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 $\mathbb{C}\ell (8)$ by incorporating a $U(1{)}_{em}$ gauge symmetry. The algebra $\mathbb{C}\ell (8)$ corresponds to the algebra of complex linear maps from the (complexification of the) Cayley-Dickson algebra of sedenions, $\mathbb{S}$, to itself. Previous work represented three generations of fermions with $SU(3{)}_C$ colour symmetry permuted by an $S_3$ 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 $S_3$, corresponding to automorphisms of $\mathbb{S}$, into $\mathbb{C}\ell (8)$, we include an $S_3$-invariant $U(1)$ that correctly assigns electric charge. First-generation states are represented in terms of two even $\mathbb{C}\ell (8)$ semi-spinors, obtained from two minimal left ideals, related to each other via the order-two $S_3$ symmetry. The remaining two generations are obtained by applying the $S_3$ symmetry of order-three to the first generation. In this model, the gauge symmetries, $SU(3{)}_C\times U(1{)}_{em}$, are $S_3$-invariant and preserve the semi-spinors. As a result of the generalised embedding of the $S_3$ automorphisms of $\mathbb{S}$ into $\mathbb{C}\ell (8)$, the three generations are now linearly independent.

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References in corpus (20)

• Three generations, two unbroken gauge symmetries, and one eight-dimensional algebra
• Generations: Three Prints, in Colour
• $SU(5)$ GUTs with $A_4$ modular symmetry
• Twin Modular $S_4$ with $SU(5)$ GUT
• Three fermion generations with two unbroken gauge symmetries from the complex sedenions
• Division algebraic symmetry breaking
• Spin(11,3), particles and octonions
• One generation of standard model Weyl representations as a single copy of $\mathbb{R}\otimes \mathbb{C}\otimes \mathbb{H}\otimes \mathbb{O}$
• An Exceptional G(2) Extension of the Standard Model from the Correspondence with Cayley-Dickson Algebras Automorphism Groups
• Octonion Internal Space Algebra for the Standard Model
• A topological model of composite preons from the minimal ideals of two Clifford algebras
• Three generations of colored fermions with $S_3$ family symmetry from Cayley-Dickson sedenions
• An octonionic construction of $E_8$ and the Lie algebra magic square
• Quantum theory without classical time: octonions, and a theoretical derivation of the fine structure constant 1/137
• CKM matrix parameters from the exceptional Jordan algebra
• A minimal inverse seesaw model with $S_4$ flavour symmetry
• Clifford algebra Cl(0,6) approach to beyond the standard model and naturalness problems
• Superselection of the weak hypercharge and the algebra of the Standard Model
• Causal Fermion Systems and Octonions
• A unified sedenion model for the origins of three generations of charged and neutral leptons, flavor mixing, mass oscillations and small masses of neutrinos

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