Is the Dirac monopole quantization condition out by 1/2? Consider an electron with electric charge $e$ that is moved around a closed horizontal circular path $C$ centered around a magnetic monopole with magnetic charge $g$. Assume that the monopole produces a radial magnetic field $\vec{B}$ perpendicular to the surface of a sphere $S$ inscribed in the path $C$.
The electron wavefunction acquires a phase shift $\phi$ given by
$$\phi=\frac{e}{\hbar}\int_C\vec{A}\cdot d\vec{x},$$
where $\vec{A}$ is the electromagnetic vector potential defined by $\vec{B}=\nabla\times\vec{A}$.
Using Stokes' theorem we find that the phase change $\phi$ is given by the sum of the magnetic flux through the upper hemisphere $U$ and the lower hemisphere $L$:
$$\phi=\frac{e}{\hbar}\int_U \vec{B}\cdot d\vec{S}+\frac{e}{\hbar}\int_L \vec{B}\cdot d\vec{S}.$$
The magnetic flux on the entire sphere $S$ due to the magnetic charge $g$ at its center is given by
$$\int_S \vec{B}.d\vec{S} = \mu_0 g.$$
Therefore the phase shift of the electron wavefunction $\phi$ is given by
$$\phi=\frac{\mu_0 e g}{\hbar}.$$
Now the standard definition of the Dirac quantization condition assumes that the electron wavefunction is unchanged after each orbit around the magnetic monopole. However the magnetic field of the monopole can interact with the electron causing its magnetic moment and therefore its spin to align radially. This will cause the electron spin to rotate as the electron orbits the monopole. As the electron is a fermion one spin rotation ($2\pi$) flips the sign of its wavefunction. It takes two spin rotations ($4\pi$) for the electron wavefunction to get back to its original state. Therefore the Dirac quantization condition should be given by
$$\frac{\mu_0 e g}{\hbar}=4\pi n$$
or
$$\frac{e g}{4\pi\epsilon_0\hbar c^2}=n,$$
where $\epsilon_0 \mu_0 = 1/c^2$ and $n$ is an integer.
Is this correct?
Correction
I agree with Meng Cheng that Stokes’ Theorem should only be applied to one hemisphere. I agree with ACuriousMind that spin shouldn’t enter into the argument. Putting those two conditions together I still come to the conclusion that the Dirac monopole condition is out by a factor $1/2$.
 A: The Dirac quantization condition does not say the wavefunction can not change when the charged particle goes around the monopole (actually, it is meaningless to say a point-like particle going around another point-like object in 3+1, but that's a minor issue). Rather, there must be a well-defined change of the phase when the particle moves around any closed path. The point is that there are two ways to calculate the change of phase. Say the path is the equator of the sphere centered at the monopole. You can either calculate the phase by finding the total magnetic flux through the northern hemisphere, or the southern hemisphere, and you should get the same answer. This gives the quantization condition. The spin of the electron does not matter. It might pick up a phase, but the electron picks up some phase moving in the (orbital) magnetic field anyway.
Note that an error in your reasoning is the wrong use of Stokes theorem. The expression for $\phi$ should only contain the northern or southern hemisphere, but not both.
