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![Page 1: Quantum Gravity and Quantum Entanglement (lecture 2) Dmitri V. Fursaev Joint Institute for Nuclear Research Dubna, RUSSIA Talk is based on hep-th/0602134.](https://reader035.fdocuments.in/reader035/viewer/2022062801/56649e4e5503460f94b45354/html5/thumbnails/1.jpg)
Quantum Gravity and Quantum Entanglement (lecture 2)
Dmitri V. Fursaev
Joint Institute for Nuclear ResearchDubna, RUSSIA
Talk is based on hep-th/0602134 hep-th/0606184 Dubna, July 26, 2007
Helmholtz International Summer School onModern Mathematical PhysicsDubna July 22 – 30, 2007
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definition of entanglement entropy
A a
1
2
1 2 2 1
1 1 1 1 2 2 2 2
/
1 2/
( , | , )
( | ) ( , | , ),
( | ) ( , | , ),
, ,
ln , ln
a
A
H T
H T
A a B b
A B A a B a
a b A a A b
Tr Tr
S Tr S Tr
eS S
Tr e
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some results of 1st lecture
• entanglement entropy in relativistic QFT’s
• path-integral method of calculation of entanglement entropy
• entropy of entanglement in a fundamental gravity theory
- the value of the entropy is given by the “Bekenstein-Hawking formula” (area of the surface playing the role of the area of the horizon)
( )
4g
A BS
G
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effective action approach to EE in a QFT
-effective action is defined on manifolds with cone-like singularities
- “inverse temperature”
1 1 1 2
1 2
( ) lim lim 1 ln ( , )
( , )
ln ( , )
2
nnS T Tr Z T
n
Z T Tr
Z T
n
- “partition function”
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effective action on a manifold with conical singularities is the gravity action (even if the manifold is locally flat)
(2)2(2 ) ( )R B
curvature at the singularity is non-trivial:
derivation of entanglement entropy in a flat space has to do with gravity effects!
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entanglement entropy in a fundamental theory
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CONJECTURE(Fursaev, hep-th/0602134)
3
4FUNDN
cs
G
FUNDs - entanglement entropy per unit area for degrees of freedom of the fundamental theory in a flat space
( 4)d
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Open questions:
● Does the definition of a “separating surface” make sense in a quantum
gravity theory (in the presence of “quantum geometry”)?
● Entanglement of gravitational degrees of freedom?
● Can the problem of UV divergences in EE be solved by the standard
renormalization prescription? What are the physical constants which
should be renormalized?
the geometry was “frozen” till now:
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assumption
... ...fundamental low energy
dof dof
the Ising model:
“fundamental” dof are the spin variables on the lattice
low-energies = near-critical regime
low-energy theory = QFT (CFT) of fermions
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at low energies integration over fundamental degrees of freedom is equivalent to the integration over all lowenergy fields, including fluctuations of the space-time metric
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B
1
B
2
This means that:
(if the boundary of the separating surface is fixed)
the geometry of the separating surface is determined by a quantum problem
B
B
Bfluctuations of are induced by fluctuations of the space-time geometry
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entanglement entropy in the semiclassical approximation
[ , ]
4 3
( ) [ ][ ] , ( ) [ , ] [ ] [ , ],
1 1[ ] ,
16 8
( ) ln ( ) [ , ],
n n
I gmatter
M M
Z T Dg D e Z T I g I g I g
I g R gd x K hd yG G
F T Z T I g
a standard procedure
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( , , )
4
1 1 1 2
( , )
2(2 ) ( ),
( )( , , ) ( , , ) (2 ) ,
8
lim lim 1 ln ( , ) ,
( )
4
( )
n
I g
B
regular
M
regular
nn g m
m
g
Z T e
R gd x R A B
A BI g I g
G
S Tr Z T S Sn
S
A BS
G
A B
fix n and “average” over all possible positionsof the separating surface on
- entanglement entropy of quantum matter
- pure gravitational part of entanglement entropy
- some average area
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“Bekenstein-Hawking” formula for the“gravitational part” of the entropy
Note:
- the formula says nothing about the nature of the degrees of freedom
- “gravitational” entanglement entropy and entanglement entropy of quantum matter fields (EE of QFT) come together;
- EE of QFT is a quantum correction to the gravitational part;
-the UV divergence of EE of QFT is eliminated by renormalization of the Newton coupling;
( )
4g
A BS
G
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renormalization
2
4
4 4
g m
g
div finm m m
divm
divm
bare ren
S S S
AS
G
S S S
AS
A AS
G G
the UV divergences in the entropy areremoved by the standard renormalization of thegravitational couplings;
the result is finite and is expressed entirely in terms of low-energy variables and effective constants like G
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what are the conditions on the separating surface?
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conditions for the separating surface
( )( 2 )( , , ) ( , , ) 8
( )( ) ( 2 )( 2 ) 88
2
( , ) ,
,
( ) 0, ( ) 0
regularA B
I g I g G
B B
A BA BGG
B
Z T e e e
e e
A B A B
the separating surface is a minimal(least area) co-dimension 2 hypersurface
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, ,
2 2
0
,
1, 0,
0,
0.
iji j
ij
n
p
X X X
n p
n p np
k n
k p
- induced metric on the surface
- normal vectors to the surface
- traces of extrinsic curvatures
Equations
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NB: we worked with Euclidean version of the theory (finite
temperature), stationary space-times was implied;
In the Lorentzian version of the theory space-times: the
surface is extremal;
Hint: In non-stationary space-times the fundamental
entanglement may be associated to extremal surfaces
A similar conclusion in AdS/CFT context is in (Hubeny,
Rangami, Takayanagi, hep-th/0705.0016)
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2, , ;
; ;
0
0
iji j
B
t
A d y X X X
X
t
B
a Killing vector field
- a constant time hypersurface (a Riemannian manifold)
is a co-dimension 1 minimal surface on a constant-time hypersurface
Stationary spacetimes: a simplification
the statement is true for the Lorentzian theory as well !
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the black hole entropy is a particular case
for stationary black holes the cross-section of the black
hole horizon with a constant-time hypersurface is a
minimal surface:
all constant time hypersurfaces intersect the horizon at
a bifurcation surface which has vanishing extrinsic
curvatures due to its symmetry
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remarks
● the equation for the separating surface ㅡ may have a different form in generalizations of the Einstein GR (the dilaton gravity, the Gauss-Bonnet gravity and etc)
● one gets a possibility to relate variations of entanglemententropy to variations of physical observables
● one can test whether EE in quantum gravity satisfy inequalities for the von Neumann entropy
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some examples of variation formulae for EE
S M l
M
l
S l
- change of the entropy per unit length (for a cosmic string)
- string tension
-change of the entropy under the shift of a point particle
-mass of the particle
- shift distance
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subadditivity
1 2 1 2
1 2 1 2
| | , lnS S S S S S Tr
S S S
1 2 strong subadditivity
1 2 1 2 1 2S S S S
equalities are applied to the von Neumann entropyand are based on the concavity property
check of inequalities for the von Neumann entropy
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entire system is in a mixed state due to the presence of a black hole
B
2
1 black hole
1 2S S S
1 2
2 1
( ) ( ),
4 4 BH
BH
A B A BS S S
G G
S S S S
Araki-Lieb inequality:
- entropy of the entire system
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strong subadditivity: 1 2 1 2 1 2S S S S
a b
c d
fa b
c d
f1 2
1 2
1 2
1 2 1 2
,
( ) ( )
ad bc
ad bc af fd bf fc
af bf fd fc ab dc
S A S A
S S A A A A A A
A A A A A A S S
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rest of the talk
● the Plateau problem
● entanglement entropy in AdS/CFT: “holographic formula”
● some examples: EE in SYM and in 2D CFT’s
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the Plateau Problem (Joseph Plateau, 1801-1883)
It is a problem of finding a least area surface (minimal surface)for a given boundary
soap films:1 2
1
1 2
( )k h p p
k
h
p p
- the mean curvature
- surface tension
-pressure difference across the film
- equilibrium equation
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the Plateau Problem there are no unique solutions in general
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the Plateau Problem simple surfaces
The structure of part of a DNA double helix
catenoid is a three-dimensional shape made by rotating a catenary curve (discovered by L.Euler in 1744) helicoid is a ruled surface, meaning that it is a trace of a line
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the Plateau Problem
Costa’s surface (1982)
other embedded surfaces (without self intersections)
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the Plateau Problem
A minimal Klein bottle with one end
Non-orientable surfaces
A projective plane with three planar ends. From far away the surface looks like the three coordinate plane
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the Plateau Problem
Non-trivial topology: surfaces with hadles
a surface was found by Chen and Gackstatter
a singly periodic Scherk surface approaches two orthogonal planes
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the Plateau Problem a minimal surface may be unstable against small perturbations
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more evidences:
entanglement entropy in QFT’s with gravity duals
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Consider the entanglement entropy in conformal
theories (CFT’s) which admit a description in terms of
anti-de Sitter (AdS) gravity one dimension higher
N=4 super Yang-Mills 5 5AdS S
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Holographic Formula for the Entropy
B
5AdS
B
4d space-time manifold (asymptotic boundary of AdS)
(bulk space)
separating surface
extension of the separating surface in the bulk
(now: there is no gravity in the boundary theory, can be arbitrary)B
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Holographic Formula for the Entropy
A
( 1)4 d
AS
G
Ryu and Takayanagi,hep-th/0603001, 0605073
CFT which admit a dual description in terms of the Anti-de Sitter (AdS) gravity one dimension higher
( 1)dG
Let be the extension of the separating surface in d-dim. CFT
1) is a minimal surface in (d+1) dimensional AdS space
2) “holographic formula” holds: is the area of
is the gravity couplingin AdS
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a simple example
B
22 2 25 42
3
2
3 2
2 25 5
32
5
( )
4
, ( ( ))
lds dz ds
z
lA A
a
A l NS A A
G a G a
lN SU N
G
2
2
1
– is IR cutoffa
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the holographic formula enables one to
compute entanglement entropy in strongly
coupled theories by using geometrical
methods
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entanglement in 2D CFT
1ln sin3
Lc LS
a L
ground state entanglement for asystem on a circle
1L is the length of
c – is a central charge
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example in d=2:CFT on a circle
0
0
0
2 2 2 2 2 2 2
2 211
2 2 10
1
3
3
cosh sinh
2
cosh 1 2sinh sin
ln sin4 3
3
2
CFT
ds l d dt d
l
Lds ds
LLA
l L
Le
a
LA cS e
G L
lc
G
- AdS radius
A is the length of the geodesic in AdS
- UV cutoff
-holographic formula reproducesthe entropy for a ground stateentanglement
- central charge in d=2 CFT
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Some other developments
● D.Fursaev, hep-th/0606184 (proof of the holographic formula)
• R. Emparan, hep-th/0603081 (application of the holographic formula to interpretation of the entropy of a braneworld black hole as an entaglement entropy)
• M. Iwashita, T. Kobayashi, T. Shiromizu, hep-th/0606027 (Holographic entanglement entropy of de Sitter braneworld)
• T.Hirata, T.Takayanagi, hep-th/0608213 (AdS/CFT and the strong subadditivity formula)
• M. Headrick and T.Takayanagi, hep-th/0704.3719 (Holographic proof of the strong subadditivity of entanglement entropy)
• V.Hubeny, M. Rangami, T.Takayanagi, hep-th/0705.0016 (A covariant holographic entanglement entropy proposal )
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conclusions and future questions
• there is a deep connection between quantum entanglement and gravity which goes beyond the black hole physics;
• entanglement entropy of fundamental degrees of freedom in quantum gravity is associated to the area of minimal surfaces;
• more checks of entropy inequalities are needed to see whether the conjecture really works;
• variation formulae for entanglement entropy, relation to changes of physical observables (analogs of black hole variation formulae)