$\DeclareMathOperator{\Hom}{Hom}\DeclareMathOperator{\Gal}{Gal}$In pure mathematics, the idea of duality pops up all over the place, and most generally (as far as I know) it can be defined in category theory in terms of the opposite category $C^{op}$. In each of these cases, there is usually a satisfying conceptual explanation behind the duality, and these often are very important in the theory. For example:
- Vector spaces $V$ over a field $K$ have a dual space $V^*=\Hom(V,K)$. For finite dimensional spaces, there is a natural isomorphism $V\cong V^{**}$ which shows up a lot in linear algebra.
- The Fundamental Theorem of Galois Theory is essentially the statement of duality between the intermediate fields of a Galois extension $K/F$, i.e. fields $L$ such that $F\subseteq L\subseteq K$, and subgroups of the Galois group $\Gal(K/F)$.
Recently, I came across the curious duality between the electricity and magnetism in classical physics. As I know very little about it, this duality seems to be very mysterious to me as I am missing a good explanation for why this should be true.
I am looking for a conceptual, mathematical explanation for:
- In what sense this duality is true: is there a way to define a map from the "space of all electric fields" to the "space of all magnetic fields" which reverses the direction of arrows, whatever that means? If not, in what sense precisely is this duality a special case of duality in mathematics in general?
- Why this duality holds. The Wikipedia article mentions that in special relativity, the Lorentz transformation makes electric fields into magnetic fields. However I do not understand this, as I thought the Lorentz transformations are just coordinate changes for different choices of reference frames, and I don't really get how this is related to the duality in question.
Thanks in advance!