area scaling from entanglement in flat space quantum field theory
DESCRIPTION
Area scaling from entanglement in flat space quantum field theory. Introduction Area scaling of quantum fluctuations Unruh radiation and Holography. Black hole thermodynamics. J. Beckenstein (1973). S. Hawking (1975). S = ¼ A. S A. T H. out. in. V. V. An ‘artificial’ horizon. - PowerPoint PPT PresentationTRANSCRIPT
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Area scaling from entanglement in flat space quantum field theory
•Introduction
•Area scaling of quantum fluctuations
•Unruh radiation and Holography
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Black hole thermodynamicsJ. Beckenstein (1973)
S. Hawking (1975)
S ATH
S = ¼ A
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An ‘artificial’ horizon.
VV in
out
xdrΟO d
V
V )(
00outin Tr
)( VinOTr
0
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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
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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)
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Other Thermodynamic quantities
Heat capacity: 2:: VinV ETrC
More generally: 2VinOTr
A?
A?
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A different viewpoint
inout
xdrOO d
V
V )(
00 VO
00outin Tr
)( VinOTr
0
=
No accessRestricted measurements
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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
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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
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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)
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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
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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
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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:)(:
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Intermediate summary
0O0 V
V
VTr(inOV)
0O0 2V
Tr(inOV2)
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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)
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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)
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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
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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
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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)=
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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}
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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)
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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)
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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*.
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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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Fin