

Tr(e-HR(HR-HR)2])Tr(in(HR-HR)2) =(HR-HR)2 =

Heat capacity in Rindler space: C A

(N-N)2Isothermal compressibility:

Area scaling of fluctuations(R. Brustein and A.Y. , 2003)

0(OV)20

V

0OiV Oj

V 0 S(V)

S(V)

0)( xO

Assumptions:

U.V. cutoff

0)()( yx OO

Area scaling of correlation functions

OiV Oj

V = V V Oi(x)Oj(y) ddx ddy

= V V Fij(|x-y|) ddx ddy

= D() Fij() d

D()= V V (xy) ddx ddy

= GVVd-1 – GSS(V)d+O(d+1) Since F() = eiqcos F (q) ddqand F (q) ~ q

F(x)=2f(x)

OiV Oj

V = - ∂ (D()/d-1)d-1 ∂f() d

Introduce U.V. cutoff short~ 1/distances

∂ (D()/d-1)S

Intermediate summary II

V

Area scaling of Fluctuations due to entanglement

V

Statistical ensembledue to restriction of d.o.f

Unruh radiation andArea dependent thermodynamics

Evidence for bulk-boundary correspondence

V1

OV1OV2 S(B(V1)B(V2))

OV

1 OV

2

V2

OV

1 OV

2

V1 V2 OV1OV2- OV1OV2

Pos. of V2

Pos. of V2

A working example0

1

])([])([

dV

d

dV

d xdxJExpxdxJExp

V V

dd

dyxddyx 110

1)()(

V V

dd

dyxddyx )()(

V V

ddd yxddyx

11

V V

ddd yxddyx

31

V V

ddd yxddyx

113

1 V

mdd

d

nn

V

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

V

mdd

d

nn

V

xdxdTrTr m 11

10

1......... 1

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

Summary

V

Area scaling of Fluctuations due to entanglement

Unruh radiation andArea dependent thermodynamics

V

Boundary theory for fluctuations

Statistical ensembledue to restriction of d.o.f

V

Proof that 0|OV |0=Tr(inOV)

outina aA 0

baba

baba

outin

AbaA

AbaA

Tr

,,

,,,,

00

00

Start with the vacuum state:

Find in:

cb

Vcb

c ba

Vba

Vin

cObAA

cOAbaAcOTr

,,

,,

)(

Hence:

On the other hand:

aObAA

aObAA

aAOAbO

Vab

Vab

aV

bV

00

Proof that Sin=Sout

AA

AbaA

AbaA

Tr

baba

baba

outin

†

,,

,,,,

00

00

outina aA 0

T

,,

,,,,

00

00

AA

AA

cAbaAc

cc

Tr

bbb

baba

c

inout

)()( kout

kin TrTr