Decomposing Electromagnetism's Four-potential into Quantizable and non-Quantizable Parts via Conserved Spacelike Projections of the Four-current

Abstract

The classical electromagnetic four-potential has non-dynamical and physically redundant gauge field degrees of freedom as well as purely dynamical ones.  As only the latter can be quantized, separating these out is highly desirable.  Analysis of the physically nonredundant electric and magnetic fields reveals that it is their transverse parts which are purely dynamical, and that these have an inhomogeneous driving term which is the transverse part of the current density vector. The remaining non-dynamical, non-radiative longitudinal component of the electric field is a coulombic homogeneous closed quadrature over the charge density.  It likewise turns out that the transverse spatial part of a Lorentz-condition constrained electromagnetic four-potential is purely dynamical and has the very same inhomogeneous driving term, which is the inherently conserved transverse spatial projection of the full conserved four-current.  The complementary remainder of the full four-current turns out to be determined by the charge density alone, and the remaining part of the Lorentz-condition constrained four-potential is a non-dynamical homogeneous closed quadrature over it.  The decomposition of a Lorentz-condition constrained four-potential into quantizable dynamical and non-quantizable, non-dynamical homogeneous closed quadrature parts is, however, by no means unique; any conserved spacelike projection of the four-current will serve as the inhomogeneous driving term of the dynamical part of such a decomposition.