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I've been doing some research on magnetic monopoles and I always end up seeing an expression for Dirac string vector potential commonly as:

$$\textbf{A}=g\frac{1+\cos\theta}{r\sin\theta} \hat{\phi}$$

I would like to work out this conclusion, but all I ever find on internet are verifications or the fact that in the presence of a magnetic monopole $\mathbf B$ can't be written as the curl of $\mathbf A$. I don't know if it's something obvious indeed, but Dirac's paper Quantized singularities in the electromagnetic field gave me no clue. If someone could give me a hand or a reference, it'd be appreciated.

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Dirac's paper is the original source, as in Wu and Yang's paper which makes use of this vector potential he is cited. You simply need to look more carefully.

He takes $\mathbf A = hc/e \cdot \mathbf \kappa$ and $A_0 = -h/e \cdot \kappa_0$ and claims that

$$\nabla \times \mathbf \kappa = \frac{e}{hc}\mathbf H, \quad \nabla \kappa_0 - \frac{\partial \kappa_0}{\partial t} = \frac{e}{h}\mathbf E.$$

For a single monopole, the magnetic field is radial and of magnitude $r^{-2}$ and it can be shown using the above equations that,

$$\kappa_0 = 0, \quad \kappa_r = \kappa_\theta = 0, \quad \kappa_\phi = \frac{1}{2r} \tan \frac12 \theta.$$

Such an $A^\mu$ leads to a magnetic field $B_r = \frac{1}{2r^2}$. If we take your vector potential, we also find that upon taking the curl, $B_r = -\frac{g}{r^2}$ so they are physically speaking the same solution.

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