Periodic charge oscillations in the Proca theory

TL;DR

This paper investigates periodic charge oscillations in the Proca theory, revealing oscillating electric fields, a paradox in charge dynamics, and shock wave phenomena affecting the system's analyticity.

Contribution

It provides a detailed analysis of charge dynamics in the Proca theory, highlighting oscillating electric fields and shock wave effects, which are novel insights into massive vector field behavior.

Findings

Periodic oscillations of the mean electric field observed.

Identification of a paradox in the mean 4-current.

Existence of a shock wave influencing system analyticity.

Abstract

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.