On expansion of equal-time relativistic two-body wave equations in powers of 1/c to higher orders

TL;DR

This paper extends the Foldy--Wouthuysen method to derive higher-order $1/c$ expansions of equal-time relativistic two-body wave equations, providing detailed Hamiltonians up to order $1/c^4$ for unequal masses.

Contribution

It develops a systematic approach to expand two-body relativistic wave equations to higher orders in $1/c$, including the handling of extra terms and transformations.

Findings

Derived Hamiltonian to order $1/c^4$ for unequal masses

Identified and addressed extra terms in higher-order expansions

Applied method to Breit and Salpeter equations for illustration

Abstract

Based on an extension of the Foldy--Wouthuysen method to two-body equations, the problem of expansion of equal-time relativistic equations for two Dirac particles in powers of $1/c$ to higher orders is considered. For the case of two particles with unequal masses, the transformed Hamiltonian in a general even-even form is obtained to order $1/c^4$. It is found that certain extra terms, which can be eliminated by an additional unitary transformation, arise in the expansion in higher orders, depending on the order of application of the generating functions in the first iteration. As examples for illustration, the Breit equation and the Salpeter equation with the Breit interaction are taken and their reduction to approximate forms including all the $1/c^{4}$-order terms is carried out using the method under consideration. The obtained results may be applied for the nonrelativistic expansion of two-body wave equations with various interaction potentials to higher orders, for the investigation of their features and symmetries, and may also be useful in the study of light atoms.