Towards Relativistic Atomic Physics. II. Collective and Relative Relativistic Variables for a System of Charged Particles plus the Electro-Magnetic Field

In this second paper we complete the classical description of an isolated system of “charged positive-energy particles, with Grassmann-valued electric charges and mutual Coulomb interaction, plus a transverse electro-magnetic field” in the rest-frame instant form of dynamics. In particular we show how to determine a collective variable associated with the internal 3-center of mass on the instantaneous 3-spaces, to be eliminated with the constraints ${\vec{\mathcal{K}}}_{(int)}\approx 0$. Here ${\vec {\cal K}}_{(int)}$ is the Lorentz boost generator in the unfaithful internal realization of the Poincare’ group and its vanishing is the gauge fixing to the rest-frame conditions ${\vec{\mathcal{P}}}_{(int)}\approx 0$. We show how to find this collective variable for the following isolated systems: a) charged particles with a Coulomb plus Darwin mutual interaction; b) transverse radiation field; c) charged particles with a mutual Coulomb interaction plus a transverse electro-magnetic field. Then we define the Dixon multipolar expansion for the open particle subsystem. We also define the relativistic electric dipole approximation of atomic physics in the rest-frame instant form and we find the a possible relativistic generalization of the electric dipole representation.