Abstract
Abstract
The conventional Dirac oscillator is not Majorana closed at any fixed nonzero frequency. Charge conjugation reverses the sign of the oscillator coupling,
H
DO
c
(
ω
)
=
H
DO
(
−
ω
)
, so a single fixed-frequency sector does not preserve the self-conjugate subspace in one, two, or three spatial dimensions. This mapping follows directly from charge conjugation, but it does not by itself determine which consistency requirement fails, how the obstruction is manifested spectrally, or which oscillator couplings remain admissible. Dynamical preservation of the self-conjugate subspace is the decisive compatibility requirement and is logically independent of both formal reality in a Majorana representation and Hermiticity. Applying these three conditions to candidate oscillator couplings shows that an earlier real-coefficient ‘Majorana oscillator’ prescription fails Hermiticity rather than reality. The spectral manifestation of the obstruction depends qualitatively on dimension: an unpaired level at
E
=
+
m
c
2
in one dimension becomes an infinitely degenerate, orbitally chiral intruder family in two dimensions and the spin–orbit-resolved
n
r
=
0
,
ϰ
0
family in three dimensions. Two admissible alternatives are then examined: a doubled
(
+
ω
,
−
ω
)
Hamiltonian with exact particle–hole symmetry and a real Lorentz-scalar confining interaction that preserves Majorana reality without doubling. A linear scalar profile in one dimension yields a normalizable zero mode and
E
N
=
±
2
N
ℏ
c
κ
, whereas central scalar confinement in two and three dimensions gives coupled radial equations in which charge conjugation pairs opposite angular channels. We introduce phase-fixed and phase-independent leakage functionals to quantify Majorana-symmetry breaking, thereby converting the compatibility question into an experimentally accessible dynamical test. Together, these results distinguish the standard Dirac oscillator from oscillator constructions that are dynamically compatible with Majorana self-conjugacy.