Graded extension of SO(2,1) Lie algebra and the search for exact solutions of Dirac equation by point canonical transformations

TL;DR

This paper extends the SO(2,1) algebra to a graded supersymmetry algebra, enabling the mapping of various relativistic potentials into the Dirac-Oscillator form through new point canonical transformations.

Contribution

It introduces a graded extension of SO(2,1) and demonstrates its realization in spinor space, linking relativistic potentials via point canonical transformations.

Findings

Graded algebra acts as supersymmetry algebra for relativistic potentials.

New transformation maps multiple relativistic potentials to Dirac-Oscillator.

Extension unifies various potentials within a relativistic framework.

Abstract

SO(2,1) is the symmetry algebra for a class of three-parameter problems that includes the oscillator, Coulomb and Morse potentials as well as other problems at zero energy. All of the potentials in this class can be mapped into the oscillator potential by point canonical transformations. We call this class the "oscillator class". A nontrivial graded extension of SO(2,1) is defined and its realization by two-dimensional matrices of differential operators acting in spinor space is given. It turns out that this graded algebra is the supersymmetry algebra for a class of relativistic potentials that includes the Dirac-Oscillator, Dirac-Coulomb and Dirac-Morse potentials. This class is, in fact, the relativistic extension of the oscillator class. A new point canonical transformation, which is compatible with the relativistic problem, is formulated. It maps all of these relativistic potentials into the Dirac-Oscillator potential.