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 parameter, 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.In this paper, we examine Tanimura's derivation in the framework of the proper time method in relativistic mechanics, and use the technique of Hojman and Shepley to study the unconstrained commutation relations. We show that Tanimura's result then corresponds to the five-dimensional electromagnetic theory previously derived from a 1 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.
The requirement of gauge invariance for the Schwinger-DeWitt equations, interpreted as a manifestly covariant quantum theory for the evolution of a system in spacetime, implies the existence of afire-dimensional pre-Maxwell field on the manifold of spacetime and "proper time" z. The Maxwell theory is contained in this theory; integration of the field equations over r restores the Maxwell equations with the usual interpretation of the sources. Following Schwinger's techniques, we study the Green's functions for the five-dimensional hyperbolic field equations for both signatures + [corresponding to 0(4, 1) or 0(3, 2) symmetry of the field equations] of the proper time derivative. The classification of the Green's functions follows that of the four-dimensional theory for "massive" fields, for which the "mass" squared may be positive or negative, respectively. The Green's functions for the five-dimensionat fietd are then given by the Fourier trans-.form over the "mass" parameter, We derive the Green's functions corresponding to the principal part 3 e and the homogeneous fimction A1; all of the Green's functions can be expressed in terms of these, as for the usual field equations with definite mass. In the 0(3, 2) case, the principal part function has support for x 2 >t v 2, corresponding to spacelike propagation, as well as along the light cone x 2 = 0 (for z = 0). There can be no transmission of information in spaeelike directions, with this propagator, since the Maxwell fleld, obtained by integration over z, does not eonta& this component of the support. Measurements are characterized by such an integration. The spacelike _field therefore can dynamically establish spaeelike correlations.
The kinematics of pre-Maxwell electrodynamics is examined and interpretations of these fields is found through an examination of the associated Lorentz force and the structure of the energy-momentum tensor.Key words: relativistic quantum theory, off-shell gauge fields, pre-Maxwell fields, higher dimensional gauge fields.In the framework of a covariant relativistic quantum theory In this letter we shall examine the kinematics of the pre-Maxwell electrodynamics and find interpretations of the new fields through an examination of the Lorentz force and the structure of the energy-momentum tensor of the field. 1 arXiv:1604.06745v1 [physics.gen-ph]
Despite the many successes of the relativistic quantum theory developed by Horwitz, et. al., certain difficulties persist in the associated covariant classical mechanics. In this paper, we explore these difficulties through an examination of the classical Coulomb problem in the framework of off-shell electrodynamics. As the local gauge theory of a covariant quantum mechanics with evolution parameter τ , off-shell electrodynamics constitutes a dynamical theory of spacetime events, interacting through five τdependent pre-Maxwell potentials. We present a straightforward solution of the classical equations of motion, for a test event traversing the field induced by a "fixed" event (an event moving uniformly along the time axis at a fixed point in space). This solution is seen to be unsatisfactory, and reveals the essential difficulties in the formalism at the classical level. We then offer a new model of the particle current -as a certain distribution of the event currents on the worldline -which eliminates these difficulties and permits comparison of classical off-shell electrodynamics with the standard Maxwell theory. In this model, the "fixed" event induces a Yukawa-type potential, permitting a semi-classical identification of the pre-Maxwell time scale λ with the inverse mass of the intervening photon. Numerical solutions to the equations of motion are compared with the standard Maxwell solutions, and are seen to coincide when λ > 10 −6 seconds, providing an initial estimate of this parameter. It is also demonstrated that the proposed model provides a natural interpretation for the photon mass cut-off required for the renormalizability of the off-shell quantum electrodynamics.
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