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Euclidean Jordan Algebras, Hidden Actions, and $J$-Kepler Problems

For a {\em simple Euclidean Jordan algebra}, let $\mathfrak{co}$ be its conformal algebra, $\mathscr P$ be the manifold consisting of its semi-positive rank-one elements, $C^\infty(\mathscr P)$ be the space of complex-valued smooth functions on $\mathscr P$. An explicit action of $\mathfrak{co}$ on $C^\infty(\mathscr P)$, referred to as the {\em hidden action} of $\mathfrak{co}$ on $\mathscr P$, is exhibited. This hidden action turns out to be mathematically responsible for the existence of the Kepler problem and its recently-discovered vast generalizations, referred to as $J$-Kepler problems. The $J$-Kepler problems are then reconstructed and re-examined in terms of the unified language of Euclidean Jordan algebras. As a result, for a simple Euclidean Jordan algebra, the minimal representation of its conformal group can be realized either as the Hilbert space of bound states for its $J$-Kepler problem or as $L^2({\mathscr P}, {1\over r}\mathrm{vol})$, where $\mathrm{vol}$ is the volume form on $\mathscr P$ and $r$ is the inner product of $x\in \mathscr P$ with the identity element of the Jordan algebra.