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 [Formula: see text], where [Formula: see text] denotes the Dirac index and [Formula: see text] the isospin index, so that the Hamiltonian acts on the tensor-product space [Formula: see text] 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 SU(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 [Formula: see text] 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. For an aligned planar background, the commutator term yields the closed-form isospin splitting [Formula: see text] thereby making the internal-Zeeman mechanism explicit. To render this result visually transparent, we also include a direct spectral representation of the exact branches as functions of the splitting parameter [Formula: see text]. Furthermore, we show that the planar Dirac oscillator admits a direct correspondence with effective graphene Hamiltonians: the same non-minimal confinement rule generates the graphene Dirac-oscillator form once the relativistic mass scale is traded for the appropriate gap parameter. In turn, retaining the internal valley or layer doublet extends this correspondence to the gauge-field case, where matrix-valued effective connections provide a natural implementation of the non-Abelian model and the commutator term controls a possible lifting of internal degeneracies. Overall, the resulting framework separates universal commutator-driven effects from backgrounddependent kinematic shifts and thus provides a controlled setting for studying relativistic bound states in Yang–Mills backgrounds and in graphene-based Dirac materials with effective non-Abelian structures.