The Thomas program and the canonical proper-time theory

In a seminal paper, Thomas [1] suggested that a many-particle relativistic dynamics can be constructed provided that we give up the assumption of invariant world-lines. In the first part of this paper, we show that, using only retarded potentials, a completely consistent action-at-a-distance classical electrodynamics can be constructed, which provides the correct dissipative force (back reaction) in the equations of motion. In particular, we can account for radiation reaction without the Lorentz-Dirac equation, which requires self-energy (divergence), advanced potentials and mass renormalization. What is equally interesting is that the Bakamjian and Thomas [2] program is a special case and, the calculations could have been completed by them, had they considered the problem. The general theory provides a new invariance group which is related to the Lorentz group by a scale transformation. This theory also provides a natural (and unique) definition of simultaneity for all observers, completing Thomas program. As an unexpected side benefit, the theory is noninvariant under time reversal and requires no assumptions about the structure of the charged particles.