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The Question
Outline
TheCorrespondence
Operators andClassicalTrajectories
What is Chaos?
The PowerSpectrum
Poincare Section
Largest LyapunovExponent
Poincarerecurrences
Where to go?
Chaos in the Gauge/GravityCorrespondence
Leopoldo A. Pando Zayas
César Terrero-Escalante 1007.0277Pallab Basu 1103.4107, ongoing
University of Michigan
Great Lakes Strings, April 2011
The Question
Outline
TheCorrespondence
Operators andClassicalTrajectories
What is Chaos?
The PowerSpectrum
Poincare Section
Largest LyapunovExponent
Poincarerecurrences
Where to go?
The main question
Semiclassical physics is the only window we arecurrently exploiting in AdS/CFT: ClassicalTrajectories.From trajectories to phase space (Poincare,Hamilton): What is the meaning on the full PhaseSpace?Integrable trajectories are a set of measure zero inmost dynamical systems.What are the CFT entries under chaos in theAdS/CFT dictionary: Lyapunov Exponent,Kolmogorov-Sinai Entropy, Fractal Dimension?
The Question
Outline
TheCorrespondence
Operators andClassicalTrajectories
What is Chaos?
The PowerSpectrum
Poincare Section
Largest LyapunovExponent
Poincarerecurrences
Where to go?
Outline
The Gauge/Gravity Correpondence: Regge, ’t Hooft,MaldacenaOperator/State correspondence in the AdS/CFTMore Operator/State: Regge trajectories, BMN, GKPFrom Trajectory to Phase SpaceBeyond Integrability.What is chaos?What is the meaning of Chaos in the Gauge/GravityCorrespondenceoutlook
The Question
Outline
TheCorrespondence
Operators andClassicalTrajectories
What is Chaos?
The PowerSpectrum
Poincare Section
Largest LyapunovExponent
Poincarerecurrences
Where to go?
Regge Trajectories
Hadronic Physics was a string theory already!Regge trajectories are best explain by spectrum of arotating string: J ∼ M2.
The Question
Outline
TheCorrespondence
Operators andClassicalTrajectories
What is Chaos?
The PowerSpectrum
Poincare Section
Largest LyapunovExponent
Poincarerecurrences
Where to go?
Regge Trajectories in the Gauge/GravityCorrespondence
rr0
0123
The Question
Outline
TheCorrespondence
Operators andClassicalTrajectories
What is Chaos?
The PowerSpectrum
Poincare Section
Largest LyapunovExponent
Poincarerecurrences
Where to go?
AdS/CFT Correspondence
ZFT [J ] = ZString[φ].
• Parameters:g2YM
= 4π gstringN =
�
S5
F5
RS5 = RAdS5 = (g2YM
N)1/4ls
• Same superconformal symmetrySO(2, 4)× SO(6) ⊂ SU(2, 2|4)
• States on AdS = Operators in the CFTSingle Particle state ↔ single Trace TrFX...Multi-Particle ↔ Multitrace Tr(FX...X)Tr(F...X..X)Chiral primaries (protected, BPS) ↔ Supergravity modesOperator with large quantum numbers ↔ Stringy states
The Question
Outline
TheCorrespondence
Operators andClassicalTrajectories
What is Chaos?
The PowerSpectrum
Poincare Section
Largest LyapunovExponent
Poincarerecurrences
Where to go?
Berenstein-Maldacena-NastaseAlmost BPS operators
OJ =
1√JNJ
Tr ZJ .
Z = φ1 + iφ2.
∆− J =∞�
−∞Nn
�1 +
λ
J2n2.
OJ
n,−n =1
√JNJ+2
J�
l=0
e2πinJ lTr
�φZ lψZJ−l
�.
The Question
Outline
TheCorrespondence
Operators andClassicalTrajectories
What is Chaos?
The PowerSpectrum
Poincare Section
Largest LyapunovExponent
Poincarerecurrences
Where to go?
The Question
Outline
TheCorrespondence
Operators andClassicalTrajectories
What is Chaos?
The PowerSpectrum
Poincare Section
Largest LyapunovExponent
Poincarerecurrences
Where to go?
Twist-two Operators Past and Present• Twist-two operators Gross-Wilczek (DIS)
TrΨ∇(a1 . . .∇an)Ψ, ∆− S = f(λ) lnn.
• Twist-two operators in AdS/CFT: GKP, Kruczenski,Makeenko
TrΦ∇(a1 . . .∇an)Φ,
∆− S =
√λ
πlnS.
• A long production about the precise spectrum of theseoperators.
The Question
Outline
TheCorrespondence
Operators andClassicalTrajectories
What is Chaos?
The PowerSpectrum
Poincare Section
Largest LyapunovExponent
Poincarerecurrences
Where to go?
L = −4R2
2πα�
ρ0�
0
dρ�
cosh2 ρ− (φ)2 sinh2 ρ, coth2 ρ0 = φ = ω2,
E = 4R2
2πα�
ρ0�
0
dρcosh2 ρ�
cosh2 ρ− ω2 sinh2 ρ
S = 4R2
2πα�
ρ0�
0
dρω sinh2 ρ�
cosh2 ρ− ω2 sinh2 ρ(1)
∆− S =
√λ
πlnS.
The Question
Outline
TheCorrespondence
Operators andClassicalTrajectories
What is Chaos?
The PowerSpectrum
Poincare Section
Largest LyapunovExponent
Poincarerecurrences
Where to go?
General Properties
Classical conserved quantities ≡ Quantum NumbersWhat matters are the conserved quantities not thetrajectoriesQ: What is the meaning of the trajectory moregenerally, is it just to give the quantum numbers?It is natural to study the whole phase space, thespace of all possible trajectories
The Question
Outline
TheCorrespondence
Operators andClassicalTrajectories
What is Chaos?
The PowerSpectrum
Poincare Section
Largest LyapunovExponent
Poincarerecurrences
Where to go?
Ring String in Schwarzschild-AdS5
L =1
2
√ggabGµν∂aX
µ∂bXν .
ds2 = −fdt2 +dr2
f(r)+ r2
�dθ2 + sin2 θdψ2 + cos2 θdφ2
�,
f(r) = 1 +r2
b2−
w4M
r2
t = t(τ), r = r(τ), θ = θ(τ), φ = φ(τ), ψ = ασ,
L =1
2f t2 −
r2
2f−
r2
2
�θ2 + cos2 θφ2
�+
r2
2α2 sin2 θ
The Question
Outline
TheCorrespondence
Operators andClassicalTrajectories
What is Chaos?
The PowerSpectrum
Poincare Section
Largest LyapunovExponent
Poincarerecurrences
Where to go?
Picture
The Question
Outline
TheCorrespondence
Operators andClassicalTrajectories
What is Chaos?
The PowerSpectrum
Poincare Section
Largest LyapunovExponent
Poincarerecurrences
Where to go?
Hr =f
2p2r +
1
2r2p2θ+
l2
2r2 cos2 θ+
α2
2r2 sin2 θ −
E2
2f,
r = −fpr,
pr =E2
2f2f � +
f �
2p2r −
1
r3p2θ−
l2
r3 cos2 θ+ α2r sin2 θ,
θ = −1
r2pθ,
pθ =l2
r2sin θ
cos3 θ+ α2r2 sin θ cos θ,
Hr = 0, Constraint (2)
The Question
Outline
TheCorrespondence
Operators andClassicalTrajectories
What is Chaos?
The PowerSpectrum
Poincare Section
Largest LyapunovExponent
Poincarerecurrences
Where to go?
What is chaos?
No water-tight definition: Sensitivity to the initialconditionsPower SpectrumPoincaré sections: Breaking of theKolmogorov-Arnold-Moser tori.Largest Lyapunov Exponent.
The Question
Outline
TheCorrespondence
Operators andClassicalTrajectories
What is Chaos?
The PowerSpectrum
Poincare Section
Largest LyapunovExponent
Poincarerecurrences
Where to go?
The power spectrum of Henon-Heiles
H =1
2m
�p2x + p2y
�+
k
2
�x2 + y2
�+ λ
�x2 y −
1
3y3�
(3)
The Question
Outline
TheCorrespondence
Operators andClassicalTrajectories
What is Chaos?
The PowerSpectrum
Poincare Section
Largest LyapunovExponent
Poincarerecurrences
Where to go?
Power Spectrum: Ring string inSchwarzschild-AdS
The Question
Outline
TheCorrespondence
Operators andClassicalTrajectories
What is Chaos?
The PowerSpectrum
Poincare Section
Largest LyapunovExponent
Poincarerecurrences
Where to go?
H =1
2m
�p2x + p2y
�+
k
2
�x2 + y2
�+ λ
�x2 y −
1
3y3�
(4)
The Question
Outline
TheCorrespondence
Operators andClassicalTrajectories
What is Chaos?
The PowerSpectrum
Poincare Section
Largest LyapunovExponent
Poincarerecurrences
Where to go?
• Poincaré sections and the destruction of the KAM torus
The Question
Outline
TheCorrespondence
Operators andClassicalTrajectories
What is Chaos?
The PowerSpectrum
Poincare Section
Largest LyapunovExponent
Poincarerecurrences
Where to go?
Lyapunov Exponent
δ �X(t) = e�λ tδ �X(0).
The Question
Outline
TheCorrespondence
Operators andClassicalTrajectories
What is Chaos?
The PowerSpectrum
Poincare Section
Largest LyapunovExponent
Poincarerecurrences
Where to go?
Largest Lyapunov Exponet: Precision
The Question
Outline
TheCorrespondence
Operators andClassicalTrajectories
What is Chaos?
The PowerSpectrum
Poincare Section
Largest LyapunovExponent
Poincarerecurrences
Where to go?
Largest Lyapunov Exponet: Precision
The Question
Outline
TheCorrespondence
Operators andClassicalTrajectories
What is Chaos?
The PowerSpectrum
Poincare Section
Largest LyapunovExponent
Poincarerecurrences
Where to go?
The distance between two strings
The Question
Outline
TheCorrespondence
Operators andClassicalTrajectories
What is Chaos?
The PowerSpectrum
Poincare Section
Largest LyapunovExponent
Poincarerecurrences
Where to go?
Poincare recurrences and Unitarity
Initial perturbation a thermal system will be dampedby thermal dissipation as long as the time scale istoo short to resolve possible gaps in the spectrum(Heisenberg times).Heisenberg time tH = 1/ω- discreteness. For t � tHthe spectrum is approximately continuous.
A(t) = eitHA(0)e−itH ,
GE(t) = e−S(E)�
Ei,Ej≤E
|Aij |2ei(Ei−Ej)t.
The Question
Outline
TheCorrespondence
Operators andClassicalTrajectories
What is Chaos?
The PowerSpectrum
Poincare Section
Largest LyapunovExponent
Poincarerecurrences
Where to go?
Poincare recurrences
If the matrix elements of some operator A in theenergy basis have frequency with Γ, the correlatorwill decay with characteristic lifetime of order Γ−1:GE(t) Standard dissipative behavior in Γ−1 � t � tH
For t > tH most phases in GE(t) would havecompleted a period and the function GE(t) startsshowing irregularities; it is a quasiperiodic function oftime. Despite thermal damping, it returns arbitrarilyclose to the initial value over periods of the order ofthe recurrence time.The new conditions: We look at strings and observethe enhancing effect of some other charge.
e−N2−→ e−N
2/J
2∼ O(1). (5)
The Question
Outline
TheCorrespondence
Operators andClassicalTrajectories
What is Chaos?
The PowerSpectrum
Poincare Section
Largest LyapunovExponent
Poincarerecurrences
Where to go?
Summary
Following the trajectories of nearby strings −→
Positive Lyapunov Exponent.Distance between two strings ≡ Correlation functionof the corresponding operatorsLate-time behavior of correlation functions ≡
Poincaré RecurrencesLyapunov Exponent ↔ Poincaré recurrence timePoincaré recurrences ≡ Unitarity, Information Loss
The Question
Outline
TheCorrespondence
Operators andClassicalTrajectories
What is Chaos?
The PowerSpectrum
Poincare Section
Largest LyapunovExponent
Poincarerecurrences
Where to go?
Outlook
Toward a concrete Quantitative argument involvingunitarity in black hole physics.
tPR ≥1
λ.(6)
The integrability/chaos balance for point particlesand for strings: KAM-theorem, Anosov story.Analytic Non-integrability: Differential Galois Theory.Quantum Chaos
The Question
Outline
TheCorrespondence
Operators andClassicalTrajectories
What is Chaos?
The PowerSpectrum
Poincare Section
Largest LyapunovExponent
Poincarerecurrences
Where to go?
Fractality
The Question
Outline
TheCorrespondence
Operators andClassicalTrajectories
What is Chaos?
The PowerSpectrum
Poincare Section
Largest LyapunovExponent
Poincarerecurrences
Where to go?
Fractality
The Question
Outline
TheCorrespondence
Operators andClassicalTrajectories
What is Chaos?
The PowerSpectrum
Poincare Section
Largest LyapunovExponent
Poincarerecurrences
Where to go?
Fractality
