Non-equilibrium dynamics of dipole-charged fields in the Proca theory

We discuss the dynamics of field configurations encoded in the certain class of electric (magnetic) dipole-charged states in the Proca theory of the real massive vector field. We construct such states so as to ensure that the long-distance structure of the mean electromagnetic field in them is initially set by the formula describing the electromagnetic field of the electric (magnetic) dipole. We analyze then how such a mean electromagnetic field evolves in time. We find that far away from the center of the initial field configuration, the long-range component of the mean electromagnetic field harmonically oscillates, which leads to the phenomenon of the periodic oscillations of the electric (magnetic) dipole moment. We also find that near the center of the initial field configuration, the mean electromagnetic field escapes from its initial arrangement and a spherical shock wave propagating with the speed of light appears in the studied system. A curious configuration of the axisymmetric mean electric field is found to accompany the mean magnetic field in magnetic dipole-charged states.

Automated summary

In this paper, we discuss the dynamics of field configurations in the Proca theory of the real massive vector field, specifically focusing on a certain class of electric (magnetic) dipole-charged states. We construct these states to ensure that the long-distance structure of the mean electromagnetic field is initially set by the formula describing the electromagnetic field of the electric (magnetic) dipole. We analyze the evolution of this mean electromagnetic field over time and observe the phenomena of harmonic oscillations of the electric (magnetic) dipole moment far from the center of the initial field configuration, as well as the emergence of a spherical shock wave propagating at the speed of light near the center. Additionally, we discover a unique axisymmetric mean electric field configuration accompanying the mean magnetic field in magnetic dipole-charged states.