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
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