Periodic charge oscillations in the Proca theory

We consider the Proca theory of the real massive vector field. There is a locally conserved 4-current operator in such a theory, which one may use to define the charge operator. Accordingly, there are charged states in which the expectation value of the charge operator is non-zero. We take a close look at the charge operator and study the dynamics of the certain class of charged states. For this purpose, we discuss the mean electric field and 4-current in such states. The mean electric field has the periodically oscillating Coulomb component, whose presence explains the periodic charge oscillations. A complementary insight at such a phenomenon is provided by the mean 4-current, whose discussion leads to the identification of a certain paradox. Last but not least, we show that there is a shock wave propagating in the studied system, which affects analyticity of the mean electric field and 4-current.