Two Times for Freudenthal

TL;DR

This paper explores the algebraic structure of two-time physics, connecting it with Jordan algebras and Freudenthal triple systems, and analyzes the implications of gauge fixing on the associated algebraic constraints in various physical systems.

Contribution

It establishes a link between two-time physics and algebraic structures like Jordan algebras and FTS, providing new insights into the phase space and gauge fixing procedures.

Findings

Extended phase space has a reduced FTS structure based on a semi-simple cubic Jordan algebra.

Gauge fixing imposes algebraic-differential constraints on the quartic tensor, limiting the phase space to specific nilpotent orbits.

Results apply to both relativistic and non-relativistic physical systems, illustrating the algebraic framework's versatility.

Abstract

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).