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Helmholtz International Summer School on Modern Mathematical Physics Dubna July 22 – 30, 2007. Quantum Gravity and Quantum Entanglement (lecture 2). Dmitri V. Fursaev Joint Institute for Nuclear Research Dubna, RUSSIA. Talk is based on hep-th/0602134 - PowerPoint PPT Presentation
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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
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
e S STr e
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)
( )4gA BSG
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”
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!
entanglement entropy in a fundamental theory
CONJECTURE(Fursaev, hep-th/0602134)
3
4FUNDN
csG
FUNDs - entanglement entropy per unit area for degrees of freedom of the fundamental theory in a flat space
( 4)d
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 whichshould be renormalized?
the geometry was “frozen” till now:
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
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
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
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
( , , )
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 gG
S Tr Z T S Sn
S
A BSG
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
“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;
( )4gA BSG
renormalization
2
4
4 4
g m
g
div finm m m
divm
divm
bare ren
S S S
ASG
S S S
AS
A ASG 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
what are the conditions on the separating surface?
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
, ,
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
NB: we worked with Euclidean version of the theory (finitetemperature), stationary space-times was implied;
In the Lorentzian version of the theory space-times: thesurface 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)
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 !
the black hole entropy is a particular case
for stationary black holes the cross-section of the blackhole horizon with a constant-time hypersurface is aminimal surface:
all constant time hypersurfaces intersect the horizon at a bifurcation surface which has vanishing extrinsic curvatures due to its symmetry
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
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
subadditivity
1 2 1 2
1 2 1 2
| | , lnS S S S S S TrS 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
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 SG G
S S S S
Araki-Lieb inequality:
- entropy of the entire system
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 AS S A A A A A A
A A A A A A S S
rest of the talk
● the Plateau problem
● entanglement entropy in AdS/CFT: “holographic formula”
● some examples: EE in SYM and in 2D CFT’s
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 pk
hp p
- the mean curvature
- surface tension
-pressure difference across the film
- equilibrium equation
the Plateau Problem there are no unique solutions in general
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
the Plateau Problem
Costa’s surface (1982)
other embedded surfaces (without self intersections)
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
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
the Plateau Problem a minimal surface may be unstable against small perturbations
more evidences:
entanglement entropy in QFT’s with gravity duals
Consider the entanglement entropy in conformaltheories (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
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
Holographic Formula for the Entropy
A
( 1)4 d
ASG
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
a simple example
B
22 2 25 42
3
2
3 2
2 25 5
32
5
( )
4
, ( ( ))
lds dz dsz
lA Aa
A l NS A AG a G a
l N SU NG
2
2
1
– is IR cutoffa
the holographic formula enables one to compute entanglement entropy in stronglycoupled theories by using geometrical methods
entanglement in 2D CFT
1ln sin3
Lc LSa L
ground state entanglement for asystem on a circle
1L is the length of
c – is a central charge
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 332
CFT
ds l d dt d
lL ds ds
LLA
l LLea
LA cS eG LlcG
- 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
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 )
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)