Relativistic Landau levels for a fermion-antifermion pair interacting through Dirac oscillator interaction

We introduce a unique model for a fermion-antifermion pair interacting via Dirac oscillator coupling in the presence of an external uniform magnetic field. This model is based on an exact solution of the corresponding form of a fully-covariant two-body Dirac equation (one-time). The dynamic symmetry of the system allows to study in $2+1$ dimensions and we choose the interaction of the particles with the external uniform magnetic field in the symmetric gauge. The corresponding equation leads $4×4$ dimensional matrix equation for such a static composite system. For spin antisymmetric state of the fermion-antifermion pair, we perform an exact solution of the matrix equation and obtain relativistic Landau levels of a fermion-antifermion pair interacting via Dirac oscillator coupling. The results show that such a composite system behaves like a single relativistic quantum oscillator carrying total rest mass of the particles. We discuss several interesting features of this system and show that the obtained energy spectrum agrees well with the previously announced results for one-body systems. We think that the introduced model in this manuscript has a great potential for many theoretical and experimental applications.