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Area scaling from entanglement in flat space quantum field theory
•Introduction
•Area scaling of quantum fluctuations
•Unruh radiation and Holography
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)
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)
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:)(:
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)
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