Some questions about gauge theory

Question

Let's talk quantum mechanics. I have a charged particle moving on a sphere. It has a wave function $\psi$. At time $t=0$, there is no magnetic flux piercing the sphere. Instantaneously, I introduce a magnetic monopole at the center of the sphere.

1. Since line bundles have integer Chern number, if I increase the charge of the monopole steadily from zero, the flux will be non-existent until I hit a unit flux quantum, and at that point the field lines will violently pierce the sphere. Increasing the flux even more will have no effect until we reach the second flux quantum, etc...

2. At this point, the line bundle containing $\psi$ becomes nontrivial. That means that $\psi$ will vanish at some point on the sphere. This means, I am guaranteed some spot where I have probability zero of finding the particle. I'll call this the blind spot.

Are any of these weird predictions consistent with the formalism of gauge theory as presented in the Standard Model? (I know we don't have magnetic monopoles in nature, but suppose that at the year 2100, we discover some on Alpha Centauri, and then bring them into the lab for this experiment.) Explicitly,

• Could we observe this flux resistance?
• Would we be able to measure a blind spot?

Answer 1

None of these two phenomena are predictions in any sense:

1. You cannot increase the charge of the monopole "steadily from zero". Dirac quantization means the electric as well as the magnetic charge are quantized, if both exist, as the Aharonov-Bohm effect due to the unphysical Dirac string must vanish. See the Wikipedia article and this answer of mine.

2. The vanishing of a wavefunction at a point is physically meaningless, as the probability to detect a particle at a point is zero in any case, since points are of zero measure. And the wavefunction vanishing at a point tells you in general nothing about the probability to detect the particle in a region containing that point, since there are no bounds on the local growth behaviour of $L^2$ functions.