Nonlocal Effects in Quantum Gravity

Transcription

1 Nonlocal Effects in Quantum Gravity Suvrat Raju International Centre for Theoretical Sciences 29th Meeting of the IAGRG IIT Guwahati 20 May 2017

2 Collaborators Based on work with 1 Kyriakos Papadodimas (CERN) 2 Sudip Ghosh (ICTS) 3 Souvik Banerjee (Groningen) 4 Jan-Willem Bryan (Groningen) Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG 17 2 / 40

3 References 1 S. Ghosh and S. Raju, The Breakdown of String Perturbation Theory for Many External Particles, Phys. Rev. Lett. 118 (2017), S. Banerjee, J. W. Bryan, K. Papadodimas and S. Raju, A Toy Model of Black Hole Complementarity, 2016, JHEP, 16, 01 (2016). 3 K. Papadodimas and S. Raju, Comments on the Necessity and Implications of State-Dependence in the Black Hole Interior, Phys. Rev. D, 93, (2016). 4 K. Papadodimas and S. Raju, Local Operators in the Eternal Black Hole, Phys. Rev. Lett., 115, (2015). 5 K. Papadodimas and S. Raju, State-Dependent Bulk-Boundary Maps and Black Hole Complementarity, Phys. Rev. D, 89, (2014). 6 K. Papadodimas and S. Raju, The Black Hole Interior in AdS/CFT and the Information Paradox, Phys. Rev. Lett., 112, (2014). Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG 17 3 / 40

4 Summary Generally accepted that locality breaks down in quantum gravity at the Planck scale [φ(t, x), φ(t, x + l pl )] =? Will argue that locality also breaks down at macroscopic distances for very high point correlators φ(x 1 ), φ(x 2 )... [φ(x i ), φ(x i+1 )]... φ(x n 1 )φ(x n ) 0 even for spacelike x p for sufficiently large n. Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG 17 4 / 40

5 The picture of a smooth spacetime breaks down if it is probed at too many points. Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG 17 5 / 40

6 Outline 1 A controlled example in AdS 2 Some lessons 3 Breakdown of string perturbation theory at large n 4 Application to the information paradox Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG 17 6 / 40

7 Outline 1 A controlled example in AdS 2 Some lessons 3 Breakdown of string perturbation theory at large n 4 Application to the information paradox Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG 17 7 / 40

8 Nonlocality for high-point correlators can be examined in a simple and precise setting in empty anti-de Sitter space. [S. Banerjee, J.W. Bryan, K. Papadodimas, S. R.,JHEP 2016] Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG 17 8 / 40

9 Setup Consider anti-de Sitter space, ds 2 = ( r 2 l 2 ads + 1)dt 2 + dr 2 r r 2 dω 2 l 2 ads This metric satisfies G µν 3 g l 2 µν = 0. ads Define N l ads l pl. Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG 17 9 / 40

10 Setup Consider a band, in time, of size T π on the boundary. h Equations of motion relate φ( D) O(B) Within local approximation, operators in D commute with operators on B. Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG / 40

11 Nonlocality in empty AdS Claim Operator at center of D can be written as complicated polynomial of operators in B even though they are uniformly spatially separated! h Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG / 40

12 Expansion with IR-UV cutoffs h N φ(d) = c nm n m n,m=0 Here we have used that 1 Energies in AdS are quantized. 2 We can throw away matrix elements for energies about M pl. Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG / 40

13 Reeh-Schlieder theorem h The Reeh-Schlieder theorem tells us where X is a simple polynomial. n = X n [φ( D)] 0 This is true in any quantum field theory. Not itself a sign of nonlocality. Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG / 40

14 Consequence of Reeh-Schlieder Reeh-Schlieder implies N φ(d) = c nm X n [φ( D)] 0 0 X m[φ( D)] n,m=0 h Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG / 40

15 Projector onto the vacuum Use complicated polynomials in B to approximate P 0 = 0 0 P[φ( D)]. P = lim α e αh n c n=0 ( αh) n But H is an operator in D. So with ( ) lads α = l ads log ; n c = l ads log l pl l pl P is a very complicated polynomial in D. n! ( lads l pl ) Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG / 40

16 Empty AdS complementarity h Combining the previous formulas we get φ(d) = nm c nm X n [φ( D)]P[φ( D)]X m[φ( D)] where X n are explicit simple polynomials, and P is a complicated polynomial. Explicitly realizes nonlocality in complicated correlators and also consistent with approximate locality in simple correlators. Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG / 40

17 Outline 1 A controlled example in AdS 2 Some lessons 3 Breakdown of string perturbation theory at large n 4 Application to the information paradox Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG / 40

18 Gauss Law nonlocality The Hamiltonian in AdS is a boundary term nonlocal Gauss law commutators. H = lim N T (r)d 2 Ω r Implies natural nonlocal commutators of size 1 N. Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG / 40

19 Breakdown of perturbation theory To enhance 1 N commutators to O (1) requires breakdown of 1 N perturbation theory in AdS. Breakdown of gravitational perturbation theory is necessary for significant nonlocal effects. Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG / 40

20 Flat space Gauss law Hamiltonian is a boundary term even in flat-space H = 1 d 2 s k (g ik,i g ii,k ) 16πG [ADM, de Witt,Regge, Teitelboim..., 62 74] Suggests nonlocal commutators controlled by E M pl Large-scale nonlocality breakdown of E M pl perturbation theory. Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG / 40

21 A Path Integral Argument A semi-classical spacetime is a saddle point of the QG path-integral. Z = e S Dg µν Breakdown of perturbation theory change in saddle-point different semi-classical metric with different notion of locality. Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG / 40

22 Outline 1 A controlled example in AdS 2 Some lessons 3 Breakdown of string perturbation theory at large n 4 Application to the information paradox Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG / 40

23 String perturbation theory in flat space breaks down for a large number of particles, even if they have low energy. [Sudip Ghosh, S. R.,PRL 2017] Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG / 40

24 String Perturbation theory limits Consider a limit where string-scale and Planck scale are widely separated, number of particles is large, and energy-per-particle is small. g 2 s = n 2πld 2 pl (2π α ) d 2 0 log(e α ) log(n) γ 0 String perturbation theory breaks down n that satisfies Requires (d 2)γ 1. log(g s ) log(n) = 1 ((d 2)γ 1) 2 n g 2 (d 2)γ 1 s Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG / 40

25 Growth of String Amplitudes To show this, we show that massless string amplitudes in the bosonic string and superstring grow at least as fast as ( ) log M (d 2)n 2d 2 pl M(k 1... k n p 1... p n ) 2 2 = 1 n log(n) M(k 1... k n p 1... p n ) n! 2 2 for large n. This is a lower bound. M (d 2)n d 2 pl Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG / 40

26 Growth from volume of moduli space Amplitude can be written as M gs n d(w.p.)(detp 1 P 1) 1 d 2 (det ) 2 W1 (ζ 1 )... W n (ζ n ) M n [D Hoker, Giddings, 87] V g,n = d(w.p) M g+n (4π2 ) 2g+n 3 (2g + n 3)! n,g [Zograf, Mirzakhani, ] Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG / 40

27 Factorial growth from numerics Possible to numerically estimate the string-scattering amplitude. γ = 0, a=41.7, b =0.52 γ=1/24, a=47.1, b =0.75 Log(M n) n log( M n ) = log( M n ) (n 2) log(4πg s ) + n log(d 2) = a + bn + log((n 3)!) Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG / 40

28 Outline 1 A controlled example in AdS 2 Some lessons 3 Breakdown of string perturbation theory at large n 4 Application to the information paradox Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG / 40

29 Cloning Paradox This loss of locality is precisely sufficient to resolve various versions of the information paradox. Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG / 40

30 Page Curve The entropy of the emitted black-hole radiation varies with time as follows S Smax Tf t [Page, 1993] Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG / 40

31 Nice slices Nice slices can capture the incoming matter and a large fraction of the outgoing radiation. Quantization on nice slices seems to lead to a cloning paradox. [Susskind, Thorlacius, Uglum, 93] Ψ Ψ Ψ? Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG / 40

32 Complementarity and Locality But, if φ(x in ) = P(φ(x 2 ), φ(x 3 ),..., φ(x n )), then no cloning paradox. Called black hole complementarity Complementarity ensures no violation of QM; subtle loss of locality instead. Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG / 40

33 Are the effects of nonlocality important for the observables required to frame this paradox? Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG / 40

34 Black Hole Evaporation in d-dimensions Recall that in d-dimensions T ( ) 1 M d 2 d 3 pl M S ( M pl R h ) d 2 1 R h To observe the cloning contradiction, we need to measure at least some connected S-point correlators in the emitted Hawking radiation. Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG / 40

35 Breakdown of perturbation theory Corresponds to S-matrix elements with S-insertions and typical momentum of order T But perturbation theory breaks down for (2 d) log T M pl log(n) Reached precisely at n = S ( ) 1 = 1 + O log(n) So S-point correlators may receive non-perturbative corrections. Suggests a version of complementarity. Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG / 40

36 Three Subsystems A related paradox is the strong-subadditivity paradox. Think of three subsystems 1 The radiation emitted long ago A 2 The Hawking quanta just being emitted B 3 Its partner falling into the BH C Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG / 40

37 Entropy of A Say the black hole is formed by the collapse of a pure state. Consider the entropy of system A S A = Trρ A ln ρ A This follows the Page curve. S Smax Tf t Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG / 40

38 Strong Subadditivity contradiction? Now, consider an old black hole, beyond its Page time where S A is decreasing. We must have since B is purifying A. S AB < S A Second, the pair B, C is related to the Bogoliubov transform of the vacuum of the infalling observer, and almost maximally entangled. S BC < S C However, strong subadditivity tells us S A + S C S AB + S BC We seem to have a violation at O(1). [Mathur, AMPS, ] Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG / 40

39 Complementarity and Strong Subadditivity In the presence of φ(x C ) = P(φ(x A1 ), φ(x A2 ),...) A, B, C are not independent subsystems. So, strong subadditivity inapplicable. [Kyriakos Papadodimas, S.R., ] Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG / 40

40 Summary Robust principles of gravity suggest non-local effects φ(x C ) = P(φ(x A1 ), φ(x A2 ),..., φ(x An )) These effects must preserve approximate locality in low-point correlation functions. Such a relation can be explicitly realized in empty AdS. The breakdown of string-perturbation theory for a large number of particles indicates the existence of a similar effect in flat space. This effect is precisely what is required to resolve some versions of the information paradox. Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG / 40