Causality and quantization of time-delay systems: A two-body model problem

What is it about?

A model problem consisting of two isolated particles with a mutual interaction depending upon retarded and advanced positions and constrained to one-dimensional motion is analyzed herein. This system has a number of features in common with a previously considered one-particle double-delay model problem. The equations of motion have closed-form solutions and the systems are both deterministic and causal. In both cases, standard low-order approximations to the exact problem, in general, have spurious solutions, which must be recognized and removed in order to extract the physically meaningful parts and to proceed reasonably with quantization. In addition, the two-particle system has specific characteristics. The center-of-momentum motion separates out. The dependence of the solutions on the mass ratio of the two particles can be examined. Further, not only does the relative motion have the ordinary solution, in which the two particles move out of phase, it can also have an extraordinary solution, in which they move in phase with each other. Finally, the total generalized linear momentum and Hamiltonian can be evaluated and seen to be constants of the motion.

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Why is it important?

This work examines the separation of the relativistic 2-body problem in terms of the coordinates for center-of-momentum and relative motion. This cannot be done exactly for the Fokker-Wheeler-Feynman (FWF) theory. Nonetheless, the total generalized linear momentum and Hamiltonian can be evaluated and seen to be constants of the motion.

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