Different electromagnetic physical representations of the Dirac's oscillator according with its spatial dimension

Abstract: Dirac's oscillator (DO) is one of the most studied systems in the Relativistic Quantum Mechanics and in the physical-mathematics. In particular, we show that this system has an unique property which it has not ever seen in other known systems: According to its spatial dimensionality, DO represent physical systems with very different electromagnetic nature. So far in the literature, it has been proved using the covariant method the gauge invariance of the Dirac's oscillator potential. It has also shown that in (3+1)dimensions the DO represents a relativistic and electrically neutral fermion with magnetic dipole momentum, into a dielectric medium with spherical symmetry and under the effect of an electric field which depends of the radial distance. In this work,and using the same methodology, we show that (2+1) dimensional DO represents a 1/2-spin relativistic fermion under the effect of a uniform and perpendicular external magnetic field; whereas in (1+1) dimensions DO reproduces a relativistic and electrically charged fermion interacting with a linear electric field. Additionally, we prove that DO does not have chiral invariance, independent of its dimensionality, due to the interaction potential which breaks explicitly the chiral symmetry $U(1)_R \times U(1)_L$ but it preserves the global gauge symmetry $U(1)$.