Non-Abelian Extensions of the Dirac Oscillator: A Theoretical Approach

We formulate the Dirac oscillator covariantly in the presence of external non-Abelian gauge fields. More precisely, the matter field is written as ${\mathrm{\Psi }}_{\alpha A}(x)$, where $\alpha$ denotes the Dirac index and $A$ the isospin index, so that the Hamiltonian acts on the tensor-product space ${\mathbb{C}}^4\otimes {\mathbb{C}}^2$ in the fundamental representation. Starting from the gauge-covariant Dirac equation, we then implement the oscillator interaction through the standard non-minimal substitution and promote the construction to an $\mathrm{S}\mathrm{U}(2)$ background. In this way, we derive the associated non-Abelian field-strength tensor and isolate the commutator contribution, which has no Abelian analogue. Consequently, the generalized Pauli interaction ${\sigma }^{\mu \nu }{\mathcal{F}}_{\mu \nu }$ produces matrix-valued spin–isospin couplings. At the same time, the Abelian sector reduces to the conventional Moshinsky–Szczepaniak Dirac oscillator, whose exactly solvable spectrum provides a natural benchmark for the extended theory.