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

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 χ(r) which rotates the string to another one is a smooth function everywhere in space, except their respective strings. On the strings, χ(r) is a multi-valued function. Consequently, some misunderstandings in the literature are clarified.