Quantum gravity and quantum chaos

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


Quantum chaos and quantum gravity

Quantum chaos $ Quantum gravity


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)


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:

hV (t)V (0)i ⇠ exp (�t/tr )

hW (t)W (t)V (0)V (0)i = hWW ihVV i+O(exp (�t/tr )


Quasinormal modes

Gravitational dual of relaxation to thermal equilibrium:

Quasinormal modes of black hole(Horowitz, Hubeny)


Quasinormal modes-LIGO


Quasinormal modes and transport

Quasinormal modes ! holographic description of transportcoe�cients

⌘/s = 1

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”)


Butterfly e↵ect

Another hallmark of chaos: sensitive dependence on initial conditions

The butterfly e↵ect

Classically, |�q(t)| ⇠ e�Lt |�q(0)|

�L a Lyapunov exponent


Quantum butterfly e↵ect

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

(with Douglas Stanford, Juan Maldacena)


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)


Quantum diagnostics

Distance between quantum states does not change with time underunitary evolution

Semiclassically

@q(t)

@q(0)= {q(t), p(0)}PB ! 1

i~ [q(t), p(0)]

(Larkin, Ovchinnikov)

C (t) = �h[q(t), p(0)]2i


Quantum diagnostics

W (t) = e iHtW (0)e�iHt

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

C (t) = �h[W (t),V (0)]2i, increases with time(Almheiri, Marolf, Polchinski, Sully, Stanford)

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

D(t) = hW (t)V (0)W (t)V (0)i, decreases with time


Quantum diagnostics


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, hV (0)W (t)W (t)V (0)i


Measuring D


D(t) = hW (t)V (0)W (t)V (0)i

To measure D one must evolve forward, then backward, in time. Orchange 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)


Thermofield Double State

� �

��

� �

��

� �

��

|TFDi = 1pZ

Pn exp(��En/2)|nLi|nRi

Maldacena, Israel

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



W(t4)|TFD⟩

t4

=t3

W(t4)|TFD⟩

t4

| 0i = V (t3

)W (t4

)|TFDi



W(t4)|TFD⟩

t4

=t3

W(t4)|TFD⟩

t4

V(t3)W(t4)|TFD⟩ W(t2)V(t1)|TFD⟩

t4

t3 t1

t2

pv

u=0v=0

pu4 3




t4

t3 t1

t2

pv

u=0v=0

pu4 3

D = h | 0i = hout|ini

out-of-time-ordered correlator !

global time ordered scattering process



D = h | 0i = hout|ini = S = e i�

For gravitational scattering the phase shift � ⇠ GNs

Translations of t correspond to global boosts

s ⇠ T 2 exp (2⇡� t)

In AdS/CFT, � ⇠ 1

N2

exp (2⇡� t)

D ⇠ c0

� c1

N2

exp (2⇡� t) + . . .




t4

t3 t1

t2

pv

u=0v=0

pu4 3

D ⇠ c0

� c1

N2

exp (2⇡� t) + . . .

Independent of V ,W . Universality of gravity

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

Sharp diagnostic of horizon physics



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

h(x)

u=0v=0v=0



D ⇠ c0

� c1

N2

exp (2⇡� t) + . . .

exp (2⇡� t) ! exp (�Lt)(Kitaev)

�L = 2⇡� = 2⇡T , universal

�L ⇠ kbT/~ ⇠ 1/tsc


Scrambling time

1 2 3 4

0.2

0.4

0.6

0.8

1.0

Deviation becomes appreciable at the scrambling time, t⇤

c1

N2

exp (2⇡� t⇤) ⇠ 1

t⇤ =�2⇡ logN2 = �

2⇡ log S

Fast Scrambling Conjecture: logarithmic growth fastest possible.(Sekino, Susskind)


Very high energy processes

s ⇠ T 2 exp (2⇡� t)

At late times s becomes enormous

t = 2t⇤, s = E 2

cm ⇠ N4

Enough energy to make a macroscopic black hole!



D(t) = hV (0)W (t)V (0)W (t)i

A range of bulk momenta are produced by V ,W , described byboundary to bulk propagators

D is actually given by the integral over hout|ini at di↵erent momenta,weighted by boundary to bulk propagators.

D is dominated by momenta for which GNs ⇠ 1

Very large s causes a strongly inelastic collision which gives a smallvalue of hout|ini, which make a small contribution to D.



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

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

Depend on V ,W . Not universal

There are some situations where it seems possible to isolated inelasticprocesses

1 2 3 4

0.2

0.4

0.6

0.8

1.0


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

�(s, b) ⇠ GNse�µb/b

d�2

2 , µ ⇠ 1

�

D(x , t) ⇠ c0

� c1

N2

e2⇡� t�µx

Chaos appreciable when x = vBt, vB =q

d2(d�1)

The “Butterfly velocity,” (entanglement saturation, (Liu, Suh))


Stringy corrections

At GNs ⇠ 1 stringy e↵ects are important

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

Roughly, replace GNs with string S-matrix

Flat space Regge limit, s ! s1+↵0t = s1�↵0k2

bI ⇠plog s

bI ⇠pt

Susskind; Peet-Thorlacius-Mezelumanian

Di↵usion of chaos, as well as ballistic spread


Stringy corrections in AdS

In AdS, s ! s1� 1p

�(c

1

+c2

k2

)

, 1p�= ( `s

`AdS)2

A spectrum: �L(k)

�L(0) =2⇡� ! 2⇡

� (1� c1p�)

Stringy e↵ects slow down the growth of chaos


Scattering bound

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

Based on unitarity, causality, analyticity


A bound on chaos

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

�L 2⇡� , The Einstein gravity value

In the spirit of the KSS ⌘/s � 1

4⇡ conjecture

A numerically precise refinement of the Fast Scrambling Conjecture


A bound on chaos

A bound on chaos: �L 2⇡� +O( 1

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


Outline of argument

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

F (t) = tr[yV (0)yW (t)yV (0)yW (t)], y4 = 1

Z e��H


Properties of F

1 2 3 4

0.2

0.4

0.6

0.8

1.0

By large N factorization, F (0) = Fd +O( 1

N2

)

Fd = tr[y2Vy2V ]tr[y2W (t)y2W (t)], (assume hVW i = 0 )


Properties of F

F (t + i⌧) is real for ⌧ = 0.

F (t + i⌧) is analytic in the half strip

β/4

-β/4

τ

0 t


Properties of F

β/4

-β/4

τ

0 t

Assert: |F (t + i⌧)| Fd +O( 1

N2

) in entire half strip.

Use maximum modulus principle

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

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

So F (t)/Fd 1 +O( 1

N2

) in whole strip.


SYK model

What systems saturate the chaos bound?

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

H =P

Jijkl �i�j�k�l

�L ! 2⇡� �J,N ! 1

Toward a solvable model of holography...(Sachdev; Polchinski, Rosenhaus; Maldacena, Stanford...)


Yoichiro Nambu 1921-2015


Quantum gravity and quantum chaos

Documents

Transcript of Quantum gravity and quantum chaos