State Dependent Operators and the Information Paradox in AdS/CFT - PowerPoint PPT Presentation

State Dependent Operators and the Information Paradox in AdS/CFT Suvrat Raju International Centre for Theoretical Sciences Tata Institute of Fundamental Research Bangalore Strings 2014 Princeton, 25 June 2014

Collaborators and References This talk is based on work with Kyriakos Papadodimas. ◮ An Infalling Observer in AdS/CFT, arXiv:1211.6767 ◮ The Black Hole Interior in AdS/CFT and the Information Paradox, arXiv:1310.6334 ◮ State-Dependent Bulk-Boundary Maps and Black Hole Complementarity, arXiv:1310.6335 And also on work in progress with Kyriakos, Prashant Samantray (postdoc at ICTS-TIFR, Bangalore) and Souvik Banerjee (postdoc at U. Groningen) Suvrat Raju (ICTS-TIFR) Information Paradox and AdS/CFT Strings 2014 2 / 31

Summary Effective field theory predicts that quantum gravity effects are confined to a Planck-scale region near the singularity. Recent work suggests that to resolve the information paradox, one must drop this robust assumption: “quantum effects radically alter the structure of the horizon.” [Mathur, Almheiri, Marolf, Polchinski, Sully, Stanford, Bousso] I will describe how our construction of the black hole interior in AdS/CFT(see talk by Kyriakos) successfully addresses all these recent arguments. Then I will discuss the “state dependence” of our proposal, and describe work in progress. Suvrat Raju (ICTS-TIFR) Information Paradox and AdS/CFT Strings 2014 3 / 31

Outline Review of the BH Interior in AdS/CFT 1 State Dependent Operators and the Information Paradox 2 Non-Equilibrium States 3 Open Questions 4 Suvrat Raju (ICTS-TIFR) Information Paradox and AdS/CFT Strings 2014 4 / 31

Need for Mirror Operators Apart from usual single-trace operators, new modes are required to construct a local field behind the horizon. Suvrat Raju (ICTS-TIFR) Information Paradox and AdS/CFT Strings 2014 5 / 31

Properties of the Mirror Operators More precisely, the condition for smoothness of the horizon is that there should exist new operators � O ( t , Ω) , satisfying � Ψ |O ( t 1 , Ω 1 ) . . . � O ( t ′ 1 , Ω ′ 1 ) . . . � O ( t ′ l , Ω ′ l ) . . . O ( t n , Ω n ) | Ψ � � l + i β = Z − 1 e − β H O ( t 1 , Ω 1 ) . . . O ( t n , Ω n ) O ( t ′ 2 , Ω ′ l ) β Tr � � � 1 + i β t ′ 2 , Ω ′ . . . O . 1 In Fourier space, we need � O ω satisfying � Ψ |O ω 1 . . . � 1 . . . � O ω ′ O ω ′ l . . . O ω n | Ψ � = e − β l ) † . . . ( O ω ′ 2 ( ω ′ 1 + ...ω ′ l ) � Ψ |O ω 1 . . . O ω n ( O ω ′ 1 ) † | Ψ � . This equation is deceptively simple. On the RHS, the tilde-operators have been moved to the right and reversed. Suvrat Raju (ICTS-TIFR) Information Paradox and AdS/CFT Strings 2014 6 / 31

Construction of the Mirror Operators Given a basis equilibrium state, | Ψ � , we can construct the mirror operators to satisfy the following linear equations. − βω � 2 O ω 1 . . . O ω n ( O ω ) † | Ψ � . O ω O ω 1 . . . O ω n | Ψ � = e Denote all products of O ω i that appear above as A 1 . . . A D . This constitutes all reasonable low energy excitations of | Ψ � . Clearly D ≪ dim ( H ) = e N 2 , and so for generic states we can solve these equations. Explicitly, with − β H β H 2 ( O ω ) † e 2 | Ψ � , | v m � = A m | Ψ � ; | u m � = A m e g mn = � v m | v n � , define O ω = g mn | u m �� v n | . � Suvrat Raju (ICTS-TIFR) Information Paradox and AdS/CFT Strings 2014 7 / 31

State Dependence To fix these operators, we need to fix the “base state” | Ψ � and then consider reasonable experiments about this state. After this, these operators act as ordinary linear operators. One can multiply them, take expectation values etc. � Ψ | � O ω 1 O ω 2 � O ω 3 . . . O ω n | Ψ � However, if we make a big change in the state, then one has to use different operators on the boundary to describe the field “at the same point” behind the horizon. Somewhat unusual, but perhaps to be expected. Suvrat Raju (ICTS-TIFR) Information Paradox and AdS/CFT Strings 2014 8 / 31

Using Mirrors to Remove the Firewall Our explicit construction contradicts arguments in support of the structure at the BH horizon which can be sharply paraphrased as follows. General reasoning (from counting, strong subadditivity of entropy, genericity of commutators etc.) suggest that the � O do not exist in the CFT I will now discuss how our explicit construction of the � O sidesteps all of these arguments. This is useful both to understand the hidden assumptions in these arguments and to understand some intriguing facets of our construction. Suvrat Raju (ICTS-TIFR) Information Paradox and AdS/CFT Strings 2014 9 / 31

Resolving the Strong Subadditivity Paradox The first argument for structure at the BH horizon was based on strong subadditivity of entropy. For an “old black hole”, S AB < S A . For a smooth horizon, S BC = 0. But, thermality of Hawking radiation implies S B = S C > 0. Seems to violate Strong Subadditivity at O(1)! S A + S C ≤ S AB + S BC . Suvrat Raju (ICTS-TIFR) Information Paradox and AdS/CFT Strings 2014 10 / 31

Resolution to the SSE Paradox A C B Our resolution is that A , B , C are not independent. Explicitly, in our construction [ O ω , � O ω ′ ] � = 0 . This is consistent with old notions of complementarity: dof in the interior of the black hole have an overlap with the dof far away. Called A = R B by some authors. [Verlinde 2 , Bousso, Maldacena, Susskind] [Nomura, Weinberg, Varela] Suvrat Raju (ICTS-TIFR) Information Paradox and AdS/CFT Strings 2014 11 / 31

The Generic Commutator For generic embeddings of the interior in the exterior, the non-zero commutator is easily measurable at O [ 1 ] . More precisely, consider some operator O ω , and try and define O ω = U † O † � ω U , for a randomly selected U . Since the Hilbert space is e N 2 dimensional, the matrix elements of − N 2 [ O ω , � 2 ) . O ω ′ ] will be very small ( e But � Ψ | [ O ω , � O ω ′ ][ O ω , � O ω ′ ] † | Ψ � = O ( 1 ) , because the exponential suppression of the matrix elements is offset by the size of the matrix ( e N 2 × e N 2 ) . Suvrat Raju (ICTS-TIFR) Information Paradox and AdS/CFT Strings 2014 12 / 31

The Commutator and Superluminal Propagation This suggests an unacceptable loss of locality. With such commutators, one could send messages across the horizon. The generic order 1 commutator was a powerful argument against the use of complementarity to remove the firewall. Suvrat Raju (ICTS-TIFR) Information Paradox and AdS/CFT Strings 2014 13 / 31

Suppressing the Commutator Our construction resolves this in a clever way. Within low point correlators, − βω ′ [ O ω , � O ω A p ( O ω ′ ) † | Ψ � O ω ′ ] A p | Ψ � = e 2 − βω ′ O ω A p ( O ω ) † | Ψ � = 0 ! − e 2 While the commutator does not vanish, it is undetectable in low point correlators. We denote this by O ω ′ ] . [ O ω , � = 0 . Resolves a central objection to the use of complementarity! Suvrat Raju (ICTS-TIFR) Information Paradox and AdS/CFT Strings 2014 14 / 31

The Counting Argument − 1 c † O † � Set � ω = G 2 ω : the normalized creation operator behind the ω horizon. Then, c † [ � c ω , � ω ] A p | Ψ � = A p | Ψ � , and so � � � c ω c † � ω = 1 ? c † 1 + � ω � c ω But creating a particle behind the horizon in the Hartle-Hawking state is like destroying a particle in front of it. c † c † [ H cft , � ω ] = − ω � ω . Since the growth of number of states with energy in the CFT is c † monotonic, � ω cannot have a left inverse? Suvrat Raju (ICTS-TIFR) Information Paradox and AdS/CFT Strings 2014 15 / 31

Resolving the Counting Argument H E − ω H E c † The action of � c ω , � ω is correct only on | Ψ � and its descendants produced by excitations with bounded energy and insertions. ω ] . c † [ � c ω , � = 1 c † ⇒ [ � c ω , � ω ] A p | Ψ � = A p | Ψ � , for any light operator A p . No contradiction with Linear Algebra! Suvrat Raju (ICTS-TIFR) Information Paradox and AdS/CFT Strings 2014 16 / 31

The N a � = 0 Paradox General arguments suggest that for a fixed operator � O ω , the microcanonical expectation of the number operator, � N a � , for the infalling observer is O [ 1 ] . [Marolf, Polchinski] But, G ω N a = ( 1 − e − βω n ) − 1 � � � � � ω − e − βω O ω − e − βω 2 � 2 � O † O † O ω ω � � � � � ω − e − βω O ω − e − βω O † � � 2 O † 2 O ω + . ω However, our operators satisfy − βω βω � 2 ( O ω ) † | Ψ � ; � O † 2 O ω | Ψ � . O ω | Ψ � = e ω | Ψ � = e Therefore N a | Ψ � = 0! Our construction has the explicit property that the infalling observer measures no particles at the horizon. Suvrat Raju (ICTS-TIFR) Information Paradox and AdS/CFT Strings 2014 17 / 31

Interim Summary The use of an appropriately state-dependent mapping between boundary operators and local bulk operators addresses all the recent information theoretic arguments for structure at the horizon. Suvrat Raju (ICTS-TIFR) Information Paradox and AdS/CFT Strings 2014 18 / 31