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Two Times for Freudenthal

Alexander Kamenshchik, Alessio Marrani, Federica Muscolino

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).

Two Times for Freudenthal

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 . The -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).
Paper Structure (48 sections, 168 equations, 2 tables)

This paper contains 48 sections, 168 equations, 2 tables.

Table of Contents

  1. Introduction
  2. The 2T, 'extended' phase space
  3. The 'extended' phase space as the Freudenthal triple system $\mathfrak{F}(\mathfrak{J}_{1,d-1})$
  4. Remark
  5. Invariants and Orbits
  6. 2T physics: Examples
  7. Relativistic massless particle
  8. The maximal linearly realized algebra: $\mathfrak{so}\left( 1,d-1\right)\oplus\mathfrak{so}(1,1)$
  9. The sign of $\mathcal{I}_2$
  10. Relativistic massless particle in maximally symmetric space
  11. The maximal linearly realized algebra: $\mathfrak{so}\left( 1,d-1\right)$
  12. The sign of $\mathcal{I}_2$
  13. Relativistic massive particle in Minkowski space
  14. The maximal linearly realized algebra: $\mathfrak{so}(1,d-1)$
  15. The sign of $\mathcal{I}_2$
  16. ...and 33 more sections