NTU,June24,2011 WeiLi FlatSpaceHolographyfromEntanglement

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Flat Space Holography from Entanglement

Wei Li

IPMU, Tokyo U, Japan

NTU, June 24, 2011

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Reference

Holography and Entanglement of Flat Space Phys. Rev. Lett. 106, 141301 (2011)

with Tadashi Takayanagi

Wei Li IPMU, Tokyo U, Japan

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Holographic duality.

Holography: A deep principle and powerful tool in quantum gravity

’t hooft ’93, Susskind ’95

Success: AdS/CFT

Maldacena ’98

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What is the holographic dual of flat space?

(1, d)-dim Minkowski space:

ds

²

R^1,d

= dρ

²

+ ρ

²

ds

_dS²

d

Quantum gravity in R^1,d

dSd: boundary theory =?

Question: What is holographic dual living on dS

_d

?

Handles

1 Entanglement entropies

2 Correlation Functions

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What is the holographic dual of flat space?

Disclaimer: In this talk, focus on Euclidean version first.

Lorentzian: (1, d)-dim Minkowski space:

ds

²

R^1,d

= dρ

²

+ ρ

²

ds

_dS² _d

Question: What is holographic dual living on dS

d

?

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What is the holographic dual of flat space?

Disclaimer: In this talk, focus on Euclidean version first.

Euclidean: (1 + d )-dim Eulidean space:

ds

²

R^1+d

= dρ

²

+ ρ

²

ds

_S²d

Question: What is holographic dual living on S

^d

?

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1

Introdution

2

Entanglement Entropy Definition and properties Holographic E.E. conjecture Holo E.E. for flat space

3

Correlation Functions GKP Procedure

Holographic correlators for flat space.

4

Summary

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Outline

1

Introdution

2

Entanglement Entropy Definition and properties Holographic E.E. conjecture Holo E.E. for flat space

3

Correlation Functions GKP Procedure

Holographic correlators for flat space.

4

Summary

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Entanglement entropy: definition

In a Quantum Mechanical system:

Statistical Entropy (of total system):

S

stat

= −Tr (ρ

tot

ln ρ

tot

)

von Neumann ’55

Entanglement Entropy of (a sub-system)

bB B A

1 Divide the spatial region: Total = A + B 2 Reduced density matrix: ρA≡ TrB ρtot

3 Entanglement entropy:

SE .E .^A ≡ −TrA (ρAln ρA)

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Entanglement entropy: properties

Entanglement entropy: S

_{E .E .}^A

≡ −Tr

_A

(ρ

A

ln ρ

A

)

1 measures information seen by observer in A — she has no access to B

2 measure quantum entanglement between A and B:

In general, it is hard to compute E.E. in field theory.

Question: How to compute E.E. holographically?

Then we can use E.E. to study holography.

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Holographic EE conjecture

Entanglement entropy: S_{E .E .}^A ≡ −TrA(ρAln ρA)

bB B A

Putting the system on boundary:

B

A

γ

Bulk

Bulk is divided into two parts by γ: A’ and B’

=⇒ A’ has no access to B’

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Holographic EE conjecture (cont.)

Bulk is divided into two parts by γA: A’ and B’=⇒A’ has no access to B’

B

A

γ

Bulk

Conjecture: Holographically Ryu+Takayanagi ’06

SE .E .^A =Area(γA)

4G_N^bulk γA: minimal surface in bulk given by ∂γA= ∂A

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“Area Law” for local QFT

Entanglement entropy measures

::::::::correlation

:::::::between

::A

::::and

:B.

Forground stateinlocaltheory: Bombelli+Koul+Lee+Sorkin ’86, Srednicki ’93

SE .E .^A ∼ L a

d −2

∼ Area(∂A) “Area Law”

Reason::::::::::::Entanglement_::::::happen_:::::most_::::::strongly_::::near_:::the_::::::::boundary

bB B

A

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Holographic EE of AdS space

Setup

1 AdS metric in Poincare coordinate:ds_EAdS²

d +1= `^{2 dz}²^+d~_z₂^x² 2 Boundary system: {x⁰, . . . , x^{d −1}}

3 Spatial slice: {x¹, . . . , x^{d −1}} =⇒ Divide it into A and B 4 Minimal surface γ_A= γ_B

For AdS

Area(γA) = `^{d −1}·Area(∂γA)

a^{d −2} + . . . (lattice spacing a ∼ z)

∂γA= ∂A =⇒ Holographic E.E. obeys“Area Law”

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Holographic EE of Flat space

Setup

1 Metric of R^{d +1}: ds²

R^{d +1}= d ρ²+ ρ²ds_S²d

2 Boundary system: S^d at ρ = ρ∞

3 Spatial slice: S^{d −1} =⇒ Divide S^{d −1} into A and B byS^{d −2} 4 Minimal surface γA= γB is volume insideS^{d −2}

ɲ

!^"

#^"$%

&

'

䃒&

!^"(%

S_A^{hol .} = Vd −1· ρ^{d −1}_∞ sin^α₂d −1

4G_N^{(d +1)}

Holographic E.E. obeys “Volume Law”

Reason: For flat space, Area(γ

_A

) → Vol (A) as A→ 0

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“Area Law” v.s. “Volume Law”

Entanglement entropy measures_::::::::correlation_:::::::between_::A_::::and_:B.

Forground stateinlocaltheory: Bombelli+Koul+Lee+Sorkin ’86, Srednicki ’93

SE .E .^A ∼ Area(∂A)

bB B A

Boundary theory of Flat space:

SE .E .^A ∼ Vol (A)

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What does “Volume Law” mean?

Two possibilites

1 The system is in a highly-excited state (E ∼ UV cutoff scale) Alba+Fagotti+Calabrese ’09 2 The system is highly non-local.

Holographic computation “Volume Law” valid for all states!

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What does “Volume Law” mean?

Two possibilites

1 The system is in a highly-excited state (E ∼ UV cutoff scale) Alba+Fagotti+Calabrese ’09 2 The system is highly non-local.

Holographic computation “Volume Law” valid for all states!

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Where does boundary theory live?

Matching holographic and QFT results:

1 Holographic E.E.:

S_{E .E .}^{hol .} = N ·(ρ∞)^{d −1}· (sin^α

2)^{d −1} G_N^{(d +1)}

2 QFT E.E. from “Volume Law”:

SE .E .∼ L a

d −1

a: lattice spacing and L: size of A UV/IR relation: E ∼ ¹_a∼ ρ

ConsiderQFT on unit-S^d and use dim-less ˜ρ∞≡^ρ^∞_` (`: length unit):

a = 1

˜ ρ∞

=⇒ S_{E .E .}^{hol .} = N ·(ρ∞)^{d −1}· (sin^α₂)^{d −1}

G_N^{(d +1)} ∼ `

lpl

d −1

· L a

d −1

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Discussion: Bulk cutoff and symmetry of bndy QFT

Matching holographic and QFT computation of E.E.

a = 1

˜ ρ∞

=⇒ S_{E .E .}^{hol .} = N ·(ρ∞)^{d −1}· (sin^α₂)^{d −1} G_N^{(d +1)}

∼ `

G_N^{(d +1)}

!d −1

· L a

d −1

Number of DOF (per site) in boundary theory:

n ∼ `

G_N^{(d +1)}

!d −1

Bulk metric ds²= `²(d ˜ρ²+ ˜ρ²d Ω²_d) has a scaling symmetry:

(`, ˜ρ) ∼ (λ`,ρ˜

λ) ← physical distance of boundary remains invariant.

+ Boundary QFT has symmetry:

(n, a) ∼ (λ^{d −1}n, λa) ← total number of DOF remains invariant.

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Decoupling of gravity at boundary

Holographic dual lives onunit-S^d at ρ = ρ∞→ ∞ UV cutoff ¹_a ∼ ˜ρ∞

The Newton’s constant on the boundary

G_N^{(d )}= (d − 1) · G_N^{(d +1)}

(ρ∞)^{d −1} −→ 0 as ρ∞→ ∞

=⇒ Gravity decouples on the boundary: boundary theory is a (non-gravitational) QFT!

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Computing E.E. from QFT side

Replica method

S = −limN→1

∂

∂N ZN

(Z1)^N ZN: Z onN-fold coverof S^d

Inverse Replica S = limN→1

∂

∂N Z1

N

(Z1)^N¹ Z1

N : Z onZN orbifoldof S^d Schwinger representation to compute Z

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Example of non-local theory

Use toy model: free scalar QFT on unit-S

^d

Sboundary= Z

d Ωd[φ ·f (−∆)· φ]

Choosef (x )s.t. S_{hol .}^A obeys“Volume Law”

Sboundary= Z

d Ωd

h φ ·e

√−∆

· φi (c.f. polynomial f (x ) always gives “Area Law”)

Question: What can this toy model teach us?

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What does boundary theory look like?

Lessons from toy model: non-localFree scalar QFT

Sboundary= Z

d Ωd

h φ ·e

√

−∆· φi

has“Volume Law”entanglement entropy

Boundary theory hasmany DOFand isstrongly interacting 1 Increasing number of DOF doesn’t change scaling of E.E.

2 Addinglocalandappropriate non-localinteractions doesn’t change scaling behavior of E.E.

Conjecture: holographic dual of flat space is a non-local QFT!

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Comparison with AdS

AdS

dsAdS² _{d +1} =dr²

r² +r²d~x²

Boundary energy scale E∼ ∂x∼bulk radial distance r Flat space (ρ = log r )

ds_R²d +1=dr²

r² +(logr )²d Ω²d

Boundary energy scale E∼ ∂Ω∼bulk radial distance log r

1 e^∂^Ω∼ r ∼ ∂x

2 Boundary energy scale E (∼¹_a) ∼ radial distance ˜ρ

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Outline

1

Introdution

2

Entanglement Entropy Definition and properties Holographic E.E. conjecture Holo E.E. for flat space

3

Correlation Functions GKP Procedure

Holographic correlators for flat space.

4

Summary

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GKP Procedure

Gubser+Klebanov+Polyakov ’98, Witten ’98

1 Consider, e.g. a scalar field theory in the bulk

2 Impose Dirichlet boundary condition: φ(ρ∞, Ω) = Φ(Ω) 3 Bulk-to-boundary relation

he^{R dΩ}^d^{Φ· ˆ}^Oi_Sd = Z_Rd +1[φ(ρ∞, Ω) = Φ(Ω)]

4 Boundary correlation functions:

hO(Ω1) . . . O(Ωn)i = ∂

∂Φ(Ω1)· · · ∂

∂Φ(Ωn)he^{R dΩ}^d^{Φ· ˆ}^Oi_Sd

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Holographic Correlation Functions

2-point function for Massless scalar

1 Naive computation

h ˆO(x₁) ˆO(x₂)i = N · ρ^{d −1}∞

G_N^{(d +1)}

1 (1 − cos ∆θ)^{d +1}²

2 Add counter-term ∼ (ρ∞)^{d −1} 3 h ˆO(x1) ˆO(x2)i → 0

2-point function for Massive scalar

1 . . .

2 . . .

3 h ˆO(x₁) ˆO(x₂)i → 0

. . .

After adding non-local counter-terms, all correlation functions vanish:

h ˆO(x1) . . . ˆO(xn)i = 0

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Puzzle: All correlators are trivial — Is boundary theory empty?

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Boundary theory is not empty but highly-entangled

Holographic E.E.

S_A^{hol .}= Vd −1·ρ^{d −1}∞ sin^α₂d −1

4G_N^{(d +1)}

+ When A→ 0, S^A ∼ Vol (A) and is maximized.

For correlator h ˆO(x1) . . . ˆO(xn)i, choose subsystem A = x1∪ . . . ∪ xn

+ S^A is maximal

=⇒ρA→ ρx₁⊗ . . . ⊗ ρx_n and eachρx_i is maximally entangled.

=⇒ All correlation functions vanish:

h Ô(x₁). . . Ô(x_n)i ≡ Tr[ρAO(xˆ ₁). . . Ô(x_n)] = 0 Resolution of Puzzle: All correlators are trivial — it means that boundary theory is highly-entangled.

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How to reproduce bulk physics?

Question: If all boundary correlators are trivial, how to reproduce bulk physics?

Answer: Use entanglement entropy!

1 Entanglement entropy contain infinite amount of information (from different choices of subsystem A)

2 Suitable for non-local theory

(definable for any theory with path integral formulation)

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Discussion: Discrete Model

Toy model based on free scalar theory has non-zero correlators =⇒

Question: How to constructnon-localQFT withzero correlators?

Hint: Discrete model: randomized Heisenberg system H = JX

hi ,ji

σi· σj hi , ji : randomchoices of pairs

1 Correlators are all zero

2 Entanglement entropy obeys “Volume Law”

Question: What is thecontinuum limitof this model?

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Outline

1

Introdution

2