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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.06637 |
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| _version_ | 1866910044013985792 |
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| author | Boumali, Abdelmalek Jafari, Nosratollah |
| author_facet | Boumali, Abdelmalek Jafari, Nosratollah |
| contents | We study the Klein--Gordon (KG) oscillator in a doubly special relativity (DSR) framework, where the mass-shell condition is deformed through a linear--fractional (Möbius-type) modification of the Casimir invariant. This is induced by a nonlinear map from physical momenta $p^μ$ to auxiliary Lorentz-covariant variables $π^μ$. In $(1+1)$ dimensions, the deformation is controlled by a constant covector $a_μ$, yielding inequivalent realizations depending on whether $a_μ$ is timelike, spacelike, or lightlike. Implementing the KG oscillator via a reverted-product nonminimal coupling, we obtain exact closed-form spectra and explicit eigensolutions for both particle and antiparticle branches across all three geometries.
Timelike and lightlike deformations produce identical spectra characterized by a Planck-suppressed additive displacement. This breaks the exact $E\leftrightarrow -E$ symmetry via a term linear in $E$, interpretable as a branch-independent reparametrization of the energy origin. Conversely, the spacelike deformation is strictly isospectral to the undeformed oscillator but generates complex-shifted wavefunctions and a non-Hermitian spatial operator. We provide a compact $\mathcal{PT}$-symmetric and pseudo-Hermitian formulation by constructing an explicit similarity map $\mathcal{S}$ to a Hermitian oscillator, deriving the metric operator $η=\mathcal{S}^\dagger \mathcal{S}$, and establishing biorthonormal relations.
Finally, we compare quantitatively with the Magueijo--Smolin (DSR2) model: the squared-denominator invariant leads to a larger Planck-suppressed displacement at fixed $m/E_{Pl}$, highlighting the denominator power's role in controlling spectral shifts. Representative plots illustrate the dependence on deformation ratio, oscillator strength, and excitation level. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_06637 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Klein--Gordon oscillator with linear--fractional deformed Casimirs in doubly special relativity Boumali, Abdelmalek Jafari, Nosratollah Quantum Physics High Energy Physics - Theory We study the Klein--Gordon (KG) oscillator in a doubly special relativity (DSR) framework, where the mass-shell condition is deformed through a linear--fractional (Möbius-type) modification of the Casimir invariant. This is induced by a nonlinear map from physical momenta $p^μ$ to auxiliary Lorentz-covariant variables $π^μ$. In $(1+1)$ dimensions, the deformation is controlled by a constant covector $a_μ$, yielding inequivalent realizations depending on whether $a_μ$ is timelike, spacelike, or lightlike. Implementing the KG oscillator via a reverted-product nonminimal coupling, we obtain exact closed-form spectra and explicit eigensolutions for both particle and antiparticle branches across all three geometries. Timelike and lightlike deformations produce identical spectra characterized by a Planck-suppressed additive displacement. This breaks the exact $E\leftrightarrow -E$ symmetry via a term linear in $E$, interpretable as a branch-independent reparametrization of the energy origin. Conversely, the spacelike deformation is strictly isospectral to the undeformed oscillator but generates complex-shifted wavefunctions and a non-Hermitian spatial operator. We provide a compact $\mathcal{PT}$-symmetric and pseudo-Hermitian formulation by constructing an explicit similarity map $\mathcal{S}$ to a Hermitian oscillator, deriving the metric operator $η=\mathcal{S}^\dagger \mathcal{S}$, and establishing biorthonormal relations. Finally, we compare quantitatively with the Magueijo--Smolin (DSR2) model: the squared-denominator invariant leads to a larger Planck-suppressed displacement at fixed $m/E_{Pl}$, highlighting the denominator power's role in controlling spectral shifts. Representative plots illustrate the dependence on deformation ratio, oscillator strength, and excitation level. |
| title | Klein--Gordon oscillator with linear--fractional deformed Casimirs in doubly special relativity |
| topic | Quantum Physics High Energy Physics - Theory |
| url | https://arxiv.org/abs/2603.06637 |