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2026
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| Acceso en línea: | https://doi.org/10.5281/zenodo.19109893 |
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| _version_ | 1866901510070206464 |
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| author | Pommier, Jean-Patrick Pommier |
| author_facet | Pommier, Jean-Patrick Pommier |
| contents | <p>We prove that the real Clifford algebra Cl(12, 2) contains a unique compact su(2) subalgebra of so(12, 2) that simultaneously commutes with an su(3) color subalgebra and with the Lorentz algebra so(3, 1) while having generators disjoint from the Lorentz sector. The three generators {e4e11, e4e12, e11e12} - all real bivectors - are identifed by exhaustive search over a 29-dimensional centralizer.</p> <p>Two of three are mixed bivectors of the graded tensor product Cl(4, 2) ⊗ˆ Cl(8, 0). A single u(1)B−L factor (e5e6 + e7e8 + e9e10)/ √</p> <p>3 completes the gauge group to su(3) × su(2) × u(1) ⊂ so(12, 2) with the correct quantum numbers on the 16- dimensional Fock space of Cl(8, 0). The entire gauge group lives in the real algebra; no complexifcation is needed. Complexifcation enters only for the chiral projector PL = 1/2 (1 + iΓ14), which projects the fermion field in the Lagrangian ex-actly as γ5 = iγ0γ1γ2γ3 does in the Stan-dard Model. We establish a no-go theorem showing that chiral discrimination and Lie-algebra closure are mutually exclusive for any even-dimensional Clifgord algebra, cleanly separating the gauge question (real) from the chirality question (complex). The Weinberg angle at unication is sin^2(θW )= 3/8. No compact su(2)R exists, the obstruction being the Lorentzian signature of the conformal factor Cl(4, 2). All results are verified numerically in the 128-dimensional spinor representation.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19109893 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
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| spellingShingle | The unique su(2)L sub-algebra in the real Clifford algebra Cl(12, 2): Pommier, Jean-Patrick Pommier <p>We prove that the real Clifford algebra Cl(12, 2) contains a unique compact su(2) subalgebra of so(12, 2) that simultaneously commutes with an su(3) color subalgebra and with the Lorentz algebra so(3, 1) while having generators disjoint from the Lorentz sector. The three generators {e4e11, e4e12, e11e12} - all real bivectors - are identifed by exhaustive search over a 29-dimensional centralizer.</p> <p>Two of three are mixed bivectors of the graded tensor product Cl(4, 2) ⊗ˆ Cl(8, 0). A single u(1)B−L factor (e5e6 + e7e8 + e9e10)/ √</p> <p>3 completes the gauge group to su(3) × su(2) × u(1) ⊂ so(12, 2) with the correct quantum numbers on the 16- dimensional Fock space of Cl(8, 0). The entire gauge group lives in the real algebra; no complexifcation is needed. Complexifcation enters only for the chiral projector PL = 1/2 (1 + iΓ14), which projects the fermion field in the Lagrangian ex-actly as γ5 = iγ0γ1γ2γ3 does in the Stan-dard Model. We establish a no-go theorem showing that chiral discrimination and Lie-algebra closure are mutually exclusive for any even-dimensional Clifgord algebra, cleanly separating the gauge question (real) from the chirality question (complex). The Weinberg angle at unication is sin^2(θW )= 3/8. No compact su(2)R exists, the obstruction being the Lorentzian signature of the conformal factor Cl(4, 2). All results are verified numerically in the 128-dimensional spinor representation.</p> |
| title | The unique su(2)L sub-algebra in the real Clifford algebra Cl(12, 2): |
| url | https://doi.org/10.5281/zenodo.19109893 |