On the gauge transformation for the rotation of the singular string in the Dirac monopole theory

TL;DR

This paper investigates the gauge transformation that rotates the Dirac string in monopole theory, revealing complex behaviors of the gauge function and clarifying misconceptions in existing literature.

Contribution

It derives an explicit analytical expression for the gauge function and analyzes its properties, providing new insights into the gauge transformation of the Dirac string.

Findings

The gauge function has complex behaviors depending on the side of the plane crossed.

Clarification of misunderstandings in the literature regarding the Dirac string rotation.

Analytical expression for the gauge function $\chi( extbf{r})$ derived and analyzed.

Abstract

In the Dirac theory of the quantum-mechanical interaction of a magnetic monopole and an electric charge, the vector potential is singular from the origin to infinity along certain direction - the so called Dirac string. Imposing the famous quantization condition, the singular string attached to the monopole can be rotated arbitrarily by a gauge transformation, and hence is not physically observable. By deriving its analytical expression and analyzing its properties, we show that the gauge function $\chi({\bf r})$ which rotates the string to another one has quite complicated behaviors depending on which side from which the position variable ${\bf r}$ gets across the plane expanded by the two strings. Consequently, some misunderstandings in the literature are clarified.