A Chiral Adelic Dirac Operator and the Spectral Realization of the Riemann Zeros
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arXiv
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| Format: | Preprint |
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2025
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| author | Hateley, James C. |
| author_facet | Hateley, James C. |
| contents | This paper develops a chiral adelic operator framework in which the functional--equation symmetry of global $L$--functions is realized directly in the spectrum of a Dirac--type Hamiltonian. Working on the idèle class space, we place a real--place Floquet Hamiltonian into an off--diagonal chiral form to obtain a global adelic Dirac operator with an exact involutive symmetry implemented by real reflection and idelic inversion. Arithmetic information is incorporated through a prime--indexed mass deformation built from spherical Hecke operators; when the coefficient functions are even, the perturbed operator preserves the chiral symmetry and produces isolated $\pm$--paired eigenvalues inside the spectral gaps of the Floquet background. These eigenvalues appear as jump discontinuities of the Dirac spectral shift function, while a separated adelic trace formula expresses the trace as a product of a Floquet orbital factor and a prime--indexed Euler--factor--type term whose logarithmic derivatives yield a prime--orbit expansion reminiscent of the explicit formula. This structure motivates a Dirac reinterpretation of the Hilbert--Pólya idea, identifying the nontrivial zeros of $ζ(s)$ not with the raw spectrum of a single operator but with the spectral--shift discontinuities of a chiral adelic Dirac system under controlled prime--indexed deformations, with finite--prime truncations providing computable models that converge distributionally and enable numerical exploration of arithmetic spectral flow. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_18309 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Chiral Adelic Dirac Operator and the Spectral Realization of the Riemann Zeros Hateley, James C. Mathematical Physics 11M26, 81Q15, 34L40, 47A55, 47A40, 81Q10 This paper develops a chiral adelic operator framework in which the functional--equation symmetry of global $L$--functions is realized directly in the spectrum of a Dirac--type Hamiltonian. Working on the idèle class space, we place a real--place Floquet Hamiltonian into an off--diagonal chiral form to obtain a global adelic Dirac operator with an exact involutive symmetry implemented by real reflection and idelic inversion. Arithmetic information is incorporated through a prime--indexed mass deformation built from spherical Hecke operators; when the coefficient functions are even, the perturbed operator preserves the chiral symmetry and produces isolated $\pm$--paired eigenvalues inside the spectral gaps of the Floquet background. These eigenvalues appear as jump discontinuities of the Dirac spectral shift function, while a separated adelic trace formula expresses the trace as a product of a Floquet orbital factor and a prime--indexed Euler--factor--type term whose logarithmic derivatives yield a prime--orbit expansion reminiscent of the explicit formula. This structure motivates a Dirac reinterpretation of the Hilbert--Pólya idea, identifying the nontrivial zeros of $ζ(s)$ not with the raw spectrum of a single operator but with the spectral--shift discontinuities of a chiral adelic Dirac system under controlled prime--indexed deformations, with finite--prime truncations providing computable models that converge distributionally and enable numerical exploration of arithmetic spectral flow. |
| title | A Chiral Adelic Dirac Operator and the Spectral Realization of the Riemann Zeros |
| topic | Mathematical Physics 11M26, 81Q15, 34L40, 47A55, 47A40, 81Q10 |
| url | https://arxiv.org/abs/2511.18309 |