The black hole information paradox and the fate of the infalling - - PowerPoint PPT Presentation

The black hole information paradox and the fate of the infalling observer

Kyriakos Papadodimas CERN and University of Groningen Quantum Gravity in Paris 2017

Motivations

Black hole information paradox What happens when crossing the horizon? Reconstructing the black hole interior in AdS/CFT

Motivations

I will describe a proposal for describing the black hole interior, which suggests a possible approach towards resolving the information paradox Key principles: i) Locality in quantum gravity is approximate ii) State dependence of physical observables More recent developments: connection to ER/EPR, a toy model of complementarity in AdS/CFT

Basic info paradox

Hawking computation predicts thermal radiation Photons thermal and independent (no correlations) |Ψstar ⇒ ρthermal (∗) Information Loss? In Quantum Mechanics time evolution is Unitary |Ψfinal = e−iHt|Ψinitial Inconsistent with (∗).

Normal “burning“

Radiation appears to be thermal Small correlations between photons (of size e−S) Accurate measurement of correlations ⇒ full information of initial state No information loss problem

Resolution of basic version of info paradox

∃ quantum corrections to Hawking’s computation e−SBH deviations from Hawking’s predictions for simple

• bservables (example: 2-point

correlations between photons) ⇒ sufficient to restore unitarity Reminder: for solar mass BH SBH ≈ 1077

Hawking ⇒ ρthermal Hawking + ”corrections“ ⇒ |Ψpure How different does radiation look?

Pure vs Mixed states

E |Ψ =

N

• i

ci|Ei vs ρmicro = 1 N I N = eS = number of eigenstates ≫ 1 ci = random coefficients Theorem: In a large quantum system, for most pure states, and simple observables A, we have Ψ|A|Ψ = Tr(ρmicroA) + O(e−S)

Pure vs Mixed states

E |Ψ =

N

• i

ci|Ei vs ρmicro = 1 N I N = eS = number of eigenstates ≫ 1 ci = random coefficients Theorem: In a large quantum system, for most pure states, and simple observables A, we have Ψ|A|Ψ = Tr(ρmicroA) + O(e−S) (not true for complicated observables n ≈ S) Ψ|A1...An|Ψ = Tr(ρmicroA1...An) + O(e−(S−n))

[S.Lloyd] Define Amicro = Tr(ρmicroA) We also define the average over pure states in HE Ψ|A|Ψ ≡

• [dµΨ]Ψ|A|Ψ

where [dµΨ] is the Haar measure. Then for any observable A we have Ψ|A|Ψ = Amicro and variance ≡ (Ψ|A|Ψ2)−(Amicro)2 = 1 eS + 1

• A2micro − (Amicro)2

”reasonable“ observables have the same expectation value in most pure states, up to exponentially small corrections.

Hawking : ρthermal |Ψpure Small number of photons ⇒ Predictions agree up to O(e−SBH) Need to measure correlator between O(SBH) photons to get information of state Hawking computation is reliable for simple observables but not for complicated ones

Comments

◮ Basic version of info paradox, where we only talk about

radiation at infinity, can in principle be resolved: Hawking predicts thermal radiation. Exponentially small deviations e−SBH to simple observables can restore unitarity

◮ We do not know how to calculate these corrections, but we do

expect them on general grounds so there is no paradox.

◮ Computing these corrections, and understanding the

microscopic mechanism of information transfer is a bigger problem (S-matrix of Quantum Gravity) but is not really a ”paradox“

◮ So far we have not said anything about the BH interior...

Modern info paradox, infalling observer

Curvature at horizon R2 ∼ 1 (GM)4 General Relativity/Equivalence Principle, predicts: for a large BH ⇒ will not notice anything when crossing horizon What if we include Quantum Mechanics? Problem with Entanglement Dramatic modification of horizon/interior?

Entanglement Reminder

Two sub-systems A, B then Hfull = HA ⊗ HB Typical state |Ψ =

ij cij|iA ⊗ |jB does not factorize = ”is

entangled“ Example: two spins Non-entangled state |Ψ = | ↑A ⊗ | ↑B Entangled state (EPR) |Ψ = | ↑A ⊗ | ↑B + | ↓A ⊗ | ↓B √ 2

Ground state of QFT is entangled

φ(0, x) φ(0, y) = 1 |x − y|2

x t R L F P

|0M = 1 √ Z

• ω

∞

• n=0

e−πωn|nL ⊗ |nR

Smooth spacetime needs entanglement

x t R L F P

1 √ Z

• ω

∞

• n=0

e−πωneiθn|nL ⊗ |nR Tµν = 0 Rindler Horizon excited

Monogamy of entanglement

A B C

A, B, C independent systems H = HA ⊗ HB ⊗ HC Strong subadditivity of Entanglement Entropy SAB + SBC ≥ SA + SC

Hawking pair production Particles of each pair highly entangled Entanglement required for smoothness of horizon

Modern info Paradox

Mathur [2009], Almheiri, Marolf, Polchinski, Sully (AMPS) [2012] General Relativity: smooth horizon, B entangled with C Quantum Mechanics: information preserved, B entangled with A B violates monogamy? Mathur’s theorem: small corrections cannot fix the problem (?)

A

B

c

SA

SBH

• ld black hole

Which one survives, Unitarity or Smooth Horizon?

Giving up B-C entanglement? Firewall, fuzball proposals ⇒ Tµν at horizon is very large, BH interior geometry is completely modified (maybe no interior at all) Infalling observer ”burns“ upon impact on the horizon. Dramatic modification of General Relativity/Effective Field Theory

• ver macroscopic scales, due to quantum effects

Chaos vs entanglement

Black Holes are Chaotic Quantum Systems

B

c

How can typical states have specific entanglement between B, C which is needed for smoothness?

A

Correct entanglement fragile under perturbations due to chaotic nature of system [Shenker, Stanford]

Summary

◮ The modern version of the info paradox, is intimately related

to the smoothness of the horizon and to what happens to the infalling observer.

◮ We have a conflict between QM and General Relativity

because it seems impossible to have the entanglement of quantum fields, needed for smoothness, near the horizon.

◮ Is there a way out?

AdS/CFT

◮ AdS/CFT: non-perturbative definition of Quantum Gravity by

dual gauge theory

◮ Black Holes in AdS ⇔ Quark-Gluon-Plasma states in QFT ◮ BH formation + evaporation ⇔ deconfinement +

hadronization

◮ Very strong argument in favor of Unitarity

AdS

Non-perturbative Black Hole S-matrix encoded in CFT correlators Manifestly Unitary

Black Hole interior in AdS/CFT?

AdS

Suppose we completely solve the CFT (know all correlators exactly) How do we describe the black hole interior? Well-defined question, but conceptual/mathematical framework is missing What computation in the CFT do we have to do to probe BH interior?

◮ AdS/CFT very successful for certain black hole questions ◮ But until recently, understanding of BH interior was limited ◮ In last few years we developed a framework for the

holographic description of the BH interior [K.P. and S. Raju, also with S.Banerjee and J.W.Bryan] based on JHEP 1310 (2013) 212, PRL 112 (2014) 5, Phys.Rev.

D89 (2014), PRL 115 (2015), JHEP 1605 (2016)

◮ We identified CFT operators relevant for BH interior ◮ This framework seems to resolve the tension of entanglement

in the modern version of the info paradox

◮ It is important to make further checks and to expand into a

complete mathematical framework

Local observables in AdS

φ(x) =

• dY K(x, Y ) O(Y )

O= local CFT operator dual to bulk field φ K =known kernel [Hamilton, Kabat, Lifschytz, Lowe]

Local observables in AdS

φ(x) =

• dY K(x, Y ) O(Y )

O= local CFT operator dual to bulk field φ K =known kernel [Hamilton, Kabat, Lifschytz, Lowe] Locality in bulk is approximate:

• 1. True in 1/N perturbation theory
• 2. [φ(P1), φ(P2)] = 0 only up to e−N2 accuracy
• 3. Locality may break down for high-point functions

For smooth horizon effective field theory requires: I) b commute with b AND II) b entangled with b b ⇔ O

• b

⇔ ? Which CFT operators O correspond to b?

◮ Smoothness of BH horizon and existence of interior,

translated into concrete mathematical problem: can we find CFT operators O with desired properties. i) for every single trace operator O there is a O ii) O’s and O’s must commute ii) O’s and O’s must be entangled (must have specific 2-point functions)

Black holes in AdS/CFT

◮ What are

Oω operators in the CFT?

Black holes in AdS/CFT

◮ What are

Oω operators in the CFT?

◮ [AMPSS, MP] paradox: if typical CFT states have smooth

horizon, using [H, O†

ω] = −ω

O†

ω we find an inconsistency with

basic properties of the spectrum of states in the CFT

[H, O†

ω] = −ω

O†

ω

[ Oω, O†

ω] = 1

• O†

ω = “creation operator”

⇒ O†

ω should not annihilate (typical) states of the CFT.

On the other hand [H, O†

ω] = −ω

O†

ω

implies that O†

ω lowers the energy so it maps CFT states of energy

E to E − ω. But in the CFT, we have S(E) > S(E − ω). ⇒ if O†

ω is an ordinary linear operator, it must have a nontrivial

kernel. Inconsistent with statement that O†

ω does not annihilate states.

⇒ The CFT does not contain O†

ω operators and cannot describe

the BH interior (?)

In the last few years the BH-info/firewall paradox has been refined and turned into a concrete problem, phrased within AdS/CFT. It seems to suggest that most states of the CFT in the deconfined phase are dual to a black hole with a singular horizon.

In the last few years the BH-info/firewall paradox has been refined and turned into a concrete problem, phrased within AdS/CFT. It seems to suggest that most states of the CFT in the deconfined phase are dual to a black hole with a singular horizon. In my work with S.Raju we demonstrated the existence of the O

• perators, thus resolving this paradox.

Interesting new property of these operators: state-dependence.

A construction of the BH interior

[KP and Suvrat Raju]

◮ If we take a CFT state |Ψ of O(N2) energy, we expect that

at late times it will thermalize. Ψ|O1(x1)...On(xn)|Ψ ≈ Z−1Tr(e−βHO1(x1)...On(xn))

◮ This is true only for simple observables n ≪ N ◮ Thermalization of pure state ⇒ must have the notion of a

small algebra of observables

◮ In a large N gauge theory, natural small “algebra” A =

products of few, single trace operators

Defining the “small Hilbert space HΨ”

Suppose we have a typical BH microstate |Ψ and bulk observer Consider possible simple experiments the observer can perform within EFT. To describe those, we do not need the entire Hilbert space of the CFT, but rather a smaller subspace. If φ(x) is a bulk field, the states we need to use are φ(x)|Ψ φ(x1)φ(x2)|Ψ, ... φ(x1)...φ(xn)|Ψ, ... and their linear combinations, where the number of insertions n does not scale with N and the points xi are not too spread-out in time.

T

In the CFT BH microstate → typical QGP state |Ψ Bulk field φ related to boundary single-trace operator O A= “algebra” of small products of single-trace operators A = span of{O(t1, x1), O(t1, x1)O(t2, x2), ...}

We define HΨ = A|Ψ = {span of : O(t1, x1)...O(tn, xn))|Ψ} Simple EFT experiments in the bulk, around BH |Ψ take place within HΨ The interior operators O will be defined to act only on this subspace. HΨ is similar to “code subspace”

Important point:

T

HΨ = A|Ψ = {span of : O(t1, x1)...O(tn, xn))|Ψ} already contains the states describing the BH interior! (i.e. states we would get in bulk EFT by acting with b) entanglement, compare with Reeh-Schlieder theorem in QFT The CFT operators that represent the BH interior will act within the subspace HΨ We will call them mirror operators and denote them by O. Notice that O, O must commute.

What is special when |Ψ is a BH microstate, which allows the “small Hilbert space” HΨ = A|Ψ to be big enough to accomodate the action of operators O which commute with O?

A typical BH microstate |Ψ cannot be annihilated by (nonvanishing) elements of the small algebra A This implies that the representation of A on HΨ has qualitative differences when |Ψ is a BH microstate, compared to -say- when |Ψ is the vacuum. Physical interpretation: The state |Ψ appears to be entangled when probed by the algebra A.

Consider the Hilbert space of two spins, and A = operators acting

• n the first.

If the two spins are in the state |Ψ = | ↑↑ In this case there is no entanglement and indeed the previous condition is violated since s(1)

+ |Ψ = 0

while s(1)

+ = 0

On the other hand consider the state |Ψ = 1 √ 2(| ↑↑ + | ↓↓) Now there is entanglement and, relatedly, there is no non-vanishing

• perator acting on the first spin that annihilates the state.

D t x t x D

Reeh-Schlieder theorem: Minkowski vacuum |0M cannot be annihilated by acting with local operators in D. ⇒ In |0M local operator algebras are entangled

A construction of the BH interior

◮ Even though we are in a single CFT in a pure state, the small

algebra of single trace operators probes the pure state |Ψ as if it were an entangled state Ψ|..|Ψ ≈ Tr[e−βH...] ↔ |TFD =

• E

e−βE/2 √ Z |E⊗|E

◮ The O(N2) d.o.f. of the CFT play the role of the “heat bath”

for the small algebra

◮ Whatever operators the single trace operators are entangled

with, will play the role of O behind the horizon.

◮ How do we identify these operators concretely? ◮ Key algebraic property: the small algebra cannot annihilate

the pure state |Ψ

Tomita-Takesaki modular theory

Algebra, cannot annihilate state. ⇒ the representation of the algebra is reducible, and the algebra has a nontrivial commutant acting on the same space. Define antilinear map SA|Ψ = A†|Ψ and ∆ = S†S J = S∆−1/2 Then the operators

• O = JOJ

i) commute with O ii) are correctly entangled with O These are the operators that we need for the Black Hole interior.

The modular Hamiltonian

The operator ∆ = S†S is a positive, hermitian operator and can be written as ∆ = e−K where K = modular Hamiltonian for the small algebra Using large N and the KMS condition for thermal correlators in equilibrium states K = β(HCFT − E0)

In practice

• Oω|Ψ = e− βω

2 O†

ω|Ψ

• OωO....O|Ψ = O...O

Oω|Ψ [H, Oω]O....O|Ψ = ω OωO....O|Ψ

These equations define the operators O on a subspace HΨ ⊂ HCFT, which is relevant for EFT around BH microstate |Ψ Equations admit solution because the algebra A cannot annihilate the state |Ψ

Bulk field inside BH φ(t, r, Ω) = ∞ dω

• Oω fω(t, Ω, r) +

Oωgω(t, Ω, r) + h.c.

• Correlation functions of these operators

Ψ|φ(t1, r1, Ω1)...φ(tn, rn, Ωn)|Ψ reproduce those of effective field theory in the exterior/interior of the black hole AdS/CFT: Smooth spacetime at the horizon, no firewall At the same time, Unitarity OK Unitarity + smoothness of horizon reconciled!

What about previous paradoxes?

Non-locality

• O were constructed based on the fact that we restricted our

attention to a “small algebra” of O’s. The construction breaks down if the “small algebra” is enlarged to include all operators [O, O] = 0 only in simple correlators, not as operator equation Operators O = complicated combinations of O [φ(P), φ(Q)] O(e−S) [φ(P), Φcomplex(Q)] = O(1) The hilbert space of Quantum Gravity does not factorize as Hinside ⊗ Houtside 1) Solves problem of Monogamy of Entanglement (and avoids Mathur’s theorem) 2) Is consistent with bulk EFT

◮ Remember that we do not know how to define what would be

“absolutely local” observables in quantum gravity

◮ The distinction between simple and complicated operators was

important for emergence of approximate locality

◮ What could be an example of a complicated operator violating

locality?

◮ If complementarity is a general feature of quantum gravity,

can we understand it in a simpler setting without a black hole?

State-dependence

◮ Interior operators defined by

• Oω|Ψ = e− βω

2 O†

ω|Ψ

• OωO....O|Ψ = O...O

Oω|Ψ

◮ Solution depends on reference state |Ψ ◮ Operators cannot be upgraded to “globally defined” operators ◮ Solves Chaos vs Entanglement problem ◮ Unusual in Quantum Mechanics, needs further study

“Derivation” of ER = EPR

[K.P and S.R. (1503.08825)] Entanglement & Wormholes (Maldacena, Susskind, Raamsdonk) H = HL + HR |TFD =

• E

e−βE/2 √ Z |EL ⊗ |ER

ER=EPR

Time-shifted wormholes

[K.P and S.R. (1502.06692)] |ΨT ≡ eiHLT |TFD

T

Strong evidence in favor of state-dependence

Thermalization in gauge theories

t

O O O

i /2

~

A new class of non-equilibrium states |Ψ′ = U( O) |Ψ = e− βH

2 U(O)e βH 2 |Ψ

A toy model of complementarity

[S. Banerjee, J-W. Brian, K.P., S. Raju, arXiv:1603.02812 ] AdS in global coordinates

h

CFT on Sd−1 × R T < π rD = tan π−T

2

• ◮ Spherical horizon in bulk

◮ Without gravity we would have:

H = HD ⊗ HD Commuting algebras A(D), A(D)

◮ Including gravity: Gauge fixing, gauss law

commutators, etc. But at large N we still expect approximate bulk decomposition.

◮ What is CFT interpretation of this

decomposition?

A toy model of complementarity

[S. Banerjee, J-W. Brian, K.P., S. Raju, arXiv:1603.02812 ] AdS in global coordinates

h

CFT on Sd−1 × R T < π rD = tan π−T

2

• ◮ Spherical horizon in bulk

◮ Without gravity we would have:

H = HD ⊗ HD Commuting algebras A(D), A(D)

◮ Including gravity: Gauge fixing, gauss law

commutators, etc. But at large N we still expect approximate bulk decomposition.

◮ What is CFT interpretation of this

decomposition?

◮ Simple vs complicated operators

Summary

◮ Reconstructing the BH interior from the CFT remains an

• utstanding problem in the AdS/CFT correspondence

◮ It may have implications for BH information paradox ◮ A proposal: state-dependence operators, realization of

complementarity

◮ Simple vs complicated operators and locality ◮ More explicit description of interior operators?

THANK YOU