quantum gravity and quantum chaos
TRANSCRIPT
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:
hV (t)V (0)i ⇠ exp (�t/tr )
hW (t)W (t)V (0)V (0)i = hWW ihVV i+O(exp (�t/tr )
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 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”)
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)| ⇠ e�Lt |�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 ofthis 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 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
Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 12 / 41
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
Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 13 / 41
Quantum diagnostics
W (t) = e iHtW (0)e�iHt
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
Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 14 / 41
Measuring D
W (t) = e iHtW (0)e�iHt
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)
Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 15 / 41
Holographic calculation of D
Holographic calculation of D, and related quantities.(SS, Stanford; Kitaev)
D = hV (t1
)W (t2
)V (t3
)W (t4
)i
D = h | 0i where
| i = W (t2
)V (t1
)|TFDi, | 0i = V (t3
)W (t4
)|TFDi
|TFDi is the Thermofield Double State
Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 16 / 41
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
Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 17 / 41
Holographic calculation of D
W(t4)|TFD⟩
t4
=t3
W(t4)|TFD⟩
t4
| 0i = V (t3
)W (t4
)|TFDi
Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 18 / 41
Holographic calculation of D
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
Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 19 / 41
Holographic calculation of D
V(t3)W(t4)|TFD⟩ W(t2)V(t1)|TFD⟩
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
Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 20 / 41
Holographic calculation of D
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) + . . .
Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 21 / 41
Holographic calculation of D
V(t3)W(t4)|TFD⟩ W(t2)V(t1)|TFD⟩
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
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)
h(x)
u=0v=0v=0
Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 23 / 41
Holographic calculation of D
D ⇠ c0
� c1
N2
exp (2⇡� t) + . . .
exp (2⇡� t) ! exp (�Lt)(Kitaev)
�L = 2⇡� = 2⇡T , universal
�L ⇠ kbT/~ ⇠ 1/tsc
Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 24 / 41
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)
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
t = 2t⇤, s = E 2
cm ⇠ N4
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) = 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.
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 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
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
�(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))
Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 29 / 41
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
Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 30 / 41
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
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 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
Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 33 / 41
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
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 alsodiagnoses chaos.
F (t) = tr[yV (0)yW (t)yV (0)yW (t)], y4 = 1
Z e��H
Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 35 / 41
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 )
Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 36 / 41
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
Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 37 / 41
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.
Stephen Shenker (Stanford University) Quantum gravity and quantum chaos Nambu Symposium 38 / 41
Outline of argument
|F (t + i⌧)/Fd | 1 +O( 1
N2
) in whole strip.
Such a function obeys the chaos bound: Schwarz-Pick theorem.
Example F (t)/Fd = 1� ✏e�Lt .
If �L > 2⇡� then the decrease in |F/Fd | changes to an increase for
some |⌧ | < �/4.
So �L 2⇡� +O( 1
N2
) The chaos bound.
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 =P
Jijkl �i�j�k�l
�L ! 2⇡� �J,N ! 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