Supersymmetry and the Dirac equation
We discuss in detail two supersymmetries of the 4-dimensional Dirac operator \({\mathbb{D}}^ 2 \)where \({\mathbb{D}}=\partial -ieA\), namely the usual chiral supersymmetry and a separate complex supersymmetry. Using SUSY methods developed to categorize solvable potentials in 1-dimensional quantum mechanics we systematically study the cases where the spectrum, eigenfunctions, and S-matrix of \({\mathbb{D}}^ 2 \)can be obtained analytically. We relate these solutions to the solutions of the ordinary massive Dirac equation in external fields. We show that whenever a Schrödinger equation for a potential V(x) is exactly solvable, then there always exists a corresponding static scalar field \(\phi\) (x) for which the Jackiw-Rebbi type \((1+1)\)-dimensional Dirac equation is exactly solvable with V(x) and \(\phi\) (x) being related by \(V(x)=\phi\) \(2(x)+\phi '(x)\). We also discuss and exploit the supersymmetry of the path integral representation for the fermion propagator in an external field.
supersymmetries of the 4-dimensional Dirac operator
scientific article; zbMATH DE number 4062187