Relativistic Center-of-Mass Motion and the Electromagnetic Interaction of Systems of Charged Particles

Abstract

Effects arising from relativistic corrections to the c.m. motion for particles interacting with an external electromagnetic field and also for their interparticle interaction are considered. The Hamiltonian discribing the electromagnetic interaction to order $\frac{v^2}{c^2}$, including Coulomb, magnetic, and spin-orbit effects, between two slowly moving charged particles is derived and the correct choice of c.m. dynamical variables is shown to be different from the usual nonrelativistic form. With the modified treatment of the c.m. motion introduced by considerations of relativistic invariance, the Hamiltonian is put in a form which exhibits, except for one term, a clean separation of over-all c.m. motion and internal dynamics and, but for this term, is of the required relativistic form to order $\frac{v^2}{c^2}$. The extra term, which spoils the above results, is demonstrated to be removed by correct treatment of the Thomas precession of the internal orbital angular momentum. Alternatively, the desired c.m. separation can be achieved by a modification of the free c.m. relativistic variables induced by the interaction. The relativistic variables for an $N$-body system with arbitrary interaction are briefly considered.