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

TL;DR

This paper derives a classical effective potential in the rest frame that includes Coulomb and Darwin terms, representing the static and dynamic effects of one-photon exchange in scalar electrodynamics, using a semiclassical phase space approach.

Contribution

It introduces a method to express Lienard-Wiechert potentials as phase space functions and derives a unique effective scalar potential incorporating Coulomb and Darwin interactions.

Findings

Effective potential includes Coulomb and Darwin terms.

Re-expresses potentials as phase space functions in the radiation gauge.

Matches known 1/c^2 Darwin component expression.

Abstract

In the semiclassical approximation in which the electric charges of scalar particles are described by Grassmann variables ($Q_i^2=0, Q_iQ_j\ne 0$), 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 Q_i^2. 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/c^2 component agrees for 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.