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

TL;DR

This paper shows that the unique Hermitian ordering of the Dirac Hamiltonian with spatially varying mass affects the spectral properties, leading to observable modifications in inhomogeneous backgrounds.

Contribution

It demonstrates that the relativistic Dirac operator's consistent ordering introduces a universal deformation of the spectral quantization condition.

Findings

A logarithmic-gradient term modifies the effective kinetic operator.

Spectral shifts are mode-dependent and enhanced near mass-inversion points.

The ordering choice impacts observable spectral properties in inhomogeneous systems.

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.