A covariant extrapolation of the noncovariant two particle Wheeler–Feynman Hamiltonian from the Todorov equation and Dirac’s constraint mechanics

Abstract

A semirelativistic expansion in powers of 1/c2 is canonically matched through order (1/c4) of the two-particle total Hamiltonian of Wheeler–Feynman vector and scalar electrodynamics to a similar expansion of the center of momentum (c.m.) total energy of two interacting particles obtained from covariant generalized mass shell constraints derived with the use of the classical Todorov equation and Dirac’s Hamiltonian constraint mechanics. This determines through order 1/c4 the direct interaction used in the covariant Todorov constraint equation. We show that these interactions are momentum independent in spite of the extensive and complicated momentum dependence of the potential energy terms in the Wheeler–Feynman Hamiltonian. The invariant expressions for the relativistic reduced mass and energy of the fictitious particle of relative motion used in the Todorov equation are also dynamically determined through this order by this same procedure. The resultant covariant Todorov equation then not only reproduces the noncovariant Wheeler–Feynman dynamics through order 1/c4 but also implicitly provides a rather simple covariant extrapolation of it to all orders of 1/c2.