The Open Relativistic Two-body Problem

TL;DR

This paper develops a covariant operator framework for the relativistic two-body problem with external potentials, deriving equations for scalar and spinor particles, and applies it to positronium in magnetic fields.

Contribution

It introduces a covariant formulation of the relativistic two-body problem with external fields, including a new basis for Dirac equations and decoupling of spinor components.

Findings

Derived exact covariant operator equations for two-body dynamics.

Identified conditions for covariant reformulation in phase space.

Applied framework to analyze low-energy positronium in magnetic fields.

Abstract

The open relativistic two-body problem, when two interacting particles also are in external potentials, is considered in terms of the principle of the least action. Based on the consistent modification of the relativistic version Newton's third law in external fields, the exact covariant operator equations, which govern dynamics of either a scalar particle or each of the components of the 16 component spinor of spin-$1/2$ fermions, depending on the particle type, are derived in the center-mass and relative motion variables, beyond the consideration in the Breit frame only. The class of external fields and interaction potentials, when the two-body problem can be covariantly reformulated in (3+1) phase space of relative motion variables, uncoupled from the center-mass motion of such a system, is outlined. In the case of fermions the new $\gamma$-matrix basis generating the Dirac-like equation for the 16 component spinors in the (3+1) phase space is found. The chosen basis allows us to decouple the derived Dirac-like equation into four independent equations governing the dynamics of the four-component spinors. The developed approach is examined in studying the low-energy positronium states in magnetic fields.