Wheeler-Feynman Electrodynamics Research Articles

Direct interaction along light cones at the quantum level

Here, we point out that interactions with time delay can be described at the quantum level using a multi-time wave function , i.e. a wave function depending on one spacetime variable per particle. In particular, such a wave function makes it possible to implement direct interaction along light cones (not mediated by fields), as in the Wheeler–Feynman formulation of electrodynamics. Our results are as follows. (1) We derive a covariant two-particle integral equation and discuss it in detail. (2) It is shown how this integral equation (or equivalently, a system of two integro-differential equations) can be understood as defining the time evolution of ψ in a consistent way. (3) We demonstrate that the equation has strong analogies with Wheeler–Feynman electrodynamics and therefore suggests a possible new quantization of that theory. (4) We propose two natural ways how the two-particle equation can be extended to N particles. It is shown that exactly one of them leads to the usual Schrödinger equation with Coulomb-type pair potentials if time delay effects are neglected.

Equations of motion for variational electrodynamics

We extend the variational problem of Wheeler–Feynman electrodynamics by generalizing the electromagnetic functional to a local space of absolutely continuous trajectories possessing a derivative (velocities) of bounded variation. We show here that the Gateaux derivative of the generalized functional defines two partial Lagrangians for variations in our generalized local space, one for each particle. We prove that the critical-point conditions of the generalized variational problem are: (i) the Euler–Lagrange equations must hold Lebesgue-almost-everywhere and (ii) the momentum of each partial Lagrangian and the Legendre transform of each partial Lagrangian must be absolutely continuous functions, generalizing the Weierstrass–Erdmann conditions.

Solutions of the Wheeler-Feynman equations with discontinuous velocities.

We generalize Wheeler-Feynman electrodynamics with a variational boundary value problem for continuous boundary segments that might include velocity discontinuity points. Critical-point orbits must satisfy the Euler-Lagrange equations of the action functional at most points, which are neutral differential delay equations (the Wheeler-Feynman equations of motion). At velocity discontinuity points, critical-point orbits must satisfy the Weierstrass-Erdmann continuity conditions for the partial momenta and the partial energies. We study a special setup having the shortest time-separation between the (infinite-dimensional) boundary segments, for which case the critical-point orbit can be found using a two-point boundary problem for an ordinary differential equation. For this simplest setup, we prove that orbits can have discontinuous velocities. We construct a numerical method to solve the Wheeler-Feynman equations together with the Weierstrass-Erdmann conditions and calculate some numerical orbits with discontinuous velocities. We also prove that the variational boundary value problem has a unique solution depending continuously on boundary data, if the continuous boundary segments have velocity discontinuities along a reduced local space.

Delay Equations of the Wheeler–Feynman Type

We present an approximate model of Wheeler–Feynman electrodynamics, for which uniqueness of solutions is proved. It is simple enough to be instructive but close enough to Wheeler–Feynman electrodynamics such that we can discuss its natural type of initial data, constants of motion, and stable orbits with regard to Wheeler–Feynman electrodynamics.

On Irreversibility and Radiation in Classical Electrodynamics of Point Particles

The direct interaction theory of electromagnetism, also known as Wheeler-Feynman electrodynamics, is often misinterpreted and found unappealing because of its reference to the absorber and, more importantly, to the so-called absorber condition.Here we remark that the absorber condition is indeed questionable and presumably not relevant for the explanation of irreversible radiation phenomena in our universe. What is relevant and deserves further scrutiny is the emergent effective description of a source particle in an environment. We therefore rephrase what we consider the relevant calculation by Wheeler and Feynman and comment on the status of the theory.

On the existence of dynamics in Wheeler–Feynman electromagnetism

Wheeler–Feynman electrodynamics (WF) is an action-at-a-distance theory about world-lines of charges that in contrary to the textbook formulation of classical electrodynamics is free of ultraviolet singularities and is capable of explaining the irreversible nature of radiation. In WF, the world-lines of charges obey the so-called Fokker–Schwarzschild–Tetrode (FST) equations, a coupled set of nonlinear and neutral differential equations that involve time-like advanced as well as retarded arguments of unbounded delay. Using a reformulation of this theory in terms of Maxwell–Lorentz electrodynamics without self-interaction that we have introduced in a preceding work, we are able to establish the existence of conditional solutions. These conditional solutions solve the FST 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 world-lines.

VARIATIONAL ELECTRODYNAMICS OF ATOMS

We generalize Wheeler-Feynman electrodynamics with a variational problem for trajectories that are required to merge continuously into given past and future boundary segments. We prove that the boundary-value problem is well posed for two classes of boundary data. The well-posed solution in general has velocity discontinuities, henceforth a broken extremum. Along regular segments, broken extrema satisfy the Euler-Lagrange neutral difierential delay equations with state-dependent deviating arguments. At points where velocities are discontinuous, broken extrema satisfy the Weierstrass-Erdmann conditions that energies and momenta are continuous. Electromagnetic flelds of the flnite trajectory segments are derived quantities that can be extended to a bounded region B of space- time. Extrema with a flnite number N of velocity discontinuities have extended flelds deflned in B with the possible exception of N spherical surfaces, and satisfy the integral laws of classical electrodynamics for most surfaces and curves inside B. As an application, we study the hydrogenoid atomic model with mass ratio varying by three orders of magnitude to include hydrogen, muonium and positronium. For each model we construct globally bounded trajectories with vanishing far-flelds using periodic perturbations of circular orbits. Our model uses solutions of the neutral difierential delay equations along regular segments and a variational approximation for the head-on collisional segments. Each hydrogenoid model predicts a discrete set of flnitely measured neighbourhoods of periodic orbits with vanishing far-flelds right at the correct atomic magnitude and in quantitative and qualitative agreement with experiment and quantum mechanics. The spacings between consecutive discrete angular momenta agree with Planck's constant within thirty-percent, while orbital frequencies agree with a corresponding spectroscopic line within a few percent.

Finite element boundary value integration of Wheeler–Feynman electrodynamics

The electromagnetic two-body problem is solved as a boundary value problem associated to an action functional. We show that the functional is Fréchet differentiable and that its conditions for criticality are the mixed-type neutral differential delay equations with state-dependent delay of Wheeler–Feynman electrodynamics. We construct a finite element method that finds C1-smooth solutions when suitable past and future positions of the particles are given as boundary data. The numerical trajectories satisfy a variational problem defined in a finite-dimensional Hermite functional space of C1 piecewise-polynomials. The numerical variational problem is solved using a combination of Newton’s method intercalated with boundary adjustments to ensure that the velocity of the solution is continuous with the boundary data. We recover the known circular orbits and compute several other novel trajectories of the Wheeler–Feynman electrodynamics. We also discuss the local convexity of the functional close to the new found trajectories and the possibility of solutions with less regularity.

Double-Slit and Electromagnetic Models to Complete Quantum Mechanics

We analyze a realistic microscopic model for electronic scattering with the neutral differential delay equations of motion of point charges of the Wheeler-Feynman electrodynamics. We propose a microscopic model according to the electrodynamics of point charges, complex enough to describe the essential physics. Our microscopic model reaches a simple qualitative agreement with the experimental results as regards interference in double-slit scattering and in electronic scattering by crystals. We discuss our model in the light of existing experimental results, including a qualitative disagreement found for the double-slit experiment. We discuss an approximation for the complex neutral differential delay equations of our model using piecewise-defined (discontinuous) velocities for all charges and piecewise-constant-velocities for the scattered charge. Our approximation predicts the De Broglie wavelength as an inverse function of the incoming velocity and in the correct order of magnitude. We explain the scattering by crystals in the light of the same simplified modeling with Einstein-local interactions. We include a discussion of the qualitative properties of the neutral-delay-equations of electrodynamics to stimulate future experimental tests on the possibility to complete quantum mechanics with electromagnetic models.

Minimizers with discontinuous velocities for the electromagnetic variational method

The electromagnetic two-body problem has neutral differential delay equations of motion that, for generic boundary data, can have solutions with discontinuous derivatives. If one wants to use these neutral differential delay equations with arbitrary boundary data, solutions with discontinuous derivatives must be expected and allowed. Surprisingly, Wheeler-Feynman electrodynamics has a boundary value variational method for which minimizer trajectories with discontinuous derivatives are also expected, as we show here. The variational method defines continuous trajectories with piecewise defined velocities and accelerations, and electromagnetic fields defined by the Euler-Lagrange equations on trajectory points. Here we use the piecewise defined minimizers with the Liénard-Wierchert formulas to define generalized electromagnetic fields almost everywhere (but on sets of points of zero measure where the advanced/retarded velocities and/or accelerations are discontinuous). Along with this generalization we formulate the generalized absorber hypothesis that the far fields vanish asymptotically almost everywhere and show that localized orbits with far fields vanishing almost everywhere must have discontinuous velocities on sewing chains of breaking points. We give the general solution for localized orbits with vanishing far fields by solving a (linear) neutral differential delay equation for these far fields. We discuss the physics of orbits with discontinuous derivatives stressing the differences to the variational methods of classical mechanics and the existence of a spinorial four-current associated with the generalized variational electrodynamics.

Electromagnetic two-body problem: recurrent dynamics in the presence of state-dependent delay

We study the electromagnetic two-body problem of classical electrodynamics as a prototype dynamical system with state-dependent delays. The equations of motion are analysed with reference to motion along a straight line in the presence of an electrostatic field. We consider the general electromagnetic equations of motion for point charges with advanced and retarded interactions and study two limits, (a) retarded-only interactions (Dirac electrodynamics) and (b) half-retarded plus half-advanced interactions (Wheeler–Feynman electrodynamics). A fixed point is created where the electrostatic field balances the Coulombian attraction, and we use local analysis near this fixed point to derive necessary conditions for a Hopf bifurcation. In case (a), we study a Hopf bifurcation about an unphysical fixed point and find that it is subcritical. In case (b), there is a Hopf bifurcation about a physical fixed point and we study several families of periodic orbits near this point. The bifurcating periodic orbits are illustrated and simulated numerically, by introducing a surrogate dynamical system into the numerical analysis which transforms future data into past data by exploiting the periodicity, thus obtaining systems with only delays.

Variational principle for the Wheeler–Feynman electrodynamics

We adapt the formally defined Fokker action into a variational principle for the electromagnetic-two-body problem. We introduce properly defined boundary conditions to construct a functional of a finite orbital segment into the reals. The boundary conditions for the variational principle are an end point along each trajectory plus the respective segment of trajectory for the other particle inside the light cone of each end point. We show that the conditions for an extremum of our functional are the mixed-type neutral equations with implicit state-dependent delay of the electromagnetic-two-body problem. We put the functional on a natural Banach space and show that the functional is Frechét differentiable. We develop a method to calculate the second variation for C2 orbital perturbations, in general, and, in particular, about circular orbits of large enough radii. We prove that our functional has a local minimum at circular orbits of large enough radii, at variance with the limiting Kepler action that has a minimum at circular orbits of arbitrary radii. Our results suggest a bifurcation at some O(1) radius below which the circular orbits become saddle-point extrema. We give a precise definition for the distributional-like integrals of the Fokker action and discuss a generalization to a Sobolev space H02 of trajectories where the equations of motion are satisfied almost everywhere. Last, we discuss the existence of solutions for the state-dependent delay equations with slightly perturbated arcs of circle as the boundary conditions and the possibility of nontrivial solenoidal orbits.

Action at a Distance and Cosmology: A Historical Perspective

▪ Abstract The first law of theoretical physics, the Newtonian law of gravitation, relies on the concept of action at a distance. The success of this law led to the concept being applied to electricity and magnetism, which were next to be explored in depth. Here the action at a distance had a limited success and ultimately had to be abandoned in favor of the increasingly more popular field theory. Nevertheless, in the 1940s, an attempt was made to revive the concept of action at a distance in a relativistically invariant way by Wheeler & Feynman (1945 , 1949) . It inspired a series of investigations in both electrodynamics and gravity in which the field concept was not used but the interaction was described as taking place directly between particles. As it impinged very intimately on cosmology, Hoyle was keenly interested in it. This review discusses the work by Hoyle, the author, and others on the development of electrodynamics and gravitation as direct particle theories. In this review, the author discusses how the work was started and went through stages of increasing sophistication, e.g., extending the Wheeler-Feynman electrodynamics to curved spacetime, its consequences in different cosmologies, and the issues arising from its quantization. The resolution of ultraviolet divergences in quantum electrodynamics is also briefly discussed. The parallel development of a Machian theory of gravitation followed the lead from electrodynamics. In both theories one sees a strong link between the large-scale structure of the universe and local physics, as might be expected from an action-at-a-distance framework.

Working with Fred on action at a distance

This paper reviews the work the author carried out with Fred Hoyle on the development of electrodynamics and gravitation as direct particle theories. In this account the author reviews how the work was started, and went through stages of increasing sophistication, e.g., extending the Wheeler-Feynman electrodynamics to curved spacetime, its consequences in different cosmologies, and the issues arising from its quantization. The resolution of ultraviolet divergences in quantum electrodynamics is also briefly discussed. The parallel development of a Machian theory of gravitation followed the lead from electrodynamics. In both theories one sees a strong link between the large scale structure of the universe and local physics, as might be expected from an action-at-a-distance framework. It is recalled why Fred considered this an important aspect of a physical theory.

Methods of numerical analysis of 1-dimensional 2-body problem in Wheeler–Feynman electrodynamics

Numerical methods for solution of differential equations with deviating arguments describing 1-dimensional ultra-relativistic scattering of 2 identical charged particles in classical electrodynamics with half-retarded/halfadvanced interaction (Wheeler and Feynman, 1949) are developed. A bifurcation of solutions and violation of their reflectional symmetries in the region of velocities v>0.937c are found in numerical analysis.

METHODS OF NUMERICAL ANALYSIS OF ONE-DIMENSIONAL TWO-BODY PROBLEM IN WHEELER-FEYNMAN ELECTRODYNAMICS

Numerical methods for solutions of differential equations with deviating arguments describing one-dimensional ultra-relativistic scattering of two identical charged particles in Wheeler-Feynman electrodynamics with half-retarded/half-advanced interaction are developed. Utilization of the methods for the physical problem analysis leads to the discovery of a bifurcation of solutions and breaking of their reflectional symmetry for particles asymptotic velocity v>0.937c in their center-of-mass frame.

On the structure of solutions of 1-dimensional 2-body problem in Wheeler-Feynman electrodynamics

The problem of 1-dimensional ultra-relativistic scattering of 2 identical charged particles in classical electrodynamics with retarded and advanced interactions is investigated.

Contact Transformations in Classical Mechanics

Transformations of coordinates of points in an infinite-dimensional graded vector space, the so-called contact transformations, are examined. An infinite jet prolongation of the extended configuration space of N spinless particles is the subspace of this vector space. The dynamical equivalence among Lagrangian N-body systems connected by an invertible jet transformation is established. As an example of the invertible jet transformations, a class of gauge transformations of Lagrangian variables is investigated. The method of contact transformations is applied to the Wheeler-Feynman electrodynamics for two point charges.

Hamiltonian formulation of the two-body problem in wheeler-feynman electrodynamics

A problem of electromagnetic interaction of two charged relativistic particles is considered in the Wheeler and Feynman approach to classical electrodynamics. We formulate this theory in such a way that its Hamiltonian description becomes available. This description is a kind of constrained Hamiltonian mechanics. The obtained Hamiltonian equations have more simple form than the Lagrangian ones. A canonical quantization of this problem can also be developed.

Canonical formalism of action-at-a-distance electrodynamics and many-particle potential among charged particles

The second post-Coulombian Lagrangian of Wheeler–Feynman electrodynamics for a many-particle system is treated according to a canonical formalism of a singular Lagrangian with higher derivatives. The canonical equations are given in terms of a reduced Hamiltonian with Dirac brackets, but they are transformed to be expressed in terms of ordinary Poisson brackets by redefinition of canonical variables. The reduced Hamiltonian includes a characteristic form of three-particle and four-particle potentials. Finally a direct pathway to the reduced Hamiltonian is presented via first-order formalism of the Maxwell theory with charged particles.