The Rest-Frame Darwin Potential from the Lienard–Wiechert Solution in the Radiation Gauge

Abstract

In the semiclassical approximation in which the electric charges of scalar particles are described by Grassmann variables (Q2i=0, QiQj≠0), it is possible to re-express the Lienard–Wiechert potentials and electric fields in the radiation gauge as phase space functions, because the difference among retarded, advanced, and symmetric Green functions is of order Q2i. By working in the rest-frame instant form of dynamics, the elimination of the electromagnetic degrees of freedom by means of suitable second classs contraints leads to the identification of the Lienard–Wiechert reduced phase space containing only N charged particles with mutual action-at-a-distance vector and scalar potentials. A Darboux canonical basis of the reduced phase space is found. This allows one to re-express the potentials for arbitrary N as a unique effective scalar potential containing the Coulomb potential and the complete Darwin one, whose 1/c2 component agrees with the known expression. The effective potential gives the classical analogue of all static and non-static effects of the one-photon exchange Feynman diagram of scalar electrodynamics.