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

Abstract

For a {\em simple Euclidean Jordan algebra}, let $\mathfrak{co}$ be its conformal algebra, $\mathcal{P}$ be the manifold consisting of its semi-positive rank-one elements, $C^{\infty }(\mathcal{P})$ be the space of complex-valued smooth functions on $\mathcal{P}$. An explicit action of $\mathfrak{co}$ on $C^{\infty }(\mathcal{P})$, referred to as the {\em hidden action} of $\mathfrak{co}$ on $\mathcal{P}$, 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 $L^2(\mathcal{P},\frac{1}{r}\mathrm{vol})$, where $\mathrm{vol}$ is the volume form on $\mathcal{P}$ and $r$ is the inner product of $x\in \mathcal{P}$ with the identity element of the Jordan algebra.

37 pages

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