Can the Position Variable be a Canonical Coordinate in a Relativistic Many-Particle Theory?

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Abstract
The time-symmetrical interaction of charges (Wheeler-Feynman electrodynamics) is shown in principle to yield second-order Newtonian-type equations of motion under the restriction that the motions be analytic extensions of free-particle motions. The means for explicitly generating the electrodynamic equations of motion describing invariant world-lines are given. These physically relevant equations do not fit into Dirac's (1949) formulation of Hamiltonian relativistic particle dynamics, where either world-line invariance is given up, or only trivially straight world lines can be described (``zero-interaction theorem'' of Currie, Jordan, Sudarshan, 1963). The misfit is due to the requirement in Dirac's scheme that position x be canonical. Under the Lie-Königs theorem, however, Hamiltonian statements of dynamics with invariant world lines remain possible when suitable Q(x, ẋ) are introduced instead of x as canonical position variables. A group of necessary conditions on the structure of any dynamics that permits x to be canonical are worked out to indicate how stringent is this permission in general.

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