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

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 a 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 [Formula: see text] which rotates the string to another one is a smooth function everywhere in space, except their respective strings. On the strings, [Formula: see text] is a multi-valued function. Consequently, some misunderstandings in the literature are clarified.