Looking for Carroll Particles in the Two-Time Spacetime

We make an attempt to describe Carroll particles with a non-vanishing value of energy (i.e. the Carroll particles which always stay in rest) in the framework of two time physics, developed in the series of papers by I. Bars and his co-authors. In the spacetime with one additional time dimension and one additional space dimension one can localize the symmetry which exists between generalized coordinates and their conjugate momenta. Such a localization implies the introduction of the gauge fields, which in turn implies the appearance of some first-class constraints. Choosing different gauge-fixing conditions and solving the constraints one obtains different time parameters, Hamiltonians, and generally, physical systems in the standard one time spacetime. We find a set of gauge fixing conditions which gives the description of a Carroll particle in the one time world. Besides, we construct the quantum theory of such a particle using an unexpected correspondence between our parametrization and that obtained by Bars for the hydrogen atom in 1999.