Journal of Physical Studies 18(1), Article 1004 [9 pages] (2014) DOI: https://doi.org/10.30970/jps.18.1004 ON THE CHRAPLYVY TRANSFORMATION AND SOME FEATURES OF ITS APPLICATION FOR HIGHER-ORDER EXPANSIONS Alexei Turovsky Bogolyubov Institute for Theoretical Physics, NAS of Ukraine, Kyiv, UA-03680, Ukraine

Based on Chraplyvy's generalization of the Foldy-Wouthuysen method to the two-body case, the problem of expansion of relativistic equations for two particles in powers of $1/c$ to higher orders is considered. For the case of particles with unequal masses, the transformed Hamiltonian is obtained up to order $1/c^4$. It is found that, depending on the order of application of the generating functions for the first iteration, one can get two different forms for the higher-order part of the transformed two-body Hamiltonian, which, however, can be converted one into the other by an additional unitary transformation. Ambiguities like this in the procedure are discussed. As an example for illustration, the Hermitian part of the three-dimensional Bethe-Salpeter equation with the Breit interaction is taken and its reduction to an approximate form including all the $1/c^{4}$-order terms is carried out using the method under consideration. The results obtained here can be applied for a nonrelativistic expansion of two-body wave equations with various interaction potentials, for the study of the fine and hyperfine structure of the levels in hydrogen-like atoms. They can be of general theoretical interest as well.

PACS number(s): 03.65.Pm, 03.65.Ca, 02.30.Mv, 31.30.jy

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