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[en] The geometrical relativistic superparticle action is analyzed from the theoretical group point of view. To this end an alternative technique of quantization, outlined by the authors in a previous work and based on the correct interpretation of the square-root Hamiltonian, is used. We show that the obtained spectrum of physical states and the Fock construction in this previous work consist of squeezed states with the even and odd representations with the lowest weights λ = 1/4 and λ= 3/4 corresponding to four possible (nontrivial) fractional representations for the group decomposition of the spin structure. The conserved currents are computed, and a new relativistic wave equation is proposed and explicitly solved for the time-dependent case. The relation between the relativistic Schrödinger equation and the time-dependent harmonic oscillator is analyzed and discussed.