Electrodynamics with magnetic monopoles: Photon wave mechanical theory

Abstract

The microscopic Maxwell-Lorentz equations show an intrinsic symmetry if magnetic charge and current densities are included. Starting from the symmetrized set of field equations a propagator approach is used to describe the transverse (photon) electrodynamics in space-time. Two dyadic propagators are needed: (i) a transverse electric propagator with a near-field part and (ii) a magnetic propagator with a midfield part. The first quantized photon wave mechanical theory, based on the analytical parts of the two transverse Riemann-Silberstein vectors, is extended to include magnetic monopole dynamics. The dynamical equations for the photon helicity eigenvectors and the local energy conservation in the photon field are discussed. The Dirac string concept and the fiber bundle approach are avoided using the double-potential formalism of Cabibbo and Ferrari. The transverse ($T$) parts of the electric (${\mathbf{A}}_{T}^{e}$) and magnetic (${\mathbf{A}}_{T}^{m}$) vector potentials relate in a simple manner to our propagator theory if this is formulated in terms of the Huygens scalar propagator and the transverse electric and magnetic current densities. A transformation of the transverse vector potentials, which is nonlocal in space and time, allows one to express the transverse parts of the electric and magnetic fields as curls of combinations of the original (${\mathbf{A}}_{T}^{e},\phantom{\rule{0.16em}{0ex}}{\mathbf{A}}_{T}^{m}$) and the transformed (${\mathsc{A}}_{T}^{e},\phantom{\rule{0.16em}{0ex}}{\mathsc{A}}_{T}^{m}$) transverse vector potentials. The momentum of the particle-photon system is discussed, and it is shown that the electromagnetic parts of the canonical electric (charge $e$) and magnetic (charge $g$) particle momenta are given by $e({\mathbit{A}}_{T}^{e}+{\mathsc{A}}_{T}^{m})$ and $g({\mathbit{A}}_{T}^{m}\ensuremath{-}{\mathsc{A}}_{T}^{e})$, respectively. The angular momentum of the particle-photon system is analyzed and contact is made to the well-known nonretarded Saha-Wilson orbital angular momentum for an ($e,g$) pair. The near field of a magnetic monopole is studied based on the overlooked fact that the magnetic near field contains both longitudinal ($L$) (with $\mathbf{\ensuremath{\nabla}}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{B}}_{L}=0$) and transverse vector field components. The sum of the two parts always is Einstein retarded, with proper account of the limitation caused by the lack of complete spatial photon localization. Finally, the relativistic electron-photon Hamiltonian in an external (prescribed) magnetic monopole field is discussed, paying particular attention to a determination of the transformed transverse magnetic vector potential, ${\mathsc{A}}_{T}^{m}$, and its positive-frequency part.