Algebraic realisation of three fermion generations with $$S_3$$ S 3 family and unbroken gauge symmetry from $$\mathbb {C}\ell (8)$$ C ℓ ( 8 )
Abstract Building on previous work, we extend an algebraic realisation of three fermion generations within the complex Clifford algebra $$\mathbb {C}\ell (8)$$ C ℓ ( 8 ) by incorporating a $$U(1)_{em}$$ U ( 1 ) em gauge symmetry. The algebra $$\mathbb {C}\ell (8)$$ C ℓ ( 8 ) corresponds to the algeb...
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2024-10-01
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| Series: | European Physical Journal C: Particles and Fields |
| Online Access: | https://doi.org/10.1140/epjc/s10052-024-13476-0 |
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| author | Liam Gourlay Niels Gresnigt |
| author_facet | Liam Gourlay Niels Gresnigt |
| author_sort | Liam Gourlay |
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| description | Abstract Building on previous work, we extend an algebraic realisation of three fermion generations within the complex Clifford algebra $$\mathbb {C}\ell (8)$$ C ℓ ( 8 ) by incorporating a $$U(1)_{em}$$ U ( 1 ) em gauge symmetry. The algebra $$\mathbb {C}\ell (8)$$ C ℓ ( 8 ) corresponds to the algebra of complex linear maps from the (complexification of the) Cayley–Dickson algebra of sedenions, $$\mathbb {S}$$ S , to itself. Previous work represented three generations of fermions with $$SU(3)_C$$ S U ( 3 ) C colour symmetry permuted by an $$S_3$$ S 3 symmetry of order-three, but failed to include a U(1) generator that assigns the correct electric charge to all states. Furthermore, the three generations suffered from a degree of linear dependence between states. By generalising the embedding of the discrete group $$S_3$$ S 3 , corresponding to automorphisms of $$\mathbb {S}$$ S , into $$\mathbb {C}\ell (8)$$ C ℓ ( 8 ) , we include an $$S_3$$ S 3 -invariant U(1) that correctly assigns electric charge. First-generation states are represented in terms of two even $$\mathbb {C}\ell (8)$$ C ℓ ( 8 ) semi-spinors, obtained from two minimal left ideals, related to each other via the order-two $$S_3$$ S 3 symmetry. The remaining two generations are obtained by applying the $$S_3$$ S 3 symmetry of order-three to the first generation. In this model, the gauge symmetries, $$SU(3)_C\times U(1)_{em}$$ S U ( 3 ) C × U ( 1 ) em , are $$S_3$$ S 3 -invariant and preserve the semi-spinors. As a result of the generalised embedding of the $$S_3$$ S 3 automorphisms of $$\mathbb {S}$$ S into $$\mathbb {C}\ell (8)$$ C ℓ ( 8 ) , the three generations are now linearly independent. |
| format | Article |
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| issn | 1434-6052 |
| language | English |
| publishDate | 2024-10-01 |
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| series | European Physical Journal C: Particles and Fields |
| spelling | doaj-art-059b46fb9ecd4fd7ad7c78a486cc79352025-08-20T02:31:04ZengSpringerOpenEuropean Physical Journal C: Particles and Fields1434-60522024-10-01841011210.1140/epjc/s10052-024-13476-0Algebraic realisation of three fermion generations with $$S_3$$ S 3 family and unbroken gauge symmetry from $$\mathbb {C}\ell (8)$$ C ℓ ( 8 )Liam Gourlay0Niels Gresnigt1Department of Physics, Xi’an Jiaotong-Liverpool UniversityDepartment of Physics, Xi’an Jiaotong-Liverpool UniversityAbstract Building on previous work, we extend an algebraic realisation of three fermion generations within the complex Clifford algebra $$\mathbb {C}\ell (8)$$ C ℓ ( 8 ) by incorporating a $$U(1)_{em}$$ U ( 1 ) em gauge symmetry. The algebra $$\mathbb {C}\ell (8)$$ C ℓ ( 8 ) corresponds to the algebra of complex linear maps from the (complexification of the) Cayley–Dickson algebra of sedenions, $$\mathbb {S}$$ S , to itself. Previous work represented three generations of fermions with $$SU(3)_C$$ S U ( 3 ) C colour symmetry permuted by an $$S_3$$ S 3 symmetry of order-three, but failed to include a U(1) generator that assigns the correct electric charge to all states. Furthermore, the three generations suffered from a degree of linear dependence between states. By generalising the embedding of the discrete group $$S_3$$ S 3 , corresponding to automorphisms of $$\mathbb {S}$$ S , into $$\mathbb {C}\ell (8)$$ C ℓ ( 8 ) , we include an $$S_3$$ S 3 -invariant U(1) that correctly assigns electric charge. First-generation states are represented in terms of two even $$\mathbb {C}\ell (8)$$ C ℓ ( 8 ) semi-spinors, obtained from two minimal left ideals, related to each other via the order-two $$S_3$$ S 3 symmetry. The remaining two generations are obtained by applying the $$S_3$$ S 3 symmetry of order-three to the first generation. In this model, the gauge symmetries, $$SU(3)_C\times U(1)_{em}$$ S U ( 3 ) C × U ( 1 ) em , are $$S_3$$ S 3 -invariant and preserve the semi-spinors. As a result of the generalised embedding of the $$S_3$$ S 3 automorphisms of $$\mathbb {S}$$ S into $$\mathbb {C}\ell (8)$$ C ℓ ( 8 ) , the three generations are now linearly independent.https://doi.org/10.1140/epjc/s10052-024-13476-0 |
| spellingShingle | Liam Gourlay Niels Gresnigt Algebraic realisation of three fermion generations with $$S_3$$ S 3 family and unbroken gauge symmetry from $$\mathbb {C}\ell (8)$$ C ℓ ( 8 ) European Physical Journal C: Particles and Fields |
| title | Algebraic realisation of three fermion generations with $$S_3$$ S 3 family and unbroken gauge symmetry from $$\mathbb {C}\ell (8)$$ C ℓ ( 8 ) |
| title_full | Algebraic realisation of three fermion generations with $$S_3$$ S 3 family and unbroken gauge symmetry from $$\mathbb {C}\ell (8)$$ C ℓ ( 8 ) |
| title_fullStr | Algebraic realisation of three fermion generations with $$S_3$$ S 3 family and unbroken gauge symmetry from $$\mathbb {C}\ell (8)$$ C ℓ ( 8 ) |
| title_full_unstemmed | Algebraic realisation of three fermion generations with $$S_3$$ S 3 family and unbroken gauge symmetry from $$\mathbb {C}\ell (8)$$ C ℓ ( 8 ) |
| title_short | Algebraic realisation of three fermion generations with $$S_3$$ S 3 family and unbroken gauge symmetry from $$\mathbb {C}\ell (8)$$ C ℓ ( 8 ) |
| title_sort | algebraic realisation of three fermion generations with s 3 s 3 family and unbroken gauge symmetry from mathbb c ell 8 c l 8 |
| url | https://doi.org/10.1140/epjc/s10052-024-13476-0 |
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