Reduction of Relativistic Two-Particle Wave Equations to Approximate Forms. I

An extension of the Foldy-Wouthuysen method to two-particle equations is developed. The relativistic Hamiltonian contains even-even, even-odd, odd-even, and odd-odd terms. To remove terms of the three latter types and thereby to convert the Hamiltonian into an even-even operator, canonical transformations are applied. A prescription for the proper choice of generating functions for such transformations is given, and an approximate expression (to the order $c^{-2\mathrm{}}$) for the transformed Hamiltonian is obtained. The even-even character of the transformed Hamiltonian makes it possible to separate out the quadrupole of equations for the four $\psi$-components describing both of the particles in positive energy states only. As an example the Breit equation is taken and its reduced form, obtained by the proposed procedure, is compared with its approximate form as obtained (by Breit himself) using the method of large components. A discussion leads to the result that the case $m_{\mathrm{I}}=m_{\mathrm{II}}$ is singular and requires a further development of the present scheme.