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 on a fifth (scalar) electromagnetic potential, and the invariant evolution parameter cannot be consistently identified with the proper time of the particle motion. Moreover, the derivation cannot be made reparameterization invariant; the scalar potential causes violations of the mass-shell constraint which this invariance should guarantee. Tanimura's derivation is examined in the framework of the proper time method in relativistic mechanics, and the technique of Hojman and Shepley is used to study the unconstrained commutation relations. It is shown that Tanimura's result then corresponds to the five-dimensional electromagnetic theory previously derived from a Stueckelberg-type quantum theory in which one gauges the invariant parameter in the proper time method. This theory provides the final step in Feynman's program of deriving the Maxwell theory from commutation relations; the Maxwell theory emerges as the "correlation limit" of a more general gauge theory, in which it is properly contained.