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)