Quadratic Algebra Approach to Relativistic Quantum Smorodinsky-Winternitz Systems

TL;DR

This paper derives the relativistic energy spectra of Smorodinsky-Winternitz systems using quadratic algebra methods, linking non-relativistic superintegrable systems to their relativistic counterparts through symmetry algebra analysis.

Contribution

It extends the quadratic algebra approach to relativistic quantum systems, specifically the Smorodinsky-Winternitz models, connecting non-relativistic algebraic structures to relativistic equations.

Findings

Relativistic energy spectra obtained for four Smorodinsky-Winternitz systems.

Established the equivalence of quadratic algebras in relativistic and non-relativistic contexts.

Presented the symmetry algebra of the Dirac equation for these systems.

Abstract

There exist a relation between the Klein-Gordon and the Dirac equations with scalar and vector potentials of equal magnitude (SVPEM) and the Schrodinger equation. We obtain the relativistic energy spectrum for the four Smorodinsky-Winternitz systems from the quasi-Hamiltonian and the quadratic algebras obtained by Daskaloyannis in the non-relativistic context. We point out how results obtained in context of quantum superintegrable systems and their polynomial algebras may be applied to the quantum relativistic case. We also present the symmetry algebra of the Dirac equation for these four systems and show that the quadratic algebra obtained is equivalent to the one obtained from the quasi-Hamiltonian.