Two Times for Freudenthal

We investigate the algebraic structure of the two-time physics introduced some time ago by I. Bars and his co-authors, clarifying its relations with quadratic and cubic Jordan algebras, as well as with reduced Freudenthal triple systems (FTS) based on them. In particular, the extended' phase space introduced by Bars can be endowed with the structure of a reduced FTS constructed over a semi-simple cubic Jordan algebra (named Lorentzian spin factor), characterized by a primitive, invariant symmetric tensor of rank $4$. The $Sp(2,\mathbb{R})$-gauge fixing procedure typical of two-time physics yields algebraic-differential constraints on the quartic polynomial associated to such a tensor, implying that only two (isomorphic) nilpotent orbits of the non-transitive action of the automorphism group of the Lorentzian spin factor are spanned by the conjugated variables which coordinatize the extended’ phase space. We illustrate our results in relativistic, manifestly Lorentz-covariant physical systems, as well as in non-relativistic systems (such as the non-relativistic massive particle, the hydrogen atom, and the Carroll particle with non-vanishing energy).