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 (Q 2 i = 0, Q i Q j = 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 2 i . By working in the restframe 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 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. C

Dynamical body frames, orientation-shape variables and canonical spin bases for the nonrelativistic N-body problem

“…[11] and then applied to the isolated system composed by N scalar charged particles plus the electromagnetic field [12,13].…”

“…In particular, a 12 In a way analoguous to the angular momentum composition in quantum mechanics; note that this spin clustering is independent of the center-of-mass clustering associated with the Jacobi coordinates: the existence of these two unrelated clusterings might prove to be a useful and flexible tool in molecular physics. dynamical definition of vibration, which replaces the S = 0 C-horizontal cross section of the static approach 13 , is based on the requirement that the components of the angular velocity vanish. Actually, the angular velocities with respect to the dynamical body frames become now measurable quantities, in agreement with the phenomenology of extended deformable bodies (e.g.…”

From Clock Synchronization to Dark Matter as a Relativistic Inertial Effect

“…In Refs. [27,28] a fully relativistic formulation of classical atomic physics in the rest-frame instant form of dynamics was given with the electro-magnetic field in the radiation gauge and with the electric charges Q i of the positiveenergy particles being Grassmann-valued (Q 2 i = 0, Q i Q j = Q j Q i for i = j) to regularize the electro-magnetic self-energies on the world-lines of particles. In the language of QED this is both a ultraviolet regularization (no loop contributions) and an infrared one (no brehmstrahlung), so that only the onephoton exchange diagram contributes and its static and non-static effects are replaced by potentials in a formulation based on the Cauchy problem.…”

“…[27] (the second paper is devoted to spinning particles) it is shown that the use of the Lienard-Wiechert solution (see the third paper in Ref. [27]) with "no incoming radiation field" allows one to arrive at a description of N charged particles dressed with a Coulomb cloud and mutually interacting through the Coulomb potential augmented with the full relativistic Darwin potential. This happens independently from the choice of the Green function (retarded, advanced, symmetric,..) due to the Grassmann regularization.…”

“…(54) like in Ref. [27] for the extraction of the Darwin potential from the LienardWiechert solution (see Eqs. (5.1)-(5.21) of Ref.…”