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The mathematical model for classical electromagnetism just doesn't forbid magnetic monopoles by construction. Consider an arbitrary vector field $X$ in 3d. Such a vector field is totally characterized by its divergence and curl. Suppose the following is true: $$\nabla \cdot X = \sigma, \quad \nabla \times X = Y$$ Then knowing $\sigma$ and $Y$ ...


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First of all some clarifications to make sure we are on the same page. The phrase "$A_\nu$ is local" is slightly misleading as it implies that A is only defined in some (small) neighborhood of the total manifold/space. This is not true. A is defined everywhere on the manifold. The correct expression is that "the theory is invariant under a local symmetry" ie ...



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