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Maxwell equations are invariant under the duality transformation. The electric and magnetic fields are defined in terms of these potentials. How do these potentials transform under duality?

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On the level of the field strength tensor $F$, duality is the map $F\mapsto {\star}F$, where ${\star}$ is the Hodge dual. Maxwell's equations in vacuum are $$ \mathrm{d}F = 0 \quad \text{and} \quad \mathrm{d}{\star}F = 0$$ and by the Poincaré lemma we can therefore locally find 1-forms $A$ and $A'$ such that $\mathrm{d}A = F$ and $\mathrm{d}A' = {\star}F$. The components of $A$ - the four-potential - are the usual scalar and vector components of electrodynamics. Since the duality exchanges $F$ and ${\star}F$, it acts on the potential $A$ by exchanging it with $A'$.

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  • $\begingroup$ Thanks but how A and A' are related? $\endgroup$ – Maxwell Mar 15 at 19:05

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