On the Lorentz Group SO(3,1), Geometrical Supersymmetric Action for Particles, and Square Root Operators

DIEGO JULIO CIRILO-LOMBARDO

Physics of Particles and Nuclei Letters

SPRINGER/PLENUM PUBLISHERS

Año: 2006 p. 416 - 423

1547-4771

In this work the problem of the square-root quantum operators is analyzed from the theoretical group point of view. To this end, we considered the relativistic geometrical action of a particle in the superspace in order to quantize it and to obtain the spectrum of physical states with the Hamiltonian remaining in the natural square-root form. The generators of group SO (3, 1) are introduced and the quantization of this model is performed completely. The obtained spectrum of physical states and the Fock construction for the physical states from the Hamiltonian operator in square-root form was proposed, explicitly constructed, and compared with the spectrum and Fock construction obtained from the Hamiltonian in the standard form (i.e., quadratic in momenta). We show that the only states that the square-root Hamiltonian can operate with correspond to the representations with the lowest weights ë = 1/4 and 3/4 with four possible (nontrivial) fractional representations for the group decomposition of the spin structure.