Green's functions for off-shell electromagnetism and spacelike correlations

Abstract

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.

Green functions for wave propagation on a five-dimensional manifold and the associated gauge fields generated by a uniformly moving point source

“…Such GF's have been found [7,10] through a 5-fold Fourier transform of the wave equation. The GF obeys the wave-equation of a point source…”

On the definition and evolution of states in relativistic classical and quantum mechanics

“…(46) have been worked out in Ref. 20. For both 4 + 1 and 3 + 2 signatures, the principal part Green's functions (the nomenclature for the Green's functions is the same as for those of the massive vector meson fields, since the method that we use is to Fourier transform with respect to z, apply the usual techniques, then invert the transform; the variable complementary to r plays the role of the mass squared, but can have either sign) have support on the light cone of the form 6(x 2) 6(~).…”

The Particle as a Statistical Ensemble of Events in Stueckelberg–Horwitz–Piron Electrodynamics

“…The classical antiparticle is identified here by its negative energy and no charge conjugation operation is necessary. Looking ahead to the current defined in (20), one sees that the electric charge Q = d 3 x j 0 (x, τ) will similarly reverse sign on this section of the trajectory. Although the laboratory apparatus observes the two events at x 0 = ct 0 as simultaneous, they may be distinguished in SHP theory by other τ-dependent interactions.…”