The Extra Time

Carroll symmetry describes particles that cannot move — they have energy but zero velocity. In ordinary physics, this is a degenerate limit. In Bars’s two-time (2T) physics framework, it is a natural sector.

arXiv:2603.16276 shows that Carroll particles with nonzero energy emerge naturally within 2T physics. The framework adds one extra time dimension and one extra space dimension, then gauges a phase-space symmetry that exchanges generalized coordinates with their conjugate momenta. Different gauge choices in the extended spacetime yield apparently different one-time systems as projections. The Carroll particle is one such projection.

The technical construction connects to Freudenthal triple systems built over a semisimple cubic Jordan algebra — specifically, the Lorentzian spin factor. This algebraic structure unifies the extended phase space of 2T physics with the Jordan-algebraic formulation of quantum mechanics, linking a kinematic symmetry (Carroll) to an algebraic framework (Jordan) through a dimensional-extension principle (2T).

The underlying idea: systems that look physically distinct in ordinary spacetime can be gauge-equivalent descriptions of the same dynamics in a higher-dimensional phase space. A particle at rest (Carroll) and a particle in motion (Galilean) are not different physics but different gauge slices of the same extended system. The restriction to one time dimension is what makes them look different.

Whether the extra time dimension is physically real or a mathematical device is beside the point. The unification it produces — connecting Carroll kinematics to Jordan algebras through phase-space gauge symmetry — reveals structural relationships invisible from the perspective of a single time coordinate.