On Feynman's Approach to the Foundations of Gauge Theory

In 1948, Feynman showed Dyson how the Lorentz force law and homogeneous Maxwell equations could be derived from commutation relations among Euclidean coordinates and velocities, without reference to an action or variational principle. When Dyson published the work in 1990, several authors noted that the derived equations have only Galilean symmetry and so are not actually the Maxwell theory. In particular, Hojman and Shepley proved that the existence of commutation relations is a strong assumption, sufficient to determine the corresponding action, which for Feynman’s derivation is of Newtonian form. In a recent paper, Tanimura generalized Feynman’s derivation to a Lorentz covariant form with scalar evolution parameters, and obtained an expression for the Lorentz force which appears to be consistent with relativistic kinematics and relates the force to the Maxwell field in the usual manner. However, Tanimura’s derivation does not lead to the usual Maxwell theory either, because the force equation depends o...