Hamiltonian Formulation of Action-at-a-Distance in Electrodynamics

The Wheeler-Feynman scheme of classical electrodynamics, using half-retarded+half-advanced fields between pairs of interacting charges, is written in a differential form which expresses the motion of the charges at any inertial observer's single present time through infinite-order differential equations of motion. The conversion of field variables to mechanical ones in this way does not imply an infinite number of degrees of freedom; rather it is shown how to construct a Hamiltonian, depending solely on ordinary particle-position coordinates and certain canonically conjugate momenta, which is both the energy and the generator of the ``correct'' motions. The latter are taken to be those continuously developable from free-particle motions as the strength of interaction increases from the value zero. It is just this criterion that establishes particle-position+conjugate momentum to be sufficient for the description of motion. The highness of the order of the equations of motion apparently gets to be expressed through high powers of momenta in the Hamiltonian. An example is sketched, illustrating a general algorithm, in which the particles are treated nonrelativistically and their interactions are treated to order e4 and c−4, corresponding to fourth-order equations of motion. The general form of the Hamiltonian is a double power series expansion in e2 and c−1; when the c−1 expansion is terminated covariance is sacrificed, but the possibility of collecting all powers of e2 is gained.