Nonlocal Effects in Quantum Gravity

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

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

References 1 S. Ghosh and S. Raju, The Breakdown of String Perturbation Theory for Many External Particles, Phys. Rev. Lett. 118 (2017), 131602. 2 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, 084049 (2016). 4 K. Papadodimas and S. Raju, Local Operators in the Eternal Black Hole, Phys. Rev. Lett., 115, 211601 (2015). 5 K. Papadodimas and S. Raju, State-Dependent Bulk-Boundary Maps and Black Hole Complementarity, Phys. Rev. D, 89, 086010 (2014). 6 K. Papadodimas and S. Raju, The Black Hole Interior in AdS/CFT and the Information Paradox, Phys. Rev. Lett., 112, 051301 (2014). Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG 17 3 / 40

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

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

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

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

Setup Consider anti-de Sitter space, ds 2 = ( r 2 l 2 ads + 1)dt 2 + dr 2 r 2 + 1 + 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

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 17 10 / 40

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 17 11 / 40

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 17 12 / 40

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 17 13 / 40

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 17 14 / 40

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 17 15 / 40

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 17 16 / 40

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 17 18 / 40

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 17 19 / 40

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 17 20 / 40

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 17 21 / 40

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 17 23 / 40

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 17 24 / 40

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 17 25 / 40

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, 2008 2013] Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG 17 26 / 40

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) 400 300 200 100 20 40 60 80 100 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 17 27 / 40

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 17 29 / 40

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 17 30 / 40

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 17 31 / 40

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 17 32 / 40

Are the effects of nonlocality important for the observables required to frame this paradox? Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG 17 33 / 40

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 17 34 / 40

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 17 35 / 40

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 17 36 / 40

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 17 37 / 40

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, 2009 12] Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG 17 38 / 40

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., 2013 15] Suvrat Raju (ICTS-TIFR) Nonlocality in quantum gravity IAGRG 17 39 / 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 17 40 / 40