Electroweak Structure and Three Fermion Generations in Clifford Algebra with S3 Family Symmetry
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908956429910016 |
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| author | Gresnigt, Niels |
| author_facet | Gresnigt, Niels |
| contents | We construct an explicit algebraic realisation of three fermion generations within a single Clifford algebra, transforming under the full Standard Model $SU(3)_C\times SU(2)_L\times U(1)_Y$ gauge group, in which an intrinsic $S_3$ family symmetry permutes three algebraically distinguished but gauge-equivalent fermion sectors without replicating the gauge bosons. Fermionic states are represented by minimal left ideals of the complex Clifford algebra $\mathbb{C}\ell(10)$, while the three-generation structure arises from an embedded discrete $S_3$ symmetry acting on the space of algebraic spinors. The Standard Model gauge generators are identified as elements commuting with this $S_3$ action and act on physical states via the adjoint (commutator) action. The resulting spectrum reproduces the correct Standard Model quantum numbers for three linearly independent generations of fermions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_07857 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Electroweak Structure and Three Fermion Generations in Clifford Algebra with S3 Family Symmetry Gresnigt, Niels General Physics We construct an explicit algebraic realisation of three fermion generations within a single Clifford algebra, transforming under the full Standard Model $SU(3)_C\times SU(2)_L\times U(1)_Y$ gauge group, in which an intrinsic $S_3$ family symmetry permutes three algebraically distinguished but gauge-equivalent fermion sectors without replicating the gauge bosons. Fermionic states are represented by minimal left ideals of the complex Clifford algebra $\mathbb{C}\ell(10)$, while the three-generation structure arises from an embedded discrete $S_3$ symmetry acting on the space of algebraic spinors. The Standard Model gauge generators are identified as elements commuting with this $S_3$ action and act on physical states via the adjoint (commutator) action. The resulting spectrum reproduces the correct Standard Model quantum numbers for three linearly independent generations of fermions. |
| title | Electroweak Structure and Three Fermion Generations in Clifford Algebra with S3 Family Symmetry |
| topic | General Physics |
| url | https://arxiv.org/abs/2601.07857 |