Quantum gravity and quantum chaos

Stephen Shenker

Stanford University

Nambu Symposium

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 1 / 41 Yoichiro Nambu 1921-2015

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 2 / 41 Quantum chaos and quantum gravity

Quantum chaos Quantum gravity $

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 3 / 41 Black holes are thermal

Black holes are thermal (Bekenstein, Hawking)

Chaos underlies thermal behavior in ordinary physical systems

AdS/CFT (Maldacena; Gubser, Klebanov, Polyakov, Witten)

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 4 / 41 Relaxation to equilibrium

One hallmark of chaos is relaxation to thermal equilibrium

Described by a relaxation time tr

Diagnosed by a time ordered or retarded correlation function:

V (t)V (0) exp ( t/t ) h i⇠ r W (t)W (t)V (0)V (0) = WW VV + (exp ( t/t ) h i h ih i O r

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 5 / 41 Quasinormal modes

Gravitational dual of relaxation to thermal equilibrium:

Quasinormal modes of black hole (Horowitz, Hubeny)

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 6 / 41 Quasinormal modes-LIGO

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 7 / 41 Quasinormal modes and transport

Quasinormal modes holographic description of transport ! coecients

1 ⌘/s = 4⇡ , Einstein gravity (Policastro, Son, Starinets)

Viscosity bound ⌘/s 1 4⇡ (Kovtun, Son, Starinets)

Characteristic strong coupling time scale tsc ~/kbT ⇠ (Sachdev, “Quantum Phase Transitions”)

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 8 / 41 Butterfly e↵ect

Another hallmark of chaos: sensitive dependence on initial conditions

The butterfly e↵ect

Classically, q(t) eLt q(0) | | ⇠ | |

L a Lyapunov exponent

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 9 / 41 Quantum butterfly e↵ect

The quantum butterfly e↵ect, and its gauge/gravity dual, is the focus of this talk.

(with Douglas Stanford, Juan Maldacena)

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 10 / 41 Background

Origin of this line of development: Quantum Information

Fast approximations to random unitary operators on n qubits.

log n time scale ( log S)(Denkert et al. . . . ) ⇠

Connection to black holes (Hayden, Preskill)

Connection to gauge/gravity duality (Sekino, Susskind)

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 11 / 41 Quantum diagnostics

Distance between quantum states does not change with time under unitary evolution Semiclassically

@q(t) 1 = q(t), p(0) PB [q(t), p(0)] @q(0) { } ! i~

(Larkin, Ovchinnikov)

C(t)= [q(t), p(0)]2 h i

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 12 / 41 Quantum diagnostics

iHt iHt W (t)=e W (0)e

Chaos causes a lack of cancellation, W (t) a complicated operator

C(t)= [W (t), V (0)]2 ,increaseswithtime h i (Almheiri, Marolf, Polchinski, Sully, Stanford)

Significant time dependence from the out-of-time order correlator

D(t)= W (t)V (0)W (t)V (0) ,decreaseswithtime h i

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 13 / 41 Quantum diagnostics

iHt iHt W (t)=e W (0)e

D(t)=tr[⇢W (t)V (0)W (t)V (0)], ⇢ =exp( H)/Z

Lack of cancellation of time folds (Roberts, Stanford, Susskind)

Cancellation for time ordered correlators, V (0)W (t)W (t)V (0) h i

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 14 / 41 Measuring D

iHt iHt W (t)=e W (0)e

D(t)= W (t)V (0)W (t)V (0) h i To measure D one must evolve forward, then backward, in time. Or change the sign of H

Many body version of spin echo–Loschmidt echo. (Pastawski et al. ...)

Proposed experiments in cavity QED (Swingle, Bentsen, Schleier-Smith, Hayden)

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 15 / 41 Holographic calculation of D

Holographic calculation of D,andrelatedquantities. (SS, Stanford; Kitaev)

D = V (t )W (t )V (t )W (t ) h 1 2 3 4 i D = where h | 0i = W (t )V (t ) TFD , = V (t )W (t ) TFD | i 2 1 | i | 0i 3 4 | i TFD is the Thermofield Double State | i

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 16 / 41 Thermofield Double State

TFD = 1 exp( E /2) n n | i pZ n n | Li| R i Maldacena, IsraelP

Shifts in boundary time t correspond to boosts in global (Kruskal) coordinates. Rindler space

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 17 / 41 Holographic calculation of D

t4 t4 =

t3

W(t4)|TFD⟩ W(t4)|TFD⟩

0 = V (t )W (t ) TFD | i 3 4 | i

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 18 / 41 Holographic calculation of D

t4 t4 =

t3

W(t4)|TFD⟩ W(t4)|TFD⟩

t4 t2 v=0

u=0

u pv p 4 3

t3 t1

V(t3)W(t4)|TFD⟩ W(t2)V(t1)|TFD⟩

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 19 / 41 Holographic calculation of D

t4 t2 v=0

u=0

u pv p 4 3

t3 t1

V(t3)W(t4)|TFD⟩ W(t2)V(t1)|TFD⟩

D = = out in h | 0i h | i out-of-time-ordered correlator ! global time ordered scattering process

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 20 / 41 Holographic calculation of D

D = = out in = S = ei h | 0i h | i For gravitational scattering the phase shift G s ⇠ N Translations of t correspond to global boosts

s T 2 exp ( 2⇡ t) ⇠

In AdS/CFT, 1 exp ( 2⇡ t) ⇠ N2

D c c1 exp ( 2⇡ t)+... ⇠ 0 N2

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 21 / 41 Holographic calculation of D

t4 t2 v=0

u=0

u pv p 4 3

t3 t1

V(t3)W(t4)|TFD⟩ W(t2)V(t1)|TFD⟩

D c c1 exp ( 2⇡ t)+... ⇠ 0 N2 Independent of V , W . Universality of gravity

The onset of chaos is dual to a high energy gravitational collision near the black hole horizon. Classical gravity at large N

Sharp diagnostic of horizon physics

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 22 / 41 Holographic calculation of D

Eikonal resummation scattering o↵ a gravitational shock wave ! (‘t Hooft)

v=0 u=0 v=0 h(x)

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 23 / 41 Holographic calculation of D

D c c1 exp ( 2⇡ t)+... ⇠ 0 N2

exp ( 2⇡ t) exp ( t) ! L (Kitaev)

2⇡ L = =2⇡T , universal

L kbT /~ 1/tsc ⇠ ⇠

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 24 / 41 Scrambling time

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Deviation becomes appreciable at the scrambling time, t ⇤

c1 2⇡ 2 exp ( t ) 1 N ⇤ ⇠

t = log N2 = log S ⇤ 2⇡ 2⇡ Fast Scrambling Conjecture: logarithmic growth fastest possible. (Sekino, Susskind)

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 25 / 41 Very high energy processes

s T 2 exp ( 2⇡ t) ⇠ At late times s becomes enormous

2 4 t =2t , s = Ecm N ⇤ ⇠ Enough energy to make a macroscopic black hole!

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 26 / 41 Very high energy processes

D(t)= V (0)W (t)V (0)W (t) h i A range of bulk momenta are produced by V , W ,describedby boundary to bulk propagators

D is actually given by the integral over out in at di↵erent momenta, h | i weighted by boundary to bulk propagators.

D is dominated by momenta for which G s 1 N ⇠ Very large s causes a strongly inelastic collision which gives a small value of out in , which make a small contribution to D. h | i

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 27 / 41 Very high energy processes

The late time behavior of D is determined by the amplitude for a very high energy collison not to happen

Determined by the tails of boundary to bulk propagators, which are determined by quasinormal modes

Depend on V , W . Not universal

There are some situations where it seems possible to isolated inelastic processes

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Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 28 / 41 Ballistic propagation of chaos

Suppose V , W are localized in space, V (0, 0), W (x, t)

Bulk scattering at various impact parameters b x ⇠ Compute localized shock wave profile

d 2 µb 1 (s, b) G se /b 2 ,µ ⇠ N ⇠ c 2⇡ t µx D(x, t) c 1 e ⇠ 0 N2 d Chaos appreciable when x = vB t, vB = 2(d 1) q The “Butterfly velocity,” (entanglement saturation, (Liu, Suh))

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 29 / 41 Stringy corrections

At G s 1 stringy e↵ects are important N ⇠

Compute stringy corrections, following (Brower, Polchinski, Strassler, Tan) (SS, Stanford)

Roughly, replace GN s with string S-matrix

2 Flat space Regge limit, s s1+↵0t = s1 ↵0k ! b plog s I ⇠ b pt I ⇠ Susskind; Peet-Thorlacius-Mezelumanian

Di↵usion of chaos, as well as ballistic spread

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 30 / 41 Stringy corrections in AdS

1 2 1 (c1+c2k ) 1 ` In AdS, s s p , =( s )2 ! p `AdS

Aspectrum:L(k)

(0) = 2⇡ 2⇡ (1 c1 ) L ! p Stringy e↵ects slow down the growth of chaos

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 31 / 41 Scattering bound

In general, (perturbative) scattering can grow no faster than s (Camanho, Edelstein, Maldacena, Zhiboedov)

Based on unitarity, causality, analyticity

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 32 / 41 A bound on chaos

The scattering bound and the stringy result suggest that there should be a universal bound:

2⇡ , The Einstein gravity value L 

In the spirit of the KSS ⌘/s 1 conjecture 4⇡ A numerically precise refinement of the Fast Scrambling Conjecture

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 33 / 41 A bound on chaos

A bound on chaos: 2⇡ + ( 1 ) L  O N2 (Maldacena, SS, Stanford)

Assuming:

Large number of degrees of freedom (N2)

Large hierarchy between relaxation and scrambling times

Canonical example: large N gauge theories. V , W single trace operators

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 34 / 41 Outline of argument

Introduce a four point function F (t), a variant of D(t), that also diagnoses chaos.

4 1 H F (t)=tr[yV (0)yW (t)yV (0)yW (t)], y = Z e

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 35 / 41 Properties of F

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By large N factorization, F (0) = F + ( 1 ) d O N2 F = tr[y 2Vy 2V ]tr[y 2W (t)y 2W (t)], (assume VW =0) d h i

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 36 / 41 Properties of F

F (t + i⌧)isrealfor⌧ = 0.

F (t + i⌧)isanalyticinthehalfstrip

τ

β/4

0 t

-β/4

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 37 / 41 Properties of F

τ

β/4

0 t

-β/4

Assert: F (t + i⌧) F + ( 1 )inentirehalfstrip. | |  d O N2 Use maximum modulus principle

At early time vertical boundary by large N factorization at early time boundary

On horizontal boundaries use a Cauchy-Schwarz inequality to bound F by a time ordered correlator. This saturates at late time –key physical input– so large N factorization applies uniformly.

So F (t)/F 1+ ( 1 ) in whole strip. d  O N2 Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 38 / 41 Outline of argument

F (t + i⌧)/F 1+ ( 1 ) in whole strip. | d |  O N2 Such a function obeys the chaos bound: Schwarz-Pick theorem.

Example F (t)/F =1 ✏eLt . d If > 2⇡ then the decrease in F /F changes to an increase for L | d | some ⌧ < /4. | | So 2⇡ + ( 1 ) The chaos bound. L  O N2

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 39 / 41 SYK model

What systems saturate the chaos bound?

A variant of the Sachdev-Ye model ! (Kitaev)

H = Jijkl i j k l P 2⇡ J, N L ! !1 Toward a solvable model of holography... (Sachdev; Polchinski, Rosenhaus; Maldacena, Stanford...)

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 40 / 41 Yoichiro Nambu 1921-2015

Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 41 / 41