Algebraic realisation of three fermion generations with $$S_3$$ family and unbroken gauge symmetry from $$\mathbb {C}\ell (8)$$

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 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.