Operator ordering as an emergent geometric background in Dirac systems with spatially varying mass

Abstract

We investigate the spectral consequences of the uniquely determined Hermitian ordering of the Dirac Hamiltonian with spatially varying mass. In contrast to the nonrelativistic case, where continuous families of admissible prescriptions exist, the relativistic Dirac operator admits a single consistent ordering compatible with probability-current conservation. This requirement generates an additional logarithmic-gradient term proportional to the spatial variation of the mass profile. We show that this contribution modifies the effective kinetic operator and induces a universal deformation of the spectral quantization condition. In compact geometry, an explicit analytic computation reveals a mode-dependent second-order spectral shift that becomes strongly enhanced near the mass-inversion threshold. These results demonstrate that the consistent relativistic ordering of the Dirac operator leads to observable modifications of discrete spectra in spatially inhomogeneous scalar backgrounds.