Generalized Spin and Pseudo-Spin Symmetry: Relativistic Extension of Supersymmetric Quantum Mechanics

TL;DR

This paper extends supersymmetric quantum mechanics by introducing a new symmetry in the Dirac equation with scalar, vector, and pseudo-scalar potentials, providing formal solutions and explicit examples.

Contribution

It introduces a novel symmetry where scalar or pseudo-scalar potentials are proportional to the vector potential, expanding the framework of supersymmetric quantum mechanics.

Findings

Formal solutions for the Dirac equation under the new symmetry

Explicit analytic results for specific potential examples

Extension of supersymmetric quantum mechanics concepts

Abstract

We consider the Dirac equation in 1+1 space-time dimension with vector, scalar and pseudo-scalar coupling. In the traditional spin (or pseudo-spin) symmetry, the difference between (or sum of) the scalar and vector potentials is a constant. Here, however, we introduce an alternative symmetry where the scalar or pseudo-scalar potential is proportional to the vector potential. This leads to a model with significant extensions to supersymmetric quantum mechanics. We present a formal solution of the problem but give explicit analytic results for specific examples.