Real-time dynamics of matrix quantum mechanics beyond the classical approximation

¹ Institute for Theoretical Physics, Regensburg University, D-93053 Regensburg, Germany
² Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606 - 8502, Japan
³ Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305, USA
⁴ Nuclear and Chemical Sciences Division, Lawrence Livermore National Laboratory, Livermore, California 94550, USA
⁵ The Hakubi Center for Advanced Research, Kyoto University, Yoshida Ushinomiyacho, Sakyo-ku, Kyoto 606-8501, Japan

^* Speaker. e-mail: pavel.buividovich@physik.uni-regensburg.de. This work was supported by the S. Kowalevskaja award from the A. von Humboldt foundation.

Published online: 26 March 2018

Abstract

We describe a numerical method which allows to go beyond the classical approximation for the real-time dynamics of many-body systems by approximating the many-body Wigner function by the most general Gaussian function with time-dependent mean and dispersion. On a simple example of a classically chaotic system with two degrees of freedom we demonstrate that this Gaussian state approximation is accurate for significantly smaller field strengths and longer times than the classical one. Applying this approximation to matrix quantum mechanics, we demonstrate that the quantum Lyapunov exponents are in general smaller than their classical counterparts, and even seem to vanish below some temperature. This behavior resembles the finite-temperature phase transition which was found for this system in Monte-Carlo simulations, and ensures that the system does not violate the Maldacena-Shenker-Stanford bound λ_L < 2πT, which inevitably happens for classical dynamics at sufficiently small temperatures.