Entanglement, thermodynamics & area
Ram Brustein
‫גוריון‬-‫אוניברסיטת בן‬
Series of papers with
Amos Yarom, BGU
(also David Oaknin, UBC)
hep-th/0302186 + to appear
 Entanglement &
area thermodynamics of Rindler space
 Entanglement & area
 Entanglement & dimensional reduction
(holography)
sorry, not today!
Thermodynamics, Area, Holography
• Black Holes
• Entropy Bounds
– BEB
S
2
c
A
S
4G N
ER
– Holographic S  ALS
4GN
– Causal
EV
S
cGN
• Holographic principle:
Bekenstein, Hawking
Bekenstein
Fichler & Susskind, Bousso
Brustein & Veneziano
‘thooft, Susskind
Boundary theory with a limited #DOF/planck area
Rindler space
   
0 z
    
t z
ds 2  (aR) 2 d 2  dR 2  (dx ) 2
Lines of constant  constant acceleration
Addition of velocities in SR
v  tanh r  t  s  r
uv
w
1  uv
dv
 A  r  A  v  tanh A
d
d 2  e 2 a d 2
dx dx / d
v

 tanh( ae a )  ae a
dt dt / d
horizon
proper acceleration
Minkowski vacuum is a
TFD
Rindler thermal state
out = z < 0
(Unruh effect)
dsE2  (aR) 2 d 2  dR 2  (dx ) 2
in / out  Trout / in 
   in out
Compare two expressions for in (by writing them as a PI)
1.
2.
in = z > 0
1.

 z  0  in ( x )

 ( x,0)  

z

0

(
x
)
out

In general:


x

in

(
x
)


in
 ( x,0)   

 x  out  out ( x )

 z  0  in ( x )

 ( x ,0)  

z

0

(
x
)
out


 z  0  in ( x )

 ( x,0)  

z

0

(
x
)
out


 z  0, t  0   in ( x )



 ( x ,0)   z  0, t  0   in ( x )
 z  0  ( x )
out


 z  0, t  0  in ( x )

 ( x,0)  




z

0
,
t

0

(
x
)
in

Result

 z  0, t  0  in ( x )

 ( x,0)  




z

0
,
t

0

(
x
)
in

out
in
2.
Heff – generator of time translations
Time slicing the interval [0,b0]:
Guess:
result
1

 g 00
Results

 z  0, t  0  in ( x )

 ( x,0)  


 z  0, t  0  in ( x )
If
Then
1. The boundary conditions are the same
2. The actions are equal
3. The measures are equal
in  e
 b0 H eff
2
For half space Heff=HRindler , b 0 
a
HRindler= boost
HR
out
in
out
in
Rindler area thermodynamics
Susskind Uglum
Callan Wilczek
Kabat Strassler
De Alwis Ohta
Emparan
…
Go to “optical” space
Compute using heat kernel method
High temperature approximation
In 4D:
Volume of optical space
Compute:
Euclidean Rindler
ds 2  R 2 d 2  dR 2  (dx ) 2
Optical metric
VD2  V
Rmin  
( D / 2) ( D) D VD 2
F 
T
D/2
( D  2)
 D2
In 4D

VD  2

U
VD  2
CV 
 D2
T |T TH 
D 2
 Oin   Tr inOin 
O2  I  O
O1  O  I
‫הפוך‬
1  Tr2 
 2  Tr1 
S
S
S,T unitary
S
S
M
1
M
M
M
M
o
M M
S
S
  O
Entanglement, thermodynamics & area
Ram Brustein
‫גוריון‬-‫אוניברסיטת בן‬
Series of papers with
Amos Yarom, BGU
(also David Oaknin, UBC)
hep-th/0302186 + to appear
 Entanglement &
area thermodynamics of Rindler space
 Entanglement & area
 Entanglement & dimensional reduction
(holography)
sorry, not today!
in  e
 b0 H eff
2
For half space Heff=HRindler , b 0 
a
HRindler= boost
out
in
out
in
 Oin   Tr inOin 
O2  I  O
O1  O  I
‫הפוך‬
1  Tr2 
 2  Tr1 
(DEV)2
•System in an energy eigenstate
 energy does not fluctuate
•Energy of a sub-system fluctuates
“Entanglement energy” fluctuations
Connect to Rindler thermodynamics
For free fields
EV=
For a massless field
X
Vanishes for the whole space!
DV
DV
Geometry
F(x)
Operator
F(x) =
UV cutoff!!
In this example
Exp(-p/L)
F(x)

Lx  1 F ( x) ~ x 2( d 1)
Lx  1 F ( x) ~ L2( d 1)
DV
  
For half space r  x  y
D



0  E  0 


2


r  ( z , r )
0  E   0 


2
0  E   0


2
0  E   0 


2
Rindler specific heat

0M ( H R ) 0M  Tr e
2
@ 0
b HR

( H R )  T CV
2
2
a
2
2
0M (: H R :) 0M
0 M (: H R :) 2 0 M
 V Ld 1a 2
a2
 2

0
(
E
) 0M 
M
2
(d  1)L
(d  1) 2  d21 
(d  1)2
2d 2

d 2
2
(2  d2 )
E+ = …  contributions from the near horizon region
Other shapes
t
y
z
Heff complicated, time dependent, no simple
thermodynamics, area dependence o.k.
For area thermodynamics need – Thermofield double
Entanglement and area
|0> is not necessarily an eigenstate of
|0> is an entnangled state w.r.t. V
Show:
Non-extensive!, depends on boundary
(similar to entanglement entropy)
Proof:
Show that
DV (2 R)  0
R is the radius of the
smallest sphere containing V
is linear in boundary area
Need to evaluate
 k2 
Ia  ka
General cutoff
Numerical factors depend
on regularization
(DEV)2 for a d-dimensional sphere
Vˆ
F(x)
DV(x)=
V
Kd
K27 =
Fluctuations live on the boundary
V2
10
5
V1
-10
V1
-5
5
-5
V2
Covariance
V3
10
V1
-10
[R  dR sin( J )] cos ,[R  dR sin( J )] sin  
The “flower”
10
5
-5
5
[R  dR sin( J )] cos ,[R  dR sin( J )] sin  
10
-5
-10
Circles 5 < R < 75
R=40, dR=4, J
R=20, dR=2, J
R=10, dR=1, J
Increasing m
DE
-10
Boundary length
Boundary theory ?
Express
This is possible iff
as a double derivative and
convert to a boundary expression
~ 2
 f ij (q )



 0 
2
2
q 0
 q



which is generally true for operators of interest
di+dj = 2  logarithmic
di+dj = d  d-function
Boundary* correlation functions
Show
(massless free field, V half space, large # of fields N)
First, n-point functions of single fields
Then, show that in the large
N limit equality holds for
all correlation functions
Only contribution in leading
order in N comes from
Summary
 Entanglement &
area thermodynamics of Rindler space
 Entanglement & area
 Entanglement & dimensional reduction