Abstract
The derivation of an equation of motion (EOM) for a particle-like field source is still an open problem for a general classical field theory. However, for the case of Maxwell field theory formulated in Minkowski spacetime, the answer is believed to be known: up to small radiative corrections, the worldline of a charged particle satisfies the Lorentz EOM. This belief is based on the implicit assumption that the mathematical methods used to derive the EOM from the field theory lead to an EOM that is fundamental in the sense that the charge-center of the particle is not very much smaller than the spatial resolution of the dynamical description, so that the EOM defines, as far as is possible, a fully resolved dynamics. Standard methods for deriving an EOM do not satisfy the above assumption. They are resolution limited, leaving open the possibility that the derived EOM is not fundamental. Typically, this is due to use of a point-multipole (eg., Lienard-Wiechert) approximation for the field of the particle that becomes applicable only at some distance from the particle. In the case of Maxwell field theory, it can be proved that the Lorentz EOM is not fundamental--it provides only a lowest resolution approximation to an electrodynamics that exhibits multiple length scales. Evidence for a multiscale electrodynamics can already be found in the physics literature, for example, in Hestenes' Zitterbewegung interpretation of Dirac electron theory. In this essay I collect together this evidence in the form of an ansatz for the quantum physics of the electron. This ansatz has been used to derive a 4th order EOM for the electron that not only models electron spin dynamics, but remains well defined throughout the full range of length scales over which quantum electrodynamics applies.
Andrew H. Norton