研究经典和量子三体问题的新概念：动力学系统的基本不可逆性和时间箭头
New Concept for Studying the Classical and Quantum Three-Body Problem: Fundamental Irreversibility and Time's Arrow of Dynamical Systems
摘要：
在共形欧氏空间（黎曼流形）中建立了经典三体问题，并从数学上严格证明了它与牛顿三体问题的等价性。它表明，一个弯曲的空间与局部坐标系允许我们检测新的隐藏的对称性的内部运动的动力系统，这使我们能够减少三体问题的6 \emph{th}阶系统。一种新的方法使测地线方程组关于动力系统的演化参数（\emph{internal time}）为\emph{fundamentally irreversible}。为了描述三体系统在不同随机环境中的运动，得到了相应的随机微分方程。使用这些SDES，福克-普朗克型方程描述的联合概率分布的测地线流在相空间和配置空间。在共形欧氏空间中建立了量子三体问题。特别是，相应的波动方程已获得研究的三体束缚态，以及调查的多通道量子散射的概念的框架内的内部时间。这使我们能够解决动力学庞加莱系统中极其重要的量子经典对应问题。
英文摘要：
The article formulates the classical three-body problem in conformal-Euclidean space (Riemannian manifold), and its equivalence to the Newton three-body problem is mathematically rigorously proved. It is shown that a curved space with a local coordinate system allows us to detect new hidden symmetries of the internal motion of a dynamical system, which allows us to reduce the three-body problem to the 6\emph{th} order system. A new approach makes the system of geodesic equations with respect to the evolution parameter of a dynamical system (\emph{internal time}) \emph{fundamentally irreversible}. To describe the motion of three-body system in different random environments, the corresponding stochastic differential equations (SDEs) are obtained. Using these SDEs, Fokker-Planck-type equations are obtained that describe the joint probability distributions of geodesic flows in phase and configuration spaces. The paper also formulates the quantum three-body problem in conformal-Euclidean space. In particular, the corresponding wave equations have been obtained for studying the three-body bound states, as well as for investigating multichannel quantum scattering in the framework of the concept of internal time. This allows us to solve the extremely important quantum-classical correspondence problem for dynamical Poincaré systems.
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