Energy Condition, Modular Flow, and AdS/CFT Black Holes and Holography Workshop,TSIMF, Jan 7-11, 2019 Huajia Wang Kavli Institute for Theoretical Physics u v ∆ v ( z ) u X + B ( z ) v O z = 0 z arXiv:1806.10560; arXiv:1706.09432; JHEP 1609 038(2016) S. Balakrishnan, T. Faulkner, R. Leigh, M. Li, Z. Khandker, O. Parrikar, H. Wang
Energy Conditions What are they? unitarity of QM: positivity of total energy extended systems (QFT): local energy/momentum density constraints on energy/momentum density Z R n dx n E ( x ) ≥ 0 E = E ≥ 0 E 0
Energy Conditions Why do we care? classical: important in general relativity energy-momentum = spacetime geometry energy conditions = constraints on spacetime Einstein’s equations: G µ ν = 8 π T µ ν
Energy Conditions Why do we care? examples: Hawking, Ellis, 1973 strong energy condition (SEC) —> singularity theorem null energy condition (NEC) —> horizon area theorem ζ t t ✓ ◆ T ab − 1 T ab k a k b ≥ 0 k ζ a ζ b ≥ 0 2 Tg ab x x
Energy Conditions What do we want? in QM: QFTs in fixed background spacetime h ˆ constraints on T µ ν i ψ NEC violated by quantum effects: e.g. Casimir effect correct modification to NEC? two main conjectures: Averaged Null Energy Condition (ANEC) Quantum Null Energy Condition (QNEC)
AVERAGED NULL ENERGY CONDITION (ANEC) affine parameter t T µ ν i ψ k µ k ν � 0 d λ h ˆ R k x λ Why? violation leads to causality breakdown — supports traversable wormhole/time machine M. Morris, K. Thorne, U. Yurtsever, PRL. 61. 13. 1988 QUANTUM NULL ENERGY CONDITION (QNEC) T µ ν ( y ) i ψ k µ k ν � ∂ 2 h ˆ A ( λ ) λ S A ( λ ) ( ψ ) λ Motivation: generalized second law y A (0)
Can we prove them in QFTs? How?
A brief history of proofs… for specific types of theories: ANEC for free scalar and Maxwell fields; G. Klinkhammer, 1991; L. Ford, T. Roman, 1995; A. Folacci, 1992 ANEC for 2d massive QFTs; R. Verch, 2000 QNEC for free/super-renormalizable fields; R. Busso, Z. Fisher, J. Koeller, S. Leichenaber, A. Wall, 2015
A brief history of proofs… for holographic theories (a broad class of CFTs): proof of ANEC using AdS/CFT: W. Kelly, A. Wall, 2014 causality constraint in the bulk. proof of QNEC using AdS/CFT: J. Koeller, S. Leichenauer, 2016; C. Akers. V. Chandrasekaran, S. Leichenaber, A. Levin, A. Moghaddam, 2017 entanglement wedge nesting (EWN)
Can we do better? Proofs for generic QFT/CFTs?
Can we do better? Proofs for generic QFT/CFTs? Recent progresses…
Can we do better? Proofs for generic QFT/CFTs? ANEC in relativistic QFTs: T. Faulkner, R. Leigh, O. Parrikar, H. Wang, 2016 monotonicity of relative entropy ANEC in CFTs: T. Hartman, S. Kundu, A. Tajdini, 2016 causality of correlation functions in light-cone limit QNEC in CFTs: S. Balakrishna, T. Faulkner, Z. Khandker, H. Wang, 2017 causality of correlation function under modular flow
Plan of the talk: Review of AdS/CFT proofs (ANEC + QNEC) Summary of general field theory proofs (ANEC + QNEC) Bulk modular flow in AdS/CFT Conclusion/outlooks
Plan of the talk: Review of AdS/CFT proofs (ANEC + QNEC) Summary of general field theory proofs (ANEC + QNEC) Bulk modular flow in AdS/CFT Conclusion/outlooks
Z ∞ dx + h ˆ Proving ANEC using AdS/CFT T ++ i ψ � 0 −∞ W. Kelly, A. Wall, 2014
Z ∞ dx + h ˆ Proving ANEC using AdS/CFT T ++ i ψ � 0 −∞ W. Kelly, A. Wall, 2014 z t L bulk ≥ L bdry x S. Gao, R. Wald, 2000 L bdry L bulk “bulk respects boundary causality”
Z ∞ dx + h ˆ Proving ANEC using AdS/CFT T ++ i ψ � 0 −∞ W. Kelly, A. Wall, 2014 z t L bulk ≥ L bdry x S. Gao, R. Wald, 2000 L bdry L bulk “bulk respects boundary causality” As a GR result, can be proved by assuming that the “ANEC” in the bulk theory is satisfied
Z ∞ dx + h ˆ Proving ANEC using AdS/CFT T ++ i ψ � 0 −∞ W. Kelly, A. Wall, 2014 z t L bulk ≥ L bdry x S. Gao, R. Wald, 2000 L bdry L bulk “bulk respects boundary causality” In AdS/CFT, via Fefferman-Graham gauge expansion: ds 2 = R 2 ⇢ z d +2 �� � η ab + z d 16 π G dz 2 + dx a dx b � dR d − 1 h T ab i ψ + O , z ! 0 z 2
Z ∞ dx + h ˆ Proving ANEC using AdS/CFT T ++ i ψ � 0 −∞ W. Kelly, A. Wall, 2014 z t L bulk ≥ L bdry x S. Gao, R. Wald, 2000 L bdry L bulk “bulk respects boundary causality” Gao-Wald’s conclusion as consistent condition for holographic CFTs Z ∞ dx + h ˆ lim z → 0 boundary ANEC T ++ i ψ � 0 −∞ leading order constraint in F. G. gauge expansion
h T uu i ψ � ∂ 2 Proving QNEC using AdS/CFT u S EE J. Koeller, S. Leichenauer, 2016; C. Akers. V. Chandrasekaran, S. Leichenaber, A. Levin, A. Moghaddam, 2017
h T uu i ψ � ∂ 2 Proving QNEC using AdS/CFT u S EE J. Koeller, S. Leichenauer, 2016; C. Akers. V. Chandrasekaran, S. Leichenaber, A. Levin, A. Moghaddam, 2017 “bulk reconstruction in entanglement wedges” AdS/CFT: bulk physics can be “reconstructed” from the boundary t z D ( A ) x how much bulk region can be reconstructed from CFT A operators localized in D(A)? subregion duality
h T uu i ψ � ∂ 2 Proving QNEC using AdS/CFT u S EE J. Koeller, S. Leichenauer, 2016; C. Akers. V. Chandrasekaran, S. Leichenaber, A. Levin, A. Moghaddam, 2017 “bulk reconstruction in entanglement wedges” strong evidence: entanglement wedge t X. Dong, D. Harlow, A. Wall, 2016 z D ( A ) x ∂ a = Σ ∪ A a A RT surface Σ
h T uu i ψ � ∂ 2 Proving QNEC using AdS/CFT u S EE J. Koeller, S. Leichenauer, 2016; C. Akers. V. Chandrasekaran, S. Leichenaber, A. Levin, A. Moghaddam, 2017 “bulk reconstruction in entanglement wedges” strong evidence: entanglement wedge D ( a ) t X. Dong, D. Harlow, A. Wall, 2016 z D ( A ) x ∂ a = Σ ∪ A a A entanglement wedge = D(a) RT surface Σ
h T uu i ψ � ∂ 2 Proving QNEC using AdS/CFT u S EE J. Koeller, S. Leichenauer, 2016; C. Akers. V. Chandrasekaran, S. Leichenaber, A. Levin, A. Moghaddam, 2017 “bulk reconstruction in entanglement wedges” strong evidence: entanglement wedge D ( a ) t X. Dong, D. Harlow, A. Wall, 2016 z D ( A ) x ∂ a = Σ ∪ A a A entanglement wedge = D(a) RT surface Σ D ( a )“ ≈ ” D ( A )
h T uu i ψ � ∂ 2 Proving QNEC using AdS/CFT u S EE J. Koeller, S. Leichenauer, 2016; C. Akers. V. Chandrasekaran, S. Leichenaber, A. Levin, A. Moghaddam, 2017 D ( ˜ A ) ⊆ D ( A ) → D (˜ a ) ⊆ D ( a ) Entanglement Wedge Nesting (EWN):
h T uu i ψ � ∂ 2 Proving QNEC using AdS/CFT u S EE J. Koeller, S. Leichenauer, 2016; C. Akers. V. Chandrasekaran, S. Leichenaber, A. Levin, A. Moghaddam, 2017 D ( ˜ A ) ⊆ D ( A ) → D (˜ a ) ⊆ D ( a ) Entanglement Wedge Nesting (EWN): at the boundary: null deformation ∆ u ≥ 0 : u v D ( ˜ A ) ⊆ D ( A ) ˜ A z ∆ u A
h T uu i ψ � ∂ 2 Proving QNEC using AdS/CFT u S EE J. Koeller, S. Leichenauer, 2016; C. Akers. V. Chandrasekaran, S. Leichenaber, A. Levin, A. Moghaddam, 2017 D ( ˜ A ) ⊆ D ( A ) → D (˜ a ) ⊆ D ( a ) Entanglement Wedge Nesting (EWN): at the boundary: null deformation ∆ u ≥ 0 : u v D ( ˜ A ) ⊆ D ( A ) ˜ A z ˜ ∆ u a into the bulk: a A Σ ˜ D (˜ a ) ⊆ D ( a ) (EWN) A Σ A Σ ˜ Σ A spacelike/null A RT surfaces dynamics
h T uu i ψ � ∂ 2 Proving QNEC using AdS/CFT u S EE J. Koeller, S. Leichenauer, 2016; C. Akers. V. Chandrasekaran, S. Leichenaber, A. Levin, A. Moghaddam, 2017 D ( ˜ A ) ⊆ D ( A ) → D (˜ a ) ⊆ D ( a ) Entanglement Wedge Nesting (EWN): Σ ˜ Σ A spacelike/null A near boundary expansion: u v (F-G gauge) ˜ A z g uu = 16 π G ˜ ∆ u a dR d − 3 z d − 2 h T ab i ψ + O ( z d ) a A Σ ˜ 4 G A X i Σ A ( z ) = X i dR d − 1 z d ∂ i S EE ( A ) + O ( z d +1 ) ∂ A + Σ A 4 G dR d − 1 z d ∂ i S EE ( ˜ X i A ( z ) = X i A ) + O ( z d +1 ) A + Σ ˜ ∂ ˜
h T uu i ψ � ∂ 2 Proving QNEC using AdS/CFT u S EE J. Koeller, S. Leichenauer, 2016; C. Akers. V. Chandrasekaran, S. Leichenaber, A. Levin, A. Moghaddam, 2017 D ( ˜ A ) ⊆ D ( A ) → D (˜ a ) ⊆ D ( a ) Entanglement Wedge Nesting (EWN): Σ ˜ Σ A spacelike/null A u v z → 0 ˜ A z ∆ u → 0 ˜ ∆ u a a A Σ ˜ A h T uu i ψ � ∂ 2 u S EE � 0 Σ A boundary QNEC
Plan of the talk: Review of AdS/CFT proofs (ANEC + QNEC) Summary of general field theory proofs (ANEC + QNEC) Bulk modular flow in AdS/CFT Conclusion/outlooks
Z ∞ dx + h ˆ Proving ANEC in relativistic QFTs T ++ i ψ � 0 −∞ T. Faulkner, R. Leigh, O. Parrikar, H. Wang, 2016
Z ∞ dx + h ˆ Proving ANEC in relativistic QFTs T ++ i ψ � 0 −∞ T. Faulkner, R. Leigh, O. Parrikar, H. Wang, 2016 difficult using conventional QFT techniques surprising origin in information theory manifested by probing the entanglement structure
Z ∞ dx + h ˆ Proving ANEC in relativistic QFTs T ++ i ψ � 0 −∞ T. Faulkner, R. Leigh, O. Parrikar, H. Wang, 2016 Modular Hamiltonian: K Ψ A = − ln ρ Ψ A ⊗ 1 A c + 1 A ⊗ ln ρ Ψ A c = H Ψ A − H Ψ A c A K Ψ K Ψ A | Ψ i = 0 A : H full → H full A c
Recommend
More recommend
