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 $4$. 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 theextended' 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).

• The paper identifies the extended phase space of Bars’ two-time physics with a reduced Freudenthal triple system over a Lorentzian spin factor, whose automorphism group is Sp(2,ℝ) × SO(2,d).
• The paper shows that Sp(2,ℝ) gauge constraints force the quartic invariant I₄ and its first derivatives to vanish, placing every classical one-time model on a nilpotent orbit distinguished by the sign of a quadratic invariant I₂.
• The paper systematically relates gauge-fixed models—including relativistic, non-relativistic, Carrollian, and hydrogen systems—to their linearly and non-linearly realized symmetry algebras, while identifying open problems in translations, conjecture proofs, and quantization.

The paper "Two Times for Freudenthal" (2603.13067) by Kamenshchik, Marrani and Muscolino establishes a precise algebraic identification between the extended phase space of two-time (2T) physics à la Bars and the reduced Freudenthal triple system (FTS) constructed over the Lorentzian spin factor. This identification allows the authors to recast the $Sp(2,\mathbb{R})$ gauge-fixing procedure of 2T physics as a selection of nilpotent orbits of the conformal group acting on the FTS, and to analyze systematically how the manifest symmetry algebra of each resulting one-time (1T) physical model arises from the breaking of $\mathfrak{so}(2,d)$.

The extended phase space and its Jordan-algebraic origin

In Bars' formulation of 2T physics, particle dynamics is defined on an extended phase space with coordinates $X^M_i = (X^M, P^M)$, where $M$ ranges over $d+2$ dimensions containing one extra timelike and one extra spacelike direction. Gauging the symplectic group $Sp(2,\mathbb{R})$ introduces three first-class constraints $Q_{ij}=0$, namely $X\cdot X=0$, $P\cdot P=0$, and $X\cdot P=0$. Different gauge fixings of these constraints yield apparently distinct 1T systems — massless or massive relativistic particles, non-relativistic particles, hydrogen atoms — which are dual under local $\mathfrak{so}(2,d)$0 transformations.

The central observation of the paper is that this structure is exactly that of a reduced FTS. The rank-2 Jordan algebra $\mathfrak{so}(2,d)$1 (Minkowski space with its quadratic norm) is enlarged to the semisimple rank-3 Jordan algebra $\mathfrak{so}(2,d)$2, the Lorentzian spin factor with cubic norm $\mathfrak{so}(2,d)$3. The reduced FTS $\mathfrak{so}(2,d)$4, of real dimension $\mathfrak{so}(2,d)$5, is then identified with the extended phase space itself. Its automorphism group is

$\mathfrak{so}(2,d)$6

so the global isometry group of 2T physics is precisely the conformal group of the cubic Jordan algebra. In an appendix, the authors prove via the Frobenius norm that $\mathfrak{so}(2,d)$7 carries a Lorentzian quadratic norm of signature $\mathfrak{so}(2,d)$8, justifying the "Lorentzian spin factor" terminology.

Invariants, orbits, and the role of the gauge fixing

The action of $\mathfrak{so}(2,d)$9 on $X^M_i = (X^M, P^M)$0 is non-transitive; the space is a prehomogeneous vector space whose ring of invariants is generated by the unique primitive quartic polynomial

$X^M_i = (X^M, P^M)$1

The $X^M_i = (X^M, P^M)$2 gauge-fixing conditions $X^M_i = (X^M, P^M)$3 imply $X^M_i = (X^M, P^M)$4 together with all first derivatives of $X^M_i = (X^M, P^M)$5 vanishing. Consequently, every 1T physical system obtained from 2T physics has its phase-space coordinates confined to a single nilpotent ("small") orbit, which further splits into two sub-orbits $X^M_i = (X^M, P^M)$6 discriminated by the sign of the quadratic invariant

$X^M_i = (X^M, P^M)$7

Both orbits are realized as the same non-symmetric coset space $X^M_i = (X^M, P^M)$8. This result means that the entire dimensional reduction from the 2T to the 1T world can be characterized invariantly: the physical phase space always sits on a degenerate stratum of the quartic norm, with the residual sign of $X^M_i = (X^M, P^M)$9 serving as the only remaining discrete label.

Symmetry reduction in explicit models

For each gauged-fixed model, the paper determines two algebras: $M$0, the maximal linearly realized subalgebra of $M$1 after gauge fixing and constraint resolution, and $M$2, the subalgebra preserving the sign of $M$3. The results are:

Gauge choice	$M$4	$M$5
Massless relativistic particle	$M$6	$M$7
Massless relativistic particle in maximally symmetric space	$M$8	$M$9
Massive relativistic particle	$d+2$0	$d+2$1
Non-relativistic massive particle	$d+2$2	$d+2$3
Carroll particle with $d+2$4	$d+2$5	$d+2$6
Hydrogen atom ($d+2$7)	$d+2$8	$d+2$9

Here $Sp(2,\mathbb{R})$0 is the intersection of the Bargmann algebra with the conformal algebra, and analogously for the Carrollian case. A recurring structural feature is that time and space translation generators do not belong to $Sp(2,\mathbb{R})$1 within these parametrizations; they appear only multiplied by non-linear factors (e.g., $Sp(2,\mathbb{R})$2 for the Galilean particle), so translations are non-linearly realized on the coset $Sp(2,\mathbb{R})$3.

Two qualitative patterns emerge. First, in massless relativistic systems the sign of $Sp(2,\mathbb{R})$4 is invariant under the full $Sp(2,\mathbb{R})$5 even when the dilatation generator is only non-linearly realized (as in curved maximally symmetric backgrounds). Second, a non-vanishing mass explicitly breaks this invariance: for the massive relativistic particle, dilatations rescale $Sp(2,\mathbb{R})$6 but leave $Sp(2,\mathbb{R})$7 fixed, so sufficiently large parameters flip the sign of $Sp(2,\mathbb{R})$8; similarly, Bargmann boosts shift $Sp(2,\mathbb{R})$9 additively. For the Carroll particle with rest energy $Q_{ij}=0$0, the breaking of the $Q_{ij}=0$1 invariance is not removable even in the limit $Q_{ij}=0$2, which the authors attribute to the intrinsic degeneracy of the Carrollian particle, where the mass coincides with the Hamiltonian.

For the hydrogen atom, the paper also constructs a $Q_{ij}=0$3-dimensional generalization of the Runge-Lenz vector,

$Q_{ij}=0$4

which, together with the dualized angular momentum generators, closes an $Q_{ij}=0$5 symmetry commuting with the Hamiltonian once the constraint $Q_{ij}=0$6 is imposed. Nevertheless, within the 2T realization the maximal linearly realized subgroup remains only $Q_{ij}=0$7; the full dynamical $Q_{ij}=0$8 symmetry of the bound-state spectrum is recovered through non-linear realizations.

Conjectures and limitations

Based on the tabulated results, the authors formulate two conjectures: (i) in massless relativistic systems the sign of $Q_{ij}=0$9 is invariant under the whole $X\cdot X=0$0 regardless of whether its generators are linearly realized; (ii) a non-vanishing mass removes the $X\cdot X=0$1 factor from the stabilizer of the sign of $X\cdot X=0$2, reducing it to $X\cdot X=0$3 in non-relativistic models. These remain conjectures supported by explicit computation rather than proven statements; turning them into theorems is stated as an open task.

Several further qualifications bear on the results. The analysis is restricted to flat metrics on the extended phase space, though the framework admits generalizations. The partial gauge-fixing used for non-relativistic models projects onto subspaces not contained in $X\cdot X=0$4, so the clean chain of reductions $X\cdot X=0$5 available for relativistic systems does not hold there. Moreover, since spacetime translations are symmetries of most considered models but lie outside $X\cdot X=0$6, the authors propose extending the isometry group to $X\cdot X=0$7 with a central extension realizing generalized translations; the corresponding coadjoint-orbit stratification is left as ongoing work. Finally, the quantization question is raised but not resolved: upon quantization, ordering effects would render $X\cdot X=0$8 non-zero (expectedly proportional to $X\cdot X=0$9), moving quantum systems off the classical nilpotent orbit onto generic open orbits, a transition the authors relate to symplectic deformations of gauged supergravity but defer to future study.

Conclusion

The paper demonstrates that the extended phase space of 2T physics is canonically the reduced FTS over the Lorentzian spin factor, with $P\cdot P=0$0 as its automorphism group. The $P\cdot P=0$1 gauge-fixing constraints select a unique nilpotent orbit with $P\cdot P=0$2, bifurcated by the sign of $P\cdot P=0$3, and the interplay between linear and non-linear realizations of the conformal algebra explains the manifest symmetry content of each 1T descendant. The work provides the basis for a systematic, orbit-theoretic classification of gauge fixings and their associated duality web in 2T physics, while leaving open the proof of the sign-invariance conjectures, the inclusion of translational sectors, and the quantum-mechanical deformation of the orbit structure.