1. Classical and Quantum Chaos in the Gauge/Gravity Correspondence
Leopoldo A. Pando Zayas
ICTP, Trieste/University of Michigan
Numerical approaches to the holographic principle quantum gravity and cosmology
July 2015, Kyoto, Japan

2. Collaborators L. PZ, Int.J.Mod.Phys. D23 (2014) 12, 1442013
A. Farahi and L. PZ, Phys.Lett. B734 (2014) 31-35
D. Giataganas, L. PZ, K. Zoubos, JHEP 1401 (2014) 129
H. de Oliveira, L. PZ, E. Rodrigues, Phys.Rev.Lett. 111 (2013) 5 
L. PZ and D. Reichmann, JHEP 1304 (2013) 083
P. Basu, D.Das, A. Ghosh, L PZ, JHEP 1205 (2012) 077
P. Basu and LPZ, Phys.Rev. D84 (2011)
P. Basu and L. PZ, Phys.Lett. B700 (2011)  
L. PZ and C. Terrero-Escalante, JHEP 1009 (2010) 094 

3. Outline What is string theory?
Chaos around holographic Regge Trajectories.
What is the Wigner conjecture (surmise)?
What is quantum chaos and what does it have to do with the spectrum of hadrons in QCD and other strongly coupled theories?
The spectrum of highly excited Hadrons in holographic models.
Classical and quantum chaos during gravitational collapse.
The quantum Rosetta stone for the information paradox.

4. What has string theory been doing lately (last ten years)?
String in curved spacetimes with RR fluxes.
Semiclassical quantization: BMN, GKP and Regge Trajectories.
Prominent role of classical trajectories.

6. Holographic Regge Trajectories

7. BMN Operators

8. Shrunk string moving at the speed of light

9. GKP Operators (twist-two in QCD)

10. GKP String and Twist-Two Operators

11. Classical Chaos around Holographic Regge Trajectories
Virasoro constraints to get to the conformal gauge.

13. Spinning and winding strings

14. Regge Trajectories in Holography (r=r0)

15. Holographic Regge Trajectories

16. Winding Strings
Equations of motion

17. Direct Analysis of Phase Space
MN solution

18. MN

19. What we really need
From particular solutions to a study of the full phase space.

20. What is chaos? Sensitivity to initial conditions.
Largest Lyapunov exponent.
Poincare sections and the Kolmogorov-Arnold-Moser theorem.
Power spectrum.
Fractal dimensions.

21. Lyapunov Exponent

22. Integrability and Poincare Sections
Integrable system: Admits N-integral of motions.
Define the action-angle variables (I_i, Θ_i)
N integrals of motion become Q_i = f_i(p; q) of constant action.
Kolmogorov-Arnold-Moser (KAM) theorem: Evolutions of the tori under non-integrable perturbation.

23. Explicitly Chaotic: Poincare Sections

24. Analytic Nonintegrability for Hamiltonian Systems (Virasoro)
Given a system
Particular solution
If the nonlinear system admits some first integrals so does the variational equation.
Ziglin's theorems: Existence of a first integral of motion with the monodromy matrices around the straight line solution.
Morales-Ruiz: Monodromy to the nature of the Galois group of the NVE.

25. Kovacic’s Algorithm
Select a particular solution, that is, define the straight line solution.
Write the normal variational equation (NVE).
Check if the identity component of the differential Galois group of the NVE is Abelian, that is, apply the Kovacic’s algorithm to determine if the NVE is integrable by quadrature.

26. Analytic Non-integrability for Regge Trajectories:
The straight line solution
Properties of confining metrics:

27. Normal Variational Equation
Matthieu -- Non-integrable

28. Conclusions of the first part:
Established classically chaotic behavior around the Regge trajectories.
The Regge trajectory itself is an island of integrability in a phase space that is generically chaotic.
Typical behavior of many classical systems.

29. Wigner
Problem: Describe the general properties of the energy levels of highly excited states of heavy nuclei.
Idea: Such a complex nuclear system is described by a Hermitian Hamiltonian H and connect the results to Random Matrix Theory (RMT).
Implications: The energy levels (eigenvalues of H) are approximated by the eigenvalues of a large random matrix.

30. Quantum Chaos
The study of quantum systems whose classical limit is chaotic (Einstein 1917): Spectrum

31. Properties of Eigenvalues
Level Spacing Distribution

32. Quantum Chaos
Classically Integrable Systems: Eigenvalue distribution coincides with a sequence of uncorrelated levels (Poisson ensemble) with the same spacing (Berry-Tabor)
Classically Chaotic Systems: Eigenvalue spacing distribution coincides with the corresponding quantity for eigenvalues of a suitable ensemble of random matrices (Bohigas, Giannoni and Schmit)

33. Haq, Pandey and Bohigas (82)
Data on slow-neutron resonances of heavy nuclei as well as on proton resonances of light nuclei; the combined set of nuclear resonance-energy data of different nuclei (Nuclear Data Ensemble) as sampling of eigenvalues of GOE. The data consisted of 1407 resonance energies corresponding to 30 sequences of 27 different nuclei.

34. Pascalutsa (2003)
The experimentally measured mass spectrum of hadrons (N,Delta, Lambda, Sigma; and all the mesons up to f2(2340)) Particle Data Group Summary (2000).
Conclusion: The nearest-neighbor mass-spacing distribution of the meson and baryon spectrum is described by the Wigner surmise corresponding to the GOE.

37. Lattice QCD (Markum)
Lattice studies of QCD in the confinement and the deconfinement phases found similar results for eigenvalues of the Dirac operator, including with non-zero chemical potential and with supersymmetry (hep-lat/ , hep-lat/ )

38. How to compute the string Spectrum?

39. Classical to Quantum via Minisuperspace

41. MN

42. WQCDSakai-Sugimoto

43. Typical Example I: Bunimovich Stadium

44. Typical Example II: Sinai’s Billiard

45. MN Potential

46. WQCD Potential

47. Winding string Sector The string can unwind.
Not a clear separation of sectors.
Can we show that the typical wave function is not localized close to R=0?

48. MN Wave function

49. WQCD Wave Function

50. Holographic Spectrum
Choose an arbitrary energy range (above the range of small interactions) spanning about 400 energy levels.
We calculate the energy difference between eigenvalues in the chosen range and plot them on an histograms, and compare them to Wigner distributions

51. MN

52. WQCD

53. How good is the fit?
The root mean square (RMS) between Wigner’s distribution and data is below 10^(−3) when the distribution is normalized so the sum is one.
This excellent matching to Wigner distribution proves our main claim that the spectrum of hadrons in the MN and WQCD theories shows a quantum chaotic eigenvalue distribution.

54. Conclusions
Found a spectrum of hadrons compatible with observations and with the Wigner’s surmise.
From classical trajectories to the mass distribution of hadrons.
To discuss the structure of highly excited states we need a quantum version of AdS/CFT.

55. The black hole information loss paradox
Black hole formation and evaporation leads to many conflicts.
No-hair theorems  most information about the collapsing body is lost to the outside region.
Hawking: Black holes radiate with a perfectly thermal spectrum.
What happens to the information about the collapsing body?

56. Information loss paradox and the AdS/CFT correspondence
The information loss paradox is assumed to be implicitly resolved in the context of the AdS/CFT correspondence where a gravity theory is equated to an explicitly unitary field theory.
Perhaps the problem is a practical one … like weather prediction:
Balasubramanian, Marolf and Rozali (first prize gravity essay for 2006): Planck level resolution is needed to understand the structure of a black hole.
Hawking (Information preserving and weather forcasting, 2014).

57. Main suggestion: AdS gravitational collapse is generically turbulent.
High sensitivity to initial conditions in the classical limit of the AdS/CFT
I argue that quantum chaos must play a central role in the understanding of the black hole information loss paradox.

61. Quantitative description of turbulence during gravitational collapse in AdS

62. Main entry in AdS/CFT dictionary
Klein-Gordon Equation

64. Regime of validity Use gravity  Large N.
Use the Klein-Gordon Equation  Small dimension operators.
What about highly excited states? Should I describe them with a scalar field?
From Klein-Gordon to Vertex Operators. From classical fields to quantum fields.

65. Rosetta Stone
We should take the lessons that quantum chaos has already taught us, extend them to the context of quantum field theory in curved spacetimes and use them in conjunction with the AdS/CFT correspondence as the Rosetta Stone that would allow us to decipher the puzzle of black hole information loss paradox.
Perhaps black hole thermodynamics is nothing but the Eigenvalue Thermalization Hypothesis.