On the existence of dynamics in Wheeler–Feynman electromagnetism

Abstract

We study the equations of Wheeler-Feynman electrodynamics which is an action-at-adistance theory about world-lines of charges that interact through their corresponding advanced and retarded Liénard-Wiechert field terms. The equations are non-linear, neutral, and involve time-like advanced as well as retarded arguments of unbounded delay. Using a reformulation in terms of Maxwell-Lorentz electrodynamics without self-interaction, which we have introduced in a preceding work, we are able to establish the existence of conditional solutions. These are solutions that solve the Wheeler-Feynman equations on any finite time interval with prescribed continuations outside of this interval. As a byproduct we also prove existence and uniqueness of solutions to the Synge equations on the time half-line for a given history of charge trajectories.

A relativistic retrocausal model violating Bell's inequality

“…(For a mathematical discussion of this issue in the context of Wheeler-Feynman electromagnetism, e.g. [15,29].) We also note that while the possibility of SFPs may be mind boggling, it need not imply the possibility of logical paradoxes-there is nothing inconsistent about the time-evolutions we consider here-though the threat of potential inconsistencies is something to be addressed in the context of a more mature theory.…”

Global solutions to the electrodynamic two-body problem on a straight line

“…As discussed earlier (3), the velocitiesȧ andḃ of any potential global solution have a modulus which is bounded away from one, which implies that the implicitly defined functions t ± i in (2) are well-defined and smooth. Moreover, because ofä > 0 andb < 0, the velocities converge monotonically to asymptotic velocities (3). Without restricting generality, we will assume…”

“…General global existence of solutions on the straight line was later shown in [2]. As yet the only result towards a solution theory of the FST equations in three space-dimensions is given in [3] where, for N rigid charges and a prescribed asymptotic behavior of the trajectories for times |t| > τ , existence of solutions on [−τ, τ ] for arbitrary large 0 < τ < ∞ was shown. Apart from the different setting, the crucial difference in the latter two works lies in the prescription of the initial data.…”

“…Apart from the different setting, the crucial difference in the latter two works lies in the prescription of the initial data. In [2] it was given as asymptotic data in the remote past and in [3] as Cauchy data. Both choices seem to have advantages and disadvantages: In order to use asymptotic data, a priori, one must determine the asymptotic behavior of potential scattering solutions; until now this was only done successfully on the straight line; see [2].…”

“…The trouble with this approach is that one usually lacks sufficient control on the global behavior of the solutions in order to apply fixed-point methods globally. Because of this one is usually only able to prove existence of solutions on finite time intervals as in [3].…”