1

I'm studying the Dirac monopole in Anomalies in Quantum Field Theory by R. A. Bertlmann (see this book). In the derivation of the Dirac quantization condition, the author argues that transporting the wavefunction around the monopole results in a phase: $$\psi\to\exp\left[-i\frac{2eg}{\hbar c}2\pi\right]\psi,$$where $e$ is the electric charge and $g$ is the magnetic charge. The condition for the wavefunction to be single-valued then leads to the quantization of $eg$.

However, I’m puzzled: why must the wavefunction itself be single-valued? If we compute a probability amplitude (say, $|\psi|^2$), the global phase would cancel out and give the same physical result. So why is this single-valuedness condition necessary, especially if the overall phase seems to have no observable consequence?

Qmechanic
  • 220,844
gabrielmglh
  • 11
  • 2

2 Answers2

0

The particle travels through the Dirac string “makes a journey” and must have the same value each time it returns to the same point in space, within this “string” the particle diverges without physical consequences, HOWEVER without handling single values the probabilities that depend on $|\psi|^2$ would not be well defined.

This is my first contribution, I hope it has helped you.

DavidTejeda
  • 1
  • 1
0

The momentum $p= -i \partial_x$ of the particle would become "infinite", if the wavefunction were not single valued. This is the same reasoning why a wavefunction and its first derivative should be continuous in elementary QM (e.g. across a step potential).

Fabian
  • 2,074