Relativistic non-instantaneous action-at-a-distance interactions

TL;DR

This paper develops a comprehensive relativistic action-at-a-distance framework with interactions propagating at light speed, deriving equations of motion, exact solutions, and quantum equations, advancing the understanding of relativistic particle interactions.

Contribution

It introduces a general class of relativistic action-at-a-distance theories with specific conditions, providing exact solutions and quantum equations, which were not previously available.

Findings

Derived exact circular orbit solutions for the relativistic one-body problem.

Formulated the relativistic Hamiltonian including potential and dynamical mass components.

Obtained generalized Klein-Gordon-Fock and Dirac equations at the quantum level.

Abstract

Relativistic action-at-a-distance theories with interactions that propagate at the speed of light in vacuum are investigated. We consider the most general action depending on the velocities and relative positions of the particles. The Poincare invariant parameters that label successive events along the world lines can be identified with the proper times of the particles provided that certain conditions are impossed on the interaction terms in the action. Further conditions on the interaction terms arise from the requirement that mass be a scalar. A generic class of theories with interactions that satisfy these conditions is found. The relativistic equations of motion for these theories are presented. We obtain exact circular orbits solutions of the relativistic one-body problem. The exact relativistic one-body Hamiltonian is also derived. The theory has three components: a linearly rising potential, a Coulomb-like interaction and a dynamical component to the Poincar\'e invariant mass. At the quantum level we obtain the generalized Klein-Gordon-Fock equation and the Dirac equation.