Emergent Locality in the AdS/CFT Correspondence

Transcription

1 June 1, 2011 Emergent Locality in the AdS/CFT Correspondence arxiv: (MG, Giddings, Penedones) arxiv: (MG, Giddings) work in progress (MG, Giddings) 1

2 Acknowledgements My committee: Joe Polchinski, Harry Nelson and my advisor: Steve Giddings My collaborators: Richard Eager, Matt Roberts, João Penedones and all the members of the High Energy & Gravity Group My friends: Viva Horowitz, Dan Malinow, Valentina Zambrano, April Barber, Wayne Tang, Katie Maynard, Allison Chapin, Dave Bennet, Lauren Uhler, Lindsay Solda, Nathan Roller, Sara Adler, entering class of 2005, CDC, SB192, carpe-diem My family 2

3 Overview Black hole evaporation The AdS/CFT correspondence Scattering in the large R limit Necessity of wavepackets Construction Singularity structure Limits on locality Can we do better? Understanding Bulk Unitarity Conclusions & Open Questions 3

4 Evaporating Black Holes Classically, everything that falls in is lost forever. singularity horizon I + i 0 Hawking: Black holes aren t completely black. Hawking Radiation black holes evaporate. matter I 4

5 Evaporating Black Holes State on Σ 0 can be pure. Locality trace over sub-sector of Σ 1 inside horizon mixed state outside. Pure state evolving into mixed state violates unitarity. singularity Σ 1 horizon Σ 0 I I + i 0 5

6 Evaporating Black Holes 3 Possible Resolutions: Non-Unitary evolution ρ $ρ Remnants Black hole only evaporates to Planck scale Non-locality 6

7 Non-Unitary Evolution? ρ $ρ Information transfer requires energy. Information loss energy loss. Virtual effects Planck-scale energy non-conservation. Banks, Peskin, Susskind (1984): Nonunitarity thermal bath at T ~ M P 7

8 Remnants? Long lived Planck-scale remnant: if remnant decays and information gets out, takes t decay = S2 M P Form black holes from arbitrarily many initial states, so arbitrarily many remnant species arbitrarily large production cross-section. 8

9 Non-Locality? Physical observables must be gauge invariant. In gravity, this means observables must be diffeomorphism invariant. There are no diffeomorphism invariant local observables in gravity. (Torre 1993) δo(x) = µ µ O(x) 9

10 Holographic Principle Hints that locality should be given up. BH Entropy grows with bounding area, not volume. Bousso (1999): trying to access too many states in a fixed volume black hole formation. Look for a non-local formulation of Quantum Gravity. Area law should look for a theory in one fewer dimensions. 10

11 AdS/CFT Quantum Gravity in asymptotically Anti-de Sitter space (AdS) is conjectured to be dual to a Conformal Field Theory (CFT) living on the boundary. (Maldacena 1997) exp i AdS α φ O φ CFT = Z S [α φ ] Operator insertions in the CFT boundary conditions for fields in AdS. 11

12 AdS Geometry ds 2 = R2 cos 2 ρ dτ 2 + dρ 2 +sin 2 ρ dω 2 d 1 Universal cover of hyperboloid of radius R in M 2,d P 4 P 3!" P 4 P 3 k2 k 1 X 0 #!" X 0 P 1 P 2 P 1 P 2 12

13 AdS/CFT Dictionary Gubser, Klebanov, Polyakov; Witten (1998): Boundary conditions on fields in AdS operator insertions & VEVs in dual CFT. Fields in AdS have normalizable & non-normalizable modes: φ cos 2h ρα(τ, Ω)+ + cos 2h + ρβ(τ, Ω) Non-normalizable mode operator insertion: L CFT L CFT + α φ O φ Normalizable mode operator expectation value: O φ = β φ 13

14 Large R Limit t = Rτ r = Rρ R Approximately flat ds 2 dt 2 + dr 2 + r 2 dω 2 Normalizable frequencies ω nl ωr Normalizable wavefunctions φ nl m 2ω d 1 (ωr) d 2 1 J l+ d 2 1(ωr)Y l m(ω) 14

15 Scattering in the Flat Region Scattering in the flat region should approximate local physics in our universe. Need to construct wavepackets to localize scattering to a single flat region of AdS. 15

16 Multiple Scattering Free fields in AdS are periodic. Purely normalizable states will interact infinitely many times. Can t isolate contribution from one scattering experiment. 16

17 Interactions Near the Boundary Boundary sources infinite particle production near the boundary. Single particle states not well defined when boundary sources turned on. N = Difficult to isolate scattering in flat region from scattering near the boundary. Sources should be compact and non-overlapping to avoid infinite interactions near the boundary and normalize states. t 0 d d x g(φ t φ)= 17

18 Boundary-Compact Wavepackets Construct using Bulk-Boundary Propagator. φ f (x) = Compactly supported sources f(b) =L of size τ, θ. 1 ωr τ τ0 dbf(b)g B (b, x) τ L τ, θ 1 θ θ Scatter when sources turned off. e iωr(τ τ 0) 18

19 Wavepackets in the Scattering Region Near the center of AdS, φ f (x) φ f (0) L d 1 (x ω θ) L d 1 (0) L u t e iωu. Longitudinal width t R τ. Transverse width x 1/(ω θ). Well localized for earlier requirement. 1/ω t, x R, equivalent to 19

20 Singularity Structure Signal of interaction from intersecting wavepackets: local bulk physics! 20

21 Analytic Continuation Singularity only present in physical Lorentzian continuation. z z = (k 1 k 3 )(k 2 k 4 ) (k 1 k 2 )(k 3 k 4 ) $! # " (1 z)(1 z) = (k 1 k 4 )(k 2 k 3 ) (k 1 k 2 )(k 3 k 4 ) z z $ 21

22 Singularity Structure Signal of interaction from intersecting wavepackets: local bulk physics! z = σe ρ, z = σe ρ A(z, z) g 2 R 5 d 2j F (σ) ( ρ 2 ) β β = j 5/2 22

23 Momentum Conservation Flat space S-Matrix conserves momentum: S =1+i(2π) D δ D ki T (s, t) Momentum conserving δ-function must emerge from CFT amplitude in appropriate limit. Delta function does emerge from form of boundary compact sources and singularity structure: R n e iν lim dν R (R 2 κ 2 (ν + i) 2 ) β δn (κ ) 23

24 The S-Matrix Determine flat space scattering amplitude from residue of CFT singularity: A(z, z) g 2 R 5 d 2j F(σ) ( ρ 2 ) β it (s, t) =Kg 2 s j 1 t s j 2 u s 3 j 1 2 F s t 24

25 Examples Compute CFT correlators using AdS/CFT dictionary (D Hoker et al., 1999), read off scattering amplitudes. Scalar exchange: F(σ) σ(1 σ) T (s, t) = g2 t Graviton exchange: (1 σ)8 F(σ) σ T (s, t) =8πG 5 s 2 + ts t 25

26 Limits on Resolution Well behaved wavepackets are critical for derivation of LSZ decomposition. Bulk-Boundary propagator maps boundary-compact sources to non-compact wavepackets in the bulk. Cannot construct regular wavepackets, typically used for formal derivations of LSZ decomposition. What about Schwartz wavepackets? 26

27 Power-Law Tails NO: boundary-compact wavepackets have power-law tails in flat region. x t/ θ, u/ θ: φ f (x) φ f (0) ω t L(ω t)ˆl (x ω θ) u t φ f (x) φ f (0) ω t L(ω t)γ( ) 2π(iωu) 27

28 Recovering the S-Matrix? Tails from direct contribution interfere with scattered contribution to amplitude. Scattered contribution doesn t always go in the correct direction: haze of order L(ω t). We cannot recover the full flat space S-Matrix this way. Can we build better wavepackets? 28

29 Resonant Wavepackets Use resonant structure of AdS to build any normalizable wavefunction. φ f (x) = c nl m φ nl m (x) nl m Source compactly supported for 1/2 AdS time. Well-behaved in interaction region. What about while the source is turned on? 29

30 Multiple Interactions and Power Law Tails While source is on, two terms contribute in the large R limit. First term: wavepacket builds/decays linearly in time, φ f (x) 1+ τ π secondary interactions while wavepackets built/decay. Second term: power law tail in time. ω d 1 2 e iωrτ dωφ f (ω) ( iωr(π + τ)) Not quite a No-Go theorem, but hinders S-Matrix. 30

31 Bulk Unitarity Often claimed: Unitary evolution in the CFT unitary evolution in the bulk. Can we see bulk unitarity from the CFT? Look at pertubative bulk unitarity. At tree level, just the OPE: = 31

32 Bulk Unitarity What about loops? Seem to need new relations: = O 1 O 2 O 3 O 4 1loop = O 1 O 2 O A O B tree O A O B O 3 O 4 tree A,B=1 Are these relations present? What does this imply? 32

33 Conclusions CFT singularity signature of fine-grained locality. Boundary-compact wavepackets are not as welllocalized as flat space wavepackets. Power-law tails can hide important physics. Signature of black hole formation is exponentially suppressed 2 2 amplitude, swamped by tails. Can build better wavepackets in AdS from compact sources on the boundary, but have multiple scattering. 33

34 Unresolved Questions Is there a way to build nice wavepackets that don t have multiple scattering or power-law tails? Are power-law tails a signature of inherent non-locality? Does the CFT really capture everything in the gravitational theory? How is bulk unitarity encoded in the boundary CFT? Necessary for understanding black hole evaporation & resolving information problem. 34

35 Thank You 35