How does the existence of a monopole affect the magnetic vector potential in gauss's law for magnetism? [duplicate]

Question

Gauss's Law for magnetism is

$$ \nabla \cdot B = 0 $$ This allows us to write the magentic field $B$ as the curl of another field the $\textbf{magnetic vector potential, } A$.

$$ B=\nabla \times A $$

This adhers to $ \nabla \cdot (\nabla \times A)=0$

However, if a monopole does exist then we have

$$ \nabla \cdot B= \rho_{m} $$

Where $\rho_{m}$ is some magnetic charge density however with a magentic vector potential this violates the equation, $ \nabla \cdot (\nabla \times A) \neq 0$.

Does that mean if magnetic monopoles does exist, that the magnetic field can no longer be defined by a magnetic vector potential? In which case how was dirac able to still define the magnetic field by a magnetic vector potentials?

Answer 1

That means that, in general, you will have to write $\mathbf B = \nabla \psi + \nabla\times\mathbf A$. Then $\nabla\cdot\mathbf B = \nabla^2\psi = \rho_m$. Of course $\nabla\cdot(\nabla\times\mathbf A)=0$ for any vector field $\mathbf A$, as $\operatorname d^2=0$.