New Concept for Studying the Classical and Quantum Three-Body Problem: Fundamental Irreversibility and Time's Arrow of Dynamical Systems

TL;DR

This paper introduces a novel approach to the classical and quantum three-body problem using conformal-Euclidean space, revealing hidden symmetries, fundamental irreversibility, and new stochastic and quantum formulations that advance understanding of dynamical systems.

Contribution

It formulates the three-body problem in conformal-Euclidean space, proves its equivalence to the Newtonian version, and introduces a new irreversibility concept and quantum formulations.

Findings

Detection of hidden symmetries in three-body motion

Reduction of the problem to a 6th order system

Development of stochastic and quantum models for the system

Abstract

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 Poincar\'e systems.