Question

Today I read articles and texts about Dirac monopoles and I have been wondering about the insistence on gauge potentials. Why do they seem (or why are they) so important to create a theory about magnetic monopoles?

And more generally, why do we like gauge potentials so much?

Answer 1

It is the gauge potentials, not the fields, that determine the quantum motion of particles. In either the Schrodinger equation or the path integral, the gauge field appears, not the E and B, and for nonabelian theories, this is impossible to fix because you don't have a integral Stokes law relation.

The interaction with charged particles is that a particle moving along a path gets a phase equal to the integral of A along the path. If you want to replace the A with B, you need to use the fact that the integral of A along a closed loop is the magnetic flux enclosed by the loop, and this is a nonlocal condition. So you can't write local equations of motion for a quantum particle using E and B.

The integral relation for B states that if you make a circle, and you want the phase that the charged particle will get if it moves on this circle, you draw some surface whose boundary is the circle, and the magnetic flux through this surface is the phase.

The dirac condition is simply the statement that if you have a monopole and draw a circle around the monopole, the flux through the northern hemisphere is equal to the flux through the southern hemisphere, up to a multiple of 2pi, which is an undetectable phase change. This tells you that the magnetic charge times the electric charge must be an integer multiple of 2?.