Emergent Grand Unified Structure in a 4 x 4 Nilpotent Matrix Algebra
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915647847399424 |
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| author | Ajaib, M. Adeel |
| author_facet | Ajaib, M. Adeel |
| contents | We show that nilpotent matrices that yield the Schrodinger equation from its first order form encode the fingerprints of grand unified theories. We perform a rigorous search for all such nilpotent matrices and find that the resulting matrices naturally organize into suggestive group theoretic structures without any other a priori assumptions. The antisymmetric sector consists of three groups of sixteen matrices, each of which further splits as 16 = 12 + 4 and exhibits unique characteristics in the step potential scattering problem. The symmetric zero-diagonal sector also forms three families, mirroring the quark-lepton decomposition of the Pati-Salam model. These results may help answer why there are three families of fermions and also demonstrate that the 4 x 4 matrix algebra is a compact, nontrivial shadow of the SO(10) embedding, with fermion-like and gauge-like subspaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_02059 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Emergent Grand Unified Structure in a 4 x 4 Nilpotent Matrix Algebra Ajaib, M. Adeel High Energy Physics - Phenomenology We show that nilpotent matrices that yield the Schrodinger equation from its first order form encode the fingerprints of grand unified theories. We perform a rigorous search for all such nilpotent matrices and find that the resulting matrices naturally organize into suggestive group theoretic structures without any other a priori assumptions. The antisymmetric sector consists of three groups of sixteen matrices, each of which further splits as 16 = 12 + 4 and exhibits unique characteristics in the step potential scattering problem. The symmetric zero-diagonal sector also forms three families, mirroring the quark-lepton decomposition of the Pati-Salam model. These results may help answer why there are three families of fermions and also demonstrate that the 4 x 4 matrix algebra is a compact, nontrivial shadow of the SO(10) embedding, with fermion-like and gauge-like subspaces. |
| title | Emergent Grand Unified Structure in a 4 x 4 Nilpotent Matrix Algebra |
| topic | High Energy Physics - Phenomenology |
| url | https://arxiv.org/abs/2512.02059 |