Abstract
Abstract We study chaos in the classical limit of the matrix quantum mechanical system describing D0-brane dynamics. We determine a precise value of the largest Lyapunov exponent, and, with less precision, calculate the entire spectrum of Lyapunov exponents. We verify that these approach a smooth limit as N → ∞. We show that a classical analog of scrambling occurs with fast scrambling scaling, t ∗ ∼ log S. These results confirm the k-locality property of matrix mechanics discussed by Sekino and Susskind.
Highlights
Shenkera aStanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305, U.S.A. bYukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan cThe Hakubi Center for Advanced Research, Kyoto University, Yoshida Ushinomiyacho, Sakyo-ku, Kyoto 606-8501, Japan E-mail: guyga@stanford.edu, hanada@yukawa.kyoto-u.ac.jp, sshenker@stanford.edu Abstract: We study chaos in the classical limit of the matrix quantum mechanical system describing D0-brane dynamics
We show that a classical analog of scrambling occurs with fast scrambling scaling, t∗ ∼ log S
Hayden and Preskill [7] connected this timescale to one characteristic of black hole horizons [8, 9] t∗ ∼ R log(M/mp) ∼ R log S, where R is the Schwarzschild radius, M is the black hole mass, mp is the Planck mass and S is the Bekenstein-Hawking entropy of the black hole
Summary
The model we consider is the low-energy effective theory that lives on a stack of N D0branes [42] It can be obtained by dimensionally reducing super Yang-Mills in 9+1 dimensions to zero space dimensions. We will work in an ensemble with fixed energy E, and where the conserved angular momentum is set to zero.. We will work in an ensemble with fixed energy E, and where the conserved angular momentum is set to zero.3 Averages in this ensemble will agree with thermal averages in the thermodynamic limit N → ∞; the corresponding temperature T is given as follows. The conservation of the angular momentum Tr(XiXj −XjXi) should be taken into account, reducing the number of degrees of freedom by d(d − 1)/2.
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