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

TL;DR

This paper develops a classical relativistic framework for charged particle systems with electromagnetic fields, introducing collective variables, multipolar expansions, and a relativistic electric dipole approximation within the rest-frame instant form of dynamics.

Contribution

It provides a complete classical relativistic description of charged particles and electromagnetic fields, including collective variables, multipolar expansions, and a relativistic electric dipole approximation.

Findings

Defined a collective 3-center of mass variable in relativistic systems.

Established a relativistic electric dipole approximation for atomic physics.

Formulated the Dixon multipolar expansion for open particle subsystems.

Abstract

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 ${\vec {\cal K}}_{(int)} \approx 0$. 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 ${\vec {\cal P}}_{(int)} \approx 0$. 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.