Relativistic center-of-mass variables for composite systems with arbitrary internal interactions

Abstract

We show how a class of relativistic center-of-mass (c.m.) variables for a composite system with arbitrary internal interactions can be constructed to any order in $\frac{1}{{c}^{2}}$ by means of a nonsingular unitary transformation which arises from a study of the Lie algebra of the Poincar\'e group.The class of c.m. variables so constructed subsumes the c.m. variables previously obtained by means of the singular Gartenhaus-Schwartz transformation. We explicitly determine the c.m. variables to order $\frac{1}{{c}^{2}}$, and as an example consider both internal electromagnetic (EM) interactions, where simplifications are pointed out with regard to a previous study, and external EM interactions, where the complete form of the "correction" term to the Foldy-Wouthuysen EM interaction Hamiltonian is given. The results of some earlier studies of relativistic corrections to phenomenological potentials are also shown to be included in our results, and separability is briefly discussed.